[{"data":1,"prerenderedAt":12618},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":374,"course-wordcounts":12486,"ref-card-index":12542},[4,28,50,71,120,152,205,230,286,306,331,352],{"module":5,"moduleNumber":6,"slug":7,"lessons":8},"Foundations",1,"foundations",[9,15,21],{"title":10,"path":11,"lessonNumber":6,"topics":12,"summary":14},"What Is an Algorithm?","\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm",[5,13],"Correctness & Induction","An algorithm is a finite, mechanical recipe that transforms inputs into outputs. We pin down what counts as an algorithm, how we write one down, and the three things we always ask of it: is it correct, is it fast, and can we prove it.\n",{"title":16,"path":17,"lessonNumber":18,"topics":19,"summary":20},"Asymptotic Analysis","\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis",2,[16],"We measure an algorithm's running time as a function of its input size, then strip away machine-specific constants and lower-order terms to compare algorithms cleanly. This lesson defines the RAM model and the $O$, $\\Omega$, $\\Theta$, $o$, and $\\omega$ notations, and shows how to read the cost of loops off the page.\n",{"title":22,"path":23,"lessonNumber":24,"topics":25,"summary":27},"Recurrences and the Master Theorem","\u002Falgorithms\u002Ffoundations\u002Frecurrences",3,[26],"Recurrences","Recursive and divide-and-conquer algorithms describe their own running time with a recurrence: $T(n)$ in terms of $T$ on smaller inputs. We solve recurrences three ways — drawing the recursion tree, guessing-and-verifying by induction, and applying the Master Theorem — using merge sort as the running example.\n",{"module":29,"moduleNumber":18,"slug":30,"lessons":31},"Divide & Conquer","divide-and-conquer",[32,38,44],{"title":33,"path":34,"lessonNumber":6,"topics":35,"summary":37},"Divide and Conquer & Mergesort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort",[29,36],"Comparison Sorting","Divide and conquer breaks a problem into smaller copies of itself, solves them recursively, and stitches the answers together. We meet the paradigm through mergesort — its merge step, its loop-invariant proof, and the recursion tree that pins its cost at $\\Theta(n\\log n)$ — then glimpse Karatsuba multiplication as a second example of the same idea.",{"title":39,"path":40,"lessonNumber":18,"topics":41,"summary":43},"Quicksort","\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort",[36,42],"Probabilistic Analysis","Quicksort sorts in place by partitioning around a pivot and recursing on each side. We give Lomuto and Hoare partitioning with a correctness invariant, see why a bad pivot costs $\\Theta(n^2)$ while a balanced one gives $\\Theta(n\\log n)$, and prove that randomizing the pivot makes the expected cost $\\Theta(n\\log n)$ on every input.",{"title":45,"path":46,"lessonNumber":24,"topics":47,"summary":49},"Linear-Time Selection","\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection",[48,29],"Order Statistics","Finding the $k$-th smallest element looks like it should require sorting, but it does not. Quickselect adapts quicksort's partition to recurse on just one side, achieving expected $O(n)$. The median-of-medians algorithm guarantees a good pivot with the groups-of-five trick, pushing the worst case down to a provable $O(n)$.",{"module":51,"moduleNumber":24,"slug":52,"lessons":53},"Sorting & Order Statistics","sorting",[54,60,65],{"title":55,"path":56,"lessonNumber":6,"topics":57,"summary":59},"Heaps and Heapsort","\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort",[58,36],"Heaps","A binary heap is a tree we store flat in an array, with index arithmetic standing in for pointers. We build the max-heap property bottom-up in $O(n)$ time, sort in place in $\\Theta(n\\log n)$ by repeatedly extracting the maximum, and reuse the same structure to implement a priority queue.",{"title":61,"path":62,"lessonNumber":18,"topics":63,"summary":64},"Lower Bounds for Comparison Sorting","\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds",[36],"Every sort we have seen runs in $\\Omega(n\\log n)$, and that is no accident. Modeling a sort as a decision tree of comparisons, we show any such tree must have $n!$ leaves, forcing height $\\ge \\log_2(n!) = \\Omega(n\\log n)$ — a worst-case bound no comparison sort can ever beat.",{"title":66,"path":67,"lessonNumber":24,"topics":68,"summary":70},"Sorting in Linear Time","\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting",[69],"Linear-Time Sorting","The $\\Omega(n\\log n)$ barrier only binds algorithms that compare. By instead using keys as array indices we slip past it: counting sort runs in $\\Theta(n+k)$ and is stable, radix sort layers it digit by digit, and bucket sort averages $\\Theta(n)$ on uniform data. We see exactly when each applies.",{"module":72,"moduleNumber":73,"slug":74,"lessons":75},"Data Structures",4,"data-structures",[76,82,88,93,99,105,113],{"title":77,"path":78,"lessonNumber":6,"topics":79,"summary":81},"Elementary Data Structures","\u002Falgorithms\u002Fdata-structures\u002Felementary-structures",[80],"Linear Structures","Every container is built one of two ways: **contiguous** in an array, or **linked** through pointers. We trade cache-friendly random access against $O(1)$ splicing, derive the **amortized $O(1)$** append of a doubling dynamic array, and assemble the two ordered access disciplines — the LIFO **stack** and the FIFO **queue** (with its generalization, the **deque**) — on top of both.",{"title":83,"path":84,"lessonNumber":18,"topics":85,"summary":87},"Hash Tables","\u002Falgorithms\u002Fdata-structures\u002Fhash-tables",[86],"Hashing","A hash table implements the dictionary — insert, search, delete — in expected $O(1)$ time by scattering keys across an array with a hash function. We build up from direct addressing, handle collisions by chaining and by open addressing, analyze the load factor $\\alpha$, and see how universal hashing earns its expected-time guarantee against every input.",{"title":89,"path":90,"lessonNumber":24,"topics":91,"summary":92},"Binary Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees",[89],"A binary search tree keeps keys ordered so that every operation follows a single root-to-leaf path. We state the BST property, give search and insert, find minimum, maximum, and successor, see that an inorder walk emits the keys in sorted order, and confront the catch — every operation costs $O(h)$, and a carelessly built tree degrades to height $h = \\Theta(n)$, motivating balance.",{"title":94,"path":95,"lessonNumber":73,"topics":96,"summary":98},"AVL Trees","\u002Falgorithms\u002Fdata-structures\u002Favl-trees",[97],"Balanced Trees","An AVL tree is the first balanced BST: at every node the two subtrees' heights differ by at most $1$. A Fibonacci-style minimal-node argument forces height $h \\le 1.44\\log_2 n = O(\\log n)$, so search, insert, and delete are all $O(\\log n)$. Insertion rebalances with at most one of four rotation cases (LL, RR, LR, RL); deletion may rotate all the way to the root.",{"title":100,"path":101,"lessonNumber":102,"topics":103,"summary":104},"Balanced Search Trees","\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees",5,[97],"An ordinary BST can degrade to height $\\Theta(n)$; balanced search trees guarantee $h = O(\\log n)$ by maintaining invariants and repairing them after every update. We meet rotations, the local restructuring primitive, then red-black trees, whose color invariants force logarithmic height, and finally B-trees, which trade tall-and-thin for short-and-wide to win on disk.",{"title":106,"path":107,"lessonNumber":108,"topics":109,"summary":112},"Disjoint Sets (Union-Find)","\u002Falgorithms\u002Fdata-structures\u002Funion-find",6,[110,111],"Disjoint Sets","Amortized Analysis","The disjoint-set data structure tracks a partition of elements into groups, answering \"are these two in the same group?\" and merging groups on demand. A forest of parent pointers, sped up by union by rank and path compression, drives every operation to near-constant $O(\\alpha(n))$ amortized time — the engine behind connectivity queries and Kruskal's minimum spanning tree.",{"title":114,"path":115,"lessonNumber":116,"topics":117,"summary":119},"Fenwick & Segment Trees","\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees",7,[118],"Range Queries","A prefix-sum array answers a range sum in $O(1)$ but pays $O(n)$ per update; a plain array updates in $O(1)$ but pays $O(n)$ per range sum. Fenwick and segment trees give us _both_ in $O(\\log n)$. The Fenwick (binary indexed) tree is a tiny array keyed by the low bit; the segment tree is a general balanced tree over canonical ranges that handles any associative aggregate and, with lazy propagation, range updates too.",{"module":121,"moduleNumber":102,"slug":122,"lessons":123},"Sequences & Strings","sequences",[124,130,135,141,147],{"title":125,"path":126,"lessonNumber":6,"topics":127,"summary":129},"Two Pointers, Sliding Windows & Prefix Sums","\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows",[128],"Array Techniques","A family of array idioms that collapse an obvious $O(n^2)$ scan into a single $O(n)$ pass by maintaining an invariant as indices move. We meet two pointers (converging on a sorted array, and a fast\u002Fslow pair for in-place rewriting), the sliding window (fixed and variable size, amortized $O(n)$), and prefix sums, which answer any range-sum in $O(1)$ and count subarrays summing to $k$.",{"title":131,"path":132,"lessonNumber":18,"topics":133,"summary":134},"Monotonic Stacks & Queues","\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks",[128],"A **monotonic stack** keeps its contents sorted by popping every element that would break the order before each push — turning a family of \"previous\u002Fnext greater (or smaller) element\" questions into a single $O(n)$ scan. We derive the next-greater-element routine and its amortized analysis, fuse two such scans to measure the **largest rectangle in a histogram** in linear time, and extend the idea to a **monotonic deque** that streams the **sliding-window maximum** in $O(n)$.",{"title":136,"path":137,"lessonNumber":24,"topics":138,"summary":140},"Binary Search on the Answer","\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer",[139],"Searching","Binary search is not really about arrays — it is about locating the boundary of a **monotone predicate** $p(x)$ in $O(\\log(\\text{range}))$ probes. We first pin down the half-open `while (lo \u003C hi)` template for $\\textsc{lower\\_bound}$ and $\\textsc{upper\\_bound}$, then generalize to \"binary search on the answer\": whenever feasibility is monotone in a numeric parameter, we binary search the parameter itself, calling a feasibility check at each step.",{"title":142,"path":143,"lessonNumber":73,"topics":144,"summary":146},"String Matching: Rabin–Karp, KMP & Z","\u002Falgorithms\u002Fsequences\u002Fstring-matching",[145],"Strings","Given a text $T$ of length $n$ and a pattern $P$ of length $m$, find every occurrence of $P$ in $T$. The naive scan costs $O(nm)$; we beat it three ways. Rabin–Karp uses a **rolling hash** to test alignments in $O(1)$ amortised each, with expected $O(n+m)$. KMP precomputes a **failure function** so the scan never re-reads a text character, for worst-case $O(n+m)$. The **Z-function** gives the same bound from a different angle and converts freely to KMP's table.",{"title":148,"path":149,"lessonNumber":102,"topics":150,"summary":151},"Tries & Prefix Trees","\u002Falgorithms\u002Fsequences\u002Ftries",[145],"A **trie** stores a set of strings in a tree keyed by _characters_, so that insert, search, and prefix-test all run in $O(L)$ time — the length of the key, _independent of how many keys are stored_. Shared prefixes are stored once, which makes tries the natural structure for autocomplete, wildcard dictionaries, board word-search, and — over the alphabet $\\{0,1\\}$ — the maximum-XOR-pair problem.",{"module":153,"moduleNumber":108,"slug":154,"lessons":155},"Graphs","graphs",[156,163,168,173,178,183,188,193,199],{"title":157,"path":158,"lessonNumber":6,"topics":159,"summary":162},"Graph Representations and Traversal","\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal",[160,161],"Graph Representations","Graph Traversal","A graph captures _relationships_ — who connects to whom. We fix the vocabulary, weigh the two standard representations (adjacency list versus matrix), then meet the two explorations you'll use constantly: breadth-first search, which finds shortest paths by number of edges, and depth-first search, whose discovery and finish times reveal a graph's hidden structure. Both run in $O(V + E)$.",{"title":164,"path":165,"lessonNumber":18,"topics":166,"summary":167},"Topological Sort and Strong Connectivity","\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc",[161],"Directed acyclic graphs model dependencies: tasks that must precede other tasks. A _topological order_ lays such a graph out in a line so every edge points forward, and depth-first finish times hand it to us almost for free. We then ask the harder question for graphs _with_ cycles: which vertices can reach each other? The answer is the strongly connected components, found by a two-pass DFS.",{"title":169,"path":170,"lessonNumber":24,"topics":171,"summary":172},"Minimum Spanning Trees","\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees",[169],"Given a weighted network, how do we connect everything as cheaply as possible? The answer is a minimum spanning tree. One lemma, the cut property, justifies _every_ correct MST algorithm, and from it two famous greedy methods fall out: Kruskal's, which grows a forest edge by edge with a union-find structure, and Prim's, which grows a single tree using a priority queue.",{"title":174,"path":175,"lessonNumber":73,"topics":176,"summary":177},"Shortest Paths","\u002Falgorithms\u002Fgraphs\u002Fshortest-paths",[174],"Finding the cheapest route through a weighted network is one of the most-used algorithms in computing. A single operation — _relaxation_ — underlies them all. Dijkstra's algorithm solves the non-negative case greedily; Bellman-Ford handles negative edges and detects negative cycles; and Floyd-Warshall finds the shortest path between _every_ pair of vertices.",{"title":179,"path":180,"lessonNumber":102,"topics":181,"summary":182},"Network Flow","\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow",[179],"How much can flow through a network from source to sink? Max-flow is a surprisingly general model — once you see a problem as flow, a whole toolbox opens up. We build flow networks, find maximum flows by repeatedly pushing along augmenting paths in the residual graph, prove the max-flow min-cut theorem, and watch bipartite matching fall out as a special case.",{"title":184,"path":185,"lessonNumber":108,"topics":186,"summary":187},"Bridges & Articulation Points","\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points",[153],"A **bridge** is an edge whose removal disconnects the graph; an **articulation point** is a vertex whose removal does. Both are single points of failure in a network. A single depth-first search computes discovery times and **low-links**, and two local criteria — $low[v] > disc[u]$ for bridges, $low[v] \\ge disc[u]$ for cut vertices — find them all in $O(V+E)$.",{"title":189,"path":190,"lessonNumber":116,"topics":191,"summary":192},"Lowest Common Ancestor & Binary Lifting","\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor",[153],"Given a rooted tree, the lowest common ancestor of $u$ and $v$ is the deepest node that is an ancestor of both. A naive walk answers one query in $O(h)$; **binary lifting** precomputes the $2^k$-th ancestor of every node in $O(n\\log n)$, then answers $k$-th-ancestor and LCA queries in $O(\\log n)$ each. We derive both jumps, apply them to tree distance, and compare against the Euler-tour + RMQ and Tarjan offline alternatives.",{"title":194,"path":195,"lessonNumber":196,"topics":197,"summary":198},"2-SAT via Implication Graphs","\u002Falgorithms\u002Fgraphs\u002Ftwo-sat",8,[153],"A boolean formula whose every clause has exactly two literals can be solved in _linear_ time — even though its three-literal cousin is NP-complete. The trick is to read each clause as a pair of implications, build a directed graph on the $2n$ literals, and ask a question we already know how to answer: which literals share a strongly connected component? The formula is satisfiable iff no variable lands in the same SCC as its own negation, and the SCCs' topological order hands us a satisfying assignment for free.",{"title":200,"path":201,"lessonNumber":202,"topics":203,"summary":204},"Eulerian Tours","\u002Falgorithms\u002Fgraphs\u002Feulerian-tours",9,[153],"An **Eulerian tour** uses every _edge_ of a graph exactly once. We give the exact parity and balance conditions under which one exists (even degree for undirected graphs, in-degree equal to out-degree for directed) and Hierholzer's $O(E)$ algorithm that constructs one by splicing closed sub-tours. We contrast this sharply with the **Hamiltonian** problem (visit every _vertex_ once), which is NP-complete: visiting edges is easy, visiting vertices is hard.",{"module":206,"moduleNumber":116,"slug":207,"lessons":208},"Greedy Algorithms","greedy",[209,214,220,225],{"title":210,"path":211,"lessonNumber":6,"topics":212,"summary":213},"The Greedy Method","\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method",[206],"A greedy algorithm builds a solution one locally-best choice at a time and never looks back. We pin down the two properties that make this work — the greedy-choice property and optimal substructure — prove the canonical activity-selection algorithm correct with an exchange argument, watch greedy fail spectacularly on the 0\u002F1 knapsack, and glimpse matroids as the theory that says exactly when greed is good.",{"title":215,"path":216,"lessonNumber":18,"topics":217,"summary":219},"Scheduling & Interval Partitioning","\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals",[218],"Greedy","Three classic scheduling problems all yield to greedy algorithms — and all three turn on a single design decision: which key to sort by. Interval scheduling sorts by **finish** time to pack the most compatible jobs; interval partitioning sorts by **start** time and proves the rooms needed equal the maximum overlap **depth**; minimizing maximum lateness sorts by **deadline** and is justified by an adjacent-swap exchange argument.",{"title":221,"path":222,"lessonNumber":24,"topics":223,"summary":224},"Huffman Codes","\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes",[206],"Huffman coding is the greedy method's most beautiful application: it builds a provably optimal prefix-free binary code by repeatedly merging the two least frequent symbols. We develop prefix-free codes as binary trees, give the algorithm with a priority queue, build a Huffman tree from example frequencies, prove optimality with the same greedy-choice-plus-substructure argument, and pin the running time at $O(n\\log n)$.",{"title":226,"path":227,"lessonNumber":73,"topics":228,"summary":229},"Matroids & Exchange Arguments","\u002Falgorithms\u002Fgreedy\u002Fmatroids",[218],"The capstone of the greedy module: _why_ and _when_ a greedy algorithm is provably optimal. We recap the two correctness templates — **greedy-stays-ahead** and the **exchange argument** — then meet the **matroid** $M=(S,\\mathcal{I})$, an abstraction whose **exchange property** is exactly the structure greedy needs. The matroid–greedy theorem says sorting by weight and taking what stays independent yields a maximum-weight basis _if and only if_ the structure is a matroid. Kruskal's MST is the canonical instance; 0\u002F1 knapsack and TSP are the canonical failures.",{"module":231,"moduleNumber":196,"slug":232,"lessons":233},"Dynamic Programming","dynamic-programming",[234,239,245,250,255,260,265,270,275,280],{"title":235,"path":236,"lessonNumber":6,"topics":237,"summary":238},"Principles of Dynamic Programming","\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples",[231,26],"Dynamic programming is recursion with memory: when a recursive solution re-solves the same subproblems again and again, we solve each one once and store the answer. We pin down the two structural conditions that make this work — overlapping subproblems and optimal substructure — contrast top-down memoization with bottom-up tabulation, and distil the whole method into a five-step recipe.",{"title":240,"path":241,"lessonNumber":18,"topics":242,"summary":244},"Sequence Alignment & LCS","\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp",[231,243],"String Structures","Two strings can be compared by asking how much of one survives inside the other. The longest common subsequence (LCS) and edit distance are the two classic answers, and they are the _same_ dynamic program wearing different costs. We derive the LCS recurrence by examining the last characters, fill a worked DP table, reconstruct the subsequence, and then show edit distance as the identical $\\Theta(mn)$ pattern.",{"title":246,"path":247,"lessonNumber":24,"topics":248,"summary":249},"Longest Increasing Subsequence","\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence",[231],"Given a sequence of numbers, how long is its longest strictly increasing subsequence? A first dynamic program indexes subproblems by the element each subsequence _ends at_, giving an $O(n^2)$ solution with parent-pointer reconstruction. A sharper idea, the patience-sorting _tails_ array searched by binary search, drops the time to $O(n\\log n)$. We then fold in the variants: non-decreasing, counting, Russian-doll envelopes, and bitonic.",{"title":251,"path":252,"lessonNumber":73,"topics":253,"summary":254},"Knapsack & Subset Problems","\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack",[231],"We start from $\\textsc{Subset-sum}$ — does some sublist hit a target $t$? — and its include\u002Fexclude recurrence over a boolean table $A(i, u)$, then bolt on values to get 0\u002F1 knapsack as the same machine with $\\lor$ promoted to $\\max$. We fill both tables, recover the chosen items, and confront the surprise that the $\\Theta(nt)$ running time is only _pseudo-polynomial_ — exponential in the bit length $b$, and unimprovable unless $\\mathrm{P}=\\mathrm{NP}$ since subset-sum is $\\textsc{NP-complete}$. The fractional variant reveals the sharp line between greedy and dynamic programming.",{"title":256,"path":257,"lessonNumber":102,"topics":258,"summary":259},"Coin Change & Unbounded Knapsack","\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded",[231],"The previous lesson let each item be taken at most once. Drop that cap — items may be reused _any number of times_ — and the 0\u002F1 knapsack collapses from a two-dimensional table to a one-dimensional one, because there is no longer a prefix of \"already-used\" items to track. We meet **unbounded knapsack**, then its most famous instance, **coin change**: the minimum-coins recurrence $C[a] = 1 + \\min_c C[a-c]$, and the counting variant where the _order of the loops_ decides whether you count unordered combinations or ordered sequences — the classic bug. Greed fails in general but works for canonical coin systems.",{"title":261,"path":262,"lessonNumber":108,"topics":263,"summary":264},"Interval DP","\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp",[231],"Many problems ask for the best way to combine a contiguous range of items, and the answer is a dynamic program over subintervals $[i,j]$ that chooses a split point $k$. We derive the pattern from matrix-chain multiplication — parenthesising a product to minimize scalar multiplications in $O(n^3)$ — distil it into a reusable template filled by increasing interval length, and then meet its sharpest variant: the \"last operation\" trick behind Burst Balloons and cutting a stick, where fixing the _last_ move (not the first) makes the two sides independent.",{"title":266,"path":267,"lessonNumber":116,"topics":268,"summary":269},"Dynamic Programming on Trees","\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp",[231],"When the subproblems of a dynamic program are _rooted subtrees_, a single post-order DFS solves the whole thing in $O(n)$: each node combines the already-computed answers of its children. We meet the archetype — maximum-weight independent set on a tree — then the \"path through a node\" pattern behind tree diameter and maximum path sum, and finally **rerooting**, which computes a per-node answer for _every_ node as root in $O(n)$ with two passes.",{"title":271,"path":272,"lessonNumber":196,"topics":273,"summary":274},"Bitmask DP","\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp",[231],"When a subproblem depends not on an index or a prefix but on _which subset_ of a small ground set has been used, we can encode that subset as the bits of an integer and index a DP table by it. With $n \\le \\sim 20$ the $2^n$ subsets fit in a table, turning $\\Theta(n!)$ brute force into $O(2^n \\cdot \\text{poly}(n))$. We meet the bit tricks, the Held–Karp TSP archetype, assignment by mask, subset-sum partitioning, and submask enumeration with its $3^n$ bound.",{"title":276,"path":277,"lessonNumber":202,"topics":278,"summary":279},"DP Optimizations","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations",[231],"A correct DP recurrence is only half the battle; its naive evaluation is often a factor of $n$ slower than necessary. This capstone surveys five techniques, monotonic-queue, the convex hull trick, divide-and-conquer optimization, Knuth's optimization, and SOS DP, that each exploit _structure in the transition_ (a sliding window, linear costs, monotone optimal splits, the quadrangle inequality, or subset lattices) to shave an $O(n)$, $O(\\log n)$, or worse factor off the running time.",{"title":281,"path":282,"lessonNumber":283,"topics":284,"summary":285},"Dynamic Programming on Graphs","\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs",10,[231],"Many graph algorithms are dynamic programs in disguise: the subproblem is the _best value reachable under a restricted resource_ — intermediate vertices allowed, edges allowed, or a topological prefix — and edge _relaxation_ is the DP transition. We frame Floyd–Warshall as the archetype ($O(V^3)$ all-pairs shortest paths), Bellman–Ford as a DP over path length (the at-most-$K$-stops variant), DAG-DP in topological order ($O(V+E)$), and Warshall's transitive closure as the boolean analog.",{"module":287,"moduleNumber":202,"slug":288,"lessons":289},"Backtracking & Search","backtracking",[290,296,301],{"title":291,"path":292,"lessonNumber":6,"topics":293,"summary":295},"Backtracking: Subsets, Permutations & Combinations","\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals",[294],"Backtracking","Backtracking builds a solution one choice at a time and abandons a partial solution the moment it cannot be completed, exploring a state-space tree by depth-first search. We meet the universal choose\u002Fexplore\u002Fun-choose template, derive the canonical enumerations — subsets ($2^n$), permutations ($n!$), and combinations ($\\binom{n}{k}$) — handle duplicate elements by skipping equal siblings, and see how pruning turns an exponential search into a tractable one.",{"title":297,"path":298,"lessonNumber":18,"topics":299,"summary":300},"Constraint Search: N-Queens & Sudoku","\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search",[294],"Many hard puzzles are **constraint satisfaction problems**: assign each variable a value from its domain so that every constraint holds. Backtracking solves them by assigning variables one at a time and rejecting a partial assignment the instant a constraint breaks. We make the rejection cheap — $O(1)$ conflict checks for N-Queens via column and diagonal sets — and prune harder with **forward checking**, **MRV** ordering, and **constraint propagation**, which is what lets an exponential search actually finish.",{"title":302,"path":303,"lessonNumber":24,"topics":304,"summary":305},"Branch & Bound and Meet in the Middle","\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound",[294],"Plain backtracking prunes a search tree by _feasibility_; for _optimization_ problems we can prune far more aggressively by _value_. **Branch and bound** keeps the best complete solution found so far and discards any partial solution whose optimistic bound cannot beat it. **Meet in the middle** splits the instance in two, enumerates each half, and recombines by binary search — turning $2^n$ into $O(2^{n\u002F2}\\,n)$ and pushing exact search out to $n \\approx 40$.",{"module":307,"moduleNumber":283,"slug":308,"lessons":309},"Mathematical Algorithms","mathematical-algorithms",[310,316,321,326],{"title":311,"path":312,"lessonNumber":6,"topics":313,"summary":315},"Number Theory: GCD & Modular Arithmetic","\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics",[314],"Number Theory","This lesson opens the mathematical-algorithms module with the bedrock of computational number theory. We prove Euclid's recurrence $\\gcd(a,b)=\\gcd(b,\\,a\\bmod b)$ and its $O(\\log\\min(a,b))$ running time, extend it to recover Bézout coefficients $x,y$ with $ax+by=\\gcd(a,b)$, and build modular arithmetic on residue classes — including when a modular inverse $a^{-1}\\bmod m$ exists and how to compute it.",{"title":317,"path":318,"lessonNumber":18,"topics":319,"summary":320},"Modular Exponentiation & Primality","\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality",[314],"Computing $a^n \\bmod m$ naively costs $n$ multiplications; **repeated squaring** does it in $O(\\log n)$ by reading the bits of the exponent. We use this routine to state **Fermat's little theorem** (and the modular inverse it gives), then to test primality — trial division, the probabilistic **Fermat** and **Miller–Rabin** tests, and the deterministic witness set that settles primality for every 64-bit number.",{"title":322,"path":323,"lessonNumber":24,"topics":324,"summary":325},"Sieves & Factorization","\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization",[314],"The previous lesson tested one number for primality; here we ask for _all_ primes up to $n$ at once. The **sieve of Eratosthenes** cross-cuts composites in $O(n\\log\\log n)$, and a **linear sieve** does it in $O(n)$ while recording each number's **smallest prime factor**, which then factors any $x \\le n$ in $O(\\log x)$. From a factorization $x = \\prod p_i^{e_i}$ the multiplicative functions $\\tau$, $\\sigma$, and Euler's totient $\\varphi$ fall out immediately.",{"title":327,"path":328,"lessonNumber":73,"topics":329,"summary":330},"Combinatorics & Counting","\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics",[314],"Counting is the arithmetic of finite sets. We build up from permutations $n!$ and combinations $\\binom{n}{k}$, prove Pascal's rule by a bijection, and count multisets with stars and bars. The practical core is computing $\\binom{n}{k}\\bmod p$ in $O(1)$ from precomputed factorials and inverse factorials. We close with inclusion–exclusion and the Chinese Remainder Theorem, both of which lean on the modular inverse from the previous lesson.",{"module":332,"moduleNumber":333,"slug":334,"lessons":335},"Computational Geometry",11,"computational-geometry",[336,342,347],{"title":337,"path":338,"lessonNumber":6,"topics":339,"summary":341},"Geometric Primitives & Orientation","\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives",[340],"Geometry","Computational geometry is built on a single reliable primitive — the **orientation test**, a sign of a cross product that tells whether three points turn left, right, or lie collinear. From points-as-vectors and the dot and cross products we derive orientation, segment intersection, the shoelace area formula, and point-in-polygon tests, keeping all arithmetic **exact and integer** so that no floating-point rounding can corrupt a sign.",{"title":343,"path":344,"lessonNumber":18,"topics":345,"summary":346},"Convex Hull","\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull",[340],"The convex hull is the smallest convex polygon enclosing a point set — the rubber band snapped around the nails. We build it with Andrew's monotone chain, sorting by $(x,y)$ and sweeping a lower and upper hull while popping any non-left turn via the orientation primitive, in $O(n\\log n)$. A reduction from sorting shows that bound is optimal, and the hull unlocks diameter, smallest enclosing rectangle, and more through rotating calipers.",{"title":348,"path":349,"lessonNumber":24,"topics":350,"summary":351},"Sweep-Line Algorithms","\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line",[340],"The plane-sweep paradigm turns a static $2$-D geometry problem into a dynamic $1$-D ordered-set problem: a vertical line sweeps left to right, stopping at an $x$-sorted **event queue** while a balanced-BST **status structure** tracks the objects it currently crosses, ordered by $y$. We derive Bentley–Ottmann segment intersection in $O((n+k)\\log n)$, recover closest-pair in $O(n\\log n)$, and reduce skyline, rectangle-area, and overlap problems to $\\pm1$ event sweeps.",{"module":353,"moduleNumber":354,"slug":355,"lessons":356},"Intractability",12,"intractability",[357,363,367],{"title":358,"path":359,"lessonNumber":6,"topics":360,"summary":362},"P, NP, and Reductions","\u002Falgorithms\u002Fintractability\u002Fp-np-reductions",[361],"NP-Completeness","Most problems we have met so far have fast algorithms. A vast and important family seemingly does not. This lesson builds the vocabulary for that divide: decision problems, the class $\\mathsf{P}$ of problems we can solve quickly, the class $\\mathsf{NP}$ of problems whose solutions we can _check_ quickly, and polynomial-time reductions, the tool that lets us compare the difficulty of two problems without solving either.",{"title":361,"path":364,"lessonNumber":18,"topics":365,"summary":366},"\u002Falgorithms\u002Fintractability\u002Fnp-completeness",[361],"Some problems in $\\mathsf{NP}$ are universally hardest: every other problem in $\\mathsf{NP}$ reduces to them. This lesson defines $\\mathsf{NP}$-hard and $\\mathsf{NP}$-complete, states the Cook–Levin theorem that anchors the whole edifice on **SAT**, walks the web of reductions that grows from it, and gives the four-step recipe for proving a brand-new problem $\\mathsf{NP}$-complete.",{"title":368,"path":369,"lessonNumber":24,"topics":370,"summary":373},"Coping with NP-Hardness","\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness",[371,372],"Approximation","Heuristics","Proving a problem $\\mathsf{NP}$-hard is the beginning, not the end. The world still needs answers. This lesson surveys the four honest responses to hardness: approximation algorithms with a provable ratio (worked through a 2-approximation for vertex cover), heuristics and local search, exact exponential methods like branch and bound, and exploiting special structure in the instances you actually face.",{"id":375,"title":174,"blurb":376,"body":377,"description":12449,"extension":12450,"meta":12451,"module":153,"navigation":12453,"path":175,"practice":12454,"rawbody":12468,"readingTime":12469,"seo":12474,"sources":12475,"status":12482,"stem":12483,"summary":177,"topics":12484,"__hash__":12485},"course\u002F01.algorithms\u002F06.graphs\u002F04.shortest-paths.md","",{"type":378,"value":379,"toc":12440},"minimark",[380,399,404,1132,1339,1380,1415,1551,1840,2303,2367,2618,2622,2732,2809,3101,4037,4114,4286,4777,5029,5033,5121,5159,5260,5397,5569,5864,6139,6199,6519,6628,6693,6896,7445,7585,8012,8071,8075,8197,8585,8944,9472,11122,11187,11232,11420,11424,11812,11837,11841,12334,12436],[381,382,383,384,388,389,393,394,398],"p",{},"Every navigation app, every network router, every game pathfinder is solving the\nsame problem: given a weighted graph, find the cheapest route from one place to\nanother. ",[385,386,387],"a",{"href":158},"BFS"," already solved\nthis when every edge counts as one step. Now the edges carry ",[390,391,392],"strong",{},"weights"," (distances,\ntimes, costs), and we want to minimize the ",[395,396,397],"em",{},"total"," weight along a path. This lesson\nbuilds the shortest-path toolkit from a single primitive shared by every algorithm\nin it.",[400,401,403],"h2",{"id":402},"the-problem-and-its-primitive","The problem and its primitive",[405,406,408],"callout",{"type":407},"definition",[381,409,410,413,414,488,489,527,528,531,532,743,744,968,969,972,1007,1008,1024,1025,1040,1041,1057,1058,1061,1062,1096,1097,1024,1112,1115,1116,1131],{},[390,411,412],{},"Definition (Shortest-path distance)."," Given a weighted directed graph ",[415,416,419],"span",{"className":417},[418],"katex",[415,420,424,452],{"className":421,"ariaHidden":423},[422],"katex-html","true",[415,425,428,433,439,444,449],{"className":426},[427],"base",[415,429],{"className":430,"style":432},[431],"strut","height:0.6833em;",[415,434,438],{"className":435},[436,437],"mord","mathnormal","G",[415,440],{"className":441,"style":443},[442],"mspace","margin-right:0.2778em;",[415,445,448],{"className":446},[447],"mrel","=",[415,450],{"className":451,"style":443},[442],[415,453,455,459,464,469,474,478,483],{"className":454},[427],[415,456],{"className":457,"style":458},[431],"height:1em;vertical-align:-0.25em;",[415,460,463],{"className":461},[462],"mopen","(",[415,465,468],{"className":466,"style":467},[436,437],"margin-right:0.2222em;","V",[415,470,473],{"className":471},[472],"mpunct",",",[415,475],{"className":476,"style":477},[442],"margin-right:0.1667em;",[415,479,482],{"className":480,"style":481},[436,437],"margin-right:0.0576em;","E",[415,484,487],{"className":485},[486],"mclose",")"," with weight ",[415,490,492],{"className":491},[418],[415,493,495],{"className":494,"ariaHidden":423},[422],[415,496,498,501,506,509,513,516,519,524],{"className":497},[427],[415,499],{"className":500,"style":458},[431],[415,502,505],{"className":503,"style":504},[436,437],"margin-right:0.0269em;","w",[415,507,463],{"className":508},[462],[415,510,512],{"className":511},[436,437],"u",[415,514,473],{"className":515},[472],[415,517],{"className":518,"style":477},[442],[415,520,523],{"className":521,"style":522},[436,437],"margin-right:0.0359em;","v",[415,525,487],{"className":526},[486]," on each\nedge, the ",[390,529,530],{},"weight of a path"," ",[415,533,535],{"className":534},[418],[415,536,538,557],{"className":537,"ariaHidden":423},[422],[415,539,541,545,548,551,554],{"className":540},[427],[415,542],{"className":543,"style":544},[431],"height:0.625em;vertical-align:-0.1944em;",[415,546,381],{"className":547},[436,437],[415,549],{"className":550,"style":443},[442],[415,552,448],{"className":553},[447],[415,555],{"className":556,"style":443},[442],[415,558,560,563],{"className":559},[427],[415,561],{"className":562,"style":458},[431],[415,564,567,573,630,633,636,677,680,683,687,690,693,696,739],{"className":565},[566],"minner",[415,568,572],{"className":569,"style":571},[462,570],"delimcenter","top:0em;","⟨",[415,574,576,579],{"className":575},[436],[415,577,523],{"className":578,"style":522},[436,437],[415,580,583],{"className":581},[582],"msupsub",[415,584,588,621],{"className":585},[586,587],"vlist-t","vlist-t2",[415,589,592,616],{"className":590},[591],"vlist-r",[415,593,597],{"className":594,"style":596},[595],"vlist","height:0.3011em;",[415,598,600,605],{"style":599},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[415,601],{"className":602,"style":604},[603],"pstrut","height:2.7em;",[415,606,612],{"className":607},[608,609,610,611],"sizing","reset-size6","size3","mtight",[415,613,615],{"className":614},[436,611],"0",[415,617,620],{"className":618},[619],"vlist-s","​",[415,622,624],{"className":623},[591],[415,625,628],{"className":626,"style":627},[595],"height:0.15em;",[415,629],{},[415,631,473],{"className":632},[472],[415,634],{"className":635,"style":477},[442],[415,637,639,642],{"className":638},[436],[415,640,523],{"className":641,"style":522},[436,437],[415,643,645],{"className":644},[582],[415,646,648,669],{"className":647},[586,587],[415,649,651,666],{"className":650},[591],[415,652,654],{"className":653,"style":596},[595],[415,655,656,659],{"style":599},[415,657],{"className":658,"style":604},[603],[415,660,662],{"className":661},[608,609,610,611],[415,663,665],{"className":664},[436,611],"1",[415,667,620],{"className":668},[619],[415,670,672],{"className":671},[591],[415,673,675],{"className":674,"style":627},[595],[415,676],{},[415,678,473],{"className":679},[472],[415,681],{"className":682,"style":477},[442],[415,684,686],{"className":685},[566],"…",[415,688],{"className":689,"style":477},[442],[415,691,473],{"className":692},[472],[415,694],{"className":695,"style":477},[442],[415,697,699,702],{"className":698},[436],[415,700,523],{"className":701,"style":522},[436,437],[415,703,705],{"className":704},[582],[415,706,708,731],{"className":707},[586,587],[415,709,711,728],{"className":710},[591],[415,712,715],{"className":713,"style":714},[595],"height:0.3361em;",[415,716,717,720],{"style":599},[415,718],{"className":719,"style":604},[603],[415,721,723],{"className":722},[608,609,610,611],[415,724,727],{"className":725,"style":726},[436,437,611],"margin-right:0.0315em;","k",[415,729,620],{"className":730},[619],[415,732,734],{"className":733},[591],[415,735,737],{"className":736,"style":627},[595],[415,738],{},[415,740,742],{"className":741,"style":571},[486,570],"⟩"," is\n",[415,745,747],{"className":746},[418],[415,748,750,777],{"className":749,"ariaHidden":423},[422],[415,751,753,756,759,762,765,768,771,774],{"className":752},[427],[415,754],{"className":755,"style":458},[431],[415,757,505],{"className":758,"style":504},[436,437],[415,760,463],{"className":761},[462],[415,763,381],{"className":764},[436,437],[415,766,487],{"className":767},[486],[415,769],{"className":770,"style":443},[442],[415,772,448],{"className":773},[447],[415,775],{"className":776,"style":443},[442],[415,778,780,784,857,860,863,866,919,922,925,965],{"className":779},[427],[415,781],{"className":782,"style":783},[431],"height:1.2887em;vertical-align:-0.2997em;",[415,785,788,795],{"className":786},[787],"mop",[415,789,794],{"className":790,"style":793},[787,791,792],"op-symbol","small-op","position:relative;top:0em;","∑",[415,796,798],{"className":797},[582],[415,799,801,848],{"className":800},[586,587],[415,802,804,845],{"className":803},[591],[415,805,808,830],{"className":806,"style":807},[595],"height:0.989em;",[415,809,811,814],{"style":810},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[415,812],{"className":813,"style":604},[603],[415,815,817],{"className":816},[608,609,610,611],[415,818,820,824,827],{"className":819},[436,611],[415,821,823],{"className":822},[436,437,611],"i",[415,825,448],{"className":826},[447,611],[415,828,665],{"className":829},[436,611],[415,831,833,836],{"style":832},"top:-3.2029em;margin-right:0.05em;",[415,834],{"className":835,"style":604},[603],[415,837,839],{"className":838},[608,609,610,611],[415,840,842],{"className":841},[436,611],[415,843,727],{"className":844,"style":726},[436,437,611],[415,846,620],{"className":847},[619],[415,849,851],{"className":850},[591],[415,852,855],{"className":853,"style":854},[595],"height:0.2997em;",[415,856],{},[415,858],{"className":859,"style":477},[442],[415,861,505],{"className":862,"style":504},[436,437],[415,864,463],{"className":865},[462],[415,867,869,872],{"className":868},[436],[415,870,523],{"className":871,"style":522},[436,437],[415,873,875],{"className":874},[582],[415,876,878,910],{"className":877},[586,587],[415,879,881,907],{"className":880},[591],[415,882,885],{"className":883,"style":884},[595],"height:0.3117em;",[415,886,887,890],{"style":599},[415,888],{"className":889,"style":604},[603],[415,891,893],{"className":892},[608,609,610,611],[415,894,896,899,904],{"className":895},[436,611],[415,897,823],{"className":898},[436,437,611],[415,900,903],{"className":901},[902,611],"mbin","−",[415,905,665],{"className":906},[436,611],[415,908,620],{"className":909},[619],[415,911,913],{"className":912},[591],[415,914,917],{"className":915,"style":916},[595],"height:0.2083em;",[415,918],{},[415,920,473],{"className":921},[472],[415,923],{"className":924,"style":477},[442],[415,926,928,931],{"className":927},[436],[415,929,523],{"className":930,"style":522},[436,437],[415,932,934],{"className":933},[582],[415,935,937,957],{"className":936},[586,587],[415,938,940,954],{"className":939},[591],[415,941,943],{"className":942,"style":884},[595],[415,944,945,948],{"style":599},[415,946],{"className":947,"style":604},[603],[415,949,951],{"className":950},[608,609,610,611],[415,952,823],{"className":953},[436,437,611],[415,955,620],{"className":956},[619],[415,958,960],{"className":959},[591],[415,961,963],{"className":962,"style":627},[595],[415,964],{},[415,966,487],{"className":967},[486],". The ",[390,970,971],{},"shortest-path distance",[415,973,975],{"className":974},[418],[415,976,978],{"className":977,"ariaHidden":423},[422],[415,979,981,984,989,992,995,998,1001,1004],{"className":980},[427],[415,982],{"className":983,"style":458},[431],[415,985,988],{"className":986,"style":987},[436,437],"margin-right:0.0379em;","δ",[415,990,463],{"className":991},[462],[415,993,512],{"className":994},[436,437],[415,996,473],{"className":997},[472],[415,999],{"className":1000,"style":477},[442],[415,1002,523],{"className":1003,"style":522},[436,437],[415,1005,487],{"className":1006},[486]," is the minimum weight over all paths from ",[415,1009,1011],{"className":1010},[418],[415,1012,1014],{"className":1013,"ariaHidden":423},[422],[415,1015,1017,1021],{"className":1016},[427],[415,1018],{"className":1019,"style":1020},[431],"height:0.4306em;",[415,1022,512],{"className":1023},[436,437]," to ",[415,1026,1028],{"className":1027},[418],[415,1029,1031],{"className":1030,"ariaHidden":423},[422],[415,1032,1034,1037],{"className":1033},[427],[415,1035],{"className":1036,"style":1020},[431],[415,1038,523],{"className":1039,"style":522},[436,437]," (or\n",[415,1042,1044],{"className":1043},[418],[415,1045,1047],{"className":1046,"ariaHidden":423},[422],[415,1048,1050,1053],{"className":1049},[427],[415,1051],{"className":1052,"style":1020},[431],[415,1054,1056],{"className":1055},[436],"∞"," if none exists). The ",[390,1059,1060],{},"single-source shortest paths"," (SSSP) problem\nasks for ",[415,1063,1065],{"className":1064},[418],[415,1066,1068],{"className":1067,"ariaHidden":423},[422],[415,1069,1071,1074,1077,1080,1084,1087,1090,1093],{"className":1070},[427],[415,1072],{"className":1073,"style":458},[431],[415,1075,988],{"className":1076,"style":987},[436,437],[415,1078,463],{"className":1079},[462],[415,1081,1083],{"className":1082},[436,437],"s",[415,1085,473],{"className":1086},[472],[415,1088],{"className":1089,"style":477},[442],[415,1091,523],{"className":1092,"style":522},[436,437],[415,1094,487],{"className":1095},[486]," from a fixed source ",[415,1098,1100],{"className":1099},[418],[415,1101,1103],{"className":1102,"ariaHidden":423},[422],[415,1104,1106,1109],{"className":1105},[427],[415,1107],{"className":1108,"style":1020},[431],[415,1110,1083],{"className":1111},[436,437],[395,1113,1114],{},"every"," vertex ",[415,1117,1119],{"className":1118},[418],[415,1120,1122],{"className":1121,"ariaHidden":423},[422],[415,1123,1125,1128],{"className":1124},[427],[415,1126],{"className":1127,"style":1020},[431],[415,1129,523],{"className":1130,"style":522},[436,437],".",[381,1133,1134,1135,1150,1151,1174,1175,1178,1179,1212,1213,1230,1231,1253,1254,1257,1258,1298,1299,1338],{},"Every algorithm maintains two arrays. For each vertex ",[415,1136,1138],{"className":1137},[418],[415,1139,1141],{"className":1140,"ariaHidden":423},[422],[415,1142,1144,1147],{"className":1143},[427],[415,1145],{"className":1146,"style":1020},[431],[415,1148,523],{"className":1149,"style":522},[436,437],", an estimate ",[415,1152,1154],{"className":1153},[418],[415,1155,1157],{"className":1156,"ariaHidden":423},[422],[415,1158,1160,1164,1167,1170],{"className":1159},[427],[415,1161],{"className":1162,"style":1163},[431],"height:0.6944em;",[415,1165,523],{"className":1166,"style":522},[436,437],[415,1168,1131],{"className":1169},[436],[415,1171,1173],{"className":1172},[436,437],"d"," is\nan ",[395,1176,1177],{},"upper bound"," on ",[415,1180,1182],{"className":1181},[418],[415,1183,1185],{"className":1184,"ariaHidden":423},[422],[415,1186,1188,1191,1194,1197,1200,1203,1206,1209],{"className":1187},[427],[415,1189],{"className":1190,"style":458},[431],[415,1192,988],{"className":1193,"style":987},[436,437],[415,1195,463],{"className":1196},[462],[415,1198,1083],{"className":1199},[436,437],[415,1201,473],{"className":1202},[472],[415,1204],{"className":1205,"style":477},[442],[415,1207,523],{"className":1208,"style":522},[436,437],[415,1210,487],{"className":1211},[486],", always ",[415,1214,1216],{"className":1215},[418],[415,1217,1219],{"className":1218,"ariaHidden":423},[422],[415,1220,1222,1226],{"className":1221},[427],[415,1223],{"className":1224,"style":1225},[431],"height:0.7719em;vertical-align:-0.136em;",[415,1227,1229],{"className":1228},[447],"≥"," the true distance, shrinking\ntoward it. A predecessor ",[415,1232,1234],{"className":1233},[418],[415,1235,1237],{"className":1236,"ariaHidden":423},[422],[415,1238,1240,1243,1246,1249],{"className":1239},[427],[415,1241],{"className":1242,"style":1020},[431],[415,1244,523],{"className":1245,"style":522},[436,437],[415,1247,1131],{"className":1248},[436],[415,1250,1252],{"className":1251,"style":522},[436,437],"π"," records the previous vertex on the best path\nfound so far, forming a ",[390,1255,1256],{},"shortest-path tree",". We initialize ",[415,1259,1261],{"className":1260},[418],[415,1262,1264,1288],{"className":1263,"ariaHidden":423},[422],[415,1265,1267,1270,1273,1276,1279,1282,1285],{"className":1266},[427],[415,1268],{"className":1269,"style":1163},[431],[415,1271,1083],{"className":1272},[436,437],[415,1274,1131],{"className":1275},[436],[415,1277,1173],{"className":1278},[436,437],[415,1280],{"className":1281,"style":443},[442],[415,1283,448],{"className":1284},[447],[415,1286],{"className":1287,"style":443},[442],[415,1289,1291,1295],{"className":1290},[427],[415,1292],{"className":1293,"style":1294},[431],"height:0.6444em;",[415,1296,615],{"className":1297},[436]," and\n",[415,1300,1302],{"className":1301},[418],[415,1303,1305,1329],{"className":1304,"ariaHidden":423},[422],[415,1306,1308,1311,1314,1317,1320,1323,1326],{"className":1307},[427],[415,1309],{"className":1310,"style":1163},[431],[415,1312,523],{"className":1313,"style":522},[436,437],[415,1315,1131],{"className":1316},[436],[415,1318,1173],{"className":1319},[436,437],[415,1321],{"className":1322,"style":443},[442],[415,1324,448],{"className":1325},[447],[415,1327],{"className":1328,"style":443},[442],[415,1330,1332,1335],{"className":1331},[427],[415,1333],{"className":1334,"style":1020},[431],[415,1336,1056],{"className":1337},[436]," for every other vertex.",[381,1340,1341,1342,1345,1346,531,1349,1364,1365,1131],{},"The one operation that updates these estimates is ",[390,1343,1344],{},"relaxation",": testing\nwhether going ",[395,1347,1348],{},"through",[415,1350,1352],{"className":1351},[418],[415,1353,1355],{"className":1354,"ariaHidden":423},[422],[415,1356,1358,1361],{"className":1357},[427],[415,1359],{"className":1360,"style":1020},[431],[415,1362,512],{"className":1363},[436,437]," improves our route to ",[415,1366,1368],{"className":1367},[418],[415,1369,1371],{"className":1370,"ariaHidden":423},[422],[415,1372,1374,1377],{"className":1373},[427],[415,1375],{"className":1376,"style":1020},[431],[415,1378,523],{"className":1379,"style":522},[436,437],[1381,1382,1386],"pre",{"className":1383,"code":1384,"language":1385,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Relax}(u, v, w)$ — try the edge $(u,v)$ as a shortcut to $v$\nnumber: 1\nif $u.d + w(u, v) \u003C v.d$ then\n  $v.d \\gets u.d + w(u, v)$ \u002F\u002F cheaper route to v via u\n  $v.\\pi \\gets u$\n","algorithm",[1387,1388,1389,1395,1400,1405,1410],"code",{"__ignoreMap":376},[415,1390,1392],{"class":1391,"line":6},"line",[415,1393,1394],{},"caption: $\\textsc{Relax}(u, v, w)$ — try the edge $(u,v)$ as a shortcut to $v$\n",[415,1396,1397],{"class":1391,"line":18},[415,1398,1399],{},"number: 1\n",[415,1401,1402],{"class":1391,"line":24},[415,1403,1404],{},"if $u.d + w(u, v) \u003C v.d$ then\n",[415,1406,1407],{"class":1391,"line":73},[415,1408,1409],{},"  $v.d \\gets u.d + w(u, v)$ \u002F\u002F cheaper route to v via u\n",[415,1411,1412],{"class":1391,"line":102},[415,1413,1414],{},"  $v.\\pi \\gets u$\n",[381,1416,1417,1418,1421,1422,1425,1426,1429,1536,1537,1540,1541,1550],{},"Relaxation never produces an estimate below the true distance, and it can only\never ",[395,1419,1420],{},"lower"," an estimate. Every shortest-path algorithm below is a different\ndiscipline for ",[395,1423,1424],{},"deciding which edges to relax, and in what order",". Two facts\nmake relaxation work: the ",[390,1427,1428],{},"triangle inequality",[415,1430,1432],{"className":1431},[418],[415,1433,1435,1472,1509],{"className":1434,"ariaHidden":423},[422],[415,1436,1438,1441,1444,1447,1450,1453,1456,1459,1462,1465,1469],{"className":1437},[427],[415,1439],{"className":1440,"style":458},[431],[415,1442,988],{"className":1443,"style":987},[436,437],[415,1445,463],{"className":1446},[462],[415,1448,1083],{"className":1449},[436,437],[415,1451,473],{"className":1452},[472],[415,1454],{"className":1455,"style":477},[442],[415,1457,523],{"className":1458,"style":522},[436,437],[415,1460,487],{"className":1461},[486],[415,1463],{"className":1464,"style":443},[442],[415,1466,1468],{"className":1467},[447],"≤",[415,1470],{"className":1471,"style":443},[442],[415,1473,1475,1478,1481,1484,1487,1490,1493,1496,1499,1502,1506],{"className":1474},[427],[415,1476],{"className":1477,"style":458},[431],[415,1479,988],{"className":1480,"style":987},[436,437],[415,1482,463],{"className":1483},[462],[415,1485,1083],{"className":1486},[436,437],[415,1488,473],{"className":1489},[472],[415,1491],{"className":1492,"style":477},[442],[415,1494,512],{"className":1495},[436,437],[415,1497,487],{"className":1498},[486],[415,1500],{"className":1501,"style":467},[442],[415,1503,1505],{"className":1504},[902],"+",[415,1507],{"className":1508,"style":467},[442],[415,1510,1512,1515,1518,1521,1524,1527,1530,1533],{"className":1511},[427],[415,1513],{"className":1514,"style":458},[431],[415,1516,505],{"className":1517,"style":504},[436,437],[415,1519,463],{"className":1520},[462],[415,1522,512],{"className":1523},[436,437],[415,1525,473],{"className":1526},[472],[415,1528],{"className":1529,"style":477},[442],[415,1531,523],{"className":1532,"style":522},[436,437],[415,1534,487],{"className":1535},[486],", and ",[390,1538,1539],{},"optimal substructure",": any\nsubpath of a shortest path is itself a shortest path.",[1542,1543,1544],"sup",{},[385,1545,665],{"href":1546,"ariaDescribedBy":1547,"dataFootnoteRef":376,"id":1549},"#user-content-fn-clrs-relax",[1548],"footnote-label","user-content-fnref-clrs-relax"," The latter is what makes\ngreedy and dynamic-programming approaches both viable.",[381,1552,1553,1554,1569,1570,1586,1587,1617,1618,1634,1635,1650,1651,1691,1692,1765,1766,1787,1788,1298,1803,1824,1825,1131],{},"One relaxation step looks like this. Before, ",[415,1555,1557],{"className":1556},[418],[415,1558,1560],{"className":1559,"ariaHidden":423},[422],[415,1561,1563,1566],{"className":1562},[427],[415,1564],{"className":1565,"style":1020},[431],[415,1567,523],{"className":1568,"style":522},[436,437]," believes its best route costs\n",[415,1571,1573],{"className":1572},[418],[415,1574,1576],{"className":1575,"ariaHidden":423},[422],[415,1577,1579,1582],{"className":1578},[427],[415,1580],{"className":1581,"style":1294},[431],[415,1583,1585],{"className":1584},[436],"9","; we test the edge ",[415,1588,1590],{"className":1589},[418],[415,1591,1593],{"className":1592,"ariaHidden":423},[422],[415,1594,1596,1599,1602,1605,1608,1611,1614],{"className":1595},[427],[415,1597],{"className":1598,"style":458},[431],[415,1600,463],{"className":1601},[462],[415,1603,512],{"className":1604},[436,437],[415,1606,473],{"className":1607},[472],[415,1609],{"className":1610,"style":477},[442],[415,1612,523],{"className":1613,"style":522},[436,437],[415,1615,487],{"className":1616},[486]," of weight ",[415,1619,1621],{"className":1620},[418],[415,1622,1624],{"className":1623,"ariaHidden":423},[422],[415,1625,1627,1630],{"className":1626},[427],[415,1628],{"className":1629,"style":1294},[431],[415,1631,1633],{"className":1632},[436],"3"," against ",[415,1636,1638],{"className":1637},[418],[415,1639,1641],{"className":1640,"ariaHidden":423},[422],[415,1642,1644,1647],{"className":1643},[427],[415,1645],{"className":1646,"style":1020},[431],[415,1648,512],{"className":1649},[436,437],"'s settled estimate\n",[415,1652,1654],{"className":1653},[418],[415,1655,1657,1681],{"className":1656,"ariaHidden":423},[422],[415,1658,1660,1663,1666,1669,1672,1675,1678],{"className":1659},[427],[415,1661],{"className":1662,"style":1163},[431],[415,1664,512],{"className":1665},[436,437],[415,1667,1131],{"className":1668},[436],[415,1670,1173],{"className":1671},[436,437],[415,1673],{"className":1674,"style":443},[442],[415,1676,448],{"className":1677},[447],[415,1679],{"className":1680,"style":443},[442],[415,1682,1684,1687],{"className":1683},[427],[415,1685],{"className":1686,"style":1294},[431],[415,1688,1690],{"className":1689},[436],"5",". Since ",[415,1693,1695],{"className":1694},[418],[415,1696,1698,1717,1735,1756],{"className":1697,"ariaHidden":423},[422],[415,1699,1701,1705,1708,1711,1714],{"className":1700},[427],[415,1702],{"className":1703,"style":1704},[431],"height:0.7278em;vertical-align:-0.0833em;",[415,1706,1690],{"className":1707},[436],[415,1709],{"className":1710,"style":467},[442],[415,1712,1505],{"className":1713},[902],[415,1715],{"className":1716,"style":467},[442],[415,1718,1720,1723,1726,1729,1732],{"className":1719},[427],[415,1721],{"className":1722,"style":1294},[431],[415,1724,1633],{"className":1725},[436],[415,1727],{"className":1728,"style":443},[442],[415,1730,448],{"className":1731},[447],[415,1733],{"className":1734,"style":443},[442],[415,1736,1738,1742,1746,1749,1753],{"className":1737},[427],[415,1739],{"className":1740,"style":1741},[431],"height:0.6835em;vertical-align:-0.0391em;",[415,1743,1745],{"className":1744},[436],"8",[415,1747],{"className":1748,"style":443},[442],[415,1750,1752],{"className":1751},[447],"\u003C",[415,1754],{"className":1755,"style":443},[442],[415,1757,1759,1762],{"className":1758},[427],[415,1760],{"className":1761,"style":1294},[431],[415,1763,1585],{"className":1764},[436],", the edge is a shortcut: ",[415,1767,1769],{"className":1768},[418],[415,1770,1772],{"className":1771,"ariaHidden":423},[422],[415,1773,1775,1778,1781,1784],{"className":1774},[427],[415,1776],{"className":1777,"style":1163},[431],[415,1779,523],{"className":1780,"style":522},[436,437],[415,1782,1131],{"className":1783},[436],[415,1785,1173],{"className":1786},[436,437]," drops to ",[415,1789,1791],{"className":1790},[418],[415,1792,1794],{"className":1793,"ariaHidden":423},[422],[415,1795,1797,1800],{"className":1796},[427],[415,1798],{"className":1799,"style":1294},[431],[415,1801,1745],{"className":1802},[436],[415,1804,1806],{"className":1805},[418],[415,1807,1809],{"className":1808,"ariaHidden":423},[422],[415,1810,1812,1815,1818,1821],{"className":1811},[427],[415,1813],{"className":1814,"style":1020},[431],[415,1816,523],{"className":1817,"style":522},[436,437],[415,1819,1131],{"className":1820},[436],[415,1822,1252],{"className":1823,"style":522},[436,437]," is rewired to point back through ",[415,1826,1828],{"className":1827},[418],[415,1829,1831],{"className":1830,"ariaHidden":423},[422],[415,1832,1834,1837],{"className":1833},[427],[415,1835],{"className":1836,"style":1020},[431],[415,1838,512],{"className":1839},[436,437],[1841,1842,1846,2088],"figure",{"className":1843},[1844,1845],"tikz-figure","tikz-diagram-rendered",[1847,1848,1853],"svg",{"xmlns":1849,"width":1850,"height":1851,"viewBox":1852},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","386.552","151.586","-75 -75 289.914 113.690",[1854,1855,1858,1878,1881,1889,1892,1899,1922,1949,1952,1955,1962,1977,1980,1986,2001,2019,2045,2060],"g",{"stroke":1856,"style":1857},"currentColor","stroke-miterlimit:10;stroke-width:.4",[1854,1859,1862,1872],{"stroke":1860,"fontFamily":1861,"fontSize":1745},"none","cmr8",[1854,1863,1865],{"transform":1864},"translate(28.497 -45.592)",[1866,1867],"path",{"d":1868,"fill":1856,"stroke":1856,"className":1869,"style":1871},"M-48.147-17.790L-48.428-17.790L-48.428-22.509Q-48.428-22.724-48.490-22.819Q-48.553-22.915-48.670-22.936Q-48.787-22.958-49.033-22.958L-49.033-23.255L-47.811-23.341L-47.811-20.852Q-47.334-21.317-46.635-21.317Q-46.154-21.317-45.746-21.073Q-45.338-20.829-45.102-20.415Q-44.865-20.001-44.865-19.517Q-44.865-19.142-45.014-18.813Q-45.162-18.485-45.432-18.233Q-45.701-17.981-46.045-17.847Q-46.389-17.712-46.748-17.712Q-47.069-17.712-47.367-17.860Q-47.666-18.009-47.873-18.270L-48.147-17.790M-47.787-20.462L-47.787-18.622Q-47.635-18.325-47.375-18.145Q-47.115-17.966-46.803-17.966Q-46.377-17.966-46.110-18.185Q-45.842-18.403-45.727-18.749Q-45.611-19.095-45.611-19.517Q-45.611-20.165-45.860-20.614Q-46.108-21.063-46.705-21.063Q-47.041-21.063-47.330-20.905Q-47.619-20.747-47.787-20.462",[1870],"tikz-text","stroke-width:0.240",[1854,1873,1874],{"transform":1864},[1866,1875],{"d":1876,"fill":1856,"stroke":1856,"className":1877,"style":1871},"M-44.102-19.544Q-44.102-20.024-43.869-20.440Q-43.637-20.856-43.227-21.106Q-42.817-21.356-42.340-21.356Q-41.610-21.356-41.211-20.915Q-40.813-20.474-40.813-19.743Q-40.813-19.638-40.906-19.614L-43.356-19.614L-43.356-19.544Q-43.356-19.134-43.235-18.778Q-43.113-18.423-42.842-18.206Q-42.570-17.989-42.141-17.989Q-41.778-17.989-41.481-18.218Q-41.184-18.446-41.082-18.798Q-41.074-18.845-40.988-18.860L-40.906-18.860Q-40.813-18.833-40.813-18.751Q-40.813-18.743-40.820-18.712Q-40.883-18.485-41.022-18.302Q-41.160-18.118-41.352-17.985Q-41.543-17.852-41.762-17.782Q-41.981-17.712-42.219-17.712Q-42.590-17.712-42.928-17.849Q-43.266-17.985-43.533-18.237Q-43.801-18.489-43.951-18.829Q-44.102-19.169-44.102-19.544M-43.348-19.852L-41.387-19.852Q-41.387-20.157-41.488-20.448Q-41.590-20.739-41.807-20.921Q-42.024-21.102-42.340-21.102Q-42.641-21.102-42.871-20.915Q-43.102-20.727-43.225-20.436Q-43.348-20.145-43.348-19.852M-38.258-17.790L-40.242-17.790L-40.242-18.087Q-39.969-18.087-39.801-18.134Q-39.633-18.181-39.633-18.349L-39.633-20.942L-40.274-20.942L-40.274-21.239L-39.633-21.239L-39.633-22.173Q-39.633-22.438-39.516-22.675Q-39.399-22.911-39.205-23.075Q-39.012-23.239-38.764-23.331Q-38.516-23.423-38.250-23.423Q-37.965-23.423-37.740-23.265Q-37.516-23.106-37.516-22.829Q-37.516-22.673-37.621-22.563Q-37.727-22.454-37.891-22.454Q-38.047-22.454-38.156-22.563Q-38.266-22.673-38.266-22.829Q-38.266-23.036-38.106-23.142Q-38.203-23.165-38.297-23.165Q-38.528-23.165-38.699-23.009Q-38.871-22.852-38.957-22.616Q-39.043-22.380-39.043-22.157L-39.043-21.239L-38.074-21.239L-38.074-20.942L-39.020-20.942L-39.020-18.349Q-39.020-18.181-38.793-18.134Q-38.567-18.087-38.258-18.087L-38.258-17.790M-37.731-19.485Q-37.731-19.989-37.475-20.421Q-37.219-20.852-36.783-21.104Q-36.348-21.356-35.848-21.356Q-35.461-21.356-35.119-21.212Q-34.778-21.067-34.516-20.806Q-34.254-20.544-34.111-20.208Q-33.969-19.872-33.969-19.485Q-33.969-18.993-34.233-18.583Q-34.496-18.173-34.926-17.942Q-35.356-17.712-35.848-17.712Q-36.340-17.712-36.774-17.944Q-37.207-18.177-37.469-18.585Q-37.731-18.993-37.731-19.485M-35.848-17.989Q-35.391-17.989-35.139-18.212Q-34.887-18.435-34.799-18.786Q-34.711-19.138-34.711-19.583Q-34.711-20.013-34.805-20.351Q-34.899-20.688-35.153-20.895Q-35.406-21.102-35.848-21.102Q-36.496-21.102-36.740-20.686Q-36.985-20.270-36.985-19.583Q-36.985-19.138-36.897-18.786Q-36.809-18.435-36.557-18.212Q-36.305-17.989-35.848-17.989M-31.477-17.790L-33.457-17.790L-33.457-18.087Q-33.188-18.087-33.020-18.132Q-32.852-18.177-32.852-18.349L-32.852-20.485Q-32.852-20.700-32.914-20.796Q-32.977-20.892-33.094-20.913Q-33.211-20.935-33.457-20.935L-33.457-21.231L-32.289-21.317L-32.289-20.532Q-32.211-20.743-32.059-20.929Q-31.906-21.114-31.707-21.216Q-31.508-21.317-31.281-21.317Q-31.035-21.317-30.844-21.173Q-30.653-21.028-30.653-20.798Q-30.653-20.642-30.758-20.532Q-30.863-20.423-31.020-20.423Q-31.176-20.423-31.285-20.532Q-31.395-20.642-31.395-20.798Q-31.395-20.958-31.289-21.063Q-31.613-21.063-31.828-20.835Q-32.043-20.606-32.139-20.267Q-32.235-19.927-32.235-19.622L-32.235-18.349Q-32.235-18.181-32.008-18.134Q-31.781-18.087-31.477-18.087L-31.477-17.790M-30.172-19.544Q-30.172-20.024-29.940-20.440Q-29.707-20.856-29.297-21.106Q-28.887-21.356-28.410-21.356Q-27.680-21.356-27.281-20.915Q-26.883-20.474-26.883-19.743Q-26.883-19.638-26.977-19.614L-29.426-19.614L-29.426-19.544Q-29.426-19.134-29.305-18.778Q-29.184-18.423-28.912-18.206Q-28.641-17.989-28.211-17.989Q-27.848-17.989-27.551-18.218Q-27.254-18.446-27.153-18.798Q-27.145-18.845-27.059-18.860L-26.977-18.860Q-26.883-18.833-26.883-18.751Q-26.883-18.743-26.891-18.712Q-26.953-18.485-27.092-18.302Q-27.231-18.118-27.422-17.985Q-27.613-17.852-27.832-17.782Q-28.051-17.712-28.289-17.712Q-28.660-17.712-28.998-17.849Q-29.336-17.985-29.604-18.237Q-29.871-18.489-30.022-18.829Q-30.172-19.169-30.172-19.544M-29.418-19.852L-27.457-19.852Q-27.457-20.157-27.559-20.448Q-27.660-20.739-27.877-20.921Q-28.094-21.102-28.410-21.102Q-28.711-21.102-28.942-20.915Q-29.172-20.727-29.295-20.436Q-29.418-20.145-29.418-19.852",[1870],[1866,1879],{"fill":1860,"d":1880},"M-33.65-17.79c0-8.642-7.006-15.649-15.649-15.649s-15.649 7.007-15.649 15.65c0 8.642 7.007 15.648 15.65 15.648 8.642 0 15.648-7.006 15.648-15.649Zm-15.649 0",[1854,1882,1884],{"transform":1883},"translate(-2.312 2.9)",[1866,1885],{"d":1886,"fill":1856,"stroke":1856,"className":1887,"style":1888},"M-48.473-18.796Q-48.332-18.383-47.972-18.131Q-47.611-17.878-47.176-17.878Q-46.724-17.878-46.458-18.131Q-46.192-18.383-46.089-18.768Q-45.986-19.152-45.986-19.609Q-45.986-21.310-46.895-21.310Q-47.216-21.310-47.445-21.216Q-47.673-21.121-47.803-21.002Q-47.932-20.884-48.044-20.745Q-48.156-20.607-48.192-20.598L-48.275-20.598Q-48.319-20.598-48.350-20.629Q-48.381-20.660-48.381-20.708L-48.381-23.705Q-48.381-23.736-48.345-23.760Q-48.310-23.784-48.284-23.784L-48.244-23.784Q-47.611-23.494-46.939-23.494Q-46.267-23.494-45.625-23.784L-45.599-23.784Q-45.568-23.784-45.535-23.762Q-45.502-23.740-45.502-23.705L-45.502-23.604Q-45.502-23.600-45.511-23.582Q-45.520-23.564-45.520-23.560Q-45.836-23.165-46.306-22.943Q-46.777-22.721-47.273-22.721Q-47.682-22.721-48.064-22.831L-48.064-21.112Q-47.607-21.569-46.895-21.569Q-46.385-21.569-45.986-21.288Q-45.586-21.007-45.364-20.552Q-45.142-20.097-45.142-19.592Q-45.142-19.042-45.421-18.583Q-45.700-18.124-46.166-17.858Q-46.632-17.592-47.176-17.592Q-47.616-17.592-48-17.819Q-48.385-18.045-48.613-18.425Q-48.842-18.805-48.842-19.249Q-48.842-19.442-48.710-19.574Q-48.578-19.706-48.381-19.706Q-48.249-19.706-48.145-19.647Q-48.042-19.587-47.983-19.484Q-47.924-19.381-47.924-19.249Q-47.924-19.051-48.051-18.919Q-48.178-18.788-48.381-18.788Q-48.442-18.788-48.473-18.796",[1870],"stroke-width:0.270",[1866,1890],{"fill":1860,"d":1891},"M46.018-17.79c0-8.642-7.006-15.649-15.649-15.649S14.72-26.432 14.72-17.789c0 8.642 7.006 15.648 15.65 15.648 8.642 0 15.648-7.006 15.648-15.649Zm-15.649 0",[1854,1893,1895],{"transform":1894},"translate(77.355 2.9)",[1866,1896],{"d":1897,"fill":1856,"stroke":1856,"className":1898,"style":1888},"M-48.236-18.177Q-47.989-17.878-47.383-17.878Q-47.102-17.878-46.862-18.016Q-46.623-18.155-46.445-18.377Q-46.267-18.599-46.157-18.862Q-45.924-19.438-45.924-20.554Q-46.091-20.189-46.396-19.965Q-46.702-19.741-47.084-19.741Q-47.620-19.741-48.036-20.020Q-48.451-20.299-48.682-20.765Q-48.912-21.231-48.912-21.758Q-48.912-22.171-48.765-22.540Q-48.618-22.910-48.354-23.186Q-48.091-23.463-47.721-23.624Q-47.352-23.784-46.939-23.784Q-46.381-23.784-46.007-23.492Q-45.634-23.200-45.430-22.736Q-45.225-22.272-45.146-21.756Q-45.067-21.240-45.067-20.708Q-45.067-19.992-45.335-19.269Q-45.603-18.546-46.126-18.069Q-46.649-17.592-47.374-17.592Q-47.924-17.592-48.301-17.841Q-48.679-18.089-48.679-18.607Q-48.679-18.726-48.622-18.829Q-48.565-18.933-48.464-18.992Q-48.363-19.051-48.236-19.051Q-48.051-19.051-47.928-18.919Q-47.805-18.788-47.805-18.607Q-47.805-18.432-47.932-18.304Q-48.060-18.177-48.236-18.177M-47.040-20.005Q-46.671-20.005-46.423-20.247Q-46.174-20.488-46.058-20.846Q-45.942-21.205-45.942-21.578Q-45.942-21.688-45.950-21.741Q-45.946-21.754-45.944-21.765Q-45.942-21.776-45.942-21.793Q-45.942-22.448-46.157-22.987Q-46.372-23.525-46.939-23.525Q-47.299-23.525-47.528-23.375Q-47.757-23.226-47.873-22.969Q-47.989-22.712-48.022-22.431Q-48.055-22.149-48.055-21.776L-48.055-21.741Q-48.055-21.415-48.029-21.128Q-48.003-20.840-47.904-20.581Q-47.805-20.321-47.594-20.163Q-47.383-20.005-47.040-20.005",[1870],[1854,1900,1902,1910,1916],{"stroke":1860,"fontSize":1901},"7",[1854,1903,1905],{"transform":1904},"translate(-12.972 30.883)",[1866,1906],{"d":1907,"fill":1856,"stroke":1856,"className":1908,"style":1909},"M-48.311-18.610Q-48.311-18.921-48.204-19.241Q-48.096-19.561-47.884-20.080Q-47.809-20.282-47.809-20.422Q-47.809-20.521-47.848-20.588Q-47.887-20.654-47.976-20.654Q-48.257-20.654-48.446-20.383Q-48.636-20.111-48.725-19.779Q-48.735-19.714-48.797-19.714L-48.906-19.714Q-48.937-19.714-48.961-19.745Q-48.985-19.776-48.985-19.800L-48.985-19.827Q-48.916-20.087-48.776-20.324Q-48.636-20.562-48.426-20.719Q-48.216-20.876-47.963-20.876Q-47.781-20.876-47.626-20.805Q-47.470-20.733-47.373-20.598Q-47.276-20.463-47.276-20.282Q-47.276-20.165-47.323-20.029Q-47.539-19.499-47.652-19.157Q-47.764-18.815-47.764-18.504Q-47.764-18.262-47.650-18.103Q-47.535-17.944-47.296-17.944Q-47.053-17.944-46.814-18.118Q-46.712-18.197-46.585-18.337Q-46.459-18.477-46.442-18.566L-45.946-20.548Q-45.915-20.661-45.823-20.735Q-45.731-20.808-45.618-20.808Q-45.515-20.808-45.444-20.745Q-45.372-20.682-45.372-20.582Q-45.372-20.555-45.385-20.500L-45.884-18.518Q-45.912-18.361-45.912-18.272Q-45.912-18.145-45.859-18.045Q-45.806-17.944-45.686-17.944Q-45.563-17.944-45.474-18.036Q-45.385-18.128-45.329-18.255Q-45.273-18.381-45.223-18.557Q-45.174-18.733-45.156-18.819Q-45.126-18.880-45.078-18.880L-44.965-18.880Q-44.931-18.880-44.910-18.855Q-44.890-18.829-44.890-18.798Q-44.890-18.785-44.897-18.771Q-45.153-17.722-45.700-17.722Q-45.939-17.722-46.146-17.838Q-46.353-17.954-46.428-18.169Q-46.828-17.722-47.310-17.722Q-47.757-17.722-48.034-17.946Q-48.311-18.169-48.311-18.610M-43.806-18.210Q-43.806-18.378-43.680-18.501Q-43.553-18.624-43.386-18.624Q-43.218-18.624-43.095-18.501Q-42.972-18.378-42.972-18.210Q-42.972-18.036-43.095-17.913Q-43.218-17.790-43.386-17.790Q-43.553-17.790-43.680-17.913Q-43.806-18.036-43.806-18.210M-40.713-17.722Q-41.038-17.722-41.282-17.879Q-41.527-18.036-41.658-18.301Q-41.790-18.566-41.790-18.891Q-41.790-19.359-41.537-19.822Q-41.284-20.285-40.860-20.581Q-40.436-20.876-39.965-20.876Q-39.749-20.876-39.563-20.772Q-39.377-20.668-39.257-20.483L-38.878-22.011Q-38.843-22.114-38.843-22.199Q-38.843-22.291-39.278-22.291Q-39.363-22.319-39.363-22.404L-39.332-22.514Q-39.305-22.561-39.257-22.572L-38.276-22.647Q-38.238-22.647-38.204-22.620Q-38.170-22.592-38.170-22.544L-39.178-18.518Q-39.206-18.361-39.206-18.272Q-39.206-18.145-39.154-18.045Q-39.103-17.944-38.984-17.944Q-38.761-17.944-38.649-18.197Q-38.536-18.450-38.450-18.819Q-38.426-18.880-38.375-18.880L-38.262-18.880Q-38.228-18.880-38.206-18.851Q-38.184-18.822-38.184-18.798Q-38.184-18.785-38.191-18.771Q-38.454-17.722-38.997-17.722Q-39.247-17.722-39.455-17.845Q-39.664-17.968-39.732-18.197Q-40.207-17.722-40.713-17.722M-40.699-17.944Q-40.426-17.944-40.175-18.128Q-39.924-18.313-39.732-18.580L-39.363-20.060Q-39.401-20.220-39.481-20.357Q-39.561-20.494-39.688-20.574Q-39.814-20.654-39.978-20.654Q-40.187-20.654-40.373-20.530Q-40.559-20.405-40.698-20.213Q-40.836-20.022-40.922-19.820Q-41.034-19.526-41.118-19.181Q-41.202-18.836-41.202-18.586Q-41.202-18.330-41.074-18.137Q-40.945-17.944-40.699-17.944",[1870],"stroke-width:0.210",[1854,1911,1912],{"transform":1904},[1866,1913],{"d":1914,"fill":1856,"stroke":1856,"className":1915,"style":1909},"M-30.274-18.597L-35.107-18.597Q-35.175-18.607-35.221-18.653Q-35.267-18.699-35.267-18.771Q-35.267-18.836-35.221-18.882Q-35.175-18.928-35.107-18.938L-30.274-18.938Q-30.205-18.928-30.159-18.882Q-30.113-18.836-30.113-18.771Q-30.113-18.699-30.159-18.653Q-30.205-18.607-30.274-18.597M-30.274-20.135L-35.107-20.135Q-35.175-20.145-35.221-20.191Q-35.267-20.237-35.267-20.309Q-35.267-20.453-35.107-20.477L-30.274-20.477Q-30.113-20.453-30.113-20.309Q-30.113-20.237-30.159-20.191Q-30.205-20.145-30.274-20.135",[1870],[1854,1917,1918],{"transform":1904},[1866,1919],{"d":1920,"fill":1856,"stroke":1856,"className":1921,"style":1909},"M-26.534-18.552L-26.565-18.552Q-26.428-18.255-26.131-18.079Q-25.834-17.903-25.506-17.903Q-25.143-17.903-24.916-18.081Q-24.689-18.258-24.595-18.547Q-24.501-18.836-24.501-19.198Q-24.501-19.513-24.555-19.798Q-24.610-20.083-24.783-20.289Q-24.955-20.494-25.270-20.494Q-25.543-20.494-25.726-20.427Q-25.909-20.360-26.013-20.271Q-26.117-20.183-26.213-20.073Q-26.309-19.964-26.353-19.954L-26.432-19.954Q-26.504-19.971-26.521-20.042L-26.521-22.360Q-26.521-22.394-26.497-22.416Q-26.473-22.438-26.439-22.438L-26.411-22.438Q-26.124-22.322-25.856-22.268Q-25.588-22.213-25.311-22.213Q-25.034-22.213-24.764-22.268Q-24.494-22.322-24.214-22.438L-24.190-22.438Q-24.155-22.438-24.132-22.415Q-24.108-22.391-24.108-22.360L-24.108-22.291Q-24.108-22.264-24.128-22.244Q-24.402-21.929-24.786-21.753Q-25.171-21.577-25.584-21.577Q-25.923-21.577-26.240-21.663L-26.240-20.381Q-25.844-20.716-25.270-20.716Q-24.866-20.716-24.530-20.506Q-24.193-20.295-24-19.943Q-23.807-19.591-23.807-19.191Q-23.807-18.860-23.947-18.574Q-24.087-18.289-24.331-18.079Q-24.576-17.869-24.878-17.759Q-25.181-17.650-25.499-17.650Q-25.858-17.650-26.184-17.814Q-26.510-17.978-26.705-18.270Q-26.900-18.562-26.900-18.925Q-26.900-19.075-26.794-19.181Q-26.688-19.287-26.534-19.287Q-26.381-19.287-26.276-19.183Q-26.172-19.079-26.172-18.925Q-26.172-18.768-26.276-18.660Q-26.381-18.552-26.534-18.552",[1870],[1854,1923,1924,1931,1937,1943],{"stroke":1860,"fontSize":1901},[1854,1925,1927],{"transform":1926},"translate(66.934 30.883)",[1866,1928],{"d":1929,"fill":1856,"stroke":1856,"className":1930,"style":1909},"M-48.311-18.638Q-48.311-18.850-48.250-19.082Q-48.188-19.314-48.077-19.595Q-47.966-19.875-47.884-20.080Q-47.809-20.282-47.809-20.422Q-47.809-20.521-47.848-20.588Q-47.887-20.654-47.976-20.654Q-48.257-20.654-48.446-20.383Q-48.636-20.111-48.725-19.779Q-48.735-19.714-48.797-19.714L-48.906-19.714Q-48.937-19.714-48.961-19.745Q-48.985-19.776-48.985-19.800L-48.985-19.827Q-48.916-20.087-48.776-20.324Q-48.636-20.562-48.426-20.719Q-48.216-20.876-47.963-20.876Q-47.781-20.876-47.626-20.805Q-47.470-20.733-47.373-20.598Q-47.276-20.463-47.276-20.282Q-47.276-20.165-47.323-20.029Q-47.552-19.465-47.655-19.157Q-47.757-18.850-47.757-18.545Q-47.757-18.269-47.611-18.106Q-47.464-17.944-47.194-17.944Q-46.807-17.944-46.486-18.378Q-46.366-18.535-46.240-18.785Q-46.113-19.034-46.030-19.294Q-45.946-19.554-45.946-19.728Q-45.946-19.933-46.007-20.027Q-46.069-20.121-46.197-20.259Q-46.325-20.398-46.325-20.494Q-46.325-20.593-46.267-20.682Q-46.209-20.770-46.119-20.827Q-46.028-20.883-45.925-20.883Q-45.734-20.883-45.647-20.714Q-45.560-20.545-45.560-20.330Q-45.560-20.001-45.674-19.559Q-45.789-19.116-46.007-18.692Q-46.226-18.269-46.532-17.995Q-46.838-17.722-47.200-17.722Q-47.689-17.722-48-17.944Q-48.311-18.166-48.311-18.638",[1870],[1854,1932,1933],{"transform":1926},[1866,1934],{"d":1935,"fill":1856,"stroke":1856,"className":1936,"style":1909},"M-44.278-18.210Q-44.278-18.378-44.151-18.501Q-44.025-18.624-43.858-18.624Q-43.690-18.624-43.567-18.501Q-43.444-18.378-43.444-18.210Q-43.444-18.036-43.567-17.913Q-43.690-17.790-43.858-17.790Q-44.025-17.790-44.151-17.913Q-44.278-18.036-44.278-18.210M-41.185-17.722Q-41.509-17.722-41.754-17.879Q-41.998-18.036-42.130-18.301Q-42.261-18.566-42.261-18.891Q-42.261-19.359-42.008-19.822Q-41.755-20.285-41.332-20.581Q-40.908-20.876-40.436-20.876Q-40.221-20.876-40.035-20.772Q-39.848-20.668-39.729-20.483L-39.349-22.011Q-39.315-22.114-39.315-22.199Q-39.315-22.291-39.749-22.291Q-39.835-22.319-39.835-22.404L-39.804-22.514Q-39.776-22.561-39.729-22.572L-38.748-22.647Q-38.710-22.647-38.676-22.620Q-38.642-22.592-38.642-22.544L-39.650-18.518Q-39.677-18.361-39.677-18.272Q-39.677-18.145-39.626-18.045Q-39.575-17.944-39.455-17.944Q-39.233-17.944-39.120-18.197Q-39.007-18.450-38.922-18.819Q-38.898-18.880-38.847-18.880L-38.734-18.880Q-38.700-18.880-38.678-18.851Q-38.655-18.822-38.655-18.798Q-38.655-18.785-38.662-18.771Q-38.925-17.722-39.469-17.722Q-39.718-17.722-39.927-17.845Q-40.135-17.968-40.204-18.197Q-40.679-17.722-41.185-17.722M-41.171-17.944Q-40.898-17.944-40.646-18.128Q-40.395-18.313-40.204-18.580L-39.835-20.060Q-39.872-20.220-39.953-20.357Q-40.033-20.494-40.159-20.574Q-40.286-20.654-40.450-20.654Q-40.658-20.654-40.845-20.530Q-41.031-20.405-41.169-20.213Q-41.308-20.022-41.393-19.820Q-41.506-19.526-41.590-19.181Q-41.673-18.836-41.673-18.586Q-41.673-18.330-41.545-18.137Q-41.417-17.944-41.171-17.944",[1870],[1854,1938,1939],{"transform":1926},[1866,1940],{"d":1941,"fill":1856,"stroke":1856,"className":1942,"style":1909},"M-30.751-18.597L-35.584-18.597Q-35.652-18.607-35.698-18.653Q-35.744-18.699-35.744-18.771Q-35.744-18.836-35.698-18.882Q-35.652-18.928-35.584-18.938L-30.751-18.938Q-30.682-18.928-30.636-18.882Q-30.590-18.836-30.590-18.771Q-30.590-18.699-30.636-18.653Q-30.682-18.607-30.751-18.597M-30.751-20.135L-35.584-20.135Q-35.652-20.145-35.698-20.191Q-35.744-20.237-35.744-20.309Q-35.744-20.453-35.584-20.477L-30.751-20.477Q-30.590-20.453-30.590-20.309Q-30.590-20.237-30.636-20.191Q-30.682-20.145-30.751-20.135",[1870],[1854,1944,1945],{"transform":1926},[1866,1946],{"d":1947,"fill":1856,"stroke":1856,"className":1948,"style":1909},"M-26.844-18.104Q-26.724-17.988-26.547-17.946Q-26.369-17.903-26.153-17.903Q-25.914-17.903-25.704-18.012Q-25.494-18.122-25.340-18.304Q-25.186-18.487-25.087-18.720Q-24.920-19.147-24.920-19.967Q-25.070-19.673-25.333-19.494Q-25.596-19.314-25.914-19.314Q-26.348-19.314-26.695-19.523Q-27.042-19.731-27.240-20.092Q-27.439-20.453-27.439-20.876Q-27.439-21.211-27.309-21.500Q-27.179-21.789-26.948-22.003Q-26.717-22.216-26.418-22.327Q-26.119-22.438-25.788-22.438Q-24.930-22.438-24.574-21.724Q-24.219-21.010-24.219-20.053Q-24.219-19.636-24.347-19.208Q-24.475-18.781-24.732-18.426Q-24.988-18.070-25.350-17.860Q-25.713-17.650-26.153-17.650Q-26.608-17.650-26.926-17.838Q-27.244-18.026-27.244-18.450Q-27.244-18.600-27.145-18.699Q-27.046-18.798-26.895-18.798Q-26.827-18.798-26.760-18.771Q-26.693-18.744-26.649-18.699Q-26.605-18.655-26.577-18.588Q-26.550-18.521-26.550-18.450Q-26.550-18.320-26.630-18.222Q-26.711-18.125-26.844-18.104M-25.873-19.540Q-25.579-19.540-25.364-19.718Q-25.149-19.895-25.041-20.171Q-24.933-20.446-24.933-20.736Q-24.933-20.781-24.935-20.808Q-24.937-20.835-24.940-20.870Q-24.937-20.880-24.935-20.887Q-24.933-20.894-24.933-20.904Q-24.933-21.406-25.131-21.806Q-25.330-22.206-25.788-22.206Q-26.355-22.206-26.548-21.847Q-26.741-21.488-26.741-20.876Q-26.741-20.490-26.687-20.207Q-26.632-19.923-26.437-19.731Q-26.242-19.540-25.873-19.540",[1870],[1866,1950],{"fill":1860,"d":1951},"M-33.45-17.79h44.895",[1866,1953],{"d":1954},"m13.95-17.79-3.584-1.35 1.179 1.35-1.179 1.351Z",[1854,1956,1958],{"transform":1957},"translate(37.84 -3.533)",[1866,1959],{"d":1960,"fill":1856,"stroke":1856,"className":1961,"style":1909},"M-48.503-18.337Q-48.383-18.180-48.192-18.081Q-48-17.981-47.785-17.942Q-47.570-17.903-47.347-17.903Q-47.050-17.903-46.855-18.058Q-46.660-18.214-46.570-18.468Q-46.479-18.723-46.479-19.007Q-46.479-19.301-46.571-19.552Q-46.664-19.803-46.862-19.959Q-47.060-20.114-47.354-20.114L-47.870-20.114Q-47.898-20.114-47.923-20.140Q-47.949-20.165-47.949-20.189L-47.949-20.261Q-47.949-20.292-47.923-20.314Q-47.898-20.336-47.870-20.336L-47.429-20.367Q-47.067-20.367-46.847-20.724Q-46.626-21.082-46.626-21.471Q-46.626-21.799-46.821-22.003Q-47.016-22.206-47.347-22.206Q-47.634-22.206-47.887-22.122Q-48.140-22.039-48.304-21.851Q-48.157-21.851-48.057-21.736Q-47.956-21.622-47.956-21.471Q-47.956-21.321-48.062-21.211Q-48.168-21.102-48.325-21.102Q-48.486-21.102-48.595-21.211Q-48.704-21.321-48.704-21.471Q-48.704-21.796-48.496-22.015Q-48.287-22.233-47.971-22.336Q-47.655-22.438-47.347-22.438Q-47.029-22.438-46.701-22.334Q-46.373-22.230-46.146-22.008Q-45.919-21.786-45.919-21.471Q-45.919-21.037-46.206-20.712Q-46.493-20.388-46.927-20.241Q-46.616-20.176-46.336-20.010Q-46.055-19.844-45.878-19.586Q-45.700-19.328-45.700-19.007Q-45.700-18.597-45.944-18.287Q-46.189-17.978-46.570-17.814Q-46.951-17.650-47.347-17.650Q-47.716-17.650-48.074-17.763Q-48.431-17.875-48.675-18.125Q-48.920-18.374-48.920-18.744Q-48.920-18.915-48.803-19.027Q-48.687-19.140-48.516-19.140Q-48.407-19.140-48.316-19.089Q-48.226-19.038-48.171-18.945Q-48.116-18.853-48.116-18.744Q-48.116-18.576-48.229-18.457Q-48.342-18.337-48.503-18.337",[1870],[1854,1963,1964,1971],{"stroke":1860,"fontFamily":1861,"fontSize":1745},[1854,1965,1967],{"transform":1966},"translate(185.727 -45.592)",[1866,1968],{"d":1969,"fill":1856,"stroke":1856,"className":1970,"style":1871},"M-48.963-18.622Q-48.963-19.106-48.561-19.401Q-48.158-19.696-47.608-19.815Q-47.057-19.935-46.565-19.935L-46.565-20.224Q-46.565-20.450-46.680-20.657Q-46.795-20.864-46.992-20.983Q-47.190-21.102-47.420-21.102Q-47.846-21.102-48.131-20.997Q-48.061-20.970-48.014-20.915Q-47.967-20.860-47.942-20.790Q-47.916-20.720-47.916-20.645Q-47.916-20.540-47.967-20.448Q-48.018-20.356-48.110-20.306Q-48.201-20.255-48.307-20.255Q-48.412-20.255-48.504-20.306Q-48.596-20.356-48.647-20.448Q-48.697-20.540-48.697-20.645Q-48.697-21.063-48.309-21.210Q-47.920-21.356-47.420-21.356Q-47.088-21.356-46.735-21.226Q-46.381-21.095-46.153-20.841Q-45.924-20.587-45.924-20.239L-45.924-18.438Q-45.924-18.306-45.852-18.196Q-45.779-18.087-45.651-18.087Q-45.526-18.087-45.457-18.192Q-45.389-18.298-45.389-18.438L-45.389-18.950L-45.108-18.950L-45.108-18.438Q-45.108-18.235-45.225-18.077Q-45.342-17.919-45.524-17.835Q-45.705-17.751-45.908-17.751Q-46.139-17.751-46.291-17.923Q-46.444-18.095-46.475-18.325Q-46.635-18.044-46.944-17.878Q-47.252-17.712-47.604-17.712Q-48.115-17.712-48.539-17.935Q-48.963-18.157-48.963-18.622M-48.276-18.622Q-48.276-18.337-48.049-18.151Q-47.822-17.966-47.529-17.966Q-47.283-17.966-47.059-18.083Q-46.834-18.200-46.699-18.403Q-46.565-18.606-46.565-18.860L-46.565-19.692Q-46.830-19.692-47.115-19.638Q-47.401-19.583-47.672-19.454Q-47.944-19.325-48.110-19.118Q-48.276-18.911-48.276-18.622M-42.748-17.790L-44.733-17.790L-44.733-18.087Q-44.459-18.087-44.291-18.134Q-44.123-18.181-44.123-18.349L-44.123-20.942L-44.764-20.942L-44.764-21.239L-44.123-21.239L-44.123-22.173Q-44.123-22.438-44.006-22.675Q-43.889-22.911-43.695-23.075Q-43.502-23.239-43.254-23.331Q-43.006-23.423-42.740-23.423Q-42.455-23.423-42.231-23.265Q-42.006-23.106-42.006-22.829Q-42.006-22.673-42.111-22.563Q-42.217-22.454-42.381-22.454Q-42.537-22.454-42.647-22.563Q-42.756-22.673-42.756-22.829Q-42.756-23.036-42.596-23.142Q-42.694-23.165-42.787-23.165Q-43.018-23.165-43.190-23.009Q-43.361-22.852-43.447-22.616Q-43.533-22.380-43.533-22.157L-43.533-21.239L-42.565-21.239L-42.565-20.942L-43.510-20.942L-43.510-18.349Q-43.510-18.181-43.283-18.134Q-43.057-18.087-42.748-18.087L-42.748-17.790M-41.596-18.751L-41.596-20.942L-42.299-20.942L-42.299-21.196Q-41.944-21.196-41.701-21.429Q-41.459-21.661-41.348-22.009Q-41.236-22.356-41.236-22.712L-40.955-22.712L-40.955-21.239L-39.779-21.239L-39.779-20.942L-40.955-20.942L-40.955-18.767Q-40.955-18.446-40.836-18.218Q-40.717-17.989-40.436-17.989Q-40.256-17.989-40.139-18.112Q-40.022-18.235-39.969-18.415Q-39.916-18.595-39.916-18.767L-39.916-19.239L-39.635-19.239L-39.635-18.751Q-39.635-18.497-39.740-18.257Q-39.846-18.017-40.043-17.864Q-40.240-17.712-40.498-17.712Q-40.815-17.712-41.067-17.835Q-41.319-17.958-41.457-18.192Q-41.596-18.427-41.596-18.751M-38.916-19.544Q-38.916-20.024-38.684-20.440Q-38.451-20.856-38.041-21.106Q-37.631-21.356-37.154-21.356Q-36.424-21.356-36.026-20.915Q-35.627-20.474-35.627-19.743Q-35.627-19.638-35.721-19.614L-38.170-19.614L-38.170-19.544Q-38.170-19.134-38.049-18.778Q-37.928-18.423-37.656-18.206Q-37.385-17.989-36.955-17.989Q-36.592-17.989-36.295-18.218Q-35.998-18.446-35.897-18.798Q-35.889-18.845-35.803-18.860L-35.721-18.860Q-35.627-18.833-35.627-18.751Q-35.627-18.743-35.635-18.712Q-35.697-18.485-35.836-18.302Q-35.975-18.118-36.166-17.985Q-36.358-17.852-36.576-17.782Q-36.795-17.712-37.033-17.712Q-37.404-17.712-37.742-17.849Q-38.080-17.985-38.348-18.237Q-38.615-18.489-38.766-18.829Q-38.916-19.169-38.916-19.544M-38.162-19.852L-36.201-19.852Q-36.201-20.157-36.303-20.448Q-36.404-20.739-36.621-20.921Q-36.838-21.102-37.154-21.102Q-37.455-21.102-37.686-20.915Q-37.916-20.727-38.039-20.436Q-38.162-20.145-38.162-19.852M-33.131-17.790L-35.111-17.790L-35.111-18.087Q-34.842-18.087-34.674-18.132Q-34.506-18.177-34.506-18.349L-34.506-20.485Q-34.506-20.700-34.569-20.796Q-34.631-20.892-34.748-20.913Q-34.865-20.935-35.111-20.935L-35.111-21.231L-33.944-21.317L-33.944-20.532Q-33.865-20.743-33.713-20.929Q-33.561-21.114-33.361-21.216Q-33.162-21.317-32.936-21.317Q-32.690-21.317-32.498-21.173Q-32.307-21.028-32.307-20.798Q-32.307-20.642-32.412-20.532Q-32.518-20.423-32.674-20.423Q-32.830-20.423-32.940-20.532Q-33.049-20.642-33.049-20.798Q-33.049-20.958-32.944-21.063Q-33.268-21.063-33.483-20.835Q-33.697-20.606-33.793-20.267Q-33.889-19.927-33.889-19.622L-33.889-18.349Q-33.889-18.181-33.662-18.134Q-33.436-18.087-33.131-18.087",[1870],[1854,1972,1973],{"transform":1966},[1866,1974],{"d":1975,"fill":1856,"stroke":1856,"className":1976,"style":1871},"M-26.976-17.790L-28.956-17.790L-28.956-18.087Q-28.687-18.087-28.519-18.132Q-28.351-18.177-28.351-18.349L-28.351-20.485Q-28.351-20.700-28.413-20.796Q-28.476-20.892-28.593-20.913Q-28.710-20.935-28.956-20.935L-28.956-21.231L-27.788-21.317L-27.788-20.532Q-27.710-20.743-27.558-20.929Q-27.406-21.114-27.206-21.216Q-27.007-21.317-26.781-21.317Q-26.535-21.317-26.343-21.173Q-26.152-21.028-26.152-20.798Q-26.152-20.642-26.257-20.532Q-26.363-20.423-26.519-20.423Q-26.675-20.423-26.785-20.532Q-26.894-20.642-26.894-20.798Q-26.894-20.958-26.788-21.063Q-27.113-21.063-27.327-20.835Q-27.542-20.606-27.638-20.267Q-27.734-19.927-27.734-19.622L-27.734-18.349Q-27.734-18.181-27.507-18.134Q-27.281-18.087-26.976-18.087L-26.976-17.790M-25.671-19.544Q-25.671-20.024-25.439-20.440Q-25.206-20.856-24.796-21.106Q-24.386-21.356-23.910-21.356Q-23.179-21.356-22.781-20.915Q-22.382-20.474-22.382-19.743Q-22.382-19.638-22.476-19.614L-24.925-19.614L-24.925-19.544Q-24.925-19.134-24.804-18.778Q-24.683-18.423-24.411-18.206Q-24.140-17.989-23.710-17.989Q-23.347-17.989-23.050-18.218Q-22.753-18.446-22.652-18.798Q-22.644-18.845-22.558-18.860L-22.476-18.860Q-22.382-18.833-22.382-18.751Q-22.382-18.743-22.390-18.712Q-22.452-18.485-22.591-18.302Q-22.730-18.118-22.921-17.985Q-23.113-17.852-23.331-17.782Q-23.550-17.712-23.788-17.712Q-24.160-17.712-24.497-17.849Q-24.835-17.985-25.103-18.237Q-25.370-18.489-25.521-18.829Q-25.671-19.169-25.671-19.544M-24.917-19.852L-22.956-19.852Q-22.956-20.157-23.058-20.448Q-23.160-20.739-23.376-20.921Q-23.593-21.102-23.910-21.102Q-24.210-21.102-24.441-20.915Q-24.671-20.727-24.794-20.436Q-24.917-20.145-24.917-19.852M-19.980-17.790L-21.812-17.790L-21.812-18.087Q-21.538-18.087-21.370-18.134Q-21.202-18.181-21.202-18.349L-21.202-22.509Q-21.202-22.724-21.265-22.819Q-21.327-22.915-21.447-22.936Q-21.566-22.958-21.812-22.958L-21.812-23.255L-20.589-23.341L-20.589-18.349Q-20.589-18.181-20.421-18.134Q-20.253-18.087-19.980-18.087L-19.980-17.790M-19.437-18.622Q-19.437-19.106-19.035-19.401Q-18.632-19.696-18.081-19.815Q-17.531-19.935-17.038-19.935L-17.038-20.224Q-17.038-20.450-17.154-20.657Q-17.269-20.864-17.466-20.983Q-17.663-21.102-17.894-21.102Q-18.320-21.102-18.605-20.997Q-18.535-20.970-18.488-20.915Q-18.441-20.860-18.415-20.790Q-18.390-20.720-18.390-20.645Q-18.390-20.540-18.441-20.448Q-18.492-20.356-18.583-20.306Q-18.675-20.255-18.781-20.255Q-18.886-20.255-18.978-20.306Q-19.070-20.356-19.120-20.448Q-19.171-20.540-19.171-20.645Q-19.171-21.063-18.783-21.210Q-18.394-21.356-17.894-21.356Q-17.562-21.356-17.208-21.226Q-16.855-21.095-16.626-20.841Q-16.398-20.587-16.398-20.239L-16.398-18.438Q-16.398-18.306-16.326-18.196Q-16.253-18.087-16.124-18.087Q-15.999-18.087-15.931-18.192Q-15.863-18.298-15.863-18.438L-15.863-18.950L-15.581-18.950L-15.581-18.438Q-15.581-18.235-15.699-18.077Q-15.816-17.919-15.997-17.835Q-16.179-17.751-16.382-17.751Q-16.613-17.751-16.765-17.923Q-16.917-18.095-16.949-18.325Q-17.109-18.044-17.417-17.878Q-17.726-17.712-18.077-17.712Q-18.589-17.712-19.013-17.935Q-19.437-18.157-19.437-18.622M-18.749-18.622Q-18.749-18.337-18.523-18.151Q-18.296-17.966-18.003-17.966Q-17.757-17.966-17.533-18.083Q-17.308-18.200-17.173-18.403Q-17.038-18.606-17.038-18.860L-17.038-19.692Q-17.304-19.692-17.589-19.638Q-17.874-19.583-18.146-19.454Q-18.417-19.325-18.583-19.118Q-18.749-18.911-18.749-18.622M-13.902-17.790L-15.398-17.790L-15.398-18.087Q-14.765-18.087-14.343-18.567L-13.574-19.477L-14.566-20.677Q-14.722-20.856-14.884-20.899Q-15.046-20.942-15.351-20.942L-15.351-21.239L-13.663-21.239L-13.663-20.942Q-13.757-20.942-13.833-20.899Q-13.910-20.856-13.910-20.767Q-13.910-20.724-13.878-20.677L-13.222-19.888L-12.742-20.462Q-12.624-20.599-12.624-20.735Q-12.624-20.825-12.675-20.884Q-12.726-20.942-12.808-20.942L-12.808-21.239L-11.320-21.239L-11.320-20.942Q-11.956-20.942-12.367-20.462L-13.046-19.661L-11.960-18.349Q-11.800-18.173-11.640-18.130Q-11.480-18.087-11.175-18.087L-11.175-17.790L-12.863-17.790L-12.863-18.087Q-12.773-18.087-12.695-18.130Q-12.617-18.173-12.617-18.263Q-12.617-18.286-12.648-18.349L-13.390-19.255L-13.976-18.567Q-14.093-18.431-14.093-18.294Q-14.093-18.208-14.042-18.147Q-13.992-18.087-13.902-18.087",[1870],[1866,1978],{"fill":1860,"d":1979},"M131.376-17.79c0-8.642-7.006-15.649-15.649-15.649s-15.649 7.007-15.649 15.65c0 8.642 7.006 15.648 15.65 15.648 8.642 0 15.648-7.006 15.648-15.649Zm-15.649 0",[1854,1981,1983],{"transform":1982},"translate(162.713 2.9)",[1866,1984],{"d":1886,"fill":1856,"stroke":1856,"className":1985,"style":1888},[1870],[1854,1987,1991,1994],{"fill":1988,"stroke":1989,"style":1990},"var(--tk-soft-accent)","var(--tk-accent)","stroke-width:1.2",[1866,1992],{"d":1993},"M211.044-17.79c0-8.642-7.006-15.649-15.649-15.649s-15.649 7.007-15.649 15.65c0 8.642 7.006 15.648 15.65 15.648 8.642 0 15.648-7.006 15.648-15.649Zm-15.649 0",[1854,1995,1997],{"transform":1996},"translate(242.381 2.9)",[1866,1998],{"d":1999,"fill":1856,"stroke":1856,"className":2000,"style":1888},"M-48.912-19.157Q-48.912-19.715-48.552-20.128Q-48.192-20.541-47.616-20.813L-47.985-21.046Q-48.288-21.248-48.475-21.578Q-48.662-21.908-48.662-22.264Q-48.662-22.918-48.156-23.351Q-47.651-23.784-46.987-23.784Q-46.588-23.784-46.203-23.624Q-45.819-23.463-45.570-23.158Q-45.322-22.852-45.322-22.435Q-45.322-21.604-46.390-21.046L-45.836-20.699Q-45.489-20.471-45.278-20.102Q-45.067-19.732-45.067-19.319Q-45.067-18.941-45.225-18.623Q-45.383-18.304-45.660-18.071Q-45.937-17.838-46.280-17.715Q-46.623-17.592-46.987-17.592Q-47.453-17.592-47.899-17.779Q-48.345-17.966-48.629-18.320Q-48.912-18.673-48.912-19.157M-48.389-19.157Q-48.389-18.612-47.970-18.245Q-47.550-17.878-46.987-17.878Q-46.658-17.878-46.333-18.010Q-46.007-18.142-45.799-18.396Q-45.590-18.651-45.590-18.994Q-45.590-19.258-45.726-19.482Q-45.862-19.706-46.095-19.860L-47.339-20.642Q-47.800-20.405-48.095-20.018Q-48.389-19.631-48.389-19.157M-47.778-21.912L-46.662-21.209Q-46.438-21.332-46.234-21.521Q-46.029-21.710-45.909-21.943Q-45.788-22.176-45.788-22.435Q-45.788-22.743-45.959-22.993Q-46.131-23.244-46.407-23.384Q-46.684-23.525-46.996-23.525Q-47.445-23.525-47.818-23.279Q-48.192-23.033-48.192-22.606Q-48.192-22.202-47.778-21.912",[1870],[1854,2002,2003,2009,2014],{"stroke":1860,"fontSize":1901},[1854,2004,2006],{"transform":2005},"translate(152.054 30.883)",[1866,2007],{"d":1907,"fill":1856,"stroke":1856,"className":2008,"style":1909},[1870],[1854,2010,2011],{"transform":2005},[1866,2012],{"d":1914,"fill":1856,"stroke":1856,"className":2013,"style":1909},[1870],[1854,2015,2016],{"transform":2005},[1866,2017],{"d":1920,"fill":1856,"stroke":1856,"className":2018,"style":1909},[1870],[1854,2020,2021],{"fill":1989,"stroke":1989},[1854,2022,2023,2029,2034,2039],{"fill":1989,"stroke":1860,"fontSize":1901},[1854,2024,2026],{"transform":2025},"translate(231.96 30.883)",[1866,2027],{"d":1929,"fill":1989,"stroke":1989,"className":2028,"style":1909},[1870],[1854,2030,2031],{"transform":2025},[1866,2032],{"d":1935,"fill":1989,"stroke":1989,"className":2033,"style":1909},[1870],[1854,2035,2036],{"transform":2025},[1866,2037],{"d":1941,"fill":1989,"stroke":1989,"className":2038,"style":1909},[1870],[1854,2040,2041],{"transform":2025},[1866,2042],{"d":2043,"fill":1989,"stroke":1989,"className":2044,"style":1909},"M-27.439-18.867Q-27.439-19.308-27.136-19.629Q-26.834-19.950-26.382-20.142L-26.622-20.282Q-26.892-20.442-27.058-20.700Q-27.223-20.958-27.223-21.256Q-27.223-21.608-27.018-21.880Q-26.813-22.151-26.492-22.295Q-26.171-22.438-25.829-22.438Q-25.507-22.438-25.184-22.322Q-24.861-22.206-24.650-21.965Q-24.438-21.724-24.438-21.389Q-24.438-21.027-24.682-20.764Q-24.926-20.500-25.306-20.323L-24.906-20.087Q-24.711-19.974-24.552-19.805Q-24.393-19.636-24.306-19.427Q-24.219-19.219-24.219-18.986Q-24.219-18.583-24.453-18.279Q-24.687-17.975-25.061-17.812Q-25.436-17.650-25.829-17.650Q-26.215-17.650-26.584-17.787Q-26.953-17.923-27.196-18.200Q-27.439-18.477-27.439-18.867M-26.991-18.867Q-26.991-18.580-26.822-18.357Q-26.652-18.135-26.384-18.019Q-26.116-17.903-25.829-17.903Q-25.391-17.903-25.029-18.120Q-24.667-18.337-24.667-18.744Q-24.667-18.945-24.795-19.123Q-24.923-19.301-25.101-19.400L-26.123-19.995Q-26.362-19.885-26.560-19.719Q-26.758-19.554-26.875-19.338Q-26.991-19.123-26.991-18.867M-26.468-20.996L-25.548-20.463Q-25.241-20.623-25.039-20.856Q-24.838-21.088-24.838-21.389Q-24.838-21.628-24.983-21.818Q-25.128-22.008-25.360-22.107Q-25.593-22.206-25.829-22.206Q-26.051-22.206-26.280-22.136Q-26.509-22.066-26.666-21.909Q-26.823-21.751-26.823-21.522Q-26.823-21.208-26.468-20.996",[1870],[1854,2046,2048,2051,2054],{"stroke":2047,"style":1990},"var(--tk-warn)",[1866,2049],{"fill":1860,"d":2050},"M131.576-17.79h42.394",[1866,2052],{"fill":2047,"d":2053},"m177.437-17.79-4.753-1.802 1.586 1.802-1.586 1.803Z",[1854,2055,2057],{"transform":2056},"translate(202.667 -3.933)",[1866,2058],{"d":1960,"fill":1856,"stroke":1856,"className":2059,"style":1909},[1870],[1854,2061,2062],{"fill":1989,"stroke":1989},[1854,2063,2064,2070,2076,2082],{"fill":1989,"stroke":1860,"fontSize":1901},[1854,2065,2067],{"transform":2066},"translate(190.456 49.877)",[1866,2068],{"d":1929,"fill":1989,"stroke":1989,"className":2069,"style":1909},[1870],[1854,2071,2072],{"transform":2066},[1866,2073],{"d":2074,"fill":1989,"stroke":1989,"className":2075,"style":1909},"M-44.278-18.210Q-44.278-18.378-44.151-18.501Q-44.025-18.624-43.858-18.624Q-43.690-18.624-43.567-18.501Q-43.444-18.378-43.444-18.210Q-43.444-18.036-43.567-17.913Q-43.690-17.790-43.858-17.790Q-44.025-17.790-44.151-17.913Q-44.278-18.036-44.278-18.210M-41.708-17.951Q-41.708-18.002-41.667-18.098Q-41.458-18.508-41.303-18.850Q-41.147-19.191-41.010-19.555Q-40.874-19.919-40.764-20.309L-41.239-20.309Q-41.506-20.309-41.696-20.195Q-41.885-20.080-42.029-19.925Q-42.172-19.769-42.193-19.766L-42.302-19.766Q-42.337-19.766-42.359-19.791Q-42.381-19.817-42.381-19.848Q-42.381-19.861-42.367-19.889Q-41.769-20.808-41.171-20.808L-38.368-20.808Q-38.283-20.808-38.221-20.750Q-38.160-20.692-38.160-20.603Q-38.160-20.490-38.244-20.400Q-38.327-20.309-38.447-20.309L-39.414-20.309Q-39.510-19.851-39.510-19.420Q-39.510-18.689-39.223-18.169Q-39.189-18.108-39.189-18.050Q-39.189-17.923-39.298-17.822Q-39.407-17.722-39.537-17.722Q-39.691-17.722-39.780-17.906Q-39.869-18.091-39.903-18.327Q-39.937-18.562-39.937-18.733Q-39.937-18.983-39.905-19.229Q-39.872-19.475-39.816-19.740Q-39.759-20.005-39.688-20.309L-40.484-20.309Q-40.593-19.872-40.718-19.391Q-40.843-18.911-40.952-18.523Q-41.062-18.135-41.120-17.992Q-41.233-17.722-41.458-17.722Q-41.561-17.722-41.634-17.787Q-41.708-17.852-41.708-17.951",[1870],[1854,2077,2078],{"transform":2066},[1866,2079],{"d":2080,"fill":1989,"stroke":1989,"className":2081,"style":1909},"M-33.271-17.776Q-33.387-18.378-33.823-18.836Q-34.259-19.294-34.861-19.448Q-34.922-19.458-34.922-19.540Q-34.922-19.612-34.861-19.632Q-34.259-19.790-33.823-20.242Q-33.387-20.695-33.271-21.297Q-33.261-21.372-33.186-21.382L-33.018-21.382Q-32.981-21.375-32.955-21.345Q-32.929-21.314-32.929-21.276Q-33.022-20.801-33.282-20.398Q-33.541-19.995-33.934-19.707L-28.209-19.707Q-28.141-19.707-28.095-19.658Q-28.049-19.608-28.049-19.540Q-28.049-19.472-28.095-19.419Q-28.141-19.366-28.209-19.366L-33.934-19.366Q-33.548-19.089-33.285-18.680Q-33.022-18.272-32.929-17.797Q-32.929-17.759-32.955-17.728Q-32.981-17.698-33.018-17.691L-33.186-17.691Q-33.261-17.701-33.271-17.776",[1870],[1854,2083,2084],{"transform":2066},[1866,2085],{"d":2086,"fill":1989,"stroke":1989,"className":2087,"style":1909},"M-24.232-18.610Q-24.232-18.921-24.125-19.241Q-24.017-19.561-23.805-20.080Q-23.730-20.282-23.730-20.422Q-23.730-20.521-23.769-20.588Q-23.808-20.654-23.897-20.654Q-24.178-20.654-24.367-20.383Q-24.557-20.111-24.646-19.779Q-24.656-19.714-24.718-19.714L-24.827-19.714Q-24.858-19.714-24.882-19.745Q-24.906-19.776-24.906-19.800L-24.906-19.827Q-24.837-20.087-24.697-20.324Q-24.557-20.562-24.347-20.719Q-24.137-20.876-23.884-20.876Q-23.702-20.876-23.547-20.805Q-23.391-20.733-23.294-20.598Q-23.197-20.463-23.197-20.282Q-23.197-20.165-23.244-20.029Q-23.460-19.499-23.573-19.157Q-23.685-18.815-23.685-18.504Q-23.685-18.262-23.571-18.103Q-23.456-17.944-23.217-17.944Q-22.974-17.944-22.735-18.118Q-22.633-18.197-22.506-18.337Q-22.380-18.477-22.363-18.566L-21.867-20.548Q-21.836-20.661-21.744-20.735Q-21.652-20.808-21.539-20.808Q-21.436-20.808-21.365-20.745Q-21.293-20.682-21.293-20.582Q-21.293-20.555-21.306-20.500L-21.805-18.518Q-21.833-18.361-21.833-18.272Q-21.833-18.145-21.780-18.045Q-21.727-17.944-21.607-17.944Q-21.484-17.944-21.395-18.036Q-21.306-18.128-21.250-18.255Q-21.194-18.381-21.144-18.557Q-21.095-18.733-21.077-18.819Q-21.047-18.880-20.999-18.880L-20.886-18.880Q-20.852-18.880-20.831-18.855Q-20.811-18.829-20.811-18.798Q-20.811-18.785-20.818-18.771Q-21.074-17.722-21.621-17.722Q-21.860-17.722-22.067-17.838Q-22.274-17.954-22.349-18.169Q-22.749-17.722-23.231-17.722Q-23.678-17.722-23.955-17.946Q-24.232-18.169-24.232-18.610",[1870],[2089,2090,2093,2094,2146,2147,2181,2182,2197,2198,2249,2250,2271,2272,2287,2288,1131],"figcaption",{"className":2091},[2092],"tikz-cap","One ",[415,2095,2097],{"className":2096},[418],[415,2098,2100],{"className":2099,"ariaHidden":423},[422],[415,2101,2103,2106,2119,2122,2125,2128,2131,2134,2137,2140,2143],{"className":2102},[427],[415,2104],{"className":2105,"style":458},[431],[415,2107,2111],{"className":2108},[2109,2110],"enclosing","textsc",[415,2112,2115],{"className":2113},[436,2114],"text",[415,2116,2118],{"className":2117},[436],"Relax",[415,2120,463],{"className":2121},[462],[415,2123,512],{"className":2124},[436,437],[415,2126,473],{"className":2127},[472],[415,2129],{"className":2130,"style":477},[442],[415,2132,523],{"className":2133,"style":522},[436,437],[415,2135,473],{"className":2136},[472],[415,2138],{"className":2139,"style":477},[442],[415,2141,505],{"className":2142,"style":504},[436,437],[415,2144,487],{"className":2145},[486]," step. The edge ",[415,2148,2150],{"className":2149},[418],[415,2151,2153,2172],{"className":2152,"ariaHidden":423},[422],[415,2154,2156,2159,2162,2165,2169],{"className":2155},[427],[415,2157],{"className":2158,"style":1020},[431],[415,2160,512],{"className":2161},[436,437],[415,2163],{"className":2164,"style":443},[442],[415,2166,2168],{"className":2167},[447],"→",[415,2170],{"className":2171,"style":443},[442],[415,2173,2175,2178],{"className":2174},[427],[415,2176],{"className":2177,"style":1020},[431],[415,2179,523],{"className":2180,"style":522},[436,437]," beats ",[415,2183,2185],{"className":2184},[418],[415,2186,2188],{"className":2187,"ariaHidden":423},[422],[415,2189,2191,2194],{"className":2190},[427],[415,2192],{"className":2193,"style":1020},[431],[415,2195,523],{"className":2196,"style":522},[436,437],"'s current estimate (",[415,2199,2201],{"className":2200},[418],[415,2202,2204,2222,2240],{"className":2203,"ariaHidden":423},[422],[415,2205,2207,2210,2213,2216,2219],{"className":2206},[427],[415,2208],{"className":2209,"style":1704},[431],[415,2211,1690],{"className":2212},[436],[415,2214],{"className":2215,"style":467},[442],[415,2217,1505],{"className":2218},[902],[415,2220],{"className":2221,"style":467},[442],[415,2223,2225,2228,2231,2234,2237],{"className":2224},[427],[415,2226],{"className":2227,"style":1741},[431],[415,2229,1633],{"className":2230},[436],[415,2232],{"className":2233,"style":443},[442],[415,2235,1752],{"className":2236},[447],[415,2238],{"className":2239,"style":443},[442],[415,2241,2243,2246],{"className":2242},[427],[415,2244],{"className":2245,"style":1294},[431],[415,2247,1585],{"className":2248},[436],"), so ",[415,2251,2253],{"className":2252},[418],[415,2254,2256],{"className":2255,"ariaHidden":423},[422],[415,2257,2259,2262,2265,2268],{"className":2258},[427],[415,2260],{"className":2261,"style":1163},[431],[415,2263,523],{"className":2264,"style":522},[436,437],[415,2266,1131],{"className":2267},[436],[415,2269,1173],{"className":2270},[436,437]," falls to ",[415,2273,2275],{"className":2274},[418],[415,2276,2278],{"className":2277,"ariaHidden":423},[422],[415,2279,2281,2284],{"className":2280},[427],[415,2282],{"className":2283,"style":1294},[431],[415,2285,1745],{"className":2286},[436]," and its predecessor is rewired to ",[415,2289,2291],{"className":2290},[418],[415,2292,2294],{"className":2293,"ariaHidden":423},[422],[415,2295,2297,2300],{"className":2296},[427],[415,2298],{"className":2299,"style":1020},[431],[415,2301,512],{"className":2302},[436,437],[381,2304,2305,2306,531,2309,1617,2343,2362,2363,2366],{},"We will trace the algorithms on this small weighted digraph. Note the\n",[390,2307,2308],{},"negative edge",[415,2310,2312],{"className":2311},[418],[415,2313,2315,2333],{"className":2314,"ariaHidden":423},[422],[415,2316,2318,2321,2324,2327,2330],{"className":2317},[427],[415,2319],{"className":2320,"style":1020},[431],[415,2322,385],{"className":2323},[436,437],[415,2325],{"className":2326,"style":443},[442],[415,2328,2168],{"className":2329},[447],[415,2331],{"className":2332,"style":443},[442],[415,2334,2336,2339],{"className":2335},[427],[415,2337],{"className":2338,"style":1163},[431],[415,2340,2342],{"className":2341},[436,437],"b",[415,2344,2346],{"className":2345},[418],[415,2347,2349],{"className":2348,"ariaHidden":423},[422],[415,2350,2352,2355,2358],{"className":2351},[427],[415,2353],{"className":2354,"style":1704},[431],[415,2356,903],{"className":2357},[436],[415,2359,2361],{"className":2360},[436],"2",": it is harmless here (there is no\nnegative ",[395,2364,2365],{},"cycle","), but it is exactly the kind of edge that will break Dijkstra\nand force us toward a dynamic program.",[1841,2368,2370,2565],{"className":2369},[1844,1845],[1847,2371,2375],{"xmlns":1849,"width":2372,"height":2373,"viewBox":2374},"368.269","143.257","-75 -75 276.202 107.443",[1854,2376,2377,2380,2387,2390,2397,2400,2407,2410,2417,2420,2427,2430,2434,2447,2450,2453,2465,2468,2472,2492,2495,2498,2510,2513,2516,2528,2531,2534,2546,2549,2553],{"stroke":1856,"style":1857},[1866,2378],{"fill":1860,"d":2379},"M-45.975-21.549c0-6.285-5.095-11.38-11.38-11.38s-11.382 5.095-11.382 11.38 5.096 11.382 11.381 11.382 11.381-5.096 11.381-11.382Zm-11.38 0",[1854,2381,2383],{"transform":2382},"translate(-2.154 1.937)",[1866,2384],{"d":2385,"fill":1856,"stroke":1856,"className":2386,"style":1888},"M-56.521-22.098Q-56.301-21.712-55.545-21.712Q-55.247-21.712-54.952-21.813Q-54.658-21.914-54.464-22.127Q-54.271-22.340-54.271-22.648Q-54.271-22.876-54.449-23.023Q-54.627-23.171-54.873-23.223L-55.383-23.320Q-55.598-23.360-55.774-23.483Q-55.950-23.606-56.055-23.792Q-56.161-23.979-56.161-24.195Q-56.161-24.594-55.937-24.900Q-55.712-25.205-55.354-25.366Q-54.996-25.526-54.592-25.526Q-54.328-25.526-54.080-25.447Q-53.832-25.368-53.662-25.188Q-53.493-25.007-53.493-24.744Q-53.493-24.537-53.614-24.379Q-53.735-24.221-53.946-24.221Q-54.069-24.221-54.155-24.302Q-54.240-24.383-54.240-24.502Q-54.240-24.665-54.117-24.799Q-53.994-24.933-53.836-24.933Q-53.924-25.113-54.141-25.190Q-54.359-25.267-54.609-25.267Q-54.851-25.267-55.077-25.179Q-55.304-25.091-55.449-24.922Q-55.594-24.753-55.594-24.502Q-55.594-24.326-55.462-24.208Q-55.330-24.089-55.132-24.041L-54.627-23.944Q-54.240-23.865-53.972-23.595Q-53.704-23.324-53.704-22.942Q-53.704-22.612-53.893-22.292Q-54.082-21.971-54.359-21.773Q-54.860-21.448-55.554-21.448Q-55.866-21.448-56.167-21.534Q-56.468-21.619-56.673-21.817Q-56.877-22.015-56.877-22.322Q-56.877-22.573-56.734-22.757Q-56.591-22.942-56.350-22.942Q-56.196-22.942-56.097-22.850Q-55.998-22.757-55.998-22.612Q-55.998-22.402-56.150-22.250Q-56.301-22.098-56.521-22.098",[1870],[1866,2388],{"fill":1860,"d":2389},"M33-60.689c0-6.286-5.096-11.381-11.381-11.381s-11.381 5.095-11.381 11.381 5.095 11.381 11.38 11.381S33-54.403 33-60.688Zm-11.381 0",[1854,2391,2393],{"transform":2392},"translate(76.524 -37.203)",[1866,2394],{"d":2395,"fill":1856,"stroke":1856,"className":2396,"style":1888},"M-55.734-21.448Q-56.130-21.448-56.416-21.652Q-56.701-21.857-56.848-22.191Q-56.996-22.525-56.996-22.916Q-56.996-23.351-56.822-23.812Q-56.648-24.274-56.336-24.665Q-56.024-25.056-55.614-25.291Q-55.203-25.526-54.763-25.526Q-54.495-25.526-54.278-25.388Q-54.060-25.249-53.928-25.003Q-53.889-25.153-53.781-25.249Q-53.673-25.346-53.533-25.346Q-53.410-25.346-53.326-25.273Q-53.243-25.201-53.243-25.078Q-53.243-25.025-53.252-24.994L-53.871-22.503Q-53.928-22.305-53.928-22.107Q-53.928-21.712-53.665-21.712Q-53.379-21.712-53.245-22.035Q-53.111-22.358-52.992-22.863Q-52.983-22.894-52.959-22.918Q-52.935-22.942-52.900-22.942L-52.794-22.942Q-52.746-22.942-52.724-22.909Q-52.702-22.876-52.702-22.828Q-52.816-22.397-52.907-22.144Q-52.997-21.892-53.190-21.670Q-53.383-21.448-53.682-21.448Q-53.990-21.448-54.238-21.619Q-54.486-21.791-54.557-22.081Q-54.812-21.795-55.108-21.622Q-55.405-21.448-55.734-21.448M-55.717-21.712Q-55.387-21.712-55.077-21.953Q-54.768-22.195-54.557-22.511Q-54.548-22.520-54.548-22.538L-54.051-24.502Q-54.108-24.819-54.300-25.043Q-54.491-25.267-54.781-25.267Q-55.150-25.267-55.449-24.948Q-55.748-24.630-55.915-24.221Q-56.051-23.874-56.176-23.364Q-56.301-22.854-56.301-22.529Q-56.301-22.204-56.163-21.958Q-56.024-21.712-55.717-21.712",[1870],[1866,2398],{"fill":1860,"d":2399},"M33 17.592C33 11.306 27.904 6.21 21.619 6.21s-11.381 5.095-11.381 11.38 5.095 11.382 11.38 11.382S33 23.877 33 17.592Zm-11.381 0",[1854,2401,2403],{"transform":2402},"translate(76.992 42.265)",[1866,2404],{"d":2405,"fill":1856,"stroke":1856,"className":2406,"style":1888},"M-55.734-21.448Q-56.310-21.448-56.631-21.879Q-56.952-22.309-56.952-22.889Q-56.952-23.294-56.868-23.522L-55.989-27.020Q-55.954-27.170-55.954-27.244Q-55.954-27.381-56.521-27.381Q-56.618-27.381-56.618-27.499Q-56.618-27.556-56.587-27.627Q-56.556-27.697-56.490-27.697L-55.269-27.794Q-55.216-27.794-55.183-27.765Q-55.150-27.736-55.150-27.688L-55.150-27.653L-55.809-25.043Q-55.286-25.526-54.763-25.526Q-54.377-25.526-54.086-25.322Q-53.796-25.117-53.649-24.783Q-53.502-24.449-53.502-24.058Q-53.502-23.474-53.805-22.865Q-54.108-22.257-54.629-21.852Q-55.150-21.448-55.734-21.448M-55.717-21.712Q-55.348-21.712-55.044-22.035Q-54.741-22.358-54.583-22.753Q-54.438-23.109-54.317-23.617Q-54.196-24.124-54.196-24.445Q-54.196-24.770-54.341-25.018Q-54.486-25.267-54.781-25.267Q-55.383-25.267-55.954-24.467L-56.196-23.474Q-56.341-22.850-56.341-22.586Q-56.341-22.243-56.189-21.977Q-56.038-21.712-55.717-21.712",[1870],[1866,2408],{"fill":1860,"d":2409},"M118.758-60.689c0-6.286-5.095-11.381-11.381-11.381s-11.381 5.095-11.381 11.381 5.095 11.381 11.38 11.381c6.287 0 11.382-5.095 11.382-11.38Zm-11.381 0",[1854,2411,2413],{"transform":2412},"translate(162.73 -37.203)",[1866,2414],{"d":2415,"fill":1856,"stroke":1856,"className":2416,"style":1888},"M-56.266-22.656Q-56.266-22.261-56.053-21.986Q-55.840-21.712-55.458-21.712Q-54.913-21.712-54.407-21.947Q-53.902-22.182-53.585-22.604Q-53.564-22.639-53.502-22.639Q-53.445-22.639-53.399-22.588Q-53.353-22.538-53.353-22.485Q-53.353-22.450-53.379-22.424Q-53.726-21.949-54.289-21.698Q-54.851-21.448-55.475-21.448Q-55.906-21.448-56.255-21.650Q-56.605-21.852-56.796-22.208Q-56.987-22.564-56.987-22.990Q-56.987-23.452-56.785-23.909Q-56.583-24.366-56.227-24.735Q-55.871-25.104-55.427-25.315Q-54.983-25.526-54.513-25.526Q-54.245-25.526-53.996-25.445Q-53.748-25.363-53.581-25.185Q-53.414-25.007-53.414-24.744Q-53.414-24.507-53.564-24.329Q-53.713-24.151-53.946-24.151Q-54.086-24.151-54.192-24.245Q-54.297-24.340-54.297-24.485Q-54.297-24.687-54.150-24.841Q-54.003-24.994-53.801-24.994Q-53.906-25.135-54.111-25.201Q-54.315-25.267-54.522-25.267Q-55.058-25.267-55.455-24.838Q-55.853-24.410-56.060-23.790Q-56.266-23.171-56.266-22.656",[1870],[1866,2418],{"fill":1860,"d":2419},"M197.732-21.549c0-6.285-5.095-11.38-11.381-11.38s-11.381 5.095-11.381 11.38 5.095 11.382 11.381 11.382 11.381-5.096 11.381-11.382Zm-11.381 0",[1854,2421,2423],{"transform":2422},"translate(242.036 2.768)",[1866,2424],{"d":2425,"fill":1856,"stroke":1856,"className":2426,"style":1888},"M-56.772-22.287Q-56.772-22.419-56.745-22.538L-56.095-25.113L-57.040-25.113Q-57.149-25.113-57.149-25.232Q-57.149-25.293-57.116-25.361Q-57.084-25.429-57.022-25.429L-56.024-25.429L-55.664-26.866Q-55.629-27.007-55.517-27.095Q-55.405-27.183-55.269-27.183Q-55.146-27.183-55.062-27.108Q-54.979-27.033-54.979-26.915Q-54.979-26.858-54.987-26.831L-55.339-25.429L-54.412-25.429Q-54.363-25.429-54.335-25.396Q-54.306-25.363-54.306-25.320Q-54.306-25.254-54.339-25.183Q-54.372-25.113-54.429-25.113L-55.414-25.113L-56.068-22.503Q-56.121-22.300-56.121-22.107Q-56.121-21.712-55.862-21.712Q-55.581-21.712-55.350-21.892Q-55.119-22.072-54.948-22.344Q-54.776-22.617-54.675-22.881Q-54.667-22.907-54.645-22.924Q-54.623-22.942-54.592-22.942L-54.486-22.942Q-54.438-22.942-54.416-22.909Q-54.394-22.876-54.394-22.828Q-54.535-22.485-54.746-22.171Q-54.957-21.857-55.242-21.652Q-55.528-21.448-55.879-21.448Q-56.130-21.448-56.332-21.553Q-56.534-21.659-56.653-21.850Q-56.772-22.041-56.772-22.287",[1870],[1866,2428],{"fill":1860,"d":2429},"M-46.98-26.69 8.818-54.346",[1866,2431],{"d":2432,"style":2433},"m11.063-55.458-3.811.382 1.656.686-.456 1.734Z","stroke-linejoin:round;stroke-width:.39997600000000005",[1854,2435,2437,2440],{"fill":2436},"#fff",[1866,2438],{"stroke":1860,"d":2439},"M-29.04-41.319h10.971v-7.51H-29.04Z",[1854,2441,2443],{"transform":2442},"translate(29.815 -21.27)",[1866,2444],{"d":2445,"fill":1856,"stroke":1856,"className":2446,"style":1909},"M-54.030-21.549L-56.560-21.549L-56.560-21.829Q-55.592-21.829-55.592-22.038L-55.592-25.657Q-55.985-25.469-56.607-25.469L-56.607-25.750Q-56.190-25.750-55.826-25.851Q-55.462-25.951-55.206-26.197L-55.080-26.197Q-55.015-26.180-54.998-26.112L-54.998-22.038Q-54.998-21.829-54.030-21.829L-54.030-21.549M-51.385-21.409Q-52.021-21.409-52.385-21.754Q-52.749-22.099-52.884-22.624Q-53.019-23.149-53.019-23.774Q-53.019-24.799-52.663-25.498Q-52.308-26.197-51.385-26.197Q-50.459-26.197-50.106-25.498Q-49.754-24.799-49.754-23.774Q-49.754-23.149-49.889-22.624Q-50.024-22.099-50.387-21.754Q-50.749-21.409-51.385-21.409M-51.385-21.634Q-50.947-21.634-50.734-22.009Q-50.520-22.383-50.471-22.850Q-50.421-23.316-50.421-23.894Q-50.421-24.447-50.471-24.875Q-50.520-25.302-50.732-25.637Q-50.944-25.972-51.385-25.972Q-51.727-25.972-51.930-25.765Q-52.133-25.558-52.221-25.246Q-52.308-24.933-52.330-24.617Q-52.352-24.300-52.352-23.894Q-52.352-23.477-52.330-23.135Q-52.308-22.793-52.219-22.445Q-52.130-22.096-51.925-21.865Q-51.720-21.634-51.385-21.634",[1870],[1866,2448],{"fill":1860,"d":2449},"M-46.98-16.406 8.818 11.248",[1866,2451],{"d":2452,"style":2433},"M11.063 12.36 8.452 9.56l.456 1.733-1.656.687Z",[1854,2454,2455,2458],{"fill":2436},[1866,2456],{"stroke":1860,"d":2457},"M-25.055 5.733h6.986v-7.511h-6.986Z",[1854,2459,2461],{"transform":2460},"translate(33.801 25.781)",[1866,2462],{"d":2463,"fill":1856,"stroke":1856,"className":2464,"style":1909},"M-56.549-22.311L-56.580-22.311Q-56.443-22.014-56.146-21.838Q-55.849-21.662-55.521-21.662Q-55.158-21.662-54.931-21.840Q-54.704-22.017-54.610-22.306Q-54.516-22.595-54.516-22.957Q-54.516-23.272-54.570-23.557Q-54.625-23.842-54.798-24.048Q-54.970-24.253-55.285-24.253Q-55.558-24.253-55.741-24.186Q-55.924-24.119-56.028-24.030Q-56.132-23.942-56.228-23.832Q-56.324-23.723-56.368-23.713L-56.447-23.713Q-56.519-23.730-56.536-23.801L-56.536-26.119Q-56.536-26.153-56.512-26.175Q-56.488-26.197-56.454-26.197L-56.426-26.197Q-56.139-26.081-55.871-26.027Q-55.603-25.972-55.326-25.972Q-55.049-25.972-54.779-26.027Q-54.509-26.081-54.229-26.197L-54.205-26.197Q-54.170-26.197-54.147-26.174Q-54.123-26.150-54.123-26.119L-54.123-26.050Q-54.123-26.023-54.143-26.003Q-54.417-25.688-54.801-25.512Q-55.186-25.336-55.599-25.336Q-55.938-25.336-56.255-25.422L-56.255-24.140Q-55.859-24.475-55.285-24.475Q-54.881-24.475-54.545-24.265Q-54.208-24.054-54.015-23.702Q-53.822-23.350-53.822-22.950Q-53.822-22.619-53.962-22.333Q-54.102-22.048-54.346-21.838Q-54.591-21.628-54.893-21.518Q-55.196-21.409-55.514-21.409Q-55.873-21.409-56.199-21.573Q-56.525-21.737-56.720-22.029Q-56.915-22.321-56.915-22.684Q-56.915-22.834-56.809-22.940Q-56.703-23.046-56.549-23.046Q-56.396-23.046-56.291-22.942Q-56.187-22.838-56.187-22.684Q-56.187-22.527-56.291-22.419Q-56.396-22.311-56.549-22.311",[1870],[1866,2466],{"fill":1860,"d":2467},"M21.619-49.108V3.305",[1866,2469],{"d":2470,"style":2471},"m21.619 5.81 1.35-3.584-1.35 1.179-1.351-1.179Z","stroke-linejoin:round",[1854,2473,2474,2477],{"fill":2436},[1866,2475],{"stroke":1860,"d":2476},"M8.182-17.376H21.42v-8.345H8.182Z",[1854,2478,2479,2486],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,2480,2482],{"transform":2481},"translate(67.038 1.839)",[1866,2483],{"d":2484,"fill":1856,"stroke":1856,"className":2485,"style":1909},"M-52.027-23.125L-56.440-23.125Q-56.508-23.135-56.554-23.181Q-56.601-23.227-56.601-23.299Q-56.601-23.443-56.440-23.466L-52.027-23.466Q-51.867-23.443-51.867-23.299Q-51.867-23.227-51.913-23.181Q-51.959-23.135-52.027-23.125",[1870],[1854,2487,2488],{"transform":2481},[1866,2489],{"d":2490,"fill":1856,"stroke":1856,"className":2491,"style":1909},"M-47.780-21.549L-50.665-21.549L-50.665-21.751Q-50.665-21.781-50.638-21.809L-49.390-23.026Q-49.318-23.101-49.276-23.143Q-49.233-23.186-49.154-23.265Q-48.741-23.678-48.510-24.036Q-48.279-24.393-48.279-24.817Q-48.279-25.049-48.358-25.252Q-48.437-25.456-48.578-25.606Q-48.720-25.757-48.915-25.837Q-49.110-25.917-49.342-25.917Q-49.653-25.917-49.911-25.758Q-50.169-25.599-50.299-25.322L-50.279-25.322Q-50.111-25.322-50.004-25.211Q-49.896-25.100-49.896-24.936Q-49.896-24.779-50.005-24.666Q-50.115-24.553-50.279-24.553Q-50.439-24.553-50.552-24.666Q-50.665-24.779-50.665-24.936Q-50.665-25.312-50.457-25.599Q-50.248-25.886-49.913-26.042Q-49.578-26.197-49.223-26.197Q-48.799-26.197-48.419-26.039Q-48.040-25.880-47.806-25.563Q-47.572-25.247-47.572-24.817Q-47.572-24.506-47.712-24.237Q-47.852-23.969-48.057-23.764Q-48.262-23.559-48.625-23.277Q-48.987-22.995-49.096-22.899L-49.951-22.171L-49.308-22.171Q-49.045-22.171-48.756-22.173Q-48.467-22.174-48.249-22.183Q-48.030-22.192-48.013-22.209Q-47.951-22.274-47.914-22.441Q-47.876-22.609-47.838-22.851L-47.572-22.851",[1870],[1866,2493],{"fill":1860,"d":2494},"M33.2-60.689h59.89",[1866,2496],{"d":2497,"style":2471},"m95.596-60.689-3.585-1.35 1.179 1.35-1.179 1.35Z",[1854,2499,2500,2503],{"fill":2436},[1866,2501],{"stroke":1860,"d":2502},"M61.005-60.889h6.986V-68.4h-6.986Z",[1854,2504,2506],{"transform":2505},"translate(119.86 -40.84)",[1866,2507],{"d":2508,"fill":1856,"stroke":1856,"className":2509,"style":1909},"M-54.030-21.549L-56.560-21.549L-56.560-21.829Q-55.592-21.829-55.592-22.038L-55.592-25.657Q-55.985-25.469-56.607-25.469L-56.607-25.750Q-56.190-25.750-55.826-25.851Q-55.462-25.951-55.206-26.197L-55.080-26.197Q-55.015-26.180-54.998-26.112L-54.998-22.038Q-54.998-21.829-54.030-21.829",[1870],[1866,2511],{"fill":1860,"d":2512},"m30.172 9.785 66.654-60.843",[1866,2514],{"d":2515,"style":2433},"m98.676-52.747-3.558 1.419 1.782.203.04 1.792Z",[1854,2517,2518,2521],{"fill":2436},[1866,2519],{"stroke":1860,"d":2520},"M54.139-7.57h6.986v-7.512H54.14Z",[1854,2522,2524],{"transform":2523},"translate(112.995 12.478)",[1866,2525],{"d":2526,"fill":1856,"stroke":1856,"className":2527,"style":1909},"M-56.382-21.863Q-56.262-21.747-56.085-21.705Q-55.907-21.662-55.691-21.662Q-55.452-21.662-55.242-21.771Q-55.032-21.881-54.878-22.063Q-54.724-22.246-54.625-22.479Q-54.458-22.906-54.458-23.726Q-54.608-23.432-54.871-23.253Q-55.134-23.073-55.452-23.073Q-55.886-23.073-56.233-23.282Q-56.580-23.490-56.778-23.851Q-56.977-24.212-56.977-24.635Q-56.977-24.970-56.847-25.259Q-56.717-25.548-56.486-25.762Q-56.255-25.975-55.956-26.086Q-55.657-26.197-55.326-26.197Q-54.468-26.197-54.112-25.483Q-53.757-24.769-53.757-23.812Q-53.757-23.395-53.885-22.967Q-54.013-22.540-54.270-22.185Q-54.526-21.829-54.888-21.619Q-55.251-21.409-55.691-21.409Q-56.146-21.409-56.464-21.597Q-56.782-21.785-56.782-22.209Q-56.782-22.359-56.683-22.458Q-56.584-22.557-56.433-22.557Q-56.365-22.557-56.298-22.530Q-56.231-22.503-56.187-22.458Q-56.143-22.414-56.115-22.347Q-56.088-22.280-56.088-22.209Q-56.088-22.079-56.168-21.981Q-56.249-21.884-56.382-21.863M-55.411-23.299Q-55.117-23.299-54.902-23.477Q-54.687-23.654-54.579-23.930Q-54.471-24.205-54.471-24.495Q-54.471-24.540-54.473-24.567Q-54.475-24.594-54.478-24.629Q-54.475-24.639-54.473-24.646Q-54.471-24.653-54.471-24.663Q-54.471-25.165-54.669-25.565Q-54.868-25.965-55.326-25.965Q-55.893-25.965-56.086-25.606Q-56.279-25.247-56.279-24.635Q-56.279-24.249-56.225-23.966Q-56.170-23.682-55.975-23.490Q-55.780-23.299-55.411-23.299",[1870],[1866,2529],{"fill":1860,"d":2530},"m117.753-55.547 55.798 27.655",[1866,2532],{"d":2533,"style":2433},"m175.796-26.78-2.612-2.802.456 1.734-1.656.687Z",[1854,2535,2536,2539],{"fill":2436},[1866,2537],{"stroke":1860,"d":2538},"M147.064-41.319h6.986v-7.51h-6.986Z",[1854,2540,2542],{"transform":2541},"translate(205.92 -21.27)",[1866,2543],{"d":2544,"fill":1856,"stroke":1856,"className":2545,"style":1909},"M-55.039-22.697L-57.083-22.697L-57.083-22.978L-54.752-26.150Q-54.717-26.197-54.652-26.197L-54.516-26.197Q-54.471-26.197-54.444-26.170Q-54.417-26.143-54.417-26.098L-54.417-22.978L-53.654-22.978L-53.654-22.697L-54.417-22.697L-54.417-22.038Q-54.417-21.829-53.661-21.829L-53.661-21.549L-55.794-21.549L-55.794-21.829Q-55.039-21.829-55.039-22.038L-55.039-22.697M-54.991-25.422L-56.782-22.978L-54.991-22.978",[1870],[1866,2547],{"fill":1860,"d":2548},"m32.885 14.915 139.567-33.162",[1866,2550],{"d":2551,"style":2552},"m174.89-18.826-3.8-.485 1.46 1.041-.836 1.587Z","stroke-linejoin:round;stroke-width:.39998",[1854,2554,2555,2558],{"fill":2436},[1866,2556],{"stroke":1860,"d":2557},"M100.492 5.733h6.986v-7.511h-6.986Z",[1854,2559,2561],{"transform":2560},"translate(159.347 25.781)",[1866,2562],{"d":2563,"fill":1856,"stroke":1856,"className":2564,"style":1909},"M-54.030-21.549L-56.915-21.549L-56.915-21.751Q-56.915-21.781-56.888-21.809L-55.640-23.026Q-55.568-23.101-55.526-23.143Q-55.483-23.186-55.404-23.265Q-54.991-23.678-54.760-24.036Q-54.529-24.393-54.529-24.817Q-54.529-25.049-54.608-25.252Q-54.687-25.456-54.828-25.606Q-54.970-25.757-55.165-25.837Q-55.360-25.917-55.592-25.917Q-55.903-25.917-56.161-25.758Q-56.419-25.599-56.549-25.322L-56.529-25.322Q-56.361-25.322-56.254-25.211Q-56.146-25.100-56.146-24.936Q-56.146-24.779-56.255-24.666Q-56.365-24.553-56.529-24.553Q-56.689-24.553-56.802-24.666Q-56.915-24.779-56.915-24.936Q-56.915-25.312-56.707-25.599Q-56.498-25.886-56.163-26.042Q-55.828-26.197-55.473-26.197Q-55.049-26.197-54.669-26.039Q-54.290-25.880-54.056-25.563Q-53.822-25.247-53.822-24.817Q-53.822-24.506-53.962-24.237Q-54.102-23.969-54.307-23.764Q-54.512-23.559-54.875-23.277Q-55.237-22.995-55.346-22.899L-56.201-22.171L-55.558-22.171Q-55.295-22.171-55.006-22.173Q-54.717-22.174-54.499-22.183Q-54.280-22.192-54.263-22.209Q-54.201-22.274-54.164-22.441Q-54.126-22.609-54.088-22.851L-53.822-22.851",[1870],[2089,2566,2568,2569,1024,2584,1617,2599,2617],{"className":2567},[2092],"A small weighted digraph with one negative edge ",[415,2570,2572],{"className":2571},[418],[415,2573,2575],{"className":2574,"ariaHidden":423},[422],[415,2576,2578,2581],{"className":2577},[427],[415,2579],{"className":2580,"style":1020},[431],[415,2582,385],{"className":2583},[436,437],[415,2585,2587],{"className":2586},[418],[415,2588,2590],{"className":2589,"ariaHidden":423},[422],[415,2591,2593,2596],{"className":2592},[427],[415,2594],{"className":2595,"style":1163},[431],[415,2597,2342],{"className":2598},[436,437],[415,2600,2602],{"className":2601},[418],[415,2603,2605],{"className":2604,"ariaHidden":423},[422],[415,2606,2608,2611,2614],{"className":2607},[427],[415,2609],{"className":2610,"style":1704},[431],[415,2612,903],{"className":2613},[436],[415,2615,2361],{"className":2616},[436],", used to trace the algorithms.",[400,2619,2621],{"id":2620},"dijkstras-algorithm","Dijkstra's algorithm",[381,2623,2624,2625,2628,2629,2652,2653,2669,2670,2673,2674,2689,2690,2711,2712,2715,2716,2731],{},"When ",[395,2626,2627],{},"all edge weights are non-negative",", we can be greedy. ",[415,2630,2632],{"className":2631},[418],[415,2633,2635],{"className":2634,"ariaHidden":423},[422],[415,2636,2638,2642],{"className":2637},[427],[415,2639],{"className":2640,"style":2641},[431],"height:0.8889em;vertical-align:-0.1944em;",[415,2643,2645],{"className":2644},[2109,2110],[415,2646,2648],{"className":2647},[436,2114],[415,2649,2651],{"className":2650},[436],"Dijkstra","'s\nalgorithm grows a set ",[415,2654,2656],{"className":2655},[418],[415,2657,2659],{"className":2658,"ariaHidden":423},[422],[415,2660,2662,2665],{"className":2661},[427],[415,2663],{"className":2664,"style":432},[431],[415,2666,2668],{"className":2667,"style":481},[436,437],"S"," of vertices whose shortest distances are ",[395,2671,2672],{},"finalized",".\nAt each step it picks the non-finalized vertex ",[415,2675,2677],{"className":2676},[418],[415,2678,2680],{"className":2679,"ariaHidden":423},[422],[415,2681,2683,2686],{"className":2682},[427],[415,2684],{"className":2685,"style":1020},[431],[415,2687,512],{"className":2688},[436,437]," with the smallest estimate\n",[415,2691,2693],{"className":2692},[418],[415,2694,2696],{"className":2695,"ariaHidden":423},[422],[415,2697,2699,2702,2705,2708],{"className":2698},[427],[415,2700],{"className":2701,"style":1163},[431],[415,2703,512],{"className":2704},[436,437],[415,2706,1131],{"className":2707},[436],[415,2709,1173],{"className":2710},[436,437],", finalizes it, and relaxes its outgoing edges. A\n",[385,2713,2714],{"href":56},"min-priority queue"," keyed\nby ",[415,2717,2719],{"className":2718},[418],[415,2720,2722],{"className":2721,"ariaHidden":423},[422],[415,2723,2725,2728],{"className":2724},[427],[415,2726],{"className":2727,"style":1163},[431],[415,2729,1173],{"className":2730},[436,437]," supplies the next vertex.",[1381,2733,2735],{"className":1383,"code":2734,"language":1385,"meta":376,"style":376},"caption: $\\textsc{Dijkstra}(G, w, s)$ — SSSP for non-negative weights\nnumber: 2\nforeach vertex $v \\in V$ do\n  $v.d \\gets \\infty$\n  $v.\\pi \\gets \\text{nil}$\n$s.d \\gets 0$\n$S \\gets \\emptyset$\n$Q \\gets V$ \u002F\u002F min-PQ keyed by d\nwhile $Q \\neq \\emptyset$ do\n  $u \\gets$ $\\textsc{Extract-Min}(Q)$ \u002F\u002F closest unfinalized\n  $S \\gets S \\cup \\set{u}$ \u002F\u002F u.d now final\n  foreach $v$ adjacent to $u$ do\n    call $\\textsc{Relax}(u, v, w)$ \u002F\u002F Decrease-Key updates Q\nreturn $d$ and $\\pi$\n",[1387,2736,2737,2742,2747,2752,2757,2762,2767,2772,2777,2782,2787,2792,2797,2803],{"__ignoreMap":376},[415,2738,2739],{"class":1391,"line":6},[415,2740,2741],{},"caption: $\\textsc{Dijkstra}(G, w, s)$ — SSSP for non-negative weights\n",[415,2743,2744],{"class":1391,"line":18},[415,2745,2746],{},"number: 2\n",[415,2748,2749],{"class":1391,"line":24},[415,2750,2751],{},"foreach vertex $v \\in V$ do\n",[415,2753,2754],{"class":1391,"line":73},[415,2755,2756],{},"  $v.d \\gets \\infty$\n",[415,2758,2759],{"class":1391,"line":102},[415,2760,2761],{},"  $v.\\pi \\gets \\text{nil}$\n",[415,2763,2764],{"class":1391,"line":108},[415,2765,2766],{},"$s.d \\gets 0$\n",[415,2768,2769],{"class":1391,"line":116},[415,2770,2771],{},"$S \\gets \\emptyset$\n",[415,2773,2774],{"class":1391,"line":196},[415,2775,2776],{},"$Q \\gets V$ \u002F\u002F min-PQ keyed by d\n",[415,2778,2779],{"class":1391,"line":202},[415,2780,2781],{},"while $Q \\neq \\emptyset$ do\n",[415,2783,2784],{"class":1391,"line":283},[415,2785,2786],{},"  $u \\gets$ $\\textsc{Extract-Min}(Q)$ \u002F\u002F closest unfinalized\n",[415,2788,2789],{"class":1391,"line":333},[415,2790,2791],{},"  $S \\gets S \\cup \\set{u}$ \u002F\u002F u.d now final\n",[415,2793,2794],{"class":1391,"line":354},[415,2795,2796],{},"  foreach $v$ adjacent to $u$ do\n",[415,2798,2800],{"class":1391,"line":2799},13,[415,2801,2802],{},"    call $\\textsc{Relax}(u, v, w)$ \u002F\u002F Decrease-Key updates Q\n",[415,2804,2806],{"class":1391,"line":2805},14,[415,2807,2808],{},"return $d$ and $\\pi$\n",[381,2810,2811,2814,2815,2818,2819,2822,2823,2825,2826,2934,2935,2979,2980,3055,3056,3071,3072,3093,3094,3097,3098,1131],{},[390,2812,2813],{},"Why the greedy choice is correct."," Correctness is framed around a\nsingle ",[390,2816,2817],{},"loop invariant",": at the start of each iteration of the ",[1387,2820,2821],{},"while"," loop,\nevery ",[395,2824,2672],{}," vertex already holds its true distance,\n",[415,2827,2829],{"className":2828},[418],[415,2830,2832,2857,2880,2904],{"className":2831,"ariaHidden":423},[422],[415,2833,2835,2839,2843,2847,2850,2854],{"className":2834},[427],[415,2836],{"className":2837,"style":2838},[431],"height:0.7335em;vertical-align:-0.0391em;",[415,2840,2842],{"className":2841},[436],"∀",[415,2844,2846],{"className":2845},[436,437],"x",[415,2848],{"className":2849,"style":443},[442],[415,2851,2853],{"className":2852},[447],"∈",[415,2855],{"className":2856,"style":443},[442],[415,2858,2860,2863,2866,2869,2873,2877],{"className":2859},[427],[415,2861],{"className":2862,"style":432},[431],[415,2864,2668],{"className":2865,"style":481},[436,437],[415,2867],{"className":2868,"style":443},[442],[415,2870,2872],{"className":2871},[447],":",[415,2874],{"className":2875,"style":2876},[442],"margin-right:1em;",[415,2878],{"className":2879,"style":443},[442],[415,2881,2883,2886,2889,2892,2895,2898,2901],{"className":2882},[427],[415,2884],{"className":2885,"style":1163},[431],[415,2887,2846],{"className":2888},[436,437],[415,2890,1131],{"className":2891},[436],[415,2893,1173],{"className":2894},[436,437],[415,2896],{"className":2897,"style":443},[442],[415,2899,448],{"className":2900},[447],[415,2902],{"className":2903,"style":443},[442],[415,2905,2907,2910,2913,2916,2919,2922,2925,2928,2931],{"className":2906},[427],[415,2908],{"className":2909,"style":458},[431],[415,2911,988],{"className":2912,"style":987},[436,437],[415,2914,463],{"className":2915},[462],[415,2917,1083],{"className":2918},[436,437],[415,2920,473],{"className":2921},[472],[415,2923],{"className":2924,"style":477},[442],[415,2926,2846],{"className":2927},[436,437],[415,2929,487],{"className":2930},[486],[415,2932,1131],{"className":2933},[436],"\nIt holds initially (",[415,2936,2938],{"className":2937},[418],[415,2939,2941,2959],{"className":2940,"ariaHidden":423},[422],[415,2942,2944,2947,2950,2953,2956],{"className":2943},[427],[415,2945],{"className":2946,"style":432},[431],[415,2948,2668],{"className":2949,"style":481},[436,437],[415,2951],{"className":2952,"style":443},[442],[415,2954,448],{"className":2955},[447],[415,2957],{"className":2958,"style":443},[442],[415,2960,2962,2965],{"className":2961},[427],[415,2963],{"className":2964,"style":458},[431],[415,2966,2968,2972,2975],{"className":2967},[566],[415,2969,2971],{"className":2970,"style":571},[462,570],"{",[415,2973,1083],{"className":2974},[436,437],[415,2976,2978],{"className":2977,"style":571},[486,570],"}"," and ",[415,2981,2983],{"className":2982},[418],[415,2984,2986,3010,3028],{"className":2985,"ariaHidden":423},[422],[415,2987,2989,2992,2995,2998,3001,3004,3007],{"className":2988},[427],[415,2990],{"className":2991,"style":1163},[431],[415,2993,1083],{"className":2994},[436,437],[415,2996,1131],{"className":2997},[436],[415,2999,1173],{"className":3000},[436,437],[415,3002],{"className":3003,"style":443},[442],[415,3005,448],{"className":3006},[447],[415,3008],{"className":3009,"style":443},[442],[415,3011,3013,3016,3019,3022,3025],{"className":3012},[427],[415,3014],{"className":3015,"style":1294},[431],[415,3017,615],{"className":3018},[436],[415,3020],{"className":3021,"style":443},[442],[415,3023,448],{"className":3024},[447],[415,3026],{"className":3027,"style":443},[442],[415,3029,3031,3034,3037,3040,3043,3046,3049,3052],{"className":3030},[427],[415,3032],{"className":3033,"style":458},[431],[415,3035,988],{"className":3036,"style":987},[436,437],[415,3038,463],{"className":3039},[462],[415,3041,1083],{"className":3042},[436,437],[415,3044,473],{"className":3045},[472],[415,3047],{"className":3048,"style":477},[442],[415,3050,1083],{"className":3051},[436,437],[415,3053,487],{"className":3054},[486],"). The maintenance\nstep is the crux of the argument: suppose finalizing ",[415,3057,3059],{"className":3058},[418],[415,3060,3062],{"className":3061,"ariaHidden":423},[422],[415,3063,3065,3068],{"className":3064},[427],[415,3066],{"className":3067,"style":1020},[431],[415,3069,512],{"className":3070},[436,437]," were the first move to\nbreak the invariant. Then either ",[415,3073,3075],{"className":3074},[418],[415,3076,3078],{"className":3077,"ariaHidden":423},[422],[415,3079,3081,3084,3087,3090],{"className":3080},[427],[415,3082],{"className":3083,"style":1163},[431],[415,3085,512],{"className":3086},[436,437],[415,3088,1131],{"className":3089},[436],[415,3091,1173],{"className":3092},[436,437]," is set too ",[395,3095,3096],{},"low"," or too ",[395,3099,3100],{},"high",[3102,3103,3104,3227],"ul",{},[3105,3106,3107,3110,3111,3168,3169,3190,3191,3226],"li",{},[395,3108,3109],{},"Too low"," (",[415,3112,3114],{"className":3113},[418],[415,3115,3117,3141],{"className":3116,"ariaHidden":423},[422],[415,3118,3120,3123,3126,3129,3132,3135,3138],{"className":3119},[427],[415,3121],{"className":3122,"style":2838},[431],[415,3124,512],{"className":3125},[436,437],[415,3127,1131],{"className":3128},[436],[415,3130,1173],{"className":3131},[436,437],[415,3133],{"className":3134,"style":443},[442],[415,3136,1752],{"className":3137},[447],[415,3139],{"className":3140,"style":443},[442],[415,3142,3144,3147,3150,3153,3156,3159,3162,3165],{"className":3143},[427],[415,3145],{"className":3146,"style":458},[431],[415,3148,988],{"className":3149,"style":987},[436,437],[415,3151,463],{"className":3152},[462],[415,3154,1083],{"className":3155},[436,437],[415,3157,473],{"className":3158},[472],[415,3160],{"className":3161,"style":477},[442],[415,3163,512],{"className":3164},[436,437],[415,3166,487],{"className":3167},[486],") is impossible, because relaxation never drops\nan estimate below the true distance — every value of ",[415,3170,3172],{"className":3171},[418],[415,3173,3175],{"className":3174,"ariaHidden":423},[422],[415,3176,3178,3181,3184,3187],{"className":3177},[427],[415,3179],{"className":3180,"style":1163},[431],[415,3182,512],{"className":3183},[436,437],[415,3185,1131],{"className":3186},[436],[415,3188,1173],{"className":3189},[436,437]," is the cost of some\nreal ",[415,3192,3194],{"className":3193},[418],[415,3195,3197,3217],{"className":3196,"ariaHidden":423},[422],[415,3198,3200,3203,3206,3209,3214],{"className":3199},[427],[415,3201],{"className":3202,"style":1020},[431],[415,3204,1083],{"className":3205},[436,437],[415,3207],{"className":3208,"style":443},[442],[415,3210,3213],{"className":3211},[447,3212],"amsrm","⇝",[415,3215],{"className":3216,"style":443},[442],[415,3218,3220,3223],{"className":3219},[427],[415,3221],{"className":3222,"style":1020},[431],[415,3224,512],{"className":3225},[436,437]," walk.",[3105,3228,3229,3110,3232,3290,3291,3306,3307,1024,3322,3337,3338,3353,3354,3369,3370,3401,3402,3420,3421,3455,3456,3529,3530,1536,3587,3617,3618,3633,3634,3763,3764,3779,3780,1536,3795,3825,3826,3841,3842,3935,3936,3981,3982,4004,4005,4020,4021,4036],{},[395,3230,3231],{},"Too high",[415,3233,3235],{"className":3234},[418],[415,3236,3238,3263],{"className":3237,"ariaHidden":423},[422],[415,3239,3241,3244,3247,3250,3253,3256,3260],{"className":3240},[427],[415,3242],{"className":3243,"style":2838},[431],[415,3245,512],{"className":3246},[436,437],[415,3248,1131],{"className":3249},[436],[415,3251,1173],{"className":3252},[436,437],[415,3254],{"className":3255,"style":443},[442],[415,3257,3259],{"className":3258},[447],">",[415,3261],{"className":3262,"style":443},[442],[415,3264,3266,3269,3272,3275,3278,3281,3284,3287],{"className":3265},[427],[415,3267],{"className":3268,"style":458},[431],[415,3270,988],{"className":3271,"style":987},[436,437],[415,3273,463],{"className":3274},[462],[415,3276,1083],{"className":3277},[436,437],[415,3279,473],{"className":3280},[472],[415,3282],{"className":3283,"style":477},[442],[415,3285,512],{"className":3286},[436,437],[415,3288,487],{"className":3289},[486],"): let ",[415,3292,3294],{"className":3293},[418],[415,3295,3297],{"className":3296,"ariaHidden":423},[422],[415,3298,3300,3303],{"className":3299},[427],[415,3301],{"className":3302,"style":1020},[431],[415,3304,1252],{"className":3305,"style":522},[436,437]," be a genuine shortest path from\n",[415,3308,3310],{"className":3309},[418],[415,3311,3313],{"className":3312,"ariaHidden":423},[422],[415,3314,3316,3319],{"className":3315},[427],[415,3317],{"className":3318,"style":1020},[431],[415,3320,1083],{"className":3321},[436,437],[415,3323,3325],{"className":3324},[418],[415,3326,3328],{"className":3327,"ariaHidden":423},[422],[415,3329,3331,3334],{"className":3330},[427],[415,3332],{"className":3333,"style":1020},[431],[415,3335,512],{"className":3336},[436,437],". Walking it from ",[415,3339,3341],{"className":3340},[418],[415,3342,3344],{"className":3343,"ariaHidden":423},[422],[415,3345,3347,3350],{"className":3346},[427],[415,3348],{"className":3349,"style":1020},[431],[415,3351,1083],{"className":3352},[436,437]," (which is in ",[415,3355,3357],{"className":3356},[418],[415,3358,3360],{"className":3359,"ariaHidden":423},[422],[415,3361,3363,3366],{"className":3362},[427],[415,3364],{"className":3365,"style":432},[431],[415,3367,2668],{"className":3368,"style":481},[436,437],"), let ",[415,3371,3373],{"className":3372},[418],[415,3374,3376],{"className":3375,"ariaHidden":423},[422],[415,3377,3379,3382,3385,3388,3391,3394,3398],{"className":3378},[427],[415,3380],{"className":3381,"style":458},[431],[415,3383,463],{"className":3384},[462],[415,3386,2846],{"className":3387},[436,437],[415,3389,473],{"className":3390},[472],[415,3392],{"className":3393,"style":477},[442],[415,3395,3397],{"className":3396,"style":522},[436,437],"y",[415,3399,487],{"className":3400},[486]," be the ",[390,3403,3404,3405],{},"first\nedge that leaves ",[415,3406,3408],{"className":3407},[418],[415,3409,3411],{"className":3410,"ariaHidden":423},[422],[415,3412,3414,3417],{"className":3413},[427],[415,3415],{"className":3416,"style":432},[431],[415,3418,2668],{"className":3419,"style":481},[436,437]," — so ",[415,3422,3424],{"className":3423},[418],[415,3425,3427,3446],{"className":3426,"ariaHidden":423},[422],[415,3428,3430,3434,3437,3440,3443],{"className":3429},[427],[415,3431],{"className":3432,"style":3433},[431],"height:0.5782em;vertical-align:-0.0391em;",[415,3435,2846],{"className":3436},[436,437],[415,3438],{"className":3439,"style":443},[442],[415,3441,2853],{"className":3442},[447],[415,3444],{"className":3445,"style":443},[442],[415,3447,3449,3452],{"className":3448},[427],[415,3450],{"className":3451,"style":432},[431],[415,3453,2668],{"className":3454,"style":481},[436,437],", ",[415,3457,3459],{"className":3458},[418],[415,3460,3462,3520],{"className":3461,"ariaHidden":423},[422],[415,3463,3465,3468,3471,3474,3517],{"className":3464},[427],[415,3466],{"className":3467,"style":458},[431],[415,3469,3397],{"className":3470,"style":522},[436,437],[415,3472],{"className":3473,"style":443},[442],[415,3475,3477,3483],{"className":3476},[447],[415,3478,3480],{"className":3479},[436],[415,3481,2853],{"className":3482},[447],[415,3484,3487],{"className":3485},[436,3486],"vbox",[415,3488,3491],{"className":3489},[3490],"thinbox",[415,3492,3495,3498,3513],{"className":3493},[3494],"llap",[415,3496],{"className":3497,"style":458},[431],[415,3499,3502],{"className":3500},[3501],"inner",[415,3503,3505,3509],{"className":3504},[436],[415,3506,3508],{"className":3507},[436],"\u002F",[415,3510],{"className":3511,"style":3512},[442],"margin-right:0.0556em;",[415,3514],{"className":3515},[3516],"fix",[415,3518],{"className":3519,"style":443},[442],[415,3521,3523,3526],{"className":3522},[427],[415,3524],{"className":3525,"style":432},[431],[415,3527,2668],{"className":3528,"style":481},[436,437],". By the invariant\n",[415,3531,3533],{"className":3532},[418],[415,3534,3536,3560],{"className":3535,"ariaHidden":423},[422],[415,3537,3539,3542,3545,3548,3551,3554,3557],{"className":3538},[427],[415,3540],{"className":3541,"style":1163},[431],[415,3543,2846],{"className":3544},[436,437],[415,3546,1131],{"className":3547},[436],[415,3549,1173],{"className":3550},[436,437],[415,3552],{"className":3553,"style":443},[442],[415,3555,448],{"className":3556},[447],[415,3558],{"className":3559,"style":443},[442],[415,3561,3563,3566,3569,3572,3575,3578,3581,3584],{"className":3562},[427],[415,3564],{"className":3565,"style":458},[431],[415,3567,988],{"className":3568,"style":987},[436,437],[415,3570,463],{"className":3571},[462],[415,3573,1083],{"className":3574},[436,437],[415,3576,473],{"className":3577},[472],[415,3579],{"className":3580,"style":477},[442],[415,3582,2846],{"className":3583},[436,437],[415,3585,487],{"className":3586},[486],[415,3588,3590],{"className":3589},[418],[415,3591,3593],{"className":3592,"ariaHidden":423},[422],[415,3594,3596,3599,3602,3605,3608,3611,3614],{"className":3595},[427],[415,3597],{"className":3598,"style":458},[431],[415,3600,463],{"className":3601},[462],[415,3603,2846],{"className":3604},[436,437],[415,3606,473],{"className":3607},[472],[415,3609],{"className":3610,"style":477},[442],[415,3612,3397],{"className":3613,"style":522},[436,437],[415,3615,487],{"className":3616},[486]," was relaxed when ",[415,3619,3621],{"className":3620},[418],[415,3622,3624],{"className":3623,"ariaHidden":423},[422],[415,3625,3627,3630],{"className":3626},[427],[415,3628],{"className":3629,"style":1020},[431],[415,3631,2846],{"className":3632},[436,437]," was finalized, so\n",[415,3635,3637],{"className":3636},[418],[415,3638,3640,3664,3700,3736],{"className":3639,"ariaHidden":423},[422],[415,3641,3643,3646,3649,3652,3655,3658,3661],{"className":3642},[427],[415,3644],{"className":3645,"style":2641},[431],[415,3647,3397],{"className":3648,"style":522},[436,437],[415,3650,1131],{"className":3651},[436],[415,3653,1173],{"className":3654},[436,437],[415,3656],{"className":3657,"style":443},[442],[415,3659,448],{"className":3660},[447],[415,3662],{"className":3663,"style":443},[442],[415,3665,3667,3670,3673,3676,3679,3682,3685,3688,3691,3694,3697],{"className":3666},[427],[415,3668],{"className":3669,"style":458},[431],[415,3671,988],{"className":3672,"style":987},[436,437],[415,3674,463],{"className":3675},[462],[415,3677,1083],{"className":3678},[436,437],[415,3680,473],{"className":3681},[472],[415,3683],{"className":3684,"style":477},[442],[415,3686,2846],{"className":3687},[436,437],[415,3689,487],{"className":3690},[486],[415,3692],{"className":3693,"style":467},[442],[415,3695,1505],{"className":3696},[902],[415,3698],{"className":3699,"style":467},[442],[415,3701,3703,3706,3709,3712,3715,3718,3721,3724,3727,3730,3733],{"className":3702},[427],[415,3704],{"className":3705,"style":458},[431],[415,3707,505],{"className":3708,"style":504},[436,437],[415,3710,463],{"className":3711},[462],[415,3713,2846],{"className":3714},[436,437],[415,3716,473],{"className":3717},[472],[415,3719],{"className":3720,"style":477},[442],[415,3722,3397],{"className":3723,"style":522},[436,437],[415,3725,487],{"className":3726},[486],[415,3728],{"className":3729,"style":443},[442],[415,3731,448],{"className":3732},[447],[415,3734],{"className":3735,"style":443},[442],[415,3737,3739,3742,3745,3748,3751,3754,3757,3760],{"className":3738},[427],[415,3740],{"className":3741,"style":458},[431],[415,3743,988],{"className":3744,"style":987},[436,437],[415,3746,463],{"className":3747},[462],[415,3749,1083],{"className":3750},[436,437],[415,3752,473],{"className":3753},[472],[415,3755],{"className":3756,"style":477},[442],[415,3758,3397],{"className":3759,"style":522},[436,437],[415,3761,487],{"className":3762},[486],". Now ",[415,3765,3767],{"className":3766},[418],[415,3768,3770],{"className":3769,"ariaHidden":423},[422],[415,3771,3773,3776],{"className":3772},[427],[415,3774],{"className":3775,"style":544},[431],[415,3777,3397],{"className":3778,"style":522},[436,437]," sits on a shortest path\nto ",[415,3781,3783],{"className":3782},[418],[415,3784,3786],{"className":3785,"ariaHidden":423},[422],[415,3787,3789,3792],{"className":3788},[427],[415,3790],{"className":3791,"style":1020},[431],[415,3793,512],{"className":3794},[436,437],[390,3796,3797,3798],{},"because every weight is ",[415,3799,3801],{"className":3800},[418],[415,3802,3804,3816],{"className":3803,"ariaHidden":423},[422],[415,3805,3807,3810,3813],{"className":3806},[427],[415,3808],{"className":3809,"style":1225},[431],[415,3811,1229],{"className":3812},[447],[415,3814],{"className":3815,"style":443},[442],[415,3817,3819,3822],{"className":3818},[427],[415,3820],{"className":3821,"style":1294},[431],[415,3823,615],{"className":3824},[436],", the rest of ",[415,3827,3829],{"className":3828},[418],[415,3830,3832],{"className":3831,"ariaHidden":423},[422],[415,3833,3835,3838],{"className":3834},[427],[415,3836],{"className":3837,"style":1020},[431],[415,3839,1252],{"className":3840,"style":522},[436,437]," only adds\ncost: ",[415,3843,3845],{"className":3844},[418],[415,3846,3848,3884,3920],{"className":3847,"ariaHidden":423},[422],[415,3849,3851,3854,3857,3860,3863,3866,3869,3872,3875,3878,3881],{"className":3850},[427],[415,3852],{"className":3853,"style":458},[431],[415,3855,988],{"className":3856,"style":987},[436,437],[415,3858,463],{"className":3859},[462],[415,3861,1083],{"className":3862},[436,437],[415,3864,473],{"className":3865},[472],[415,3867],{"className":3868,"style":477},[442],[415,3870,3397],{"className":3871,"style":522},[436,437],[415,3873,487],{"className":3874},[486],[415,3876],{"className":3877,"style":443},[442],[415,3879,1468],{"className":3880},[447],[415,3882],{"className":3883,"style":443},[442],[415,3885,3887,3890,3893,3896,3899,3902,3905,3908,3911,3914,3917],{"className":3886},[427],[415,3888],{"className":3889,"style":458},[431],[415,3891,988],{"className":3892,"style":987},[436,437],[415,3894,463],{"className":3895},[462],[415,3897,1083],{"className":3898},[436,437],[415,3900,473],{"className":3901},[472],[415,3903],{"className":3904,"style":477},[442],[415,3906,512],{"className":3907},[436,437],[415,3909,487],{"className":3910},[486],[415,3912],{"className":3913,"style":443},[442],[415,3915,1468],{"className":3916},[447],[415,3918],{"className":3919,"style":443},[442],[415,3921,3923,3926,3929,3932],{"className":3922},[427],[415,3924],{"className":3925,"style":1163},[431],[415,3927,512],{"className":3928},[436,437],[415,3930,1131],{"className":3931},[436],[415,3933,1173],{"className":3934},[436,437],". Hence ",[415,3937,3939],{"className":3938},[418],[415,3940,3942,3966],{"className":3941,"ariaHidden":423},[422],[415,3943,3945,3948,3951,3954,3957,3960,3963],{"className":3944},[427],[415,3946],{"className":3947,"style":2641},[431],[415,3949,3397],{"className":3950,"style":522},[436,437],[415,3952,1131],{"className":3953},[436],[415,3955,1173],{"className":3956},[436,437],[415,3958],{"className":3959,"style":443},[442],[415,3961,1468],{"className":3962},[447],[415,3964],{"className":3965,"style":443},[442],[415,3967,3969,3972,3975,3978],{"className":3968},[427],[415,3970],{"className":3971,"style":1163},[431],[415,3973,512],{"className":3974},[436,437],[415,3976,1131],{"className":3977},[436],[415,3979,1173],{"className":3980},[436,437],", so\n",[415,3983,3985],{"className":3984},[418],[415,3986,3988],{"className":3987,"ariaHidden":423},[422],[415,3989,3991,3994],{"className":3990},[427],[415,3992],{"className":3993,"style":432},[431],[415,3995,3997],{"className":3996},[2109,2110],[415,3998,4000],{"className":3999},[436,2114],[415,4001,4003],{"className":4002},[436],"Extract-Min"," would have returned ",[415,4006,4008],{"className":4007},[418],[415,4009,4011],{"className":4010,"ariaHidden":423},[422],[415,4012,4014,4017],{"className":4013},[427],[415,4015],{"className":4016,"style":544},[431],[415,4018,3397],{"className":4019,"style":522},[436,437],", not ",[415,4022,4024],{"className":4023},[418],[415,4025,4027],{"className":4026,"ariaHidden":423},[422],[415,4028,4030,4033],{"className":4029},[427],[415,4031],{"className":4032,"style":1020},[431],[415,4034,512],{"className":4035},[436,437]," — contradiction.",[381,4038,4039,4040,4043,4044,4113],{},"The non-negativity is doing all the work in that last inequality: it guarantees\nextending a path never ",[395,4041,4042],{},"decreases"," its cost, so the closest frontier vertex can\nbe safely frozen. A single negative edge breaks ",[415,4045,4047],{"className":4046},[418],[415,4048,4050,4086],{"className":4049,"ariaHidden":423},[422],[415,4051,4053,4056,4059,4062,4065,4068,4071,4074,4077,4080,4083],{"className":4052},[427],[415,4054],{"className":4055,"style":458},[431],[415,4057,988],{"className":4058,"style":987},[436,437],[415,4060,463],{"className":4061},[462],[415,4063,1083],{"className":4064},[436,437],[415,4066,473],{"className":4067},[472],[415,4069],{"className":4070,"style":477},[442],[415,4072,3397],{"className":4073,"style":522},[436,437],[415,4075,487],{"className":4076},[486],[415,4078],{"className":4079,"style":443},[442],[415,4081,1468],{"className":4082},[447],[415,4084],{"className":4085,"style":443},[442],[415,4087,4089,4092,4095,4098,4101,4104,4107,4110],{"className":4088},[427],[415,4090],{"className":4091,"style":458},[431],[415,4093,988],{"className":4094,"style":987},[436,437],[415,4096,463],{"className":4097},[462],[415,4099,1083],{"className":4100},[436,437],[415,4102,473],{"className":4103},[472],[415,4105],{"className":4106,"style":477},[442],[415,4108,512],{"className":4109},[436,437],[415,4111,487],{"className":4112},[486],",\nso the greedy commitment becomes unsound.",[381,4115,4116,4117,4132,4133,4148,4149,3455,4200,4203,4204,1617,4237,4253,4254,4269,4270,4285],{},"On a small non-negative digraph the finalization order is exactly nondecreasing\nin distance, which is the visible signature of the greedy invariant. Notice that\n",[415,4118,4120],{"className":4119},[418],[415,4121,4123],{"className":4122,"ariaHidden":423},[422],[415,4124,4126,4129],{"className":4125},[427],[415,4127],{"className":4128,"style":1020},[431],[415,4130,385],{"className":4131},[436,437]," is finalized at distance ",[415,4134,4136],{"className":4135},[418],[415,4137,4139],{"className":4138,"ariaHidden":423},[422],[415,4140,4142,4145],{"className":4141},[427],[415,4143],{"className":4144,"style":1294},[431],[415,4146,1633],{"className":4147},[436]," via the two-hop route ",[415,4150,4152],{"className":4151},[418],[415,4153,4155,4173,4191],{"className":4154,"ariaHidden":423},[422],[415,4156,4158,4161,4164,4167,4170],{"className":4157},[427],[415,4159],{"className":4160,"style":1020},[431],[415,4162,1083],{"className":4163},[436,437],[415,4165],{"className":4166,"style":443},[442],[415,4168,2168],{"className":4169},[447],[415,4171],{"className":4172,"style":443},[442],[415,4174,4176,4179,4182,4185,4188],{"className":4175},[427],[415,4177],{"className":4178,"style":1163},[431],[415,4180,2342],{"className":4181},[436,437],[415,4183],{"className":4184,"style":443},[442],[415,4186,2168],{"className":4187},[447],[415,4189],{"className":4190,"style":443},[442],[415,4192,4194,4197],{"className":4193},[427],[415,4195],{"className":4196,"style":1020},[431],[415,4198,385],{"className":4199},[436,437],[395,4201,4202],{},"beating","\nthe direct edge ",[415,4205,4207],{"className":4206},[418],[415,4208,4210,4228],{"className":4209,"ariaHidden":423},[422],[415,4211,4213,4216,4219,4222,4225],{"className":4212},[427],[415,4214],{"className":4215,"style":1020},[431],[415,4217,1083],{"className":4218},[436,437],[415,4220],{"className":4221,"style":443},[442],[415,4223,2168],{"className":4224},[447],[415,4226],{"className":4227,"style":443},[442],[415,4229,4231,4234],{"className":4230},[427],[415,4232],{"className":4233,"style":1020},[431],[415,4235,385],{"className":4236},[436,437],[415,4238,4240],{"className":4239},[418],[415,4241,4243],{"className":4242,"ariaHidden":423},[422],[415,4244,4246,4249],{"className":4245},[427],[415,4247],{"className":4248,"style":1294},[431],[415,4250,4252],{"className":4251},[436],"4"," — the relaxation through ",[415,4255,4257],{"className":4256},[418],[415,4258,4260],{"className":4259,"ariaHidden":423},[422],[415,4261,4263,4266],{"className":4262},[427],[415,4264],{"className":4265,"style":1163},[431],[415,4267,2342],{"className":4268},[436,437]," fired before\n",[415,4271,4273],{"className":4272},[418],[415,4274,4276],{"className":4275,"ariaHidden":423},[422],[415,4277,4279,4282],{"className":4278},[427],[415,4280],{"className":4281,"style":1020},[431],[415,4283,385],{"className":4284},[436,437]," was ever extracted:",[1841,4287,4289,4575],{"className":4288},[1844,1845],[1847,4290,4294],{"xmlns":1849,"width":4291,"height":4292,"viewBox":4293},"263.246","161.749","-75 -75 197.435 121.312",[1854,4295,4296,4299,4305,4308,4315,4318,4325,4328,4335,4338,4345,4348,4352,4359,4362,4365,4372,4375,4378,4385,4388,4392,4399,4402,4405,4411,4414,4418,4425,4428,4431,4438,4445,4471,4497,4523,4549],{"stroke":1856,"style":1857},[1866,4297],{"fill":1860,"d":4298},"M-40.37-31.135c0-6.286-5.096-11.381-11.382-11.381s-11.381 5.095-11.381 11.38c0 6.287 5.095 11.382 11.381 11.382s11.381-5.095 11.381-11.381Zm-11.382 0",[1854,4300,4301],{"transform":2382},[1866,4302],{"d":4303,"fill":1856,"stroke":1856,"className":4304,"style":1888},"M-50.917-31.684Q-50.697-31.298-49.941-31.298Q-49.643-31.298-49.348-31.399Q-49.054-31.500-48.860-31.713Q-48.667-31.926-48.667-32.234Q-48.667-32.462-48.845-32.609Q-49.023-32.757-49.269-32.809L-49.779-32.906Q-49.994-32.946-50.170-33.069Q-50.346-33.192-50.451-33.378Q-50.557-33.565-50.557-33.781Q-50.557-34.180-50.333-34.486Q-50.108-34.791-49.750-34.952Q-49.392-35.112-48.988-35.112Q-48.724-35.112-48.476-35.033Q-48.228-34.954-48.058-34.774Q-47.889-34.593-47.889-34.330Q-47.889-34.123-48.010-33.965Q-48.131-33.807-48.342-33.807Q-48.465-33.807-48.551-33.888Q-48.636-33.969-48.636-34.088Q-48.636-34.251-48.513-34.385Q-48.390-34.519-48.232-34.519Q-48.320-34.699-48.537-34.776Q-48.755-34.853-49.005-34.853Q-49.247-34.853-49.473-34.765Q-49.700-34.677-49.845-34.508Q-49.990-34.339-49.990-34.088Q-49.990-33.912-49.858-33.794Q-49.726-33.675-49.528-33.627L-49.023-33.530Q-48.636-33.451-48.368-33.181Q-48.100-32.910-48.100-32.528Q-48.100-32.198-48.289-31.878Q-48.478-31.557-48.755-31.359Q-49.256-31.034-49.950-31.034Q-50.262-31.034-50.563-31.120Q-50.864-31.205-51.069-31.403Q-51.273-31.601-51.273-31.908Q-51.273-32.159-51.130-32.343Q-50.987-32.528-50.746-32.528Q-50.592-32.528-50.493-32.436Q-50.394-32.343-50.394-32.198Q-50.394-31.988-50.546-31.836Q-50.697-31.684-50.917-31.684",[1870],[1866,4306],{"fill":1860,"d":4307},"M10.844-2.682c0-6.286-5.095-11.381-11.38-11.381-6.287 0-11.382 5.095-11.382 11.38S-6.823 8.7-.537 8.7 10.844 3.603 10.844-2.682Zm-11.38 0",[1854,4309,4311],{"transform":4310},"translate(49.233 31.578)",[1866,4312],{"d":4313,"fill":1856,"stroke":1856,"className":4314,"style":1888},"M-50.130-31.034Q-50.706-31.034-51.027-31.465Q-51.348-31.895-51.348-32.475Q-51.348-32.880-51.264-33.108L-50.385-36.606Q-50.350-36.756-50.350-36.830Q-50.350-36.967-50.917-36.967Q-51.014-36.967-51.014-37.085Q-51.014-37.142-50.983-37.213Q-50.952-37.283-50.886-37.283L-49.665-37.380Q-49.612-37.380-49.579-37.351Q-49.546-37.323-49.546-37.274L-49.546-37.239L-50.205-34.629Q-49.682-35.112-49.159-35.112Q-48.773-35.112-48.482-34.908Q-48.192-34.703-48.045-34.369Q-47.898-34.035-47.898-33.644Q-47.898-33.060-48.201-32.451Q-48.504-31.843-49.025-31.438Q-49.546-31.034-50.130-31.034M-50.113-31.298Q-49.744-31.298-49.440-31.621Q-49.137-31.944-48.979-32.339Q-48.834-32.695-48.713-33.203Q-48.592-33.710-48.592-34.031Q-48.592-34.356-48.737-34.604Q-48.882-34.853-49.177-34.853Q-49.779-34.853-50.350-34.053L-50.592-33.060Q-50.737-32.436-50.737-32.172Q-50.737-31.829-50.585-31.563Q-50.434-31.298-50.113-31.298",[1870],[1866,4316],{"fill":1860,"d":4317},"M10.844-59.588c0-6.286-5.095-11.38-11.38-11.38-6.287 0-11.382 5.094-11.382 11.38S-6.823-48.207-.537-48.207s11.381-5.095 11.381-11.38Zm-11.38 0",[1854,4319,4321],{"transform":4320},"translate(48.765 -26.515)",[1866,4322],{"d":4323,"fill":1856,"stroke":1856,"className":4324,"style":1888},"M-50.130-31.034Q-50.526-31.034-50.812-31.238Q-51.097-31.443-51.244-31.777Q-51.392-32.111-51.392-32.502Q-51.392-32.937-51.218-33.398Q-51.044-33.860-50.732-34.251Q-50.420-34.642-50.010-34.877Q-49.599-35.112-49.159-35.112Q-48.891-35.112-48.674-34.974Q-48.456-34.835-48.324-34.589Q-48.285-34.739-48.177-34.835Q-48.069-34.932-47.929-34.932Q-47.806-34.932-47.722-34.859Q-47.639-34.787-47.639-34.664Q-47.639-34.611-47.648-34.580L-48.267-32.089Q-48.324-31.891-48.324-31.693Q-48.324-31.298-48.061-31.298Q-47.775-31.298-47.641-31.621Q-47.507-31.944-47.388-32.449Q-47.379-32.480-47.355-32.504Q-47.331-32.528-47.296-32.528L-47.190-32.528Q-47.142-32.528-47.120-32.495Q-47.098-32.462-47.098-32.414Q-47.212-31.983-47.303-31.730Q-47.393-31.478-47.586-31.256Q-47.779-31.034-48.078-31.034Q-48.386-31.034-48.634-31.205Q-48.882-31.377-48.953-31.667Q-49.208-31.381-49.504-31.208Q-49.801-31.034-50.130-31.034M-50.113-31.298Q-49.783-31.298-49.473-31.539Q-49.164-31.781-48.953-32.097Q-48.944-32.106-48.944-32.124L-48.447-34.088Q-48.504-34.405-48.696-34.629Q-48.887-34.853-49.177-34.853Q-49.546-34.853-49.845-34.534Q-50.144-34.216-50.311-33.807Q-50.447-33.460-50.572-32.950Q-50.697-32.440-50.697-32.115Q-50.697-31.790-50.559-31.544Q-50.420-31.298-50.113-31.298",[1870],[1866,4326],{"fill":1860,"d":4327},"M67.75-31.135c0-6.286-5.096-11.381-11.381-11.381s-11.382 5.095-11.382 11.38c0 6.287 5.096 11.382 11.382 11.382s11.38-5.095 11.38-11.381Zm-11.381 0",[1854,4329,4331],{"transform":4330},"translate(106.118 1.937)",[1866,4332],{"d":4333,"fill":1856,"stroke":1856,"className":4334,"style":1888},"M-50.662-32.242Q-50.662-31.847-50.449-31.572Q-50.236-31.298-49.854-31.298Q-49.309-31.298-48.803-31.533Q-48.298-31.768-47.981-32.190Q-47.960-32.225-47.898-32.225Q-47.841-32.225-47.795-32.174Q-47.749-32.124-47.749-32.071Q-47.749-32.036-47.775-32.010Q-48.122-31.535-48.685-31.284Q-49.247-31.034-49.871-31.034Q-50.302-31.034-50.651-31.236Q-51.001-31.438-51.192-31.794Q-51.383-32.150-51.383-32.576Q-51.383-33.038-51.181-33.495Q-50.979-33.952-50.623-34.321Q-50.267-34.690-49.823-34.901Q-49.379-35.112-48.909-35.112Q-48.641-35.112-48.392-35.031Q-48.144-34.949-47.977-34.771Q-47.810-34.593-47.810-34.330Q-47.810-34.093-47.960-33.915Q-48.109-33.737-48.342-33.737Q-48.482-33.737-48.588-33.831Q-48.693-33.926-48.693-34.071Q-48.693-34.273-48.546-34.427Q-48.399-34.580-48.197-34.580Q-48.302-34.721-48.507-34.787Q-48.711-34.853-48.918-34.853Q-49.454-34.853-49.851-34.424Q-50.249-33.996-50.456-33.376Q-50.662-32.757-50.662-32.242",[1870],[1866,4336],{"fill":1860,"d":4337},"M118.965-31.135c0-6.286-5.096-11.381-11.381-11.381s-11.382 5.095-11.382 11.38c0 6.287 5.096 11.382 11.382 11.382s11.38-5.095 11.38-11.381Zm-11.381 0",[1854,4339,4341],{"transform":4340},"translate(157.665 2.768)",[1866,4342],{"d":4343,"fill":1856,"stroke":1856,"className":4344,"style":1888},"M-51.168-31.873Q-51.168-32.005-51.141-32.124L-50.491-34.699L-51.436-34.699Q-51.545-34.699-51.545-34.818Q-51.545-34.879-51.512-34.947Q-51.480-35.015-51.418-35.015L-50.420-35.015L-50.060-36.452Q-50.025-36.593-49.913-36.681Q-49.801-36.769-49.665-36.769Q-49.542-36.769-49.458-36.694Q-49.375-36.619-49.375-36.501Q-49.375-36.444-49.383-36.417L-49.735-35.015L-48.808-35.015Q-48.759-35.015-48.731-34.982Q-48.702-34.949-48.702-34.906Q-48.702-34.840-48.735-34.769Q-48.768-34.699-48.825-34.699L-49.810-34.699L-50.464-32.089Q-50.517-31.886-50.517-31.693Q-50.517-31.298-50.258-31.298Q-49.977-31.298-49.746-31.478Q-49.515-31.658-49.344-31.930Q-49.172-32.203-49.071-32.467Q-49.063-32.493-49.041-32.510Q-49.019-32.528-48.988-32.528L-48.882-32.528Q-48.834-32.528-48.812-32.495Q-48.790-32.462-48.790-32.414Q-48.931-32.071-49.142-31.757Q-49.353-31.443-49.638-31.238Q-49.924-31.034-50.275-31.034Q-50.526-31.034-50.728-31.139Q-50.930-31.245-51.049-31.436Q-51.168-31.627-51.168-31.873",[1870],[1866,4346],{"fill":1860,"d":4347},"m-41.628-36.76 28.28-15.71",[1866,4349],{"d":4350,"style":4351},"m-11.158-53.687-3.79.56 1.687.608-.375 1.754Z","stroke-width:.399996",[1854,4353,4355],{"transform":4354},"translate(20.421 -15.426)",[1866,4356],{"d":4357,"fill":1856,"stroke":1856,"className":4358,"style":1909},"M-49.435-32.283L-51.479-32.283L-51.479-32.564L-49.148-35.736Q-49.113-35.783-49.048-35.783L-48.912-35.783Q-48.867-35.783-48.840-35.756Q-48.813-35.729-48.813-35.684L-48.813-32.564L-48.050-32.564L-48.050-32.283L-48.813-32.283L-48.813-31.624Q-48.813-31.415-48.057-31.415L-48.057-31.135L-50.190-31.135L-50.190-31.415Q-49.435-31.415-49.435-31.624L-49.435-32.283M-49.387-35.008L-51.178-32.564L-49.387-32.564",[1870],[1866,4360],{"fill":1860,"d":4361},"m-41.628-25.51 28.28 15.71",[1866,4363],{"d":4364,"style":4351},"m-11.158-8.583-2.478-2.922.375 1.753-1.687.609Z",[1854,4366,4368],{"transform":4367},"translate(20.421 19.937)",[1866,4369],{"d":4370,"fill":1856,"stroke":1856,"className":4371,"style":1909},"M-48.426-31.135L-50.956-31.135L-50.956-31.415Q-49.988-31.415-49.988-31.624L-49.988-35.243Q-50.381-35.055-51.003-35.055L-51.003-35.336Q-50.586-35.336-50.222-35.437Q-49.858-35.537-49.602-35.783L-49.476-35.783Q-49.411-35.766-49.394-35.698L-49.394-31.624Q-49.394-31.415-48.426-31.415",[1870],[1866,4373],{"fill":1860,"d":4374},"M-.537-14.263v-30.668",[1866,4376],{"d":4377},"m-.537-47.437-1.35 3.584 1.35-1.178 1.35 1.178Z",[1854,4379,4381],{"transform":4380},"translate(46.029 2.256)",[1866,4382],{"d":4383,"fill":1856,"stroke":1856,"className":4384,"style":1909},"M-48.426-31.135L-51.311-31.135L-51.311-31.337Q-51.311-31.367-51.284-31.395L-50.036-32.612Q-49.964-32.687-49.922-32.729Q-49.879-32.772-49.800-32.851Q-49.387-33.264-49.156-33.622Q-48.925-33.979-48.925-34.403Q-48.925-34.635-49.004-34.838Q-49.083-35.042-49.224-35.192Q-49.366-35.343-49.561-35.423Q-49.756-35.503-49.988-35.503Q-50.299-35.503-50.557-35.344Q-50.815-35.185-50.945-34.908L-50.925-34.908Q-50.757-34.908-50.650-34.797Q-50.542-34.686-50.542-34.522Q-50.542-34.365-50.651-34.252Q-50.761-34.139-50.925-34.139Q-51.085-34.139-51.198-34.252Q-51.311-34.365-51.311-34.522Q-51.311-34.898-51.103-35.185Q-50.894-35.472-50.559-35.628Q-50.224-35.783-49.869-35.783Q-49.445-35.783-49.065-35.625Q-48.686-35.466-48.452-35.149Q-48.218-34.833-48.218-34.403Q-48.218-34.092-48.358-33.823Q-48.498-33.555-48.703-33.350Q-48.908-33.145-49.271-32.863Q-49.633-32.581-49.742-32.485L-50.597-31.757L-49.954-31.757Q-49.691-31.757-49.402-31.759Q-49.113-31.760-48.895-31.769Q-48.676-31.778-48.659-31.795Q-48.597-31.860-48.560-32.027Q-48.522-32.195-48.484-32.437L-48.218-32.437",[1870],[1866,4386],{"fill":1860,"d":4387},"m9.82-7.861 33.44-16.72",[1866,4389],{"d":4390,"style":4391},"m45.501-25.702-3.81.395 1.659.681-.45 1.735Z","stroke-width:.39996800000000005",[1854,4393,4395],{"transform":4394},"translate(77.675 19.937)",[1866,4396],{"d":4397,"fill":1856,"stroke":1856,"className":4398,"style":1909},"M-50.945-31.897L-50.976-31.897Q-50.839-31.600-50.542-31.424Q-50.245-31.248-49.917-31.248Q-49.554-31.248-49.327-31.426Q-49.100-31.603-49.006-31.892Q-48.912-32.181-48.912-32.543Q-48.912-32.858-48.966-33.143Q-49.021-33.428-49.194-33.634Q-49.366-33.839-49.681-33.839Q-49.954-33.839-50.137-33.772Q-50.320-33.705-50.424-33.616Q-50.528-33.528-50.624-33.418Q-50.720-33.309-50.764-33.299L-50.843-33.299Q-50.915-33.316-50.932-33.387L-50.932-35.705Q-50.932-35.739-50.908-35.761Q-50.884-35.783-50.850-35.783L-50.822-35.783Q-50.535-35.667-50.267-35.613Q-49.999-35.558-49.722-35.558Q-49.445-35.558-49.175-35.613Q-48.905-35.667-48.625-35.783L-48.601-35.783Q-48.566-35.783-48.543-35.760Q-48.519-35.736-48.519-35.705L-48.519-35.636Q-48.519-35.609-48.539-35.589Q-48.813-35.274-49.197-35.098Q-49.582-34.922-49.995-34.922Q-50.334-34.922-50.651-35.008L-50.651-33.726Q-50.255-34.061-49.681-34.061Q-49.277-34.061-48.941-33.851Q-48.604-33.640-48.411-33.288Q-48.218-32.936-48.218-32.536Q-48.218-32.205-48.358-31.919Q-48.498-31.634-48.742-31.424Q-48.987-31.214-49.289-31.104Q-49.592-30.995-49.910-30.995Q-50.269-30.995-50.595-31.159Q-50.921-31.323-51.116-31.615Q-51.311-31.907-51.311-32.270Q-51.311-32.420-51.205-32.526Q-51.099-32.632-50.945-32.632Q-50.792-32.632-50.687-32.528Q-50.583-32.424-50.583-32.270Q-50.583-32.113-50.687-32.005Q-50.792-31.897-50.945-31.897",[1870],[1866,4400],{"fill":1860,"d":4401},"m9.82-54.41 33.44 16.72",[1866,4403],{"d":4404,"style":4391},"m45.501-36.569-2.601-2.81.45 1.735-1.658.68Z",[1854,4406,4408],{"transform":4407},"translate(77.675 -15.426)",[1866,4409],{"d":4370,"fill":1856,"stroke":1856,"className":4410,"style":1909},[1870],[1866,4412],{"fill":1860,"d":4413},"M10.636-62.635c34.59-9.435 63.257-1.89 86.538 21.183",[1866,4415],{"d":4416,"style":4417},"m98.954-39.688-1.595-3.482-.114 1.789-1.788.13Z","stroke-width:.39998",[1854,4419,4421],{"transform":4420},"translate(108.177 -34.028)",[1866,4422],{"d":4423,"fill":1856,"stroke":1856,"className":4424,"style":1909},"M-49.763-30.995Q-50.221-30.995-50.539-31.210Q-50.856-31.426-51.038-31.778Q-51.219-32.130-51.296-32.550Q-51.373-32.970-51.373-33.398Q-51.373-33.982-51.120-34.538Q-50.867-35.093-50.397-35.438Q-49.927-35.783-49.329-35.783Q-48.919-35.783-48.635-35.585Q-48.351-35.387-48.351-34.984Q-48.351-34.888-48.397-34.809Q-48.443-34.731-48.524-34.686Q-48.604-34.642-48.693-34.642Q-48.840-34.642-48.941-34.739Q-49.042-34.837-49.042-34.984Q-49.042-35.114-48.951-35.221Q-48.860-35.329-48.727-35.329Q-48.915-35.551-49.329-35.551Q-49.643-35.551-49.917-35.387Q-50.190-35.223-50.357-34.949Q-50.545-34.659-50.610-34.293Q-50.675-33.927-50.675-33.473Q-50.525-33.767-50.260-33.945Q-49.995-34.122-49.681-34.122Q-49.250-34.122-48.901-33.916Q-48.553-33.709-48.353-33.353Q-48.153-32.998-48.153-32.571Q-48.153-32.126-48.370-31.766Q-48.587-31.405-48.960-31.200Q-49.332-30.995-49.763-30.995M-49.763-31.248Q-49.387-31.248-49.183-31.431Q-48.980-31.614-48.917-31.897Q-48.854-32.181-48.854-32.571Q-48.854-32.957-48.908-33.237Q-48.963-33.517-49.158-33.709Q-49.353-33.900-49.722-33.900Q-50.012-33.900-50.224-33.724Q-50.436-33.548-50.544-33.275Q-50.651-33.001-50.651-32.718L-50.651-32.577L-50.651-32.536Q-50.651-32.031-50.440-31.639Q-50.228-31.248-49.763-31.248",[1870],[1866,4426],{"fill":1860,"d":4427},"M67.95-31.135h24.977",[1866,4429],{"d":4430},"m95.433-31.135-3.585-1.35 1.18 1.35-1.18 1.35Z",[1854,4432,4434],{"transform":4433},"translate(131.735 5.711)",[1866,4435],{"d":4436,"fill":1856,"stroke":1856,"className":4437,"style":1909},"M-50.956-31.682Q-50.836-31.525-50.645-31.426Q-50.453-31.326-50.238-31.287Q-50.023-31.248-49.800-31.248Q-49.503-31.248-49.308-31.403Q-49.113-31.559-49.023-31.813Q-48.932-32.068-48.932-32.352Q-48.932-32.646-49.024-32.897Q-49.117-33.148-49.315-33.304Q-49.513-33.459-49.807-33.459L-50.323-33.459Q-50.351-33.459-50.376-33.485Q-50.402-33.510-50.402-33.534L-50.402-33.606Q-50.402-33.637-50.376-33.659Q-50.351-33.681-50.323-33.681L-49.882-33.712Q-49.520-33.712-49.300-34.069Q-49.079-34.427-49.079-34.816Q-49.079-35.144-49.274-35.348Q-49.469-35.551-49.800-35.551Q-50.087-35.551-50.340-35.467Q-50.593-35.384-50.757-35.196Q-50.610-35.196-50.510-35.081Q-50.409-34.967-50.409-34.816Q-50.409-34.666-50.515-34.556Q-50.621-34.447-50.778-34.447Q-50.939-34.447-51.048-34.556Q-51.157-34.666-51.157-34.816Q-51.157-35.141-50.949-35.360Q-50.740-35.578-50.424-35.681Q-50.108-35.783-49.800-35.783Q-49.482-35.783-49.154-35.679Q-48.826-35.575-48.599-35.353Q-48.372-35.131-48.372-34.816Q-48.372-34.382-48.659-34.057Q-48.946-33.733-49.380-33.586Q-49.069-33.521-48.789-33.355Q-48.508-33.189-48.331-32.931Q-48.153-32.673-48.153-32.352Q-48.153-31.942-48.397-31.632Q-48.642-31.323-49.023-31.159Q-49.404-30.995-49.800-30.995Q-50.169-30.995-50.527-31.108Q-50.884-31.220-51.128-31.470Q-51.373-31.719-51.373-32.089Q-51.373-32.260-51.256-32.372Q-51.140-32.485-50.969-32.485Q-50.860-32.485-50.769-32.434Q-50.679-32.383-50.624-32.290Q-50.569-32.198-50.569-32.089Q-50.569-31.921-50.682-31.802Q-50.795-31.682-50.956-31.682",[1870],[1854,4439,4441],{"transform":4440},"translate(-10.518 68.22)",[1866,4442],{"d":4443,"fill":1856,"stroke":1856,"className":4444,"style":1871},"M-49.385-31.135L-51.400-31.135L-51.400-31.584L-50.873-31.584L-50.873-34.240L-51.486-34.240L-51.486-34.686L-50.873-34.686L-50.873-35.510Q-50.873-35.877-50.623-36.127Q-50.373-36.377-49.988-36.510Q-49.604-36.643-49.199-36.694Q-48.795-36.744-48.447-36.744Q-48.147-36.744-47.863-36.629Q-47.580-36.514-47.398-36.287Q-47.217-36.061-47.217-35.760Q-47.217-35.522-47.383-35.356Q-47.549-35.190-47.791-35.190Q-48.025-35.190-48.193-35.358Q-48.361-35.526-48.361-35.760Q-48.361-35.916-48.285-36.055Q-48.209-36.194-48.072-36.264Q-48.295-36.373-48.584-36.373Q-48.904-36.373-49.219-36.291Q-49.533-36.209-49.746-36.014Q-49.959-35.819-49.959-35.494L-49.959-34.686L-48.682-34.686L-48.682-34.709L-47.193-34.776L-47.193-31.584L-46.662-31.584L-46.662-31.135L-48.682-31.135L-48.682-31.584L-48.150-31.584L-48.150-34.045Q-48.150-34.205-48.264-34.240L-49.912-34.240L-49.912-31.584L-49.385-31.584L-49.385-31.135M-43.815-31.135L-45.877-31.135L-45.877-31.584L-45.350-31.584L-45.350-34.006Q-45.350-34.158-45.496-34.196Q-45.643-34.233-45.877-34.233L-45.877-34.678L-44.447-34.744L-44.447-33.951Q-44.299-34.198-44.065-34.377Q-43.830-34.557-43.553-34.651Q-43.275-34.744-42.982-34.744Q-42.541-34.744-42.244-34.637Q-41.947-34.530-41.785-34.270Q-41.623-34.010-41.623-33.576L-41.623-31.584L-41.096-31.584L-41.096-31.135L-43.158-31.135L-43.158-31.584L-42.631-31.584L-42.631-33.549Q-42.631-33.924-42.709-34.149Q-42.787-34.373-43.080-34.373Q-43.592-34.373-43.967-34.039Q-44.342-33.705-44.342-33.198L-44.342-31.584L-43.815-31.584L-43.815-31.135M-40.596-32.065Q-40.596-32.436-40.301-32.688Q-40.006-32.940-39.555-33.071Q-39.104-33.201-38.668-33.248Q-38.232-33.295-37.846-33.295L-37.846-33.549Q-37.846-33.948-38.076-34.178Q-38.307-34.408-38.701-34.408Q-39.127-34.408-39.334-34.358Q-39.174-34.213-39.174-33.959Q-39.174-33.725-39.332-33.567Q-39.490-33.408-39.725-33.408Q-39.967-33.408-40.125-33.567Q-40.283-33.725-40.283-33.959Q-40.283-34.471-39.818-34.623Q-39.354-34.776-38.701-34.776Q-38.279-34.776-37.850-34.660Q-37.420-34.545-37.129-34.274Q-36.838-34.002-36.838-33.569L-36.838-31.709Q-36.807-31.584-36.268-31.584Q-36.209-31.584-36.162-31.537Q-36.115-31.490-36.115-31.432L-36.115-31.295Q-36.115-31.229-36.162-31.182Q-36.209-31.135-36.268-31.135L-36.815-31.135Q-37.693-31.135-37.693-31.639Q-37.881-31.362-38.227-31.221Q-38.572-31.080-38.940-31.080Q-39.315-31.080-39.695-31.164Q-40.076-31.248-40.336-31.471Q-40.596-31.694-40.596-32.065M-39.588-32.065Q-39.588-31.877-39.477-31.737Q-39.365-31.596-39.190-31.522Q-39.014-31.448-38.830-31.448Q-38.600-31.448-38.371-31.528Q-38.143-31.608-37.994-31.768Q-37.846-31.928-37.846-32.166L-37.846-32.959Q-38.182-32.959-38.584-32.875Q-38.986-32.791-39.287-32.590Q-39.588-32.389-39.588-32.065M-33.635-31.135L-35.650-31.135L-35.650-31.584L-35.123-31.584L-35.123-35.951Q-35.123-36.100-35.270-36.137Q-35.416-36.174-35.650-36.174L-35.650-36.623L-34.162-36.686L-34.162-31.584L-33.635-31.584L-33.635-31.135M-30.975-31.135L-32.932-31.135L-32.932-31.584L-32.404-31.584L-32.404-34.006Q-32.404-34.158-32.541-34.196Q-32.678-34.233-32.908-34.233L-32.908-34.678L-31.443-34.744L-31.443-31.584L-30.975-31.584L-30.975-31.135M-32.686-36.037Q-32.686-36.315-32.492-36.504Q-32.299-36.694-32.029-36.694Q-31.850-36.694-31.699-36.604Q-31.549-36.514-31.461-36.367Q-31.373-36.221-31.373-36.037Q-31.373-35.768-31.566-35.574Q-31.760-35.381-32.029-35.381Q-32.303-35.381-32.494-35.573Q-32.686-35.764-32.686-36.037M-26.893-31.135L-30.229-31.135Q-30.295-31.135-30.346-31.186Q-30.397-31.237-30.397-31.303L-30.397-31.408Q-30.397-31.471-30.357-31.510L-28.037-34.319L-28.510-34.319Q-29.178-34.319-29.481-34.100Q-29.783-33.881-29.783-33.233L-30.264-33.233L-30.158-34.686L-26.928-34.686Q-26.861-34.686-26.811-34.635Q-26.760-34.584-26.760-34.518L-26.760-34.440Q-26.760-34.377-26.791-34.334L-29.111-31.541L-28.580-31.541Q-28.236-31.541-27.986-31.588Q-27.736-31.635-27.557-31.799Q-27.389-31.963-27.301-32.254Q-27.213-32.545-27.213-32.815L-26.732-32.815L-26.893-31.135M-26.045-32.936Q-26.045-33.514-25.762-33.934Q-25.479-34.354-24.994-34.565Q-24.510-34.776-23.943-34.776Q-23.502-34.776-23.166-34.664Q-22.830-34.553-22.596-34.332Q-22.361-34.112-22.236-33.781Q-22.111-33.451-22.111-33.014Q-22.111-32.858-22.264-32.830L-24.936-32.830Q-24.936-31.487-23.678-31.487Q-23.326-31.487-23.020-31.643Q-22.713-31.799-22.592-32.096Q-22.518-32.225-22.432-32.225L-22.264-32.225Q-22.111-32.190-22.111-32.057Q-22.111-32.022-22.119-32.006Q-22.303-31.530-22.768-31.305Q-23.232-31.080-23.807-31.080Q-24.404-31.080-24.914-31.283Q-25.424-31.487-25.734-31.908Q-26.045-32.330-26.045-32.936M-24.936-33.174L-22.928-33.174Q-22.928-33.713-23.176-34.061Q-23.424-34.408-23.943-34.408Q-24.287-34.408-24.512-34.242Q-24.736-34.076-24.836-33.799Q-24.936-33.522-24.936-33.174M-21.502-32.912Q-21.502-33.358-21.332-33.703Q-21.162-34.049-20.861-34.280Q-20.561-34.510-20.172-34.627Q-19.783-34.744-19.365-34.744Q-19.057-34.744-18.766-34.658Q-18.475-34.573-18.229-34.408L-18.229-35.951Q-18.229-36.100-18.375-36.137Q-18.522-36.174-18.760-36.174L-18.760-36.623L-17.272-36.686L-17.272-31.807Q-17.272-31.658-17.125-31.621Q-16.979-31.584-16.740-31.584L-16.740-31.135L-18.279-31.080L-18.279-31.455Q-18.795-31.080-19.471-31.080Q-19.881-31.080-20.248-31.199Q-20.615-31.319-20.900-31.555Q-21.186-31.791-21.344-32.137Q-21.502-32.483-21.502-32.912M-19.381-31.448Q-19.057-31.448-18.760-31.606Q-18.463-31.764-18.279-32.037L-18.279-33.905Q-18.451-34.127-18.719-34.250Q-18.986-34.373-19.272-34.373Q-19.725-34.373-19.973-34.184Q-20.221-33.994-20.305-33.682Q-20.389-33.369-20.389-32.912Q-20.389-32.463-20.322-32.156Q-20.256-31.850-20.033-31.649Q-19.811-31.448-19.381-31.448M-15.686-31.791Q-15.686-32.061-15.492-32.254Q-15.299-32.448-15.029-32.448Q-14.850-32.448-14.699-32.360Q-14.549-32.272-14.461-32.121Q-14.373-31.971-14.373-31.791Q-14.373-31.522-14.566-31.328Q-14.760-31.135-15.029-31.135Q-15.303-31.135-15.494-31.326Q-15.686-31.518-15.686-31.791M-15.686-34.030Q-15.686-34.307-15.492-34.496Q-15.299-34.686-15.029-34.686Q-14.850-34.686-14.699-34.596Q-14.549-34.506-14.461-34.360Q-14.373-34.213-14.373-34.030Q-14.373-33.760-14.566-33.567Q-14.760-33.373-15.029-33.373Q-15.303-33.373-15.494-33.565Q-15.686-33.756-15.686-34.030",[1870],[1854,4446,4447,4450],{"fill":1988},[1866,4448],{"d":4449},"M9.69 25.77H.617a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.071a4 4 0 0 0 4-4V29.77a4 4 0 0 0-4-4ZM-3.383 42.842",[1854,4451,4452,4459,4465],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,4453,4455],{"transform":4454},"translate(51.893 67.697)",[1866,4456],{"d":4457,"fill":1856,"stroke":1856,"className":4458,"style":1909},"M-50.918-31.555Q-50.727-31.289-50.122-31.289Q-49.893-31.289-49.648-31.359Q-49.404-31.429-49.242-31.584Q-49.079-31.740-49.079-31.983Q-49.079-32.157-49.230-32.265Q-49.380-32.372-49.575-32.410L-50.029-32.492Q-50.299-32.547-50.487-32.735Q-50.675-32.923-50.675-33.179Q-50.675-33.504-50.489-33.741Q-50.303-33.979-50.005-34.100Q-49.708-34.221-49.387-34.221Q-49.182-34.221-48.966-34.162Q-48.751-34.102-48.609-33.967Q-48.467-33.832-48.467-33.620Q-48.467-33.452-48.561-33.326Q-48.655-33.199-48.819-33.199Q-48.919-33.199-48.992-33.264Q-49.065-33.329-49.065-33.432Q-49.065-33.552-48.982-33.649Q-48.898-33.746-48.785-33.767Q-48.942-33.999-49.400-33.999Q-49.698-33.999-49.947-33.856Q-50.197-33.712-50.197-33.432Q-50.197-33.176-49.835-33.093L-49.373-33.011Q-49.059-32.950-48.830-32.741Q-48.601-32.533-48.601-32.225Q-48.601-31.962-48.756-31.711Q-48.912-31.460-49.148-31.309Q-49.541-31.067-50.128-31.067Q-50.556-31.067-50.906-31.222Q-51.256-31.378-51.256-31.743Q-51.256-31.938-51.140-32.082Q-51.024-32.225-50.829-32.225Q-50.710-32.225-50.629-32.154Q-50.549-32.082-50.549-31.962Q-50.549-31.805-50.655-31.689Q-50.761-31.573-50.918-31.555",[1870],[1854,4460,4461],{"transform":4454},[1866,4462],{"d":4463,"fill":1856,"stroke":1856,"className":4464,"style":1909},"M-47.264-31.555Q-47.264-31.723-47.141-31.846Q-47.018-31.969-46.843-31.969Q-46.676-31.969-46.553-31.846Q-46.430-31.723-46.430-31.555Q-46.430-31.381-46.553-31.258Q-46.676-31.135-46.843-31.135Q-47.018-31.135-47.141-31.258Q-47.264-31.381-47.264-31.555M-47.264-33.739Q-47.264-33.907-47.141-34.030Q-47.018-34.153-46.843-34.153Q-46.676-34.153-46.553-34.030Q-46.430-33.907-46.430-33.739Q-46.430-33.565-46.553-33.442Q-46.676-33.319-46.843-33.319Q-47.018-33.319-47.141-33.442Q-47.264-33.565-47.264-33.739",[1870],[1854,4466,4467],{"transform":4454},[1866,4468],{"d":4469,"fill":1856,"stroke":1856,"className":4470,"style":1909},"M-43.725-30.995Q-44.360-30.995-44.724-31.340Q-45.089-31.685-45.224-32.210Q-45.359-32.735-45.359-33.360Q-45.359-34.385-45.003-35.084Q-44.648-35.783-43.725-35.783Q-42.798-35.783-42.446-35.084Q-42.094-34.385-42.094-33.360Q-42.094-32.735-42.229-32.210Q-42.364-31.685-42.727-31.340Q-43.089-30.995-43.725-30.995M-43.725-31.220Q-43.287-31.220-43.074-31.595Q-42.860-31.969-42.810-32.436Q-42.761-32.902-42.761-33.480Q-42.761-34.033-42.810-34.461Q-42.860-34.888-43.072-35.223Q-43.284-35.558-43.725-35.558Q-44.067-35.558-44.270-35.351Q-44.473-35.144-44.560-34.832Q-44.648-34.519-44.670-34.203Q-44.692-33.886-44.692-33.480Q-44.692-33.063-44.670-32.721Q-44.648-32.379-44.559-32.031Q-44.470-31.682-44.265-31.451Q-44.060-31.220-43.725-31.220",[1870],[1854,4472,4473,4476],{"fill":1988},[1866,4474],{"d":4475},"M35.297 25.77h-9.072a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.072a4 4 0 0 0 4-4V29.77a4 4 0 0 0-4-4ZM22.225 42.842",[1854,4477,4478,4485,4491],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,4479,4481],{"transform":4480},"translate(77.63 67.872)",[1866,4482],{"d":4483,"fill":1856,"stroke":1856,"className":4484,"style":1909},"M-50.262-31.067Q-50.580-31.067-50.812-31.222Q-51.044-31.378-51.171-31.641Q-51.297-31.904-51.297-32.218Q-51.297-32.437-51.239-32.653L-50.583-35.302Q-50.535-35.476-50.535-35.544Q-50.535-35.636-50.969-35.636Q-51.051-35.664-51.051-35.749L-51.024-35.859Q-51.017-35.900-50.945-35.917L-49.968-35.992Q-49.930-35.992-49.896-35.965Q-49.862-35.937-49.862-35.889L-50.364-33.859Q-49.954-34.221-49.513-34.221Q-49.192-34.221-48.946-34.066Q-48.700-33.910-48.570-33.644Q-48.440-33.377-48.440-33.052Q-48.440-32.700-48.585-32.348Q-48.731-31.996-48.982-31.708Q-49.233-31.419-49.568-31.243Q-49.903-31.067-50.262-31.067M-50.248-31.289Q-50.040-31.289-49.850-31.415Q-49.660-31.542-49.523-31.731Q-49.387-31.921-49.301-32.123Q-49.195-32.386-49.112-32.753Q-49.028-33.121-49.028-33.353Q-49.028-33.507-49.079-33.659Q-49.130-33.811-49.243-33.905Q-49.356-33.999-49.527-33.999Q-49.804-33.999-50.050-33.818Q-50.296-33.637-50.491-33.360L-50.682-32.618Q-50.778-32.201-50.778-31.976Q-50.778-31.699-50.645-31.494Q-50.511-31.289-50.248-31.289",[1870],[1854,4486,4487],{"transform":4480},[1866,4488],{"d":4489,"fill":1856,"stroke":1856,"className":4490,"style":1909},"M-47.521-31.555Q-47.521-31.723-47.398-31.846Q-47.275-31.969-47.100-31.969Q-46.933-31.969-46.810-31.846Q-46.687-31.723-46.687-31.555Q-46.687-31.381-46.810-31.258Q-46.933-31.135-47.100-31.135Q-47.275-31.135-47.398-31.258Q-47.521-31.381-47.521-31.555M-47.521-33.739Q-47.521-33.907-47.398-34.030Q-47.275-34.153-47.100-34.153Q-46.933-34.153-46.810-34.030Q-46.687-33.907-46.687-33.739Q-46.687-33.565-46.810-33.442Q-46.933-33.319-47.100-33.319Q-47.275-33.319-47.398-33.442Q-47.521-33.565-47.521-33.739",[1870],[1854,4492,4493],{"transform":4480},[1866,4494],{"d":4495,"fill":1856,"stroke":1856,"className":4496,"style":1909},"M-42.645-31.135L-45.175-31.135L-45.175-31.415Q-44.207-31.415-44.207-31.624L-44.207-35.243Q-44.600-35.055-45.222-35.055L-45.222-35.336Q-44.805-35.336-44.441-35.437Q-44.077-35.537-43.821-35.783L-43.695-35.783Q-43.630-35.766-43.613-35.698L-43.613-31.624Q-43.613-31.415-42.645-31.415",[1870],[1854,4498,4499,4502],{"fill":1988},[1866,4500],{"d":4501},"M60.995 25.77h-9.253a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.253a4 4 0 0 0 4-4V29.77a4 4 0 0 0-4-4ZM47.742 42.842",[1854,4503,4504,4511,4517],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,4505,4507],{"transform":4506},"translate(102.827 67.697)",[1866,4508],{"d":4509,"fill":1856,"stroke":1856,"className":4510,"style":1909},"M-50.262-31.067Q-50.586-31.067-50.831-31.224Q-51.075-31.381-51.207-31.646Q-51.338-31.911-51.338-32.236Q-51.338-32.704-51.085-33.167Q-50.833-33.630-50.409-33.926Q-49.985-34.221-49.513-34.221Q-49.298-34.221-49.112-34.117Q-48.925-34.013-48.806-33.828Q-48.782-33.941-48.688-34.015Q-48.594-34.088-48.478-34.088Q-48.375-34.088-48.307-34.027Q-48.238-33.965-48.238-33.866Q-48.238-33.808-48.245-33.781L-48.727-31.863Q-48.754-31.706-48.754-31.617Q-48.754-31.490-48.703-31.390Q-48.652-31.289-48.532-31.289Q-48.310-31.289-48.197-31.542Q-48.085-31.795-47.999-32.164Q-47.975-32.225-47.924-32.225L-47.811-32.225Q-47.777-32.225-47.755-32.196Q-47.732-32.167-47.732-32.143Q-47.732-32.130-47.739-32.116Q-48.002-31.067-48.546-31.067Q-48.795-31.067-49.004-31.190Q-49.212-31.313-49.281-31.542Q-49.756-31.067-50.262-31.067M-50.248-31.289Q-49.975-31.289-49.723-31.473Q-49.472-31.658-49.281-31.925L-48.912-33.405Q-48.949-33.565-49.030-33.702Q-49.110-33.839-49.236-33.919Q-49.363-33.999-49.527-33.999Q-49.735-33.999-49.922-33.875Q-50.108-33.750-50.246-33.558Q-50.385-33.367-50.470-33.165Q-50.583-32.871-50.667-32.526Q-50.751-32.181-50.751-31.931Q-50.751-31.675-50.622-31.482Q-50.494-31.289-50.248-31.289",[1870],[1854,4512,4513],{"transform":4506},[1866,4514],{"d":4515,"fill":1856,"stroke":1856,"className":4516,"style":1909},"M-46.700-31.555Q-46.700-31.723-46.577-31.846Q-46.454-31.969-46.279-31.969Q-46.112-31.969-45.989-31.846Q-45.866-31.723-45.866-31.555Q-45.866-31.381-45.989-31.258Q-46.112-31.135-46.279-31.135Q-46.454-31.135-46.577-31.258Q-46.700-31.381-46.700-31.555M-46.700-33.739Q-46.700-33.907-46.577-34.030Q-46.454-34.153-46.279-34.153Q-46.112-34.153-45.989-34.030Q-45.866-33.907-45.866-33.739Q-45.866-33.565-45.989-33.442Q-46.112-33.319-46.279-33.319Q-46.454-33.319-46.577-33.442Q-46.700-33.565-46.700-33.739",[1870],[1854,4518,4519],{"transform":4506},[1866,4520],{"d":4521,"fill":1856,"stroke":1856,"className":4522,"style":1909},"M-44.354-31.682Q-44.234-31.525-44.043-31.426Q-43.851-31.326-43.636-31.287Q-43.421-31.248-43.198-31.248Q-42.901-31.248-42.706-31.403Q-42.511-31.559-42.421-31.813Q-42.330-32.068-42.330-32.352Q-42.330-32.646-42.422-32.897Q-42.515-33.148-42.713-33.304Q-42.911-33.459-43.205-33.459L-43.721-33.459Q-43.749-33.459-43.774-33.485Q-43.800-33.510-43.800-33.534L-43.800-33.606Q-43.800-33.637-43.774-33.659Q-43.749-33.681-43.721-33.681L-43.280-33.712Q-42.918-33.712-42.698-34.069Q-42.477-34.427-42.477-34.816Q-42.477-35.144-42.672-35.348Q-42.867-35.551-43.198-35.551Q-43.485-35.551-43.738-35.467Q-43.991-35.384-44.155-35.196Q-44.008-35.196-43.908-35.081Q-43.807-34.967-43.807-34.816Q-43.807-34.666-43.913-34.556Q-44.019-34.447-44.176-34.447Q-44.337-34.447-44.446-34.556Q-44.555-34.666-44.555-34.816Q-44.555-35.141-44.347-35.360Q-44.138-35.578-43.822-35.681Q-43.506-35.783-43.198-35.783Q-42.880-35.783-42.552-35.679Q-42.224-35.575-41.997-35.353Q-41.770-35.131-41.770-34.816Q-41.770-34.382-42.057-34.057Q-42.344-33.733-42.778-33.586Q-42.467-33.521-42.187-33.355Q-41.906-33.189-41.729-32.931Q-41.551-32.673-41.551-32.352Q-41.551-31.942-41.795-31.632Q-42.040-31.323-42.421-31.159Q-42.802-30.995-43.198-30.995Q-43.567-30.995-43.925-31.108Q-44.282-31.220-44.526-31.470Q-44.771-31.719-44.771-32.089Q-44.771-32.260-44.654-32.372Q-44.538-32.485-44.367-32.485Q-44.258-32.485-44.167-32.434Q-44.077-32.383-44.022-32.290Q-43.967-32.198-43.967-32.089Q-43.967-31.921-44.080-31.802Q-44.193-31.682-44.354-31.682",[1870],[1854,4524,4525,4528],{"fill":1988},[1866,4526],{"d":4527},"M86.512 25.77H77.44a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.072a4 4 0 0 0 4-4V29.77a4 4 0 0 0-4-4ZM73.44 42.842",[1854,4529,4530,4537,4543],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,4531,4533],{"transform":4532},"translate(128.816 67.697)",[1866,4534],{"d":4535,"fill":1856,"stroke":1856,"className":4536,"style":1909},"M-50.723-32.037Q-50.723-31.713-50.535-31.501Q-50.347-31.289-50.029-31.289Q-49.578-31.289-49.168-31.455Q-48.758-31.620-48.491-31.955Q-48.474-31.983-48.426-31.983Q-48.378-31.983-48.332-31.933Q-48.286-31.884-48.286-31.836Q-48.286-31.805-48.307-31.778Q-48.597-31.412-49.059-31.239Q-49.520-31.067-50.043-31.067Q-50.395-31.067-50.692-31.220Q-50.990-31.374-51.161-31.655Q-51.332-31.935-51.332-32.290Q-51.332-32.666-51.159-33.018Q-50.986-33.370-50.686-33.644Q-50.385-33.917-50.028-34.069Q-49.670-34.221-49.294-34.221Q-49.093-34.221-48.877-34.162Q-48.662-34.102-48.520-33.967Q-48.378-33.832-48.378-33.620Q-48.378-33.432-48.493-33.292Q-48.607-33.152-48.799-33.152Q-48.915-33.152-48.997-33.225Q-49.079-33.299-49.079-33.418Q-49.079-33.565-48.980-33.678Q-48.881-33.791-48.734-33.822Q-48.922-33.999-49.308-33.999Q-49.643-33.999-49.908-33.813Q-50.173-33.627-50.354-33.329Q-50.535-33.032-50.629-32.685Q-50.723-32.338-50.723-32.037",[1870],[1854,4538,4539],{"transform":4532},[1866,4540],{"d":4541,"fill":1856,"stroke":1856,"className":4542,"style":1909},"M-47.464-31.555Q-47.464-31.723-47.341-31.846Q-47.218-31.969-47.043-31.969Q-46.876-31.969-46.753-31.846Q-46.630-31.723-46.630-31.555Q-46.630-31.381-46.753-31.258Q-46.876-31.135-47.043-31.135Q-47.218-31.135-47.341-31.258Q-47.464-31.381-47.464-31.555M-47.464-33.739Q-47.464-33.907-47.341-34.030Q-47.218-34.153-47.043-34.153Q-46.876-34.153-46.753-34.030Q-46.630-33.907-46.630-33.739Q-46.630-33.565-46.753-33.442Q-46.876-33.319-47.043-33.319Q-47.218-33.319-47.341-33.442Q-47.464-33.565-47.464-33.739",[1870],[1854,4544,4545],{"transform":4532},[1866,4546],{"d":4547,"fill":1856,"stroke":1856,"className":4548,"style":1909},"M-43.597-32.283L-45.641-32.283L-45.641-32.564L-43.310-35.736Q-43.275-35.783-43.210-35.783L-43.074-35.783Q-43.029-35.783-43.002-35.756Q-42.975-35.729-42.975-35.684L-42.975-32.564L-42.212-32.564L-42.212-32.283L-42.975-32.283L-42.975-31.624Q-42.975-31.415-42.219-31.415L-42.219-31.135L-44.352-31.135L-44.352-31.415Q-43.597-31.415-43.597-31.624L-43.597-32.283M-43.549-35.008L-45.340-32.564L-43.549-32.564",[1870],[1854,4550,4551,4554],{"fill":1988},[1866,4552],{"d":4553},"M112.12 25.77h-9.072a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.071a4 4 0 0 0 4-4V29.77a4 4 0 0 0-4-4ZM99.047 42.842",[1854,4555,4556,4563,4569],{"fill":1856,"stroke":1860,"fontSize":1901},[1854,4557,4559],{"transform":4558},"translate(154.7 67.697)",[1866,4560],{"d":4561,"fill":1856,"stroke":1856,"className":4562,"style":1909},"M-51.109-31.723Q-51.109-31.825-51.085-31.911L-50.597-33.873L-51.379-33.873Q-51.472-33.897-51.472-33.986L-51.444-34.095Q-51.427-34.139-51.359-34.153L-50.528-34.153L-50.248-35.250Q-50.217-35.363-50.125-35.437Q-50.033-35.510-49.917-35.510Q-49.824-35.510-49.752-35.447Q-49.681-35.384-49.681-35.284Q-49.681-35.237-49.688-35.216L-49.954-34.153L-49.175-34.153Q-49.093-34.126-49.093-34.047L-49.120-33.934Q-49.127-33.890-49.195-33.873L-50.023-33.873L-50.528-31.863Q-50.556-31.706-50.556-31.617Q-50.556-31.490-50.503-31.390Q-50.450-31.289-50.330-31.289Q-50.023-31.289-49.768-31.555Q-49.513-31.822-49.380-32.157Q-49.346-32.225-49.301-32.225L-49.189-32.225Q-49.154-32.225-49.134-32.200Q-49.113-32.174-49.113-32.143Q-49.113-32.130-49.120-32.116Q-49.230-31.843-49.406-31.605Q-49.582-31.367-49.826-31.217Q-50.070-31.067-50.344-31.067Q-50.651-31.067-50.880-31.246Q-51.109-31.426-51.109-31.723",[1870],[1854,4564,4565],{"transform":4558},[1866,4566],{"d":4567,"fill":1856,"stroke":1856,"className":4568,"style":1909},"M-48.017-31.555Q-48.017-31.723-47.894-31.846Q-47.771-31.969-47.596-31.969Q-47.429-31.969-47.306-31.846Q-47.183-31.723-47.183-31.555Q-47.183-31.381-47.306-31.258Q-47.429-31.135-47.596-31.135Q-47.771-31.135-47.894-31.258Q-48.017-31.381-48.017-31.555M-48.017-33.739Q-48.017-33.907-47.894-34.030Q-47.771-34.153-47.596-34.153Q-47.429-34.153-47.306-34.030Q-47.183-33.907-47.183-33.739Q-47.183-33.565-47.306-33.442Q-47.429-33.319-47.596-33.319Q-47.771-33.319-47.894-33.442Q-48.017-33.565-48.017-33.739",[1870],[1854,4570,4571],{"transform":4558},[1866,4572],{"d":4573,"fill":1856,"stroke":1856,"className":4574,"style":1909},"M-45.031-31.343Q-45.031-31.849-44.902-32.357Q-44.772-32.864-44.534-33.326Q-44.297-33.787-43.962-34.208L-43.316-35.021L-44.129-35.021Q-44.714-35.021-45.110-35.013Q-45.507-35.004-45.530-34.984Q-45.633-34.867-45.712-34.341L-45.978-34.341L-45.732-35.865L-45.466-35.865L-45.466-35.845Q-45.466-35.777-45.390-35.734Q-45.315-35.691-45.237-35.684Q-45.045-35.660-44.850-35.654Q-44.655-35.647-44.464-35.645Q-44.273-35.643-44.074-35.643L-42.653-35.643L-42.653-35.455Q-42.663-35.407-42.673-35.397L-43.729-34.074Q-43.948-33.801-44.071-33.488Q-44.194-33.176-44.252-32.827Q-44.310-32.478-44.324-32.147Q-44.338-31.815-44.338-31.343Q-44.338-31.193-44.437-31.094Q-44.536-30.995-44.683-30.995Q-44.833-30.995-44.932-31.094Q-45.031-31.193-45.031-31.343",[1870],[2089,4576,4578,4579,4677,4678,4693,4694,4709,4710,4761,4762,1131],{"className":4577},[2092],"Dijkstra finalizes vertices in nondecreasing distance: ",[415,4580,4582],{"className":4581},[418],[415,4583,4585],{"className":4584,"ariaHidden":423},[422],[415,4586,4588,4591,4594,4597,4600,4603,4606,4609,4612,4615,4618,4621,4624,4627,4630,4633,4636,4639,4642,4645,4649,4652,4655,4658,4661,4664,4668,4671,4674],{"className":4587},[427],[415,4589],{"className":4590,"style":458},[431],[415,4592,1083],{"className":4593},[436,437],[415,4595,463],{"className":4596},[462],[415,4598,615],{"className":4599},[436],[415,4601,487],{"className":4602},[486],[415,4604,473],{"className":4605},[472],[415,4607],{"className":4608,"style":477},[442],[415,4610,2342],{"className":4611},[436,437],[415,4613,463],{"className":4614},[462],[415,4616,665],{"className":4617},[436],[415,4619,487],{"className":4620},[486],[415,4622,473],{"className":4623},[472],[415,4625],{"className":4626,"style":477},[442],[415,4628,385],{"className":4629},[436,437],[415,4631,463],{"className":4632},[462],[415,4634,1633],{"className":4635},[436],[415,4637,487],{"className":4638},[486],[415,4640,473],{"className":4641},[472],[415,4643],{"className":4644,"style":477},[442],[415,4646,4648],{"className":4647},[436,437],"c",[415,4650,463],{"className":4651},[462],[415,4653,4252],{"className":4654},[436],[415,4656,487],{"className":4657},[486],[415,4659,473],{"className":4660},[472],[415,4662],{"className":4663,"style":477},[442],[415,4665,4667],{"className":4666},[436,437],"t",[415,4669,463],{"className":4670},[462],[415,4672,1901],{"className":4673},[436],[415,4675,487],{"className":4676},[486],". Vertex ",[415,4679,4681],{"className":4680},[418],[415,4682,4684],{"className":4683,"ariaHidden":423},[422],[415,4685,4687,4690],{"className":4686},[427],[415,4688],{"className":4689,"style":1020},[431],[415,4691,385],{"className":4692},[436,437]," settles at ",[415,4695,4697],{"className":4696},[418],[415,4698,4700],{"className":4699,"ariaHidden":423},[422],[415,4701,4703,4706],{"className":4702},[427],[415,4704],{"className":4705,"style":1294},[431],[415,4707,1633],{"className":4708},[436]," via ",[415,4711,4713],{"className":4712},[418],[415,4714,4716,4734,4752],{"className":4715,"ariaHidden":423},[422],[415,4717,4719,4722,4725,4728,4731],{"className":4718},[427],[415,4720],{"className":4721,"style":1020},[431],[415,4723,1083],{"className":4724},[436,437],[415,4726],{"className":4727,"style":443},[442],[415,4729,2168],{"className":4730},[447],[415,4732],{"className":4733,"style":443},[442],[415,4735,4737,4740,4743,4746,4749],{"className":4736},[427],[415,4738],{"className":4739,"style":1163},[431],[415,4741,2342],{"className":4742},[436,437],[415,4744],{"className":4745,"style":443},[442],[415,4747,2168],{"className":4748},[447],[415,4750],{"className":4751,"style":443},[442],[415,4753,4755,4758],{"className":4754},[427],[415,4756],{"className":4757,"style":1020},[431],[415,4759,385],{"className":4760},[436,437],", beating the direct edge of weight ",[415,4763,4765],{"className":4764},[418],[415,4766,4768],{"className":4767,"ariaHidden":423},[422],[415,4769,4771,4774],{"className":4770},[427],[415,4772],{"className":4773,"style":1294},[431],[415,4775,4252],{"className":4776},[436],[381,4778,4779,4782,4783,4786,4787,531,4802,4823,4824,4839,4862,4863,4971,4972,1131],{},[390,4780,4781],{},"Running time."," Like ",[385,4784,4785],{"href":170},"Prim",", Dijkstra\ndoes ",[415,4788,4790],{"className":4789},[418],[415,4791,4793],{"className":4792,"ariaHidden":423},[422],[415,4794,4796,4799],{"className":4795},[427],[415,4797],{"className":4798,"style":432},[431],[415,4800,468],{"className":4801,"style":467},[436,437],[415,4803,4805],{"className":4804},[418],[415,4806,4808],{"className":4807,"ariaHidden":423},[422],[415,4809,4811,4814],{"className":4810},[427],[415,4812],{"className":4813,"style":432},[431],[415,4815,4817],{"className":4816},[2109,2110],[415,4818,4820],{"className":4819},[436,2114],[415,4821,4003],{"className":4822},[436]," and up to ",[415,4825,4827],{"className":4826},[418],[415,4828,4830],{"className":4829,"ariaHidden":423},[422],[415,4831,4833,4836],{"className":4832},[427],[415,4834],{"className":4835,"style":432},[431],[415,4837,482],{"className":4838,"style":481},[436,437],[415,4840,4842],{"className":4841},[418],[415,4843,4845],{"className":4844,"ariaHidden":423},[422],[415,4846,4848,4852],{"className":4847},[427],[415,4849],{"className":4850,"style":4851},[431],"height:0.8778em;vertical-align:-0.1944em;",[415,4853,4855],{"className":4854},[2109,2110],[415,4856,4858],{"className":4857},[436,2114],[415,4859,4861],{"className":4860},[436],"Decrease-Key"," operations. A binary heap gives ",[415,4864,4866],{"className":4865},[418],[415,4867,4869,4896,4938],{"className":4868,"ariaHidden":423},[422],[415,4870,4872,4875,4880,4884,4887,4890,4893],{"className":4871},[427],[415,4873],{"className":4874,"style":458},[431],[415,4876,4879],{"className":4877,"style":4878},[436,437],"margin-right:0.0278em;","O",[415,4881,4883],{"className":4882},[462],"((",[415,4885,468],{"className":4886,"style":467},[436,437],[415,4888],{"className":4889,"style":467},[442],[415,4891,1505],{"className":4892},[902],[415,4894],{"className":4895,"style":467},[442],[415,4897,4899,4902,4905,4908,4911,4920,4923,4926,4929,4932,4935],{"className":4898},[427],[415,4900],{"className":4901,"style":458},[431],[415,4903,482],{"className":4904,"style":481},[436,437],[415,4906,487],{"className":4907},[486],[415,4909],{"className":4910,"style":477},[442],[415,4912,4914],{"className":4913},[787],[415,4915,4919],{"className":4916,"style":4918},[436,4917],"mathrm","margin-right:0.0139em;","log",[415,4921],{"className":4922,"style":477},[442],[415,4924,468],{"className":4925,"style":467},[436,437],[415,4927,487],{"className":4928},[486],[415,4930],{"className":4931,"style":443},[442],[415,4933,448],{"className":4934},[447],[415,4936],{"className":4937,"style":443},[442],[415,4939,4941,4944,4947,4950,4953,4956,4962,4965,4968],{"className":4940},[427],[415,4942],{"className":4943,"style":458},[431],[415,4945,4879],{"className":4946,"style":4878},[436,437],[415,4948,463],{"className":4949},[462],[415,4951,482],{"className":4952,"style":481},[436,437],[415,4954],{"className":4955,"style":477},[442],[415,4957,4959],{"className":4958},[787],[415,4960,4919],{"className":4961,"style":4918},[436,4917],[415,4963],{"className":4964,"style":477},[442],[415,4966,468],{"className":4967,"style":467},[436,437],[415,4969,487],{"className":4970},[486],";\na Fibonacci heap improves this to ",[415,4973,4975],{"className":4974},[418],[415,4976,4978,5002],{"className":4977,"ariaHidden":423},[422],[415,4979,4981,4984,4987,4990,4993,4996,4999],{"className":4980},[427],[415,4982],{"className":4983,"style":458},[431],[415,4985,4879],{"className":4986,"style":4878},[436,437],[415,4988,463],{"className":4989},[462],[415,4991,482],{"className":4992,"style":481},[436,437],[415,4994],{"className":4995,"style":467},[442],[415,4997,1505],{"className":4998},[902],[415,5000],{"className":5001,"style":467},[442],[415,5003,5005,5008,5011,5014,5020,5023,5026],{"className":5004},[427],[415,5006],{"className":5007,"style":458},[431],[415,5009,468],{"className":5010,"style":467},[436,437],[415,5012],{"className":5013,"style":477},[442],[415,5015,5017],{"className":5016},[787],[415,5018,4919],{"className":5019,"style":4918},[436,4917],[415,5021],{"className":5022,"style":477},[442],[415,5024,468],{"className":5025,"style":467},[436,437],[415,5027,487],{"className":5028},[486],[400,5030,5032],{"id":5031},"bellman-ford-as-a-dynamic-program","Bellman-Ford as a dynamic program",[381,5034,5035,5036,5039,5040,5062,5063,5067,5068,5073,5074,1131],{},"Dijkstra breaks when edges can be ",[390,5037,5038],{},"negative"," — its greedy finalization assumes\na finalized estimate can never improve, which negative edges violate.\n",[415,5041,5043],{"className":5042},[418],[415,5044,5046],{"className":5045,"ariaHidden":423},[422],[415,5047,5049,5052],{"className":5048},[427],[415,5050],{"className":5051,"style":1163},[431],[415,5053,5055],{"className":5054},[2109,2110],[415,5056,5058],{"className":5057},[436,2114],[415,5059,5061],{"className":5060},[436],"Bellman-Ford"," trades speed for generality. It is best derived\nnot as ",[5064,5065,5066],"q",{},"relax everything repeatedly"," but as a genuine\n",[385,5069,5070],{"href":282},[390,5071,5072],{},"dynamic program",". That framing is\nworth keeping, because it explains ",[395,5075,5076,5077,5120],{},"where the ",[415,5078,5080],{"className":5079},[418],[415,5081,5083,5111],{"className":5082,"ariaHidden":423},[422],[415,5084,5086,5089,5102,5105,5108],{"className":5085},[427],[415,5087],{"className":5088,"style":458},[431],[415,5090,5092,5096,5099],{"className":5091},[566],[415,5093,5095],{"className":5094,"style":571},[462,570],"∣",[415,5097,468],{"className":5098,"style":467},[436,437],[415,5100,5095],{"className":5101,"style":571},[486,570],[415,5103],{"className":5104,"style":467},[442],[415,5106,903],{"className":5107},[902],[415,5109],{"className":5110,"style":467},[442],[415,5112,5114,5117],{"className":5113},[427],[415,5115],{"className":5116,"style":1294},[431],[415,5118,665],{"className":5119},[436]," comes from",[381,5122,5123,5126,5127,5142,5143,5158],{},[390,5124,5125],{},"The subproblem."," Bound the number of edges a walk may use. For each vertex\n",[415,5128,5130],{"className":5129},[418],[415,5131,5133],{"className":5132,"ariaHidden":423},[422],[415,5134,5136,5139],{"className":5135},[427],[415,5137],{"className":5138,"style":1020},[431],[415,5140,523],{"className":5141,"style":522},[436,437]," and each budget ",[415,5144,5146],{"className":5145},[418],[415,5147,5149],{"className":5148,"ariaHidden":423},[422],[415,5150,5152,5155],{"className":5151},[427],[415,5153],{"className":5154,"style":1163},[431],[415,5156,727],{"className":5157,"style":726},[436,437],", define",[415,5160,5163],{"className":5161},[5162],"katex-display",[415,5164,5166],{"className":5165},[418],[415,5167,5169,5209,5234],{"className":5168,"ariaHidden":423},[422],[415,5170,5172,5175,5182,5185,5188,5191,5194,5197,5200,5203,5206],{"className":5171},[427],[415,5173],{"className":5174,"style":458},[431],[415,5176,5178],{"className":5177},[436,2114],[415,5179,5181],{"className":5180},[436],"OPT",[415,5183,463],{"className":5184},[462],[415,5186,727],{"className":5187,"style":726},[436,437],[415,5189,473],{"className":5190},[472],[415,5192],{"className":5193,"style":477},[442],[415,5195,523],{"className":5196,"style":522},[436,437],[415,5198,487],{"className":5199},[486],[415,5201],{"className":5202,"style":443},[442],[415,5204,448],{"className":5205},[447],[415,5207],{"className":5208,"style":443},[442],[415,5210,5212,5215,5222,5225,5228,5231],{"className":5211},[427],[415,5213],{"className":5214,"style":2641},[431],[415,5216,5218],{"className":5217},[436,2114],[415,5219,5221],{"className":5220},[436],"cost of a cheapest ",[415,5223,1083],{"className":5224},[436,437],[415,5226],{"className":5227,"style":443},[442],[415,5229,3213],{"className":5230},[447,3212],[415,5232],{"className":5233,"style":443},[442],[415,5235,5237,5240,5243,5250,5253],{"className":5236},[427],[415,5238],{"className":5239,"style":2641},[431],[415,5241,523],{"className":5242,"style":522},[436,437],[415,5244,5246],{"className":5245},[436,2114],[415,5247,5249],{"className":5248},[436]," walk using at most ",[415,5251,727],{"className":5252,"style":726},[436,437],[415,5254,5256],{"className":5255},[436,2114],[415,5257,5259],{"className":5258},[436]," edges.",[381,5261,5262,5263,5318,5319,5361,5362,5396],{},"The full answer we want is ",[415,5264,5266],{"className":5265},[418],[415,5267,5269,5297],{"className":5268,"ariaHidden":423},[422],[415,5270,5272,5275,5281,5284,5288,5291,5294],{"className":5271},[427],[415,5273],{"className":5274,"style":458},[431],[415,5276,5278],{"className":5277},[436,2114],[415,5279,5181],{"className":5280},[436],[415,5282,463],{"className":5283},[462],[415,5285,5287],{"className":5286},[436,437],"n",[415,5289],{"className":5290,"style":467},[442],[415,5292,903],{"className":5293},[902],[415,5295],{"className":5296,"style":467},[442],[415,5298,5300,5303,5306,5309,5312,5315],{"className":5299},[427],[415,5301],{"className":5302,"style":458},[431],[415,5304,665],{"className":5305},[436],[415,5307,473],{"className":5308},[472],[415,5310],{"className":5311,"style":477},[442],[415,5313,523],{"className":5314,"style":522},[436,437],[415,5316,487],{"className":5317},[486]," for ",[415,5320,5322],{"className":5321},[418],[415,5323,5325,5343],{"className":5324,"ariaHidden":423},[422],[415,5326,5328,5331,5334,5337,5340],{"className":5327},[427],[415,5329],{"className":5330,"style":1020},[431],[415,5332,5287],{"className":5333},[436,437],[415,5335],{"className":5336,"style":443},[442],[415,5338,448],{"className":5339},[447],[415,5341],{"className":5342,"style":443},[442],[415,5344,5346,5349],{"className":5345},[427],[415,5347],{"className":5348,"style":458},[431],[415,5350,5352,5355,5358],{"className":5351},[566],[415,5353,5095],{"className":5354,"style":571},[462,570],[415,5356,468],{"className":5357,"style":467},[436,437],[415,5359,5095],{"className":5360,"style":571},[486,570],", since (as we\nprove below) shortest paths never need more than ",[415,5363,5365],{"className":5364},[418],[415,5366,5368,5387],{"className":5367,"ariaHidden":423},[422],[415,5369,5371,5375,5378,5381,5384],{"className":5370},[427],[415,5372],{"className":5373,"style":5374},[431],"height:0.6667em;vertical-align:-0.0833em;",[415,5376,5287],{"className":5377},[436,437],[415,5379],{"className":5380,"style":467},[442],[415,5382,903],{"className":5383},[902],[415,5385],{"className":5386,"style":467},[442],[415,5388,5390,5393],{"className":5389},[427],[415,5391],{"className":5392,"style":1294},[431],[415,5394,665],{"className":5395},[436]," edges.",[381,5398,5399,5402,5403,5418,5419,5434,5435,5469,5470,5500,5501,5552,5553,5568],{},[390,5400,5401],{},"The recurrence."," A cheapest walk to ",[415,5404,5406],{"className":5405},[418],[415,5407,5409],{"className":5408,"ariaHidden":423},[422],[415,5410,5412,5415],{"className":5411},[427],[415,5413],{"className":5414,"style":1020},[431],[415,5416,523],{"className":5417,"style":522},[436,437]," using at most ",[415,5420,5422],{"className":5421},[418],[415,5423,5425],{"className":5424,"ariaHidden":423},[422],[415,5426,5428,5431],{"className":5427},[427],[415,5429],{"className":5430,"style":1163},[431],[415,5432,727],{"className":5433,"style":726},[436,437]," edges either uses at\nmost ",[415,5436,5438],{"className":5437},[418],[415,5439,5441,5460],{"className":5440,"ariaHidden":423},[422],[415,5442,5444,5448,5451,5454,5457],{"className":5443},[427],[415,5445],{"className":5446,"style":5447},[431],"height:0.7778em;vertical-align:-0.0833em;",[415,5449,727],{"className":5450,"style":726},[436,437],[415,5452],{"className":5453,"style":467},[442],[415,5455,903],{"className":5456},[902],[415,5458],{"className":5459,"style":467},[442],[415,5461,5463,5466],{"className":5462},[427],[415,5464],{"className":5465,"style":1294},[431],[415,5467,665],{"className":5468},[436]," edges already, or it takes one final edge ",[415,5471,5473],{"className":5472},[418],[415,5474,5476],{"className":5475,"ariaHidden":423},[422],[415,5477,5479,5482,5485,5488,5491,5494,5497],{"className":5478},[427],[415,5480],{"className":5481,"style":458},[431],[415,5483,463],{"className":5484},[462],[415,5486,512],{"className":5487},[436,437],[415,5489,473],{"className":5490},[472],[415,5492],{"className":5493,"style":477},[442],[415,5495,523],{"className":5496,"style":522},[436,437],[415,5498,487],{"className":5499},[486]," after a cheapest\n",[415,5502,5504],{"className":5503},[418],[415,5505,5507,5519,5540],{"className":5506,"ariaHidden":423},[422],[415,5508,5510,5513,5516],{"className":5509},[427],[415,5511],{"className":5512,"style":1225},[431],[415,5514,1468],{"className":5515},[447],[415,5517],{"className":5518,"style":443},[442],[415,5520,5522,5525,5528,5531,5534,5537],{"className":5521},[427],[415,5523],{"className":5524,"style":458},[431],[415,5526,463],{"className":5527},[462],[415,5529,727],{"className":5530,"style":726},[436,437],[415,5532],{"className":5533,"style":467},[442],[415,5535,903],{"className":5536},[902],[415,5538],{"className":5539,"style":467},[442],[415,5541,5543,5546,5549],{"className":5542},[427],[415,5544],{"className":5545,"style":458},[431],[415,5547,665],{"className":5548},[436],[415,5550,487],{"className":5551},[486],"-edge walk to ",[415,5554,5556],{"className":5555},[418],[415,5557,5559],{"className":5558,"ariaHidden":423},[422],[415,5560,5562,5565],{"className":5561},[427],[415,5563],{"className":5564,"style":1020},[431],[415,5566,512],{"className":5567},[436,437],". Minimizing over both:",[415,5570,5572],{"className":5571},[5162],[415,5573,5575],{"className":5574},[418],[415,5576,5578,5617,5663,5795,5825],{"className":5577,"ariaHidden":423},[422],[415,5579,5581,5584,5590,5593,5596,5599,5602,5605,5608,5611,5614],{"className":5580},[427],[415,5582],{"className":5583,"style":458},[431],[415,5585,5587],{"className":5586},[436,2114],[415,5588,5181],{"className":5589},[436],[415,5591,463],{"className":5592},[462],[415,5594,727],{"className":5595,"style":726},[436,437],[415,5597,473],{"className":5598},[472],[415,5600],{"className":5601,"style":477},[442],[415,5603,523],{"className":5604,"style":522},[436,437],[415,5606,487],{"className":5607},[486],[415,5609],{"className":5610,"style":443},[442],[415,5612,448],{"className":5613},[447],[415,5615],{"className":5616,"style":443},[442],[415,5618,5620,5624,5631,5639,5642,5648,5651,5654,5657,5660],{"className":5619},[427],[415,5621],{"className":5622,"style":5623},[431],"height:1.8em;vertical-align:-0.65em;",[415,5625,5627],{"className":5626},[787],[415,5628,5630],{"className":5629},[436,4917],"min",[415,5632,5634],{"className":5633},[462],[415,5635,463],{"className":5636},[5637,5638],"delimsizing","size2",[415,5640],{"className":5641,"style":477},[442],[415,5643,5645],{"className":5644},[436,2114],[415,5646,5181],{"className":5647},[436],[415,5649,463],{"className":5650},[462],[415,5652,727],{"className":5653,"style":726},[436,437],[415,5655],{"className":5656,"style":467},[442],[415,5658,903],{"className":5659},[902],[415,5661],{"className":5662,"style":467},[442],[415,5664,5666,5670,5673,5676,5679,5682,5685,5688,5691,5694,5768,5771,5774,5780,5783,5786,5789,5792],{"className":5665},[427],[415,5667],{"className":5668,"style":5669},[431],"height:1.1052em;vertical-align:-0.3552em;",[415,5671,665],{"className":5672},[436],[415,5674,473],{"className":5675},[472],[415,5677],{"className":5678,"style":477},[442],[415,5680,523],{"className":5681,"style":522},[436,437],[415,5683,487],{"className":5684},[486],[415,5686,473],{"className":5687},[472],[415,5689],{"className":5690,"style":443},[442],[415,5692],{"className":5693,"style":477},[442],[415,5695,5697,5703],{"className":5696},[787],[415,5698,5700],{"className":5699},[787],[415,5701,5630],{"className":5702},[436,4917],[415,5704,5706],{"className":5705},[582],[415,5707,5709,5759],{"className":5708},[586,587],[415,5710,5712,5756],{"className":5711},[591],[415,5713,5716],{"className":5714,"style":5715},[595],"height:0.3448em;",[415,5717,5719,5722],{"style":5718},"top:-2.5198em;margin-right:0.05em;",[415,5720],{"className":5721,"style":604},[603],[415,5723,5725],{"className":5724},[608,609,610,611],[415,5726,5728,5731,5734,5737,5740,5743,5747,5750,5753],{"className":5727},[436,611],[415,5729,463],{"className":5730},[462,611],[415,5732,512],{"className":5733},[436,437,611],[415,5735,473],{"className":5736},[472,611],[415,5738,523],{"className":5739,"style":522},[436,437,611],[415,5741,487],{"className":5742},[486,611],[415,5744],{"className":5745,"style":5746},[442,611],"margin-right:0.1952em;",[415,5748,2853],{"className":5749},[447,611],[415,5751],{"className":5752,"style":5746},[442,611],[415,5754,482],{"className":5755,"style":481},[436,437,611],[415,5757,620],{"className":5758},[619],[415,5760,5762],{"className":5761},[591],[415,5763,5766],{"className":5764,"style":5765},[595],"height:0.3552em;",[415,5767],{},[415,5769],{"className":5770,"style":443},[442],[415,5772],{"className":5773,"style":477},[442],[415,5775,5777],{"className":5776},[436,2114],[415,5778,5181],{"className":5779},[436],[415,5781,463],{"className":5782},[462],[415,5784,727],{"className":5785,"style":726},[436,437],[415,5787],{"className":5788,"style":467},[442],[415,5790,903],{"className":5791},[902],[415,5793],{"className":5794,"style":467},[442],[415,5796,5798,5801,5804,5807,5810,5813,5816,5819,5822],{"className":5797},[427],[415,5799],{"className":5800,"style":458},[431],[415,5802,665],{"className":5803},[436],[415,5805,473],{"className":5806},[472],[415,5808],{"className":5809,"style":477},[442],[415,5811,512],{"className":5812},[436,437],[415,5814,487],{"className":5815},[486],[415,5817],{"className":5818,"style":467},[442],[415,5820,1505],{"className":5821},[902],[415,5823],{"className":5824,"style":467},[442],[415,5826,5828,5831,5834,5837,5840,5843,5846,5849,5852,5855,5861],{"className":5827},[427],[415,5829],{"className":5830,"style":5623},[431],[415,5832,505],{"className":5833,"style":504},[436,437],[415,5835,463],{"className":5836},[462],[415,5838,512],{"className":5839},[436,437],[415,5841,473],{"className":5842},[472],[415,5844],{"className":5845,"style":477},[442],[415,5847,523],{"className":5848,"style":522},[436,437],[415,5850,487],{"className":5851},[486],[415,5853],{"className":5854,"style":477},[442],[415,5856,5858],{"className":5857},[486],[415,5859,487],{"className":5860},[5637,5638],[415,5862,473],{"className":5863},[472],[381,5865,5866,5867,2979,5921,5975,5976,6045,6046,6061,6062,6095,6096,6111,6112,2872],{},"with base cases ",[415,5868,5870],{"className":5869},[418],[415,5871,5873,5912],{"className":5872,"ariaHidden":423},[422],[415,5874,5876,5879,5885,5888,5891,5894,5897,5900,5903,5906,5909],{"className":5875},[427],[415,5877],{"className":5878,"style":458},[431],[415,5880,5882],{"className":5881},[436,2114],[415,5883,5181],{"className":5884},[436],[415,5886,463],{"className":5887},[462],[415,5889,615],{"className":5890},[436],[415,5892,473],{"className":5893},[472],[415,5895],{"className":5896,"style":477},[442],[415,5898,1083],{"className":5899},[436,437],[415,5901,487],{"className":5902},[486],[415,5904],{"className":5905,"style":443},[442],[415,5907,448],{"className":5908},[447],[415,5910],{"className":5911,"style":443},[442],[415,5913,5915,5918],{"className":5914},[427],[415,5916],{"className":5917,"style":1294},[431],[415,5919,615],{"className":5920},[436],[415,5922,5924],{"className":5923},[418],[415,5925,5927,5966],{"className":5926,"ariaHidden":423},[422],[415,5928,5930,5933,5939,5942,5945,5948,5951,5954,5957,5960,5963],{"className":5929},[427],[415,5931],{"className":5932,"style":458},[431],[415,5934,5936],{"className":5935},[436,2114],[415,5937,5181],{"className":5938},[436],[415,5940,463],{"className":5941},[462],[415,5943,615],{"className":5944},[436],[415,5946,473],{"className":5947},[472],[415,5949],{"className":5950,"style":477},[442],[415,5952,523],{"className":5953,"style":522},[436,437],[415,5955,487],{"className":5956},[486],[415,5958],{"className":5959,"style":443},[442],[415,5961,448],{"className":5962},[447],[415,5964],{"className":5965,"style":443},[442],[415,5967,5969,5972],{"className":5968},[427],[415,5970],{"className":5971,"style":1020},[431],[415,5973,1056],{"className":5974},[436]," for\n",[415,5977,5979],{"className":5978},[418],[415,5980,5982,6036],{"className":5981,"ariaHidden":423},[422],[415,5983,5985,5988,5991,5994,6033],{"className":5984},[427],[415,5986],{"className":5987,"style":2641},[431],[415,5989,523],{"className":5990,"style":522},[436,437],[415,5992],{"className":5993,"style":443},[442],[415,5995,5997,6026,6030],{"className":5996},[447],[415,5998,6000],{"className":5999},[447],[415,6001,6003],{"className":6002},[436,3486],[415,6004,6006],{"className":6005},[3490],[415,6007,6010,6013,6023],{"className":6008},[6009],"rlap",[415,6011],{"className":6012,"style":2641},[431],[415,6014,6016],{"className":6015},[3501],[415,6017,6019],{"className":6018},[436],[415,6020,6022],{"className":6021},[447],"",[415,6024],{"className":6025},[3516],[415,6027],{"className":6028},[442,6029],"nobreak",[415,6031,448],{"className":6032},[447],[415,6034],{"className":6035,"style":443},[442],[415,6037,6039,6042],{"className":6038},[427],[415,6040],{"className":6041,"style":1020},[431],[415,6043,1083],{"className":6044},[436,437],". This is exactly relaxation: filling row ",[415,6047,6049],{"className":6048},[418],[415,6050,6052],{"className":6051,"ariaHidden":423},[422],[415,6053,6055,6058],{"className":6054},[427],[415,6056],{"className":6057,"style":1163},[431],[415,6059,727],{"className":6060,"style":726},[436,437]," of the table from row\n",[415,6063,6065],{"className":6064},[418],[415,6066,6068,6086],{"className":6067,"ariaHidden":423},[422],[415,6069,6071,6074,6077,6080,6083],{"className":6070},[427],[415,6072],{"className":6073,"style":5447},[431],[415,6075,727],{"className":6076,"style":726},[436,437],[415,6078],{"className":6079,"style":467},[442],[415,6081,903],{"className":6082},[902],[415,6084],{"className":6085,"style":467},[442],[415,6087,6089,6092],{"className":6088},[427],[415,6090],{"className":6091,"style":1294},[431],[415,6093,665],{"className":6094},[436]," is one full pass that relaxes every edge. The tabular form\nmakes the layering explicit, with row ",[415,6097,6099],{"className":6098},[418],[415,6100,6102],{"className":6101,"ariaHidden":423},[422],[415,6103,6105,6108],{"className":6104},[427],[415,6106],{"className":6107,"style":1163},[431],[415,6109,727],{"className":6110,"style":726},[436,437]," holding the best walk of length ",[415,6113,6115],{"className":6114},[418],[415,6116,6118,6130],{"className":6117,"ariaHidden":423},[422],[415,6119,6121,6124,6127],{"className":6120},[427],[415,6122],{"className":6123,"style":1225},[431],[415,6125,1468],{"className":6126},[447],[415,6128],{"className":6129,"style":443},[442],[415,6131,6133,6136],{"className":6132},[427],[415,6134],{"className":6135,"style":1163},[431],[415,6137,727],{"className":6138,"style":726},[436,437],[1381,6140,6142],{"className":1383,"code":6141,"language":1385,"meta":376,"style":376},"caption: $\\textsc{Bellman-Ford-DP}(G, w, s)$ — the explicit DP table\nnumber: 3\n$\\text{OPT}[0, s] \\gets 0$; $\\text{OPT}[0, v] \\gets \\infty$ for $v \\neq s$\nfor $k \\gets 1$ to $n - 1$ do \u002F\u002F one pass over all edges per row\n  foreach vertex $v \\in V$ do\n    $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, v]$ \u002F\u002F inherit: no extra edge\n  foreach edge $(u, v) \\in E$ do\n    if $\\text{OPT}[k-1, u] + w(u, v) \u003C \\text{OPT}[k, v]$ then\n      $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, u] + w(u, v)$\n      $v.\\pi \\gets u$\nreturn $\\text{OPT}[n-1, \\cdot]$ and $\\pi$\n",[1387,6143,6144,6149,6154,6159,6164,6169,6174,6179,6184,6189,6194],{"__ignoreMap":376},[415,6145,6146],{"class":1391,"line":6},[415,6147,6148],{},"caption: $\\textsc{Bellman-Ford-DP}(G, w, s)$ — the explicit DP table\n",[415,6150,6151],{"class":1391,"line":18},[415,6152,6153],{},"number: 3\n",[415,6155,6156],{"class":1391,"line":24},[415,6157,6158],{},"$\\text{OPT}[0, s] \\gets 0$; $\\text{OPT}[0, v] \\gets \\infty$ for $v \\neq s$\n",[415,6160,6161],{"class":1391,"line":73},[415,6162,6163],{},"for $k \\gets 1$ to $n - 1$ do \u002F\u002F one pass over all edges per row\n",[415,6165,6166],{"class":1391,"line":102},[415,6167,6168],{},"  foreach vertex $v \\in V$ do\n",[415,6170,6171],{"class":1391,"line":108},[415,6172,6173],{},"    $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, v]$ \u002F\u002F inherit: no extra edge\n",[415,6175,6176],{"class":1391,"line":116},[415,6177,6178],{},"  foreach edge $(u, v) \\in E$ do\n",[415,6180,6181],{"class":1391,"line":196},[415,6182,6183],{},"    if $\\text{OPT}[k-1, u] + w(u, v) \u003C \\text{OPT}[k, v]$ then\n",[415,6185,6186],{"class":1391,"line":202},[415,6187,6188],{},"      $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, u] + w(u, v)$\n",[415,6190,6191],{"class":1391,"line":283},[415,6192,6193],{},"      $v.\\pi \\gets u$\n",[415,6195,6196],{"class":1391,"line":333},[415,6197,6198],{},"return $\\text{OPT}[n-1, \\cdot]$ and $\\pi$\n",[381,6200,6201,6238,6239,6363,6364,531,6398,6401,6402,6447,6448,6451,6452,6485,6486,1131],{},[390,6202,6203,6204,6237],{},"Why stop after ",[415,6205,6207],{"className":6206},[418],[415,6208,6210,6228],{"className":6209,"ariaHidden":423},[422],[415,6211,6213,6216,6219,6222,6225],{"className":6212},[427],[415,6214],{"className":6215,"style":5374},[431],[415,6217,5287],{"className":6218},[436,437],[415,6220],{"className":6221,"style":467},[442],[415,6223,903],{"className":6224},[902],[415,6226],{"className":6227,"style":467},[442],[415,6229,6231,6234],{"className":6230},[427],[415,6232],{"className":6233,"style":1294},[431],[415,6235,665],{"className":6236},[436]," rounds?"," This is the theorem that licenses the whole\nalgorithm. ",[395,6240,6241,6242,6257,6258,6283,6284,6317,6318,1131],{},"If ",[415,6243,6245],{"className":6244},[418],[415,6246,6248],{"className":6247,"ariaHidden":423},[422],[415,6249,6251,6254],{"className":6250},[427],[415,6252],{"className":6253,"style":432},[431],[415,6255,438],{"className":6256},[436,437]," has no negative-cost cycle, then for all ",[415,6259,6261],{"className":6260},[418],[415,6262,6264],{"className":6263,"ariaHidden":423},[422],[415,6265,6267,6271,6274,6277,6280],{"className":6266},[427],[415,6268],{"className":6269,"style":6270},[431],"height:0.8095em;vertical-align:-0.1944em;",[415,6272,1083],{"className":6273},[436,437],[415,6275,473],{"className":6276},[472],[415,6278],{"className":6279,"style":477},[442],[415,6281,4667],{"className":6282},[436,437]," the distance\n",[415,6285,6287],{"className":6286},[418],[415,6288,6290],{"className":6289,"ariaHidden":423},[422],[415,6291,6293,6296,6299,6302,6305,6308,6311,6314],{"className":6292},[427],[415,6294],{"className":6295,"style":458},[431],[415,6297,988],{"className":6298,"style":987},[436,437],[415,6300,463],{"className":6301},[462],[415,6303,1083],{"className":6304},[436,437],[415,6306,473],{"className":6307},[472],[415,6309],{"className":6310,"style":477},[442],[415,6312,4667],{"className":6313},[436,437],[415,6315,487],{"className":6316},[486]," is achieved by a path of length ",[415,6319,6321],{"className":6320},[418],[415,6322,6324,6336,6354],{"className":6323,"ariaHidden":423},[422],[415,6325,6327,6330,6333],{"className":6326},[427],[415,6328],{"className":6329,"style":1225},[431],[415,6331,1468],{"className":6332},[447],[415,6334],{"className":6335,"style":443},[442],[415,6337,6339,6342,6345,6348,6351],{"className":6338},[427],[415,6340],{"className":6341,"style":5374},[431],[415,6343,5287],{"className":6344},[436,437],[415,6346],{"className":6347,"style":467},[442],[415,6349,903],{"className":6350},[902],[415,6352],{"className":6353,"style":467},[442],[415,6355,6357,6360],{"className":6356},[427],[415,6358],{"className":6359,"style":1294},[431],[415,6361,665],{"className":6362},[436]," The proof is a\nshort-circuiting argument: any ",[415,6365,6367],{"className":6366},[418],[415,6368,6370,6388],{"className":6369,"ariaHidden":423},[422],[415,6371,6373,6376,6379,6382,6385],{"className":6372},[427],[415,6374],{"className":6375,"style":1020},[431],[415,6377,1083],{"className":6378},[436,437],[415,6380],{"className":6381,"style":443},[442],[415,6383,3213],{"className":6384},[447,3212],[415,6386],{"className":6387,"style":443},[442],[415,6389,6391,6395],{"className":6390},[427],[415,6392],{"className":6393,"style":6394},[431],"height:0.6151em;",[415,6396,4667],{"className":6397},[436,437],[390,6399,6400],{},"walk"," of length ",[415,6403,6405],{"className":6404},[418],[415,6406,6408,6420,6438],{"className":6407,"ariaHidden":423},[422],[415,6409,6411,6414,6417],{"className":6410},[427],[415,6412],{"className":6413,"style":3433},[431],[415,6415,3259],{"className":6416},[447],[415,6418],{"className":6419,"style":443},[442],[415,6421,6423,6426,6429,6432,6435],{"className":6422},[427],[415,6424],{"className":6425,"style":5374},[431],[415,6427,5287],{"className":6428},[436,437],[415,6430],{"className":6431,"style":467},[442],[415,6433,903],{"className":6434},[902],[415,6436],{"className":6437,"style":467},[442],[415,6439,6441,6444],{"className":6440},[427],[415,6442],{"className":6443,"style":1294},[431],[415,6445,665],{"className":6446},[436],"\nvisits some vertex twice, so it contains a cycle. Excising that cycle yields a\nshorter walk of cost no greater than the original (the removed cycle has\nnon-negative cost). Repeating until no cycle remains leaves a ",[395,6449,6450],{},"simple"," path, with\nat most ",[415,6453,6455],{"className":6454},[418],[415,6456,6458,6476],{"className":6457,"ariaHidden":423},[422],[415,6459,6461,6464,6467,6470,6473],{"className":6460},[427],[415,6462],{"className":6463,"style":5374},[431],[415,6465,5287],{"className":6466},[436,437],[415,6468],{"className":6469,"style":467},[442],[415,6471,903],{"className":6472},[902],[415,6474],{"className":6475,"style":467},[442],[415,6477,6479,6482],{"className":6478},[427],[415,6480],{"className":6481,"style":1294},[431],[415,6483,665],{"className":6484},[436]," edges, that is no more expensive. So the rows of the DP stop\nchanging by row ",[415,6487,6489],{"className":6488},[418],[415,6490,6492,6510],{"className":6491,"ariaHidden":423},[422],[415,6493,6495,6498,6501,6504,6507],{"className":6494},[427],[415,6496],{"className":6497,"style":5374},[431],[415,6499,5287],{"className":6500},[436,437],[415,6502],{"className":6503,"style":467},[442],[415,6505,903],{"className":6506},[902],[415,6508],{"className":6509,"style":467},[442],[415,6511,6513,6516],{"className":6512},[427],[415,6514],{"className":6515,"style":1294},[431],[415,6517,665],{"className":6518},[436],[381,6520,6521,6524,6525,6558,6559,6586,6587,6620,6621,6624,6625,6627],{},[390,6522,6523],{},"Saving space."," The recurrence only ever reads row ",[415,6526,6528],{"className":6527},[418],[415,6529,6531,6549],{"className":6530,"ariaHidden":423},[422],[415,6532,6534,6537,6540,6543,6546],{"className":6533},[427],[415,6535],{"className":6536,"style":5447},[431],[415,6538,727],{"className":6539,"style":726},[436,437],[415,6541],{"className":6542,"style":467},[442],[415,6544,903],{"className":6545},[902],[415,6547],{"className":6548,"style":467},[442],[415,6550,6552,6555],{"className":6551},[427],[415,6553],{"className":6554,"style":1294},[431],[415,6556,665],{"className":6557},[436],", so we collapse the\ntable to a single 1-D array ",[415,6560,6562],{"className":6561},[418],[415,6563,6565],{"className":6564,"ariaHidden":423},[422],[415,6566,6568,6571,6574,6578,6582],{"className":6567},[427],[415,6569],{"className":6570,"style":458},[431],[415,6572,1173],{"className":6573},[436,437],[415,6575,6577],{"className":6576},[462],"[",[415,6579,6581],{"className":6580},[436],"⋅",[415,6583,6585],{"className":6584},[486],"]"," updated in place, recovering the familiar\nform of Bellman-Ford as ",[415,6588,6590],{"className":6589},[418],[415,6591,6593,6611],{"className":6592,"ariaHidden":423},[422],[415,6594,6596,6599,6602,6605,6608],{"className":6595},[427],[415,6597],{"className":6598,"style":5374},[431],[415,6600,5287],{"className":6601},[436,437],[415,6603],{"className":6604,"style":467},[442],[415,6606,903],{"className":6607},[902],[415,6609],{"className":6610,"style":467},[442],[415,6612,6614,6617],{"className":6613},[427],[415,6615],{"className":6616,"style":1294},[431],[415,6618,665],{"className":6619},[436]," rounds of relaxing every edge. Note ",[390,6622,6623],{},"negative\nedges are fine","; only a negative ",[395,6626,2365],{}," breaks the theorem.",[1381,6629,6631],{"className":1383,"code":6630,"language":1385,"meta":376,"style":376},"caption: $\\textsc{Bellman-Ford}(G, w, s)$ — space-saved; with cycle check\nnumber: 4\nforeach vertex $v \\in V$ do\n  $v.d \\gets \\infty$\n  $v.\\pi \\gets \\text{nil}$\n$s.d \\gets 0$\nfor $i \\gets 1$ to $n - 1$ do \u002F\u002F V-1 full-relaxation rounds\n  foreach edge $(u, v) \\in E$ do\n    call $\\textsc{Relax}(u, v, w)$\nforeach edge $(u, v) \\in E$ do \u002F\u002F extra pass: detect neg cycle\n  if $u.d + w(u, v) \u003C v.d$ then\n    return false \u002F\u002F neg cycle reachable from s\nreturn true\n",[1387,6632,6633,6638,6643,6647,6651,6655,6659,6664,6668,6673,6678,6683,6688],{"__ignoreMap":376},[415,6634,6635],{"class":1391,"line":6},[415,6636,6637],{},"caption: $\\textsc{Bellman-Ford}(G, w, s)$ — space-saved; with cycle check\n",[415,6639,6640],{"class":1391,"line":18},[415,6641,6642],{},"number: 4\n",[415,6644,6645],{"class":1391,"line":24},[415,6646,2751],{},[415,6648,6649],{"class":1391,"line":73},[415,6650,2756],{},[415,6652,6653],{"class":1391,"line":102},[415,6654,2761],{},[415,6656,6657],{"class":1391,"line":108},[415,6658,2766],{},[415,6660,6661],{"class":1391,"line":116},[415,6662,6663],{},"for $i \\gets 1$ to $n - 1$ do \u002F\u002F V-1 full-relaxation rounds\n",[415,6665,6666],{"class":1391,"line":196},[415,6667,6178],{},[415,6669,6670],{"class":1391,"line":202},[415,6671,6672],{},"    call $\\textsc{Relax}(u, v, w)$\n",[415,6674,6675],{"class":1391,"line":283},[415,6676,6677],{},"foreach edge $(u, v) \\in E$ do \u002F\u002F extra pass: detect neg cycle\n",[415,6679,6680],{"class":1391,"line":333},[415,6681,6682],{},"  if $u.d + w(u, v) \u003C v.d$ then\n",[415,6684,6685],{"class":1391,"line":354},[415,6686,6687],{},"    return false \u002F\u002F neg cycle reachable from s\n",[415,6689,6690],{"class":1391,"line":2799},[415,6691,6692],{},"return true\n",[381,6694,6695,6698,6699,6732,6733,6736,6737,6752,6753,6756,6757,6772,6779,6780,6783,6784,6827,6828,6870,6871,6895],{},[390,6696,6697],{},"Detecting negative cycles."," After ",[415,6700,6702],{"className":6701},[418],[415,6703,6705,6723],{"className":6704,"ariaHidden":423},[422],[415,6706,6708,6711,6714,6717,6720],{"className":6707},[427],[415,6709],{"className":6710,"style":5374},[431],[415,6712,5287],{"className":6713},[436,437],[415,6715],{"className":6716,"style":467},[442],[415,6718,903],{"className":6719},[902],[415,6721],{"className":6722,"style":467},[442],[415,6724,6726,6729],{"className":6725},[427],[415,6727],{"className":6728,"style":1294},[431],[415,6730,665],{"className":6731},[436]," rounds, if any edge can ",[395,6734,6735],{},"still"," be\nrelaxed, some walk was improved using ",[415,6738,6740],{"className":6739},[418],[415,6741,6743],{"className":6742,"ariaHidden":423},[422],[415,6744,6746,6749],{"className":6745},[427],[415,6747],{"className":6748,"style":1020},[431],[415,6750,5287],{"className":6751},[436,437]," edges — only possible if a ",[390,6754,6755],{},"negative\ncycle"," lets the cost descend without bound. So one extra pass is exactly the\ntest: relax everything once more, and any successful relaxation certifies a\nnegative cycle reachable from ",[415,6758,6760],{"className":6759},[418],[415,6761,6763],{"className":6762,"ariaHidden":423},[422],[415,6764,6766,6769],{"className":6765},[427],[415,6767],{"className":6768,"style":1020},[431],[415,6770,1083],{"className":6771},[436,437],[1542,6773,6774],{},[385,6775,2361],{"href":6776,"ariaDescribedBy":6777,"dataFootnoteRef":376,"id":6778},"#user-content-fn-erickson-sp",[1548],"user-content-fnref-erickson-sp"," (where ",[5064,6781,6782],{},"cheapest cost to reach a vertex"," is no\nlonger even well-defined). Dijkstra entirely lacks that capability. The\ncost is ",[415,6785,6787],{"className":6786},[418],[415,6788,6790,6815],{"className":6789,"ariaHidden":423},[422],[415,6791,6793,6796,6800,6803,6806,6809,6812],{"className":6792},[427],[415,6794],{"className":6795,"style":458},[431],[415,6797,6799],{"className":6798},[436],"Θ",[415,6801,463],{"className":6802},[462],[415,6804,468],{"className":6805,"style":467},[436,437],[415,6807],{"className":6808,"style":467},[442],[415,6810,6581],{"className":6811},[902],[415,6813],{"className":6814,"style":467},[442],[415,6816,6818,6821,6824],{"className":6817},[427],[415,6819],{"className":6820,"style":458},[431],[415,6822,482],{"className":6823,"style":481},[436,437],[415,6825,487],{"className":6826},[486],": ",[415,6829,6831],{"className":6830},[418],[415,6832,6834,6861],{"className":6833,"ariaHidden":423},[422],[415,6835,6837,6840,6852,6855,6858],{"className":6836},[427],[415,6838],{"className":6839,"style":458},[431],[415,6841,6843,6846,6849],{"className":6842},[566],[415,6844,5095],{"className":6845,"style":571},[462,570],[415,6847,468],{"className":6848,"style":467},[436,437],[415,6850,5095],{"className":6851,"style":571},[486,570],[415,6853],{"className":6854,"style":467},[442],[415,6856,903],{"className":6857},[902],[415,6859],{"className":6860,"style":467},[442],[415,6862,6864,6867],{"className":6863},[427],[415,6865],{"className":6866,"style":1294},[431],[415,6868,665],{"className":6869},[436]," passes, each relaxing all ",[415,6872,6874],{"className":6873},[418],[415,6875,6877],{"className":6876,"ariaHidden":423},[422],[415,6878,6880,6883],{"className":6879},[427],[415,6881],{"className":6882,"style":458},[431],[415,6884,6886,6889,6892],{"className":6885},[566],[415,6887,5095],{"className":6888,"style":571},[462,570],[415,6890,482],{"className":6891,"style":481},[436,437],[415,6893,5095],{"className":6894,"style":571},[486,570],"\nedges.",[405,6897,6899],{"type":6898},"remark",[381,6900,6901,6904,6905,6920,6921,3455,6961,7000,7001,2979,7083,7164,7165,7198,7199,7277,7278,7293,7294,7376,7377,7392,7393,7444],{},[390,6902,6903],{},"Example (Tracing it)."," On our digraph from ",[415,6906,6908],{"className":6907},[418],[415,6909,6911],{"className":6910,"ariaHidden":423},[422],[415,6912,6914,6917],{"className":6913},[427],[415,6915],{"className":6916,"style":1020},[431],[415,6918,1083],{"className":6919},[436,437],", the DP rows settle quickly. Row 1\nrelaxes the source edges: ",[415,6922,6924],{"className":6923},[418],[415,6925,6927,6951],{"className":6926,"ariaHidden":423},[422],[415,6928,6930,6933,6936,6939,6942,6945,6948],{"className":6929},[427],[415,6931],{"className":6932,"style":1163},[431],[415,6934,385],{"className":6935},[436,437],[415,6937,1131],{"className":6938},[436],[415,6940,1173],{"className":6941},[436,437],[415,6943],{"className":6944,"style":443},[442],[415,6946,448],{"className":6947},[447],[415,6949],{"className":6950,"style":443},[442],[415,6952,6954,6957],{"className":6953},[427],[415,6955],{"className":6956,"style":1294},[431],[415,6958,6960],{"className":6959},[436],"10",[415,6962,6964],{"className":6963},[418],[415,6965,6967,6991],{"className":6966,"ariaHidden":423},[422],[415,6968,6970,6973,6976,6979,6982,6985,6988],{"className":6969},[427],[415,6971],{"className":6972,"style":1163},[431],[415,6974,2342],{"className":6975},[436,437],[415,6977,1131],{"className":6978},[436],[415,6980,1173],{"className":6981},[436,437],[415,6983],{"className":6984,"style":443},[442],[415,6986,448],{"className":6987},[447],[415,6989],{"className":6990,"style":443},[442],[415,6992,6994,6997],{"className":6993},[427],[415,6995],{"className":6996,"style":1294},[431],[415,6998,1690],{"className":6999},[436],". Row 2 propagates one edge\nfurther, setting ",[415,7002,7004],{"className":7003},[418],[415,7005,7007,7031,7055,7073],{"className":7006,"ariaHidden":423},[422],[415,7008,7010,7013,7016,7019,7022,7025,7028],{"className":7009},[427],[415,7011],{"className":7012,"style":1163},[431],[415,7014,4648],{"className":7015},[436,437],[415,7017,1131],{"className":7018},[436],[415,7020,1173],{"className":7021},[436,437],[415,7023],{"className":7024,"style":443},[442],[415,7026,448],{"className":7027},[447],[415,7029],{"className":7030,"style":443},[442],[415,7032,7034,7037,7040,7043,7046,7049,7052],{"className":7033},[427],[415,7035],{"className":7036,"style":5447},[431],[415,7038,385],{"className":7039},[436,437],[415,7041,1131],{"className":7042},[436],[415,7044,1173],{"className":7045},[436,437],[415,7047],{"className":7048,"style":467},[442],[415,7050,1505],{"className":7051},[902],[415,7053],{"className":7054,"style":467},[442],[415,7056,7058,7061,7064,7067,7070],{"className":7057},[427],[415,7059],{"className":7060,"style":1294},[431],[415,7062,665],{"className":7063},[436],[415,7065],{"className":7066,"style":443},[442],[415,7068,448],{"className":7069},[447],[415,7071],{"className":7072,"style":443},[442],[415,7074,7076,7079],{"className":7075},[427],[415,7077],{"className":7078,"style":1294},[431],[415,7080,7082],{"className":7081},[436],"11",[415,7084,7086],{"className":7085},[418],[415,7087,7089,7113,7137,7155],{"className":7088,"ariaHidden":423},[422],[415,7090,7092,7095,7098,7101,7104,7107,7110],{"className":7091},[427],[415,7093],{"className":7094,"style":1163},[431],[415,7096,4667],{"className":7097},[436,437],[415,7099,1131],{"className":7100},[436],[415,7102,1173],{"className":7103},[436,437],[415,7105],{"className":7106,"style":443},[442],[415,7108,448],{"className":7109},[447],[415,7111],{"className":7112,"style":443},[442],[415,7114,7116,7119,7122,7125,7128,7131,7134],{"className":7115},[427],[415,7117],{"className":7118,"style":5447},[431],[415,7120,2342],{"className":7121},[436,437],[415,7123,1131],{"className":7124},[436],[415,7126,1173],{"className":7127},[436,437],[415,7129],{"className":7130,"style":467},[442],[415,7132,1505],{"className":7133},[902],[415,7135],{"className":7136,"style":467},[442],[415,7138,7140,7143,7146,7149,7152],{"className":7139},[427],[415,7141],{"className":7142,"style":1294},[431],[415,7144,2361],{"className":7145},[436],[415,7147],{"className":7148,"style":443},[442],[415,7150,448],{"className":7151},[447],[415,7153],{"className":7154,"style":443},[442],[415,7156,7158,7161],{"className":7157},[427],[415,7159],{"className":7160,"style":1294},[431],[415,7162,1901],{"className":7163},[436],", and tests the\nnegative edge ",[415,7166,7168],{"className":7167},[418],[415,7169,7171,7189],{"className":7170,"ariaHidden":423},[422],[415,7172,7174,7177,7180,7183,7186],{"className":7173},[427],[415,7175],{"className":7176,"style":1020},[431],[415,7178,385],{"className":7179},[436,437],[415,7181],{"className":7182,"style":443},[442],[415,7184,2168],{"className":7185},[447],[415,7187],{"className":7188,"style":443},[442],[415,7190,7192,7195],{"className":7191},[427],[415,7193],{"className":7194,"style":1163},[431],[415,7196,2342],{"className":7197},[436,437],", but ",[415,7200,7202],{"className":7201},[418],[415,7203,7205,7223,7250,7268],{"className":7204,"ariaHidden":423},[422],[415,7206,7208,7211,7214,7217,7220],{"className":7207},[427],[415,7209],{"className":7210,"style":1704},[431],[415,7212,6960],{"className":7213},[436],[415,7215],{"className":7216,"style":467},[442],[415,7218,1505],{"className":7219},[902],[415,7221],{"className":7222,"style":467},[442],[415,7224,7226,7229,7232,7235,7238,7241,7244,7247],{"className":7225},[427],[415,7227],{"className":7228,"style":458},[431],[415,7230,463],{"className":7231},[462],[415,7233,903],{"className":7234},[436],[415,7236,2361],{"className":7237},[436],[415,7239,487],{"className":7240},[486],[415,7242],{"className":7243,"style":443},[442],[415,7245,448],{"className":7246},[447],[415,7248],{"className":7249,"style":443},[442],[415,7251,7253,7256,7259,7262,7265],{"className":7252},[427],[415,7254],{"className":7255,"style":1741},[431],[415,7257,1745],{"className":7258},[436],[415,7260],{"className":7261,"style":443},[442],[415,7263,3259],{"className":7264},[447],[415,7266],{"className":7267,"style":443},[442],[415,7269,7271,7274],{"className":7270},[427],[415,7272],{"className":7273,"style":1294},[431],[415,7275,1690],{"className":7276},[436],", so ",[415,7279,7281],{"className":7280},[418],[415,7282,7284],{"className":7283,"ariaHidden":423},[422],[415,7285,7287,7290],{"className":7286},[427],[415,7288],{"className":7289,"style":1163},[431],[415,7291,2342],{"className":7292},[436,437]," keeps its cheaper\ndirect route. Row 3 settles ",[415,7295,7297],{"className":7296},[418],[415,7298,7300,7324,7348,7366],{"className":7299,"ariaHidden":423},[422],[415,7301,7303,7306,7309,7312,7315,7318,7321],{"className":7302},[427],[415,7304],{"className":7305,"style":1163},[431],[415,7307,4667],{"className":7308},[436,437],[415,7310,1131],{"className":7311},[436],[415,7313,1173],{"className":7314},[436,437],[415,7316],{"className":7317,"style":443},[442],[415,7319,448],{"className":7320},[447],[415,7322],{"className":7323,"style":443},[442],[415,7325,7327,7330,7333,7336,7339,7342,7345],{"className":7326},[427],[415,7328],{"className":7329,"style":5447},[431],[415,7331,4648],{"className":7332},[436,437],[415,7334,1131],{"className":7335},[436],[415,7337,1173],{"className":7338},[436,437],[415,7340],{"className":7341,"style":467},[442],[415,7343,1505],{"className":7344},[902],[415,7346],{"className":7347,"style":467},[442],[415,7349,7351,7354,7357,7360,7363],{"className":7350},[427],[415,7352],{"className":7353,"style":1294},[431],[415,7355,4252],{"className":7356},[436],[415,7358],{"className":7359,"style":443},[442],[415,7361,448],{"className":7362},[447],[415,7364],{"className":7365,"style":443},[442],[415,7367,7369,7372],{"className":7368},[427],[415,7370],{"className":7371,"style":1294},[431],[415,7373,7375],{"className":7374},[436],"15"," vs. the existing ",[415,7378,7380],{"className":7379},[418],[415,7381,7383],{"className":7382,"ariaHidden":423},[422],[415,7384,7386,7389],{"className":7385},[427],[415,7387],{"className":7388,"style":1294},[431],[415,7390,1901],{"className":7391},[436],", so no\nchange. The table has converged in fewer than ",[415,7394,7396],{"className":7395},[418],[415,7397,7399,7417,7435],{"className":7398,"ariaHidden":423},[422],[415,7400,7402,7405,7408,7411,7414],{"className":7401},[427],[415,7403],{"className":7404,"style":5374},[431],[415,7406,5287],{"className":7407},[436,437],[415,7409],{"className":7410,"style":467},[442],[415,7412,903],{"className":7413},[902],[415,7415],{"className":7416,"style":467},[442],[415,7418,7420,7423,7426,7429,7432],{"className":7419},[427],[415,7421],{"className":7422,"style":1294},[431],[415,7424,665],{"className":7425},[436],[415,7427],{"className":7428,"style":443},[442],[415,7430,448],{"className":7431},[447],[415,7433],{"className":7434,"style":443},[442],[415,7436,7438,7441],{"className":7437},[427],[415,7439],{"className":7440,"style":1294},[431],[415,7442,4252],{"className":7443},[436]," rounds; a final pass\nchanges nothing, certifying no reachable negative cycle.",[381,7446,7447,7448,7484,7485,7500,7501,7516,7517,7532,7533,7584],{},"Laid out as the DP table ",[415,7449,7451],{"className":7450},[418],[415,7452,7454],{"className":7453,"ariaHidden":423},[422],[415,7455,7457,7460,7466,7469,7472,7475,7478,7481],{"className":7456},[427],[415,7458],{"className":7459,"style":458},[431],[415,7461,7463],{"className":7462},[436,2114],[415,7464,5181],{"className":7465},[436],[415,7467,6577],{"className":7468},[462],[415,7470,727],{"className":7471,"style":726},[436,437],[415,7473,473],{"className":7474},[472],[415,7476],{"className":7477,"style":477},[442],[415,7479,523],{"className":7480,"style":522},[436,437],[415,7482,6585],{"className":7483},[486],", the layering is plain: row ",[415,7486,7488],{"className":7487},[418],[415,7489,7491],{"className":7490,"ariaHidden":423},[422],[415,7492,7494,7497],{"className":7493},[427],[415,7495],{"className":7496,"style":1163},[431],[415,7498,727],{"className":7499,"style":726},[436,437]," holds\nthe cheapest walk of at most ",[415,7502,7504],{"className":7503},[418],[415,7505,7507],{"className":7506,"ariaHidden":423},[422],[415,7508,7510,7513],{"className":7509},[427],[415,7511],{"className":7512,"style":1163},[431],[415,7514,727],{"className":7515,"style":726},[436,437]," edges, each row computed from the one above by a\nsingle full pass of relaxation. The entries that improved on each row are shaded;\nthe table stops changing after row ",[415,7518,7520],{"className":7519},[418],[415,7521,7523],{"className":7522,"ariaHidden":423},[422],[415,7524,7526,7529],{"className":7525},[427],[415,7527],{"className":7528,"style":1294},[431],[415,7530,2361],{"className":7531},[436],", well within the ",[415,7534,7536],{"className":7535},[418],[415,7537,7539,7557,7575],{"className":7538,"ariaHidden":423},[422],[415,7540,7542,7545,7548,7551,7554],{"className":7541},[427],[415,7543],{"className":7544,"style":5374},[431],[415,7546,5287],{"className":7547},[436,437],[415,7549],{"className":7550,"style":467},[442],[415,7552,903],{"className":7553},[902],[415,7555],{"className":7556,"style":467},[442],[415,7558,7560,7563,7566,7569,7572],{"className":7559},[427],[415,7561],{"className":7562,"style":1294},[431],[415,7564,665],{"className":7565},[436],[415,7567],{"className":7568,"style":443},[442],[415,7570,448],{"className":7571},[447],[415,7573],{"className":7574,"style":443},[442],[415,7576,7578,7581],{"className":7577},[427],[415,7579],{"className":7580,"style":1294},[431],[415,7582,4252],{"className":7583},[436]," bound:",[1841,7586,7588,7860],{"className":7587},[1844,1845],[1847,7589,7593],{"xmlns":1849,"width":7590,"height":7591,"viewBox":7592},"231.743","185.697","-75 -75 173.807 139.273",[1854,7594,7595,7602,7609,7616,7623,7630,7637,7644,7651,7658,7665,7676,7679,7686,7689,7695,7698,7704,7707,7713,7716,7722,7734,7746,7749,7755,7758,7764,7767,7773,7776,7782,7785,7791,7803,7815,7818,7824,7827,7833,7836,7842,7845,7851,7854],{"stroke":1856,"style":1857},[1854,7596,7598],{"transform":7597},"translate(-2.54 3.125)",[1866,7599],{"d":7600,"fill":1856,"stroke":1856,"className":7601,"style":1888},"M-59.242-65.983Q-59.242-66.036-59.233-66.062L-57.928-71.283Q-57.893-71.433-57.893-71.507Q-57.893-71.644-58.460-71.644Q-58.561-71.644-58.561-71.762Q-58.561-71.819-58.530-71.890Q-58.500-71.960-58.434-71.960L-57.212-72.057L-57.172-72.057Q-57.133-72.039-57.113-72.010Q-57.093-71.982-57.093-71.951L-57.093-71.916L-58.021-68.207Q-57.757-68.317-57.568-68.488Q-57.379-68.660-56.975-69.060Q-56.570-69.459-56.300-69.624Q-56.030-69.789-55.678-69.789Q-55.415-69.789-55.241-69.611Q-55.067-69.433-55.067-69.169Q-55.067-68.928-55.215-68.748Q-55.362-68.567-55.599-68.567Q-55.740-68.567-55.845-68.660Q-55.951-68.752-55.951-68.897Q-55.951-69.099-55.795-69.255Q-55.639-69.411-55.437-69.411Q-55.524-69.530-55.696-69.530Q-56.021-69.530-56.331-69.303Q-56.641-69.077-57.074-68.649Q-57.506-68.220-57.730-68.080Q-57.190-68.018-56.794-67.803Q-56.399-67.587-56.399-67.126Q-56.399-67.042-56.434-66.884Q-56.496-66.616-56.509-66.388Q-56.509-65.975-56.228-65.975Q-55.907-65.975-55.729-66.328Q-55.551-66.682-55.445-67.126Q-55.437-67.157-55.412-67.181Q-55.388-67.205-55.357-67.205L-55.248-67.205Q-55.204-67.205-55.182-67.172Q-55.160-67.139-55.160-67.091Q-55.300-66.533-55.553-66.122Q-55.806-65.711-56.245-65.711Q-56.641-65.711-56.893-65.972Q-57.146-66.234-57.146-66.621Q-57.146-66.717-57.111-66.919Q-57.084-67.012-57.084-67.108Q-57.084-67.341-57.245-67.499Q-57.405-67.658-57.649-67.737Q-57.893-67.816-58.108-67.838L-58.570-66.019Q-58.614-65.882-58.717-65.797Q-58.820-65.711-58.957-65.711Q-59.084-65.711-59.163-65.786Q-59.242-65.860-59.242-65.983",[1870],[1854,7603,7605],{"transform":7604},"translate(26.298 1.937)",[1866,7606],{"d":7607,"fill":1856,"stroke":1856,"className":7608,"style":1888},"M-58.895-66.361Q-58.675-65.975-57.919-65.975Q-57.621-65.975-57.326-66.076Q-57.032-66.177-56.838-66.390Q-56.645-66.603-56.645-66.911Q-56.645-67.139-56.823-67.286Q-57.001-67.434-57.247-67.486L-57.757-67.583Q-57.972-67.623-58.148-67.746Q-58.324-67.869-58.429-68.055Q-58.535-68.242-58.535-68.458Q-58.535-68.857-58.311-69.163Q-58.086-69.468-57.728-69.629Q-57.370-69.789-56.966-69.789Q-56.702-69.789-56.454-69.710Q-56.206-69.631-56.036-69.451Q-55.867-69.270-55.867-69.007Q-55.867-68.800-55.988-68.642Q-56.109-68.484-56.320-68.484Q-56.443-68.484-56.529-68.565Q-56.614-68.646-56.614-68.765Q-56.614-68.928-56.491-69.062Q-56.368-69.196-56.210-69.196Q-56.298-69.376-56.515-69.453Q-56.733-69.530-56.983-69.530Q-57.225-69.530-57.451-69.442Q-57.678-69.354-57.823-69.185Q-57.968-69.016-57.968-68.765Q-57.968-68.589-57.836-68.471Q-57.704-68.352-57.506-68.304L-57.001-68.207Q-56.614-68.128-56.346-67.858Q-56.078-67.587-56.078-67.205Q-56.078-66.875-56.267-66.555Q-56.456-66.234-56.733-66.036Q-57.234-65.711-57.928-65.711Q-58.240-65.711-58.541-65.797Q-58.842-65.882-59.047-66.080Q-59.251-66.278-59.251-66.585Q-59.251-66.836-59.108-67.020Q-58.965-67.205-58.724-67.205Q-58.570-67.205-58.471-67.113Q-58.372-67.020-58.372-66.875Q-58.372-66.665-58.524-66.513Q-58.675-66.361-58.895-66.361",[1870],[1854,7610,7612],{"transform":7611},"translate(54.455 1.937)",[1866,7613],{"d":7614,"fill":1856,"stroke":1856,"className":7615,"style":1888},"M-58.108-65.711Q-58.504-65.711-58.790-65.915Q-59.075-66.120-59.222-66.454Q-59.370-66.788-59.370-67.179Q-59.370-67.614-59.196-68.075Q-59.022-68.537-58.710-68.928Q-58.398-69.319-57.988-69.554Q-57.577-69.789-57.137-69.789Q-56.869-69.789-56.652-69.651Q-56.434-69.512-56.302-69.266Q-56.263-69.416-56.155-69.512Q-56.047-69.609-55.907-69.609Q-55.784-69.609-55.700-69.536Q-55.617-69.464-55.617-69.341Q-55.617-69.288-55.626-69.257L-56.245-66.766Q-56.302-66.568-56.302-66.370Q-56.302-65.975-56.039-65.975Q-55.753-65.975-55.619-66.298Q-55.485-66.621-55.366-67.126Q-55.357-67.157-55.333-67.181Q-55.309-67.205-55.274-67.205L-55.168-67.205Q-55.120-67.205-55.098-67.172Q-55.076-67.139-55.076-67.091Q-55.190-66.660-55.281-66.407Q-55.371-66.155-55.564-65.933Q-55.757-65.711-56.056-65.711Q-56.364-65.711-56.612-65.882Q-56.860-66.054-56.931-66.344Q-57.186-66.058-57.482-65.885Q-57.779-65.711-58.108-65.711M-58.091-65.975Q-57.761-65.975-57.451-66.216Q-57.142-66.458-56.931-66.774Q-56.922-66.783-56.922-66.801L-56.425-68.765Q-56.482-69.082-56.674-69.306Q-56.865-69.530-57.155-69.530Q-57.524-69.530-57.823-69.211Q-58.122-68.893-58.289-68.484Q-58.425-68.137-58.550-67.627Q-58.675-67.117-58.675-66.792Q-58.675-66.467-58.537-66.221Q-58.398-65.975-58.091-65.975",[1870],[1854,7617,7619],{"transform":7618},"translate(83.376 3.125)",[1866,7620],{"d":7621,"fill":1856,"stroke":1856,"className":7622,"style":1888},"M-58.108-65.711Q-58.684-65.711-59.005-66.142Q-59.326-66.572-59.326-67.152Q-59.326-67.557-59.242-67.785L-58.363-71.283Q-58.328-71.433-58.328-71.507Q-58.328-71.644-58.895-71.644Q-58.992-71.644-58.992-71.762Q-58.992-71.819-58.961-71.890Q-58.930-71.960-58.864-71.960L-57.643-72.057Q-57.590-72.057-57.557-72.028Q-57.524-71.999-57.524-71.951L-57.524-71.916L-58.183-69.306Q-57.660-69.789-57.137-69.789Q-56.751-69.789-56.460-69.585Q-56.170-69.380-56.023-69.046Q-55.876-68.712-55.876-68.321Q-55.876-67.737-56.179-67.128Q-56.482-66.520-57.003-66.115Q-57.524-65.711-58.108-65.711M-58.091-65.975Q-57.722-65.975-57.418-66.298Q-57.115-66.621-56.957-67.016Q-56.812-67.372-56.691-67.880Q-56.570-68.387-56.570-68.708Q-56.570-69.033-56.715-69.281Q-56.860-69.530-57.155-69.530Q-57.757-69.530-58.328-68.730L-58.570-67.737Q-58.715-67.113-58.715-66.849Q-58.715-66.506-58.563-66.240Q-58.412-65.975-58.091-65.975",[1870],[1854,7624,7626],{"transform":7625},"translate(111.809 1.937)",[1866,7627],{"d":7628,"fill":1856,"stroke":1856,"className":7629,"style":1888},"M-58.640-66.919Q-58.640-66.524-58.427-66.249Q-58.214-65.975-57.832-65.975Q-57.287-65.975-56.781-66.210Q-56.276-66.445-55.959-66.867Q-55.938-66.902-55.876-66.902Q-55.819-66.902-55.773-66.851Q-55.727-66.801-55.727-66.748Q-55.727-66.713-55.753-66.687Q-56.100-66.212-56.663-65.961Q-57.225-65.711-57.849-65.711Q-58.280-65.711-58.629-65.913Q-58.979-66.115-59.170-66.471Q-59.361-66.827-59.361-67.253Q-59.361-67.715-59.159-68.172Q-58.957-68.629-58.601-68.998Q-58.245-69.367-57.801-69.578Q-57.357-69.789-56.887-69.789Q-56.619-69.789-56.370-69.708Q-56.122-69.626-55.955-69.448Q-55.788-69.270-55.788-69.007Q-55.788-68.770-55.938-68.592Q-56.087-68.414-56.320-68.414Q-56.460-68.414-56.566-68.508Q-56.671-68.603-56.671-68.748Q-56.671-68.950-56.524-69.104Q-56.377-69.257-56.175-69.257Q-56.280-69.398-56.485-69.464Q-56.689-69.530-56.896-69.530Q-57.432-69.530-57.829-69.101Q-58.227-68.673-58.434-68.053Q-58.640-67.434-58.640-66.919",[1870],[1854,7631,7633],{"transform":7632},"translate(140.594 2.768)",[1866,7634],{"d":7635,"fill":1856,"stroke":1856,"className":7636,"style":1888},"M-59.146-66.550Q-59.146-66.682-59.119-66.801L-58.469-69.376L-59.414-69.376Q-59.523-69.376-59.523-69.495Q-59.523-69.556-59.490-69.624Q-59.458-69.692-59.396-69.692L-58.398-69.692L-58.038-71.129Q-58.003-71.270-57.891-71.358Q-57.779-71.446-57.643-71.446Q-57.520-71.446-57.436-71.371Q-57.353-71.296-57.353-71.178Q-57.353-71.121-57.361-71.094L-57.713-69.692L-56.786-69.692Q-56.737-69.692-56.709-69.659Q-56.680-69.626-56.680-69.583Q-56.680-69.517-56.713-69.446Q-56.746-69.376-56.803-69.376L-57.788-69.376L-58.442-66.766Q-58.495-66.563-58.495-66.370Q-58.495-65.975-58.236-65.975Q-57.955-65.975-57.724-66.155Q-57.493-66.335-57.322-66.607Q-57.150-66.880-57.049-67.144Q-57.041-67.170-57.019-67.187Q-56.997-67.205-56.966-67.205L-56.860-67.205Q-56.812-67.205-56.790-67.172Q-56.768-67.139-56.768-67.091Q-56.909-66.748-57.120-66.434Q-57.331-66.120-57.616-65.915Q-57.902-65.711-58.253-65.711Q-58.504-65.711-58.706-65.816Q-58.908-65.922-59.027-66.113Q-59.146-66.304-59.146-66.550",[1870],[1854,7638,7640],{"transform":7639},"translate(-2.312 31.353)",[1866,7641],{"d":7642,"fill":1856,"stroke":1856,"className":7643,"style":1888},"M-57.418-65.614Q-58.543-65.614-58.957-66.511Q-59.370-67.407-59.370-68.682Q-59.370-69.455-59.220-70.154Q-59.071-70.853-58.636-71.329Q-58.201-71.806-57.418-71.806Q-56.641-71.806-56.206-71.327Q-55.771-70.848-55.621-70.152Q-55.472-69.455-55.472-68.682Q-55.472-67.403-55.885-66.509Q-56.298-65.614-57.418-65.614M-57.418-65.874Q-56.900-65.874-56.649-66.385Q-56.399-66.897-56.342-67.508Q-56.285-68.119-56.285-68.827Q-56.285-69.512-56.342-70.072Q-56.399-70.633-56.652-71.090Q-56.904-71.547-57.418-71.547Q-57.823-71.547-58.060-71.270Q-58.297-70.993-58.405-70.552Q-58.513-70.110-58.537-69.717Q-58.561-69.323-58.561-68.827Q-58.561-68.321-58.537-67.893Q-58.513-67.464-58.405-66.981Q-58.297-66.498-58.058-66.186Q-57.818-65.874-57.418-65.874",[1870],[1854,7645,7647],{"transform":7646},"translate(-2.312 59.805)",[1866,7648],{"d":7649,"fill":1856,"stroke":1856,"className":7650,"style":1888},"M-55.823-65.812L-58.855-65.812L-58.855-66.128Q-57.704-66.128-57.704-66.423L-57.704-71.147Q-58.192-70.914-58.913-70.914L-58.913-71.230Q-57.783-71.230-57.221-71.806L-57.076-71.806Q-57.041-71.806-57.008-71.773Q-56.975-71.740-56.975-71.705L-56.975-66.423Q-56.975-66.128-55.823-66.128",[1870],[1854,7652,7654],{"transform":7653},"translate(-2.312 88.258)",[1866,7655],{"d":7656,"fill":1856,"stroke":1856,"className":7657,"style":1888},"M-55.823-65.812L-59.273-65.812L-59.273-66.045Q-59.273-66.058-59.242-66.089L-57.788-67.666Q-57.322-68.163-57.069-68.468Q-56.816-68.774-56.625-69.185Q-56.434-69.596-56.434-70.035Q-56.434-70.624-56.757-71.057Q-57.080-71.490-57.660-71.490Q-57.924-71.490-58.170-71.380Q-58.416-71.270-58.592-71.083Q-58.768-70.896-58.864-70.646L-58.785-70.646Q-58.583-70.646-58.440-70.510Q-58.297-70.374-58.297-70.158Q-58.297-69.952-58.440-69.813Q-58.583-69.675-58.785-69.675Q-58.987-69.675-59.130-69.818Q-59.273-69.960-59.273-70.158Q-59.273-70.620-59.036-70.993Q-58.798-71.367-58.398-71.586Q-57.999-71.806-57.550-71.806Q-57.027-71.806-56.573-71.591Q-56.118-71.375-55.845-70.976Q-55.573-70.576-55.573-70.035Q-55.573-69.640-55.744-69.286Q-55.916-68.932-56.181-68.653Q-56.447-68.374-56.898-67.989Q-57.348-67.605-57.427-67.530L-58.451-66.568L-57.634-66.568Q-56.983-66.568-56.546-66.579Q-56.109-66.590-56.078-66.612Q-56.008-66.695-55.953-66.935Q-55.898-67.174-55.858-67.442L-55.573-67.442",[1870],[1854,7659,7661],{"transform":7660},"translate(-2.312 116.711)",[1866,7662],{"d":7663,"fill":1856,"stroke":1856,"className":7664,"style":1888},"M-58.829-66.533L-58.873-66.533Q-58.671-66.216-58.284-66.058Q-57.897-65.900-57.471-65.900Q-56.935-65.900-56.696-66.335Q-56.456-66.770-56.456-67.350Q-56.456-67.930-56.702-68.370Q-56.948-68.809-57.480-68.809L-58.100-68.809Q-58.126-68.809-58.159-68.838Q-58.192-68.866-58.192-68.888L-58.192-68.989Q-58.192-69.020-58.163-69.044Q-58.135-69.068-58.100-69.068L-57.581-69.108Q-57.115-69.108-56.869-69.580Q-56.623-70.053-56.623-70.571Q-56.623-70.998-56.836-71.272Q-57.049-71.547-57.471-71.547Q-57.814-71.547-58.139-71.417Q-58.464-71.288-58.649-71.033L-58.623-71.033Q-58.420-71.033-58.284-70.892Q-58.148-70.751-58.148-70.554Q-58.148-70.356-58.282-70.222Q-58.416-70.088-58.614-70.088Q-58.816-70.088-58.954-70.222Q-59.093-70.356-59.093-70.554Q-59.093-71.143-58.590-71.474Q-58.086-71.806-57.471-71.806Q-57.093-71.806-56.691-71.666Q-56.289-71.525-56.021-71.246Q-55.753-70.967-55.753-70.571Q-55.753-70.022-56.107-69.585Q-56.460-69.147-57.001-68.963Q-56.610-68.884-56.265-68.660Q-55.920-68.436-55.709-68.095Q-55.498-67.754-55.498-67.359Q-55.498-66.977-55.661-66.654Q-55.823-66.331-56.115-66.095Q-56.408-65.860-56.755-65.737Q-57.102-65.614-57.471-65.614Q-57.919-65.614-58.350-65.775Q-58.781-65.935-59.062-66.262Q-59.343-66.590-59.343-67.047Q-59.343-67.262-59.196-67.405Q-59.049-67.548-58.829-67.548Q-58.618-67.548-58.473-67.403Q-58.328-67.258-58.328-67.047Q-58.328-66.836-58.475-66.684Q-58.623-66.533-58.829-66.533",[1870],[1854,7666,7667,7670],{"fill":1988},[1866,7668],{"d":7669},"M-44.081-24.556h25.607v-25.607h-25.607Z",[1854,7671,7673],{"transform":7672},"translate(26.14 31.353)",[1866,7674],{"d":7642,"fill":1856,"stroke":1856,"className":7675,"style":1888},[1870],[1866,7677],{"fill":1860,"d":7678},"M-15.629-24.556H9.98v-25.607H-15.63Z",[1854,7680,7682],{"transform":7681},"translate(52.28 30.39)",[1866,7683],{"d":7684,"fill":1856,"stroke":1856,"className":7685,"style":1888},"M-57.454-65.711Q-57.840-65.711-58.170-65.882Q-58.500-66.054-58.732-66.350Q-58.965-66.647-59.091-67.014Q-59.216-67.381-59.216-67.754Q-59.216-68.137-59.086-68.506Q-58.957-68.875-58.721-69.165Q-58.486-69.455-58.157-69.622Q-57.827-69.789-57.427-69.789Q-57.115-69.789-56.810-69.701Q-56.504-69.613-56.239-69.459Q-55.973-69.306-55.744-69.101Q-55.516-68.897-55.318-68.646L-55.032-68.286Q-54.764-68.712-54.424-69.046Q-54.083-69.380-53.657-69.585Q-53.230-69.789-52.765-69.789Q-52.246-69.789-51.842-69.499Q-51.438-69.209-51.218-68.734Q-50.998-68.260-50.998-67.754Q-50.998-67.376-51.128-67.003Q-51.257-66.629-51.492-66.344Q-51.728-66.058-52.062-65.885Q-52.396-65.711-52.791-65.711Q-53.393-65.711-53.936-66.025Q-54.479-66.339-54.896-66.858L-55.186-67.214Q-55.599-66.559-56.184-66.135Q-56.768-65.711-57.454-65.711M-57.533-66.054Q-56.364-66.054-55.454-67.557L-55.933-68.150Q-56.346-68.682-56.715-68.994Q-57.084-69.306-57.568-69.306Q-57.963-69.306-58.284-69.082Q-58.605-68.857-58.781-68.497Q-58.957-68.137-58.957-67.746Q-58.957-67.333-58.783-66.939Q-58.609-66.546-58.286-66.300Q-57.963-66.054-57.533-66.054M-54.764-67.943L-54.285-67.350Q-53.995-66.985-53.778-66.757Q-53.560-66.528-53.270-66.364Q-52.980-66.199-52.646-66.199Q-52.048-66.199-51.655-66.673Q-51.262-67.148-51.262-67.754Q-51.262-68.075-51.358-68.376Q-51.455-68.677-51.635-68.915Q-51.815-69.152-52.084-69.299Q-52.352-69.446-52.681-69.446Q-53.121-69.446-53.510-69.238Q-53.898-69.029-54.204-68.697Q-54.509-68.365-54.764-67.943",[1870],[1866,7687],{"fill":1860,"d":7688},"M12.824-24.556h25.607v-25.607H12.824Z",[1854,7690,7692],{"transform":7691},"translate(80.733 30.39)",[1866,7693],{"d":7684,"fill":1856,"stroke":1856,"className":7694,"style":1888},[1870],[1866,7696],{"fill":1860,"d":7697},"M41.277-24.556h25.607v-25.607H41.277Z",[1854,7699,7701],{"transform":7700},"translate(109.186 30.39)",[1866,7702],{"d":7684,"fill":1856,"stroke":1856,"className":7703,"style":1888},[1870],[1866,7705],{"fill":1860,"d":7706},"M69.73-24.556h25.607v-25.607H69.73Z",[1854,7708,7710],{"transform":7709},"translate(137.639 30.39)",[1866,7711],{"d":7684,"fill":1856,"stroke":1856,"className":7712,"style":1888},[1870],[1866,7714],{"fill":1860,"d":7715},"M-44.081 3.897h25.607V-21.71h-25.607Z",[1854,7717,7719],{"transform":7718},"translate(26.14 59.805)",[1866,7720],{"d":7642,"fill":1856,"stroke":1856,"className":7721,"style":1888},[1870],[1854,7723,7724,7727],{"fill":1988},[1866,7725],{"d":7726},"M-15.629 3.897H9.98V-21.71H-15.63Z",[1854,7728,7730],{"transform":7729},"translate(52.28 59.805)",[1866,7731],{"d":7732,"fill":1856,"stroke":1856,"className":7733,"style":1888},"M-55.823-65.812L-58.855-65.812L-58.855-66.128Q-57.704-66.128-57.704-66.423L-57.704-71.147Q-58.192-70.914-58.913-70.914L-58.913-71.230Q-57.783-71.230-57.221-71.806L-57.076-71.806Q-57.041-71.806-57.008-71.773Q-56.975-71.740-56.975-71.705L-56.975-66.423Q-56.975-66.128-55.823-66.128L-55.823-65.812M-52.800-65.614Q-53.925-65.614-54.338-66.511Q-54.751-67.407-54.751-68.682Q-54.751-69.455-54.602-70.154Q-54.452-70.853-54.017-71.329Q-53.582-71.806-52.800-71.806Q-52.022-71.806-51.587-71.327Q-51.152-70.848-51.002-70.152Q-50.853-69.455-50.853-68.682Q-50.853-67.403-51.266-66.509Q-51.679-65.614-52.800-65.614M-52.800-65.874Q-52.281-65.874-52.031-66.385Q-51.780-66.897-51.723-67.508Q-51.666-68.119-51.666-68.827Q-51.666-69.512-51.723-70.072Q-51.780-70.633-52.033-71.090Q-52.286-71.547-52.800-71.547Q-53.204-71.547-53.441-71.270Q-53.679-70.993-53.786-70.552Q-53.894-70.110-53.918-69.717Q-53.942-69.323-53.942-68.827Q-53.942-68.321-53.918-67.893Q-53.894-67.464-53.786-66.981Q-53.679-66.498-53.439-66.186Q-53.200-65.874-52.800-65.874",[1870],[1854,7735,7736,7739],{"fill":1988},[1866,7737],{"d":7738},"M12.824 3.897h25.607V-21.71H12.824Z",[1854,7740,7742],{"transform":7741},"translate(83.046 59.805)",[1866,7743],{"d":7744,"fill":1856,"stroke":1856,"className":7745,"style":1888},"M-58.904-66.818Q-58.763-66.405-58.403-66.153Q-58.042-65.900-57.607-65.900Q-57.155-65.900-56.889-66.153Q-56.623-66.405-56.520-66.790Q-56.417-67.174-56.417-67.631Q-56.417-69.332-57.326-69.332Q-57.647-69.332-57.876-69.238Q-58.104-69.143-58.234-69.024Q-58.363-68.906-58.475-68.767Q-58.587-68.629-58.623-68.620L-58.706-68.620Q-58.750-68.620-58.781-68.651Q-58.812-68.682-58.812-68.730L-58.812-71.727Q-58.812-71.758-58.776-71.782Q-58.741-71.806-58.715-71.806L-58.675-71.806Q-58.042-71.516-57.370-71.516Q-56.698-71.516-56.056-71.806L-56.030-71.806Q-55.999-71.806-55.966-71.784Q-55.933-71.762-55.933-71.727L-55.933-71.626Q-55.933-71.622-55.942-71.604Q-55.951-71.586-55.951-71.582Q-56.267-71.187-56.737-70.965Q-57.208-70.743-57.704-70.743Q-58.113-70.743-58.495-70.853L-58.495-69.134Q-58.038-69.591-57.326-69.591Q-56.816-69.591-56.417-69.310Q-56.017-69.029-55.795-68.574Q-55.573-68.119-55.573-67.614Q-55.573-67.064-55.852-66.605Q-56.131-66.146-56.597-65.880Q-57.063-65.614-57.607-65.614Q-58.047-65.614-58.431-65.841Q-58.816-66.067-59.044-66.447Q-59.273-66.827-59.273-67.271Q-59.273-67.464-59.141-67.596Q-59.009-67.728-58.812-67.728Q-58.680-67.728-58.576-67.669Q-58.473-67.609-58.414-67.506Q-58.355-67.403-58.355-67.271Q-58.355-67.073-58.482-66.941Q-58.609-66.810-58.812-66.810Q-58.873-66.810-58.904-66.818",[1870],[1866,7747],{"fill":1860,"d":7748},"M41.277 3.897h25.607V-21.71H41.277Z",[1854,7750,7752],{"transform":7751},"translate(109.186 58.843)",[1866,7753],{"d":7684,"fill":1856,"stroke":1856,"className":7754,"style":1888},[1870],[1866,7756],{"fill":1860,"d":7757},"M69.73 3.897h25.607V-21.71H69.73Z",[1854,7759,7761],{"transform":7760},"translate(137.639 58.843)",[1866,7762],{"d":7684,"fill":1856,"stroke":1856,"className":7763,"style":1888},[1870],[1866,7765],{"fill":1860,"d":7766},"M-44.081 32.35h25.607V6.742h-25.607Z",[1854,7768,7770],{"transform":7769},"translate(26.14 88.258)",[1866,7771],{"d":7642,"fill":1856,"stroke":1856,"className":7772,"style":1888},[1870],[1866,7774],{"fill":1860,"d":7775},"M-15.629 32.35H9.98V6.742H-15.63Z",[1854,7777,7779],{"transform":7778},"translate(52.28 88.258)",[1866,7780],{"d":7732,"fill":1856,"stroke":1856,"className":7781,"style":1888},[1870],[1866,7783],{"fill":1860,"d":7784},"M12.824 32.35h25.607V6.742H12.824Z",[1854,7786,7788],{"transform":7787},"translate(83.046 88.258)",[1866,7789],{"d":7744,"fill":1856,"stroke":1856,"className":7790,"style":1888},[1870],[1854,7792,7793,7796],{"fill":1988},[1866,7794],{"d":7795},"M41.277 32.35h25.607V6.742H41.277Z",[1854,7797,7799],{"transform":7798},"translate(109.186 88.258)",[1866,7800],{"d":7801,"fill":1856,"stroke":1856,"className":7802,"style":1888},"M-55.823-65.812L-58.855-65.812L-58.855-66.128Q-57.704-66.128-57.704-66.423L-57.704-71.147Q-58.192-70.914-58.913-70.914L-58.913-71.230Q-57.783-71.230-57.221-71.806L-57.076-71.806Q-57.041-71.806-57.008-71.773Q-56.975-71.740-56.975-71.705L-56.975-66.423Q-56.975-66.128-55.823-66.128L-55.823-65.812M-51.205-65.812L-54.237-65.812L-54.237-66.128Q-53.085-66.128-53.085-66.423L-53.085-71.147Q-53.573-70.914-54.294-70.914L-54.294-71.230Q-53.165-71.230-52.602-71.806L-52.457-71.806Q-52.422-71.806-52.389-71.773Q-52.356-71.740-52.356-71.705L-52.356-66.423Q-52.356-66.128-51.205-66.128",[1870],[1854,7804,7805,7808],{"fill":1988},[1866,7806],{"d":7807},"M69.73 32.35h25.607V6.742H69.73Z",[1854,7809,7811],{"transform":7810},"translate(139.951 88.258)",[1866,7812],{"d":7813,"fill":1856,"stroke":1856,"className":7814,"style":1888},"M-58.108-66.054Q-58.108-66.691-57.952-67.337Q-57.796-67.983-57.504-68.589Q-57.212-69.196-56.803-69.745L-55.986-70.853L-57.014-70.853Q-58.658-70.853-58.706-70.809Q-58.812-70.681-58.930-69.978L-59.216-69.978L-58.921-71.894L-58.631-71.894L-58.631-71.868Q-58.631-71.705-58.067-71.657Q-57.502-71.608-56.957-71.608L-55.239-71.608L-55.239-71.402Q-55.239-71.384-55.241-71.375Q-55.243-71.367-55.248-71.358L-56.535-69.609Q-56.786-69.257-56.933-68.831Q-57.080-68.405-57.146-67.941Q-57.212-67.478-57.225-67.067Q-57.238-66.656-57.238-66.054Q-57.238-65.874-57.364-65.744Q-57.489-65.614-57.669-65.614Q-57.788-65.614-57.891-65.671Q-57.994-65.729-58.051-65.832Q-58.108-65.935-58.108-66.054",[1870],[1866,7816],{"fill":1860,"d":7817},"M-44.081 60.803h25.607V35.195h-25.607Z",[1854,7819,7821],{"transform":7820},"translate(26.14 116.711)",[1866,7822],{"d":7642,"fill":1856,"stroke":1856,"className":7823,"style":1888},[1870],[1866,7825],{"fill":1860,"d":7826},"M-15.629 60.803H9.98V35.195H-15.63Z",[1854,7828,7830],{"transform":7829},"translate(52.28 116.711)",[1866,7831],{"d":7732,"fill":1856,"stroke":1856,"className":7832,"style":1888},[1870],[1866,7834],{"fill":1860,"d":7835},"M12.824 60.803h25.607V35.195H12.824Z",[1854,7837,7839],{"transform":7838},"translate(83.046 116.711)",[1866,7840],{"d":7744,"fill":1856,"stroke":1856,"className":7841,"style":1888},[1870],[1866,7843],{"fill":1860,"d":7844},"M41.277 60.803h25.607V35.195H41.277Z",[1854,7846,7848],{"transform":7847},"translate(109.186 116.711)",[1866,7849],{"d":7801,"fill":1856,"stroke":1856,"className":7850,"style":1888},[1870],[1866,7852],{"fill":1860,"d":7853},"M69.73 60.803h25.607V35.195H69.73Z",[1854,7855,7857],{"transform":7856},"translate(139.951 116.711)",[1866,7858],{"d":7813,"fill":1856,"stroke":1856,"className":7859,"style":1888},[1870],[2089,7861,7863,7864,7900,7901,7916,7917,7950,7951,7978,7979,1131],{"className":7862},[2092],"The Bellman-Ford DP table ",[415,7865,7867],{"className":7866},[418],[415,7868,7870],{"className":7869,"ariaHidden":423},[422],[415,7871,7873,7876,7882,7885,7888,7891,7894,7897],{"className":7872},[427],[415,7874],{"className":7875,"style":458},[431],[415,7877,7879],{"className":7878},[436,2114],[415,7880,5181],{"className":7881},[436],[415,7883,6577],{"className":7884},[462],[415,7886,727],{"className":7887,"style":726},[436,437],[415,7889,473],{"className":7890},[472],[415,7892],{"className":7893,"style":477},[442],[415,7895,523],{"className":7896,"style":522},[436,437],[415,7898,6585],{"className":7899},[486]," for the trace digraph. Row ",[415,7902,7904],{"className":7903},[418],[415,7905,7907],{"className":7906,"ariaHidden":423},[422],[415,7908,7910,7913],{"className":7909},[427],[415,7911],{"className":7912,"style":1163},[431],[415,7914,727],{"className":7915,"style":726},[436,437]," = cheapest ",[415,7918,7920],{"className":7919},[418],[415,7921,7923,7941],{"className":7922,"ariaHidden":423},[422],[415,7924,7926,7929,7932,7935,7938],{"className":7925},[427],[415,7927],{"className":7928,"style":1020},[431],[415,7930,1083],{"className":7931},[436,437],[415,7933],{"className":7934,"style":443},[442],[415,7936,3213],{"className":7937},[447,3212],[415,7939],{"className":7940,"style":443},[442],[415,7942,7944,7947],{"className":7943},[427],[415,7945],{"className":7946,"style":1020},[431],[415,7948,523],{"className":7949,"style":522},[436,437]," walk of ",[415,7952,7954],{"className":7953},[418],[415,7955,7957,7969],{"className":7956,"ariaHidden":423},[422],[415,7958,7960,7963,7966],{"className":7959},[427],[415,7961],{"className":7962,"style":1225},[431],[415,7964,1468],{"className":7965},[447],[415,7967],{"className":7968,"style":443},[442],[415,7970,7972,7975],{"className":7971},[427],[415,7973],{"className":7974,"style":1163},[431],[415,7976,727],{"className":7977,"style":726},[436,437]," edges; shaded cells improved that round. Rows freeze after ",[415,7980,7982],{"className":7981},[418],[415,7983,7985,8003],{"className":7984,"ariaHidden":423},[422],[415,7986,7988,7991,7994,7997,8000],{"className":7987},[427],[415,7989],{"className":7990,"style":1163},[431],[415,7992,727],{"className":7993,"style":726},[436,437],[415,7995],{"className":7996,"style":443},[442],[415,7998,448],{"className":7999},[447],[415,8001],{"className":8002,"style":443},[442],[415,8004,8006,8009],{"className":8005},[427],[415,8007],{"className":8008,"style":1294},[431],[415,8010,2361],{"className":8011},[436],[405,8013,8014],{"type":6898},[381,8015,8016,8019,8020,8023,8024,8027,8028,8070],{},[390,8017,8018],{},"Remark (A special case worth knowing)."," On a ",[390,8021,8022],{},"DAG"," there are no cycles at all, so\nwe can relax edges in ",[395,8025,8026],{},"topological order"," (from the previous lesson) in a\nsingle sweep, since every vertex's predecessors are finalized before it is reached.\nThis solves DAG shortest paths in ",[415,8029,8031],{"className":8030},[418],[415,8032,8034,8058],{"className":8033,"ariaHidden":423},[422],[415,8035,8037,8040,8043,8046,8049,8052,8055],{"className":8036},[427],[415,8038],{"className":8039,"style":458},[431],[415,8041,6799],{"className":8042},[436],[415,8044,463],{"className":8045},[462],[415,8047,468],{"className":8048,"style":467},[436,437],[415,8050],{"className":8051,"style":467},[442],[415,8053,1505],{"className":8054},[902],[415,8056],{"className":8057,"style":467},[442],[415,8059,8061,8064,8067],{"className":8060},[427],[415,8062],{"className":8063,"style":458},[431],[415,8065,482],{"className":8066,"style":481},[436,437],[415,8068,487],{"className":8069},[486],", and it works with negative\nweights too.",[400,8072,8074],{"id":8073},"floyd-warshall-all-pairs-at-once","Floyd-Warshall: all pairs at once",[381,8076,8077,8078,5318,8111,8113,8114,8170,8171,8193,8194,1131],{},"Sometimes we want ",[415,8079,8081],{"className":8080},[418],[415,8082,8084],{"className":8083,"ariaHidden":423},[422],[415,8085,8087,8090,8093,8096,8099,8102,8105,8108],{"className":8086},[427],[415,8088],{"className":8089,"style":458},[431],[415,8091,988],{"className":8092,"style":987},[436,437],[415,8094,463],{"className":8095},[462],[415,8097,512],{"className":8098},[436,437],[415,8100,473],{"className":8101},[472],[415,8103],{"className":8104,"style":477},[442],[415,8106,523],{"className":8107,"style":522},[436,437],[415,8109,487],{"className":8110},[486],[395,8112,1114],{}," pair of vertices, a full distance\nmatrix. Running Bellman-Ford from each source costs ",[415,8115,8117],{"className":8116},[418],[415,8118,8120],{"className":8119,"ariaHidden":423},[422],[415,8121,8123,8127,8130,8133,8164,8167],{"className":8122},[427],[415,8124],{"className":8125,"style":8126},[431],"height:1.0641em;vertical-align:-0.25em;",[415,8128,4879],{"className":8129,"style":4878},[436,437],[415,8131,463],{"className":8132},[462],[415,8134,8136,8139],{"className":8135},[436],[415,8137,468],{"className":8138,"style":467},[436,437],[415,8140,8142],{"className":8141},[582],[415,8143,8145],{"className":8144},[586],[415,8146,8148],{"className":8147},[591],[415,8149,8152],{"className":8150,"style":8151},[595],"height:0.8141em;",[415,8153,8155,8158],{"style":8154},"top:-3.063em;margin-right:0.05em;",[415,8156],{"className":8157,"style":604},[603],[415,8159,8161],{"className":8160},[608,609,610,611],[415,8162,2361],{"className":8163},[436,611],[415,8165,482],{"className":8166,"style":481},[436,437],[415,8168,487],{"className":8169},[486],", but\n",[415,8172,8174],{"className":8173},[418],[415,8175,8177],{"className":8176,"ariaHidden":423},[422],[415,8178,8180,8183],{"className":8179},[427],[415,8181],{"className":8182,"style":2641},[431],[415,8184,8186],{"className":8185},[2109,2110],[415,8187,8189],{"className":8188},[436,2114],[415,8190,8192],{"className":8191},[436],"Floyd-Warshall"," does better with a slick dynamic program over ",[395,8195,8196],{},"intermediate\nvertices",[381,8198,8199,8200,8237,8238,8320,8321,1024,8337,8354,8355,8400,8401,8416,8417,8498,8499,531,8501,8516,8517,8550,8551,8584],{},"Number the vertices ",[415,8201,8203],{"className":8202},[418],[415,8204,8206],{"className":8205,"ariaHidden":423},[422],[415,8207,8209,8213,8216,8219,8222,8225,8228,8231,8234],{"className":8208},[427],[415,8210],{"className":8211,"style":8212},[431],"height:0.8389em;vertical-align:-0.1944em;",[415,8214,665],{"className":8215},[436],[415,8217,473],{"className":8218},[472],[415,8220],{"className":8221,"style":477},[442],[415,8223,686],{"className":8224},[566],[415,8226],{"className":8227,"style":477},[442],[415,8229,473],{"className":8230},[472],[415,8232],{"className":8233,"style":477},[442],[415,8235,5287],{"className":8236},[436,437],". Let ",[415,8239,8241],{"className":8240},[418],[415,8242,8244],{"className":8243,"ariaHidden":423},[422],[415,8245,8247,8251],{"className":8246},[427],[415,8248],{"className":8249,"style":8250},[431],"height:1.4578em;vertical-align:-0.413em;",[415,8252,8254,8257],{"className":8253},[436],[415,8255,1173],{"className":8256},[436,437],[415,8258,8260],{"className":8259},[582],[415,8261,8263,8311],{"className":8262},[586,587],[415,8264,8266,8308],{"className":8265},[591],[415,8267,8270,8287],{"className":8268,"style":8269},[595],"height:1.0448em;",[415,8271,8273,8276],{"style":8272},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[415,8274],{"className":8275,"style":604},[603],[415,8277,8279],{"className":8278},[608,609,610,611],[415,8280,8282],{"className":8281},[436,611],[415,8283,8286],{"className":8284,"style":8285},[436,437,611],"margin-right:0.0572em;","ij",[415,8288,8290,8293],{"style":8289},"top:-3.2198em;margin-right:0.05em;",[415,8291],{"className":8292,"style":604},[603],[415,8294,8296],{"className":8295},[608,609,610,611],[415,8297,8299,8302,8305],{"className":8298},[436,611],[415,8300,463],{"className":8301},[462,611],[415,8303,727],{"className":8304,"style":726},[436,437,611],[415,8306,487],{"className":8307},[486,611],[415,8309,620],{"className":8310},[619],[415,8312,8314],{"className":8313},[591],[415,8315,8318],{"className":8316,"style":8317},[595],"height:0.413em;",[415,8319],{}," be the weight of the\nshortest path from ",[415,8322,8324],{"className":8323},[418],[415,8325,8327],{"className":8326,"ariaHidden":423},[422],[415,8328,8330,8334],{"className":8329},[427],[415,8331],{"className":8332,"style":8333},[431],"height:0.6595em;",[415,8335,823],{"className":8336},[436,437],[415,8338,8340],{"className":8339},[418],[415,8341,8343],{"className":8342,"ariaHidden":423},[422],[415,8344,8346,8350],{"className":8345},[427],[415,8347],{"className":8348,"style":8349},[431],"height:0.854em;vertical-align:-0.1944em;",[415,8351,8353],{"className":8352,"style":8285},[436,437],"j"," whose intermediate vertices all lie in\n",[415,8356,8358],{"className":8357},[418],[415,8359,8361],{"className":8360,"ariaHidden":423},[422],[415,8362,8364,8367],{"className":8363},[427],[415,8365],{"className":8366,"style":458},[431],[415,8368,8370,8373,8376,8379,8382,8385,8388,8391,8394,8397],{"className":8369},[566],[415,8371,2971],{"className":8372,"style":571},[462,570],[415,8374,665],{"className":8375},[436],[415,8377,473],{"className":8378},[472],[415,8380],{"className":8381,"style":477},[442],[415,8383,686],{"className":8384},[566],[415,8386],{"className":8387,"style":477},[442],[415,8389,473],{"className":8390},[472],[415,8392],{"className":8393,"style":477},[442],[415,8395,727],{"className":8396,"style":726},[436,437],[415,8398,2978],{"className":8399,"style":571},[486,570],". Either the shortest such path avoids vertex ",[415,8402,8404],{"className":8403},[418],[415,8405,8407],{"className":8406,"ariaHidden":423},[422],[415,8408,8410,8413],{"className":8409},[427],[415,8411],{"className":8412,"style":1163},[431],[415,8414,727],{"className":8415,"style":726},[436,437]," (then it\nis ",[415,8418,8420],{"className":8419},[418],[415,8421,8423],{"className":8422,"ariaHidden":423},[422],[415,8424,8426,8429],{"className":8425},[427],[415,8427],{"className":8428,"style":8250},[431],[415,8430,8432,8435],{"className":8431},[436],[415,8433,1173],{"className":8434},[436,437],[415,8436,8438],{"className":8437},[582],[415,8439,8441,8490],{"className":8440},[586,587],[415,8442,8444,8487],{"className":8443},[591],[415,8445,8447,8461],{"className":8446,"style":8269},[595],[415,8448,8449,8452],{"style":8272},[415,8450],{"className":8451,"style":604},[603],[415,8453,8455],{"className":8454},[608,609,610,611],[415,8456,8458],{"className":8457},[436,611],[415,8459,8286],{"className":8460,"style":8285},[436,437,611],[415,8462,8463,8466],{"style":8289},[415,8464],{"className":8465,"style":604},[603],[415,8467,8469],{"className":8468},[608,609,610,611],[415,8470,8472,8475,8478,8481,8484],{"className":8471},[436,611],[415,8473,463],{"className":8474},[462,611],[415,8476,727],{"className":8477,"style":726},[436,437,611],[415,8479,903],{"className":8480},[902,611],[415,8482,665],{"className":8483},[436,611],[415,8485,487],{"className":8486},[486,611],[415,8488,620],{"className":8489},[619],[415,8491,8493],{"className":8492},[591],[415,8494,8496],{"className":8495,"style":8317},[595],[415,8497],{},") or it routes ",[395,8500,1348],{},[415,8502,8504],{"className":8503},[418],[415,8505,8507],{"className":8506,"ariaHidden":423},[422],[415,8508,8510,8513],{"className":8509},[427],[415,8511],{"className":8512,"style":1163},[431],[415,8514,727],{"className":8515,"style":726},[436,437],", splitting into an ",[415,8518,8520],{"className":8519},[418],[415,8521,8523,8541],{"className":8522,"ariaHidden":423},[422],[415,8524,8526,8529,8532,8535,8538],{"className":8525},[427],[415,8527],{"className":8528,"style":8333},[431],[415,8530,823],{"className":8531},[436,437],[415,8533],{"className":8534,"style":443},[442],[415,8536,2168],{"className":8537},[447],[415,8539],{"className":8540,"style":443},[442],[415,8542,8544,8547],{"className":8543},[427],[415,8545],{"className":8546,"style":1163},[431],[415,8548,727],{"className":8549,"style":726},[436,437],"\npiece and a ",[415,8552,8554],{"className":8553},[418],[415,8555,8557,8575],{"className":8556,"ariaHidden":423},[422],[415,8558,8560,8563,8566,8569,8572],{"className":8559},[427],[415,8561],{"className":8562,"style":1163},[431],[415,8564,727],{"className":8565,"style":726},[436,437],[415,8567],{"className":8568,"style":443},[442],[415,8570,2168],{"className":8571},[447],[415,8573],{"className":8574,"style":443},[442],[415,8576,8578,8581],{"className":8577},[427],[415,8579],{"className":8580,"style":8349},[431],[415,8582,8353],{"className":8583,"style":8285},[436,437]," piece, each using only earlier intermediates:",[415,8586,8588],{"className":8587},[5162],[415,8589,8591],{"className":8590},[418],[415,8592,8594,8672],{"className":8593,"ariaHidden":423},[422],[415,8595,8597,8600,8663,8666,8669],{"className":8596},[427],[415,8598],{"className":8599,"style":8250},[431],[415,8601,8603,8606],{"className":8602},[436],[415,8604,1173],{"className":8605},[436,437],[415,8607,8609],{"className":8608},[582],[415,8610,8612,8655],{"className":8611},[586,587],[415,8613,8615,8652],{"className":8614},[591],[415,8616,8618,8632],{"className":8617,"style":8269},[595],[415,8619,8620,8623],{"style":8272},[415,8621],{"className":8622,"style":604},[603],[415,8624,8626],{"className":8625},[608,609,610,611],[415,8627,8629],{"className":8628},[436,611],[415,8630,8286],{"className":8631,"style":8285},[436,437,611],[415,8633,8634,8637],{"style":8289},[415,8635],{"className":8636,"style":604},[603],[415,8638,8640],{"className":8639},[608,609,610,611],[415,8641,8643,8646,8649],{"className":8642},[436,611],[415,8644,463],{"className":8645},[462,611],[415,8647,727],{"className":8648,"style":726},[436,437,611],[415,8650,487],{"className":8651},[486,611],[415,8653,620],{"className":8654},[619],[415,8656,8658],{"className":8657},[591],[415,8659,8661],{"className":8660,"style":8317},[595],[415,8662],{},[415,8664],{"className":8665,"style":443},[442],[415,8667,448],{"className":8668},[447],[415,8670],{"className":8671,"style":443},[442],[415,8673,8675,8678,8684,8688,8691,8938,8941],{"className":8674},[427],[415,8676],{"className":8677,"style":5623},[431],[415,8679,8681],{"className":8680},[787],[415,8682,5630],{"className":8683},[436,4917],[415,8685],{"className":8686,"style":8687},[442],"margin-right:-0.1667em;",[415,8689],{"className":8690,"style":477},[442],[415,8692,8694,8700,8769,8772,8775,8778,8850,8853,8856,8859,8932],{"className":8693},[566],[415,8695,8697],{"className":8696,"style":571},[462,570],[415,8698,463],{"className":8699},[5637,5638],[415,8701,8703,8706],{"className":8702},[436],[415,8704,1173],{"className":8705},[436,437],[415,8707,8709],{"className":8708},[582],[415,8710,8712,8761],{"className":8711},[586,587],[415,8713,8715,8758],{"className":8714},[591],[415,8716,8718,8732],{"className":8717,"style":8269},[595],[415,8719,8720,8723],{"style":8272},[415,8721],{"className":8722,"style":604},[603],[415,8724,8726],{"className":8725},[608,609,610,611],[415,8727,8729],{"className":8728},[436,611],[415,8730,8286],{"className":8731,"style":8285},[436,437,611],[415,8733,8734,8737],{"style":8289},[415,8735],{"className":8736,"style":604},[603],[415,8738,8740],{"className":8739},[608,609,610,611],[415,8741,8743,8746,8749,8752,8755],{"className":8742},[436,611],[415,8744,463],{"className":8745},[462,611],[415,8747,727],{"className":8748,"style":726},[436,437,611],[415,8750,903],{"className":8751},[902,611],[415,8753,665],{"className":8754},[436,611],[415,8756,487],{"className":8757},[486,611],[415,8759,620],{"className":8760},[619],[415,8762,8764],{"className":8763},[591],[415,8765,8767],{"className":8766,"style":8317},[595],[415,8768],{},[415,8770,473],{"className":8771},[472],[415,8773],{"className":8774,"style":443},[442],[415,8776],{"className":8777,"style":477},[442],[415,8779,8781,8784],{"className":8780},[436],[415,8782,1173],{"className":8783},[436,437],[415,8785,8787],{"className":8786},[582],[415,8788,8790,8841],{"className":8789},[586,587],[415,8791,8793,8838],{"className":8792},[591],[415,8794,8796,8812],{"className":8795,"style":8269},[595],[415,8797,8799,8802],{"style":8798},"top:-2.3987em;margin-left:0em;margin-right:0.05em;",[415,8800],{"className":8801,"style":604},[603],[415,8803,8805],{"className":8804},[608,609,610,611],[415,8806,8808],{"className":8807},[436,611],[415,8809,8811],{"className":8810,"style":726},[436,437,611],"ik",[415,8813,8814,8817],{"style":8289},[415,8815],{"className":8816,"style":604},[603],[415,8818,8820],{"className":8819},[608,609,610,611],[415,8821,8823,8826,8829,8832,8835],{"className":8822},[436,611],[415,8824,463],{"className":8825},[462,611],[415,8827,727],{"className":8828,"style":726},[436,437,611],[415,8830,903],{"className":8831},[902,611],[415,8833,665],{"className":8834},[436,611],[415,8836,487],{"className":8837},[486,611],[415,8839,620],{"className":8840},[619],[415,8842,8844],{"className":8843},[591],[415,8845,8848],{"className":8846,"style":8847},[595],"height:0.3013em;",[415,8849],{},[415,8851],{"className":8852,"style":467},[442],[415,8854,1505],{"className":8855},[902],[415,8857],{"className":8858,"style":467},[442],[415,8860,8862,8865],{"className":8861},[436],[415,8863,1173],{"className":8864},[436,437],[415,8866,8868],{"className":8867},[582],[415,8869,8871,8923],{"className":8870},[586,587],[415,8872,8874,8920],{"className":8873},[591],[415,8875,8877,8894],{"className":8876,"style":8269},[595],[415,8878,8879,8882],{"style":8798},[415,8880],{"className":8881,"style":604},[603],[415,8883,8885],{"className":8884},[608,609,610,611],[415,8886,8888,8891],{"className":8887},[436,611],[415,8889,727],{"className":8890,"style":726},[436,437,611],[415,8892,8353],{"className":8893,"style":8285},[436,437,611],[415,8895,8896,8899],{"style":8289},[415,8897],{"className":8898,"style":604},[603],[415,8900,8902],{"className":8901},[608,609,610,611],[415,8903,8905,8908,8911,8914,8917],{"className":8904},[436,611],[415,8906,463],{"className":8907},[462,611],[415,8909,727],{"className":8910,"style":726},[436,437,611],[415,8912,903],{"className":8913},[902,611],[415,8915,665],{"className":8916},[436,611],[415,8918,487],{"className":8919},[486,611],[415,8921,620],{"className":8922},[619],[415,8924,8926],{"className":8925},[591],[415,8927,8930],{"className":8928,"style":8929},[595],"height:0.4374em;",[415,8931],{},[415,8933,8935],{"className":8934,"style":571},[486,570],[415,8936,487],{"className":8937},[5637,5638],[415,8939],{"className":8940,"style":477},[442],[415,8942,1131],{"className":8943},[436],[381,8945,8946,8947,8998,8999,9032,9033,9083,9084,9142,9143,1024,9158,4709,9173,9224,9225,9331,9332,9347,9348,9363,9364,9471],{},"One round of the recurrence makes the mechanism concrete. Starting from\n",[415,8948,8950],{"className":8949},[418],[415,8951,8953],{"className":8952,"ariaHidden":423},[422],[415,8954,8956,8960],{"className":8955},[427],[415,8957],{"className":8958,"style":8959},[431],"height:0.888em;",[415,8961,8963,8966],{"className":8962},[436],[415,8964,1173],{"className":8965},[436,437],[415,8967,8969],{"className":8968},[582],[415,8970,8972],{"className":8971},[586],[415,8973,8975],{"className":8974},[591],[415,8976,8978],{"className":8977,"style":8959},[595],[415,8979,8980,8983],{"style":8154},[415,8981],{"className":8982,"style":604},[603],[415,8984,8986],{"className":8985},[608,609,610,611],[415,8987,8989,8992,8995],{"className":8988},[436,611],[415,8990,463],{"className":8991},[462,611],[415,8993,615],{"className":8994},[436,611],[415,8996,487],{"className":8997},[486,611]," — direct edges only — and admitting intermediates from ",[415,9000,9002],{"className":9001},[418],[415,9003,9005],{"className":9004,"ariaHidden":423},[422],[415,9006,9008,9011],{"className":9007},[427],[415,9009],{"className":9010,"style":458},[431],[415,9012,9014,9017,9020,9023,9026,9029],{"className":9013},[566],[415,9015,2971],{"className":9016,"style":571},[462,570],[415,9018,665],{"className":9019},[436],[415,9021,473],{"className":9022},[472],[415,9024],{"className":9025,"style":477},[442],[415,9027,2361],{"className":9028},[436],[415,9030,2978],{"className":9031,"style":571},[486,570],"\nproduces ",[415,9034,9036],{"className":9035},[418],[415,9037,9039],{"className":9038,"ariaHidden":423},[422],[415,9040,9042,9045],{"className":9041},[427],[415,9043],{"className":9044,"style":8959},[431],[415,9046,9048,9051],{"className":9047},[436],[415,9049,1173],{"className":9050},[436,437],[415,9052,9054],{"className":9053},[582],[415,9055,9057],{"className":9056},[586],[415,9058,9060],{"className":9059},[591],[415,9061,9063],{"className":9062,"style":8959},[595],[415,9064,9065,9068],{"style":8154},[415,9066],{"className":9067,"style":604},[603],[415,9069,9071],{"className":9070},[608,609,610,611],[415,9072,9074,9077,9080],{"className":9073},[436,611],[415,9075,463],{"className":9076},[462,611],[415,9078,2361],{"className":9079},[436,611],[415,9081,487],{"className":9082},[486,611],": the entry ",[415,9085,9087],{"className":9086},[418],[415,9088,9090],{"className":9089,"ariaHidden":423},[422],[415,9091,9093,9097],{"className":9092},[427],[415,9094],{"className":9095,"style":9096},[431],"height:0.8444em;vertical-align:-0.15em;",[415,9098,9100,9103],{"className":9099},[436],[415,9101,1173],{"className":9102},[436,437],[415,9104,9106],{"className":9105},[582],[415,9107,9109,9134],{"className":9108},[586,587],[415,9110,9112,9131],{"className":9111},[591],[415,9113,9115],{"className":9114,"style":596},[595],[415,9116,9118,9121],{"style":9117},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[415,9119],{"className":9120,"style":604},[603],[415,9122,9124],{"className":9123},[608,609,610,611],[415,9125,9127],{"className":9126},[436,611],[415,9128,9130],{"className":9129},[436,611],"13",[415,9132,620],{"className":9133},[619],[415,9135,9137],{"className":9136},[591],[415,9138,9140],{"className":9139,"style":627},[595],[415,9141],{}," drops from ",[415,9144,9146],{"className":9145},[418],[415,9147,9149],{"className":9148,"ariaHidden":423},[422],[415,9150,9152,9155],{"className":9151},[427],[415,9153],{"className":9154,"style":1294},[431],[415,9156,1745],{"className":9157},[436],[415,9159,9161],{"className":9160},[418],[415,9162,9164],{"className":9163,"ariaHidden":423},[422],[415,9165,9167,9170],{"className":9166},[427],[415,9168],{"className":9169,"style":1294},[431],[415,9171,1690],{"className":9172},[436],[415,9174,9176],{"className":9175},[418],[415,9177,9179,9197,9215],{"className":9178,"ariaHidden":423},[422],[415,9180,9182,9185,9188,9191,9194],{"className":9181},[427],[415,9183],{"className":9184,"style":1294},[431],[415,9186,665],{"className":9187},[436],[415,9189],{"className":9190,"style":443},[442],[415,9192,2168],{"className":9193},[447],[415,9195],{"className":9196,"style":443},[442],[415,9198,9200,9203,9206,9209,9212],{"className":9199},[427],[415,9201],{"className":9202,"style":1294},[431],[415,9204,2361],{"className":9205},[436],[415,9207],{"className":9208,"style":443},[442],[415,9210,2168],{"className":9211},[447],[415,9213],{"className":9214,"style":443},[442],[415,9216,9218,9221],{"className":9217},[427],[415,9219],{"className":9220,"style":1294},[431],[415,9222,1633],{"className":9223},[436],",\nand the once-unreachable ",[415,9226,9228],{"className":9227},[418],[415,9229,9231],{"className":9230,"ariaHidden":423},[422],[415,9232,9234,9237,9281,9284,9287],{"className":9233},[427],[415,9235],{"className":9236,"style":2641},[431],[415,9238,9240,9243],{"className":9239},[436],[415,9241,1173],{"className":9242},[436,437],[415,9244,9246],{"className":9245},[582],[415,9247,9249,9273],{"className":9248},[586,587],[415,9250,9252,9270],{"className":9251},[591],[415,9253,9255],{"className":9254,"style":596},[595],[415,9256,9257,9260],{"style":9117},[415,9258],{"className":9259,"style":604},[603],[415,9261,9263],{"className":9262},[608,609,610,611],[415,9264,9266],{"className":9265},[436,611],[415,9267,9269],{"className":9268},[436,611],"42",[415,9271,620],{"className":9272},[619],[415,9274,9276],{"className":9275},[591],[415,9277,9279],{"className":9278,"style":627},[595],[415,9280],{},[415,9282,473],{"className":9283},[472],[415,9285],{"className":9286,"style":477},[442],[415,9288,9290,9293],{"className":9289},[436],[415,9291,1173],{"className":9292},[436,437],[415,9294,9296],{"className":9295},[582],[415,9297,9299,9323],{"className":9298},[586,587],[415,9300,9302,9320],{"className":9301},[591],[415,9303,9305],{"className":9304,"style":596},[595],[415,9306,9307,9310],{"style":9117},[415,9308],{"className":9309,"style":604},[603],[415,9311,9313],{"className":9312},[608,609,610,611],[415,9314,9316],{"className":9315},[436,611],[415,9317,9319],{"className":9318},[436,611],"43",[415,9321,620],{"className":9322},[619],[415,9324,9326],{"className":9325},[591],[415,9327,9329],{"className":9328,"style":627},[595],[415,9330],{}," become finite by routing through vertex\n",[415,9333,9335],{"className":9334},[418],[415,9336,9338],{"className":9337,"ariaHidden":423},[422],[415,9339,9341,9344],{"className":9340},[427],[415,9342],{"className":9343,"style":1294},[431],[415,9345,665],{"className":9346},[436]," then ",[415,9349,9351],{"className":9350},[418],[415,9352,9354],{"className":9353,"ariaHidden":423},[422],[415,9355,9357,9360],{"className":9356},[427],[415,9358],{"className":9359,"style":1294},[431],[415,9361,2361],{"className":9362},[436],". The five matrices ",[415,9365,9367],{"className":9366},[418],[415,9368,9370],{"className":9369,"ariaHidden":423},[422],[415,9371,9373,9377,9415,9418,9421,9424,9427,9430,9433],{"className":9372},[427],[415,9374],{"className":9375,"style":9376},[431],"height:1.0824em;vertical-align:-0.1944em;",[415,9378,9380,9383],{"className":9379},[436],[415,9381,1173],{"className":9382},[436,437],[415,9384,9386],{"className":9385},[582],[415,9387,9389],{"className":9388},[586],[415,9390,9392],{"className":9391},[591],[415,9393,9395],{"className":9394,"style":8959},[595],[415,9396,9397,9400],{"style":8154},[415,9398],{"className":9399,"style":604},[603],[415,9401,9403],{"className":9402},[608,609,610,611],[415,9404,9406,9409,9412],{"className":9405},[436,611],[415,9407,463],{"className":9408},[462,611],[415,9410,615],{"className":9411},[436,611],[415,9413,487],{"className":9414},[486,611],[415,9416,473],{"className":9417},[472],[415,9419],{"className":9420,"style":477},[442],[415,9422,686],{"className":9423},[566],[415,9425],{"className":9426,"style":477},[442],[415,9428,473],{"className":9429},[472],[415,9431],{"className":9432,"style":477},[442],[415,9434,9436,9439],{"className":9435},[436],[415,9437,1173],{"className":9438},[436,437],[415,9440,9442],{"className":9441},[582],[415,9443,9445],{"className":9444},[586],[415,9446,9448],{"className":9447},[591],[415,9449,9451],{"className":9450,"style":8959},[595],[415,9452,9453,9456],{"style":8154},[415,9454],{"className":9455,"style":604},[603],[415,9457,9459],{"className":9458},[608,609,610,611],[415,9460,9462,9465,9468],{"className":9461},[436,611],[415,9463,463],{"className":9464},[462,611],[415,9466,4252],{"className":9467},[436,611],[415,9469,487],{"className":9470},[486,611]," below trace one round per\npermitted intermediate; the entries that improved in each round are shaded:",[1841,9473,9475,10941],{"className":9474},[1844,1845],[1847,9476,9480],{"xmlns":1849,"width":9477,"height":9478,"viewBox":9479},"498.987","264.727","-75 -75 374.240 198.545",[1854,9481,9482,9485,9492,9495,9502,9505,9512,9515,9522,9525,9528,9539,9542,9545,9556,9559,9562,9573,9576,9579,9590,9593,9597,9609,9612,9615,9627,9643,9654,9663,9672,9681,9689,9697,9705,9713,9720,9726,9732,9739,9745,9751,9757,9763,9769,9775,9781,9787,9793,9799,9805,9811,9829,9843,9851,9859,9867,9875,9883,9891,9899,9907,9910,9916,9922,9928,9934,9940,9946,9952,9958,9964,9970,9976,9982,9988,9995,10002,10008,10037,10051,10059,10067,10075,10083,10091,10099,10107,10115,10118,10124,10130,10136,10142,10148,10154,10160,10166,10172,10178,10184,10190,10196,10202,10208,10214,10252,10266,10274,10282,10290,10298,10306,10314,10322,10330,10333,10339,10345,10351,10358,10364,10370,10376,10382,10388,10394,10400,10406,10412,10418,10424,10430,10478,10492,10500,10508,10516,10524,10532,10540,10548,10556,10559,10565,10571,10577,10583,10589,10595,10601,10607,10613,10619,10625,10631,10637,10643,10649,10655,10716],{"stroke":1856,"style":1857},[1866,9483],{"fill":1860,"d":9484},"M95.068-52.153c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958Zm-9.959 0",[1854,9486,9488],{"transform":9487},"translate(131.735 -111.556)",[1866,9489],{"d":9490,"fill":1856,"stroke":1856,"className":9491,"style":1909},"M-45.293 61.658L-47.823 61.658L-47.823 61.378Q-46.855 61.378-46.855 61.169L-46.855 57.550Q-47.248 57.738-47.870 57.738L-47.870 57.457Q-47.453 57.457-47.089 57.356Q-46.725 57.256-46.469 57.010L-46.343 57.010Q-46.278 57.027-46.261 57.095L-46.261 61.169Q-46.261 61.378-45.293 61.378",[1870],[1866,9493],{"fill":1860,"d":9494},"M157.663-52.153c0-5.5-4.458-9.959-9.958-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.958-4.458 9.958-9.958Zm-9.958 0",[1854,9496,9498],{"transform":9497},"translate(194.33 -111.556)",[1866,9499],{"d":9500,"fill":1856,"stroke":1856,"className":9501,"style":1909},"M-45.293 61.658L-48.178 61.658L-48.178 61.456Q-48.178 61.426-48.151 61.398L-46.903 60.181Q-46.831 60.106-46.789 60.064Q-46.746 60.021-46.667 59.942Q-46.254 59.529-46.023 59.171Q-45.792 58.814-45.792 58.390Q-45.792 58.158-45.871 57.955Q-45.950 57.751-46.091 57.601Q-46.233 57.450-46.428 57.370Q-46.623 57.290-46.855 57.290Q-47.166 57.290-47.424 57.449Q-47.682 57.608-47.812 57.885L-47.792 57.885Q-47.624 57.885-47.517 57.996Q-47.409 58.107-47.409 58.271Q-47.409 58.428-47.518 58.541Q-47.628 58.654-47.792 58.654Q-47.952 58.654-48.065 58.541Q-48.178 58.428-48.178 58.271Q-48.178 57.895-47.970 57.608Q-47.761 57.321-47.426 57.165Q-47.091 57.010-46.736 57.010Q-46.312 57.010-45.932 57.168Q-45.553 57.327-45.319 57.644Q-45.085 57.960-45.085 58.390Q-45.085 58.701-45.225 58.970Q-45.365 59.238-45.570 59.443Q-45.775 59.648-46.138 59.930Q-46.500 60.212-46.609 60.308L-47.464 61.036L-46.821 61.036Q-46.558 61.036-46.269 61.034Q-45.980 61.033-45.762 61.024Q-45.543 61.015-45.526 60.998Q-45.464 60.933-45.427 60.766Q-45.389 60.598-45.351 60.356L-45.085 60.356",[1870],[1866,9503],{"fill":1860,"d":9504},"M157.663-6.629c0-5.5-4.458-9.958-9.958-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.958-4.459 9.958-9.959Zm-9.958 0",[1854,9506,9508],{"transform":9507},"translate(194.33 -66.031)",[1866,9509],{"d":9510,"fill":1856,"stroke":1856,"className":9511,"style":1909},"M-47.823 61.111Q-47.703 61.268-47.512 61.367Q-47.320 61.467-47.105 61.506Q-46.890 61.545-46.667 61.545Q-46.370 61.545-46.175 61.390Q-45.980 61.234-45.890 60.980Q-45.799 60.725-45.799 60.441Q-45.799 60.147-45.891 59.896Q-45.984 59.645-46.182 59.489Q-46.380 59.334-46.674 59.334L-47.190 59.334Q-47.218 59.334-47.243 59.308Q-47.269 59.283-47.269 59.259L-47.269 59.187Q-47.269 59.156-47.243 59.134Q-47.218 59.112-47.190 59.112L-46.749 59.081Q-46.387 59.081-46.167 58.724Q-45.946 58.366-45.946 57.977Q-45.946 57.649-46.141 57.445Q-46.336 57.242-46.667 57.242Q-46.954 57.242-47.207 57.326Q-47.460 57.409-47.624 57.597Q-47.477 57.597-47.377 57.712Q-47.276 57.826-47.276 57.977Q-47.276 58.127-47.382 58.237Q-47.488 58.346-47.645 58.346Q-47.806 58.346-47.915 58.237Q-48.024 58.127-48.024 57.977Q-48.024 57.652-47.816 57.433Q-47.607 57.215-47.291 57.112Q-46.975 57.010-46.667 57.010Q-46.349 57.010-46.021 57.114Q-45.693 57.218-45.466 57.440Q-45.239 57.662-45.239 57.977Q-45.239 58.411-45.526 58.736Q-45.813 59.060-46.247 59.207Q-45.936 59.272-45.656 59.438Q-45.375 59.604-45.198 59.862Q-45.020 60.120-45.020 60.441Q-45.020 60.851-45.264 61.161Q-45.509 61.470-45.890 61.634Q-46.271 61.798-46.667 61.798Q-47.036 61.798-47.394 61.685Q-47.751 61.573-47.995 61.323Q-48.240 61.074-48.240 60.704Q-48.240 60.533-48.123 60.421Q-48.007 60.308-47.836 60.308Q-47.727 60.308-47.636 60.359Q-47.546 60.410-47.491 60.503Q-47.436 60.595-47.436 60.704Q-47.436 60.872-47.549 60.991Q-47.662 61.111-47.823 61.111",[1870],[1866,9513],{"fill":1860,"d":9514},"M95.068-6.629c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0",[1854,9516,9518],{"transform":9517},"translate(131.735 -66.031)",[1866,9519],{"d":9520,"fill":1856,"stroke":1856,"className":9521,"style":1909},"M-46.302 60.510L-48.346 60.510L-48.346 60.229L-46.015 57.057Q-45.980 57.010-45.915 57.010L-45.779 57.010Q-45.734 57.010-45.707 57.037Q-45.680 57.064-45.680 57.109L-45.680 60.229L-44.917 60.229L-44.917 60.510L-45.680 60.510L-45.680 61.169Q-45.680 61.378-44.924 61.378L-44.924 61.658L-47.057 61.658L-47.057 61.378Q-46.302 61.378-46.302 61.169L-46.302 60.510M-46.254 57.785L-48.045 60.229L-46.254 60.229",[1870],[1866,9523],{"fill":1860,"d":9524},"M95.268-52.153h39.203",[1866,9526],{"d":9527},"m136.977-52.153-3.585-1.35 1.18 1.35-1.18 1.35Z",[1854,9529,9530,9533],{"fill":2436},[1866,9531],{"stroke":1860,"d":9532},"M126.366-62.312c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959m-9.959 0",[1854,9534,9536],{"transform":9535},"translate(163.033 -121.714)",[1866,9537],{"d":9510,"fill":1856,"stroke":1856,"className":9538,"style":1909},[1870],[1866,9540],{"fill":1860,"d":9541},"M147.705-41.995v22.133",[1866,9543],{"d":9544},"m147.705-17.357 1.35-3.584-1.35 1.179-1.35-1.18Z",[1854,9546,9547,9550],{"fill":2436},[1866,9548],{"stroke":1860,"d":9549},"M167.822-29.39c0-5.5-4.459-9.96-9.959-9.96s-9.958 4.46-9.958 9.96 4.459 9.958 9.958 9.958 9.959-4.459 9.959-9.959m-9.959 0",[1854,9551,9553],{"transform":9552},"translate(204.49 -88.793)",[1866,9554],{"d":9500,"fill":1856,"stroke":1856,"className":9555,"style":1909},[1870],[1866,9557],{"fill":1860,"d":9558},"M137.547-6.629H98.343",[1866,9560],{"d":9561},"m95.837-6.629 3.585 1.351-1.18-1.35 1.18-1.351Z",[1854,9563,9564,9567],{"fill":2436},[1866,9565],{"stroke":1860,"d":9566},"M126.366 3.53c0-5.5-4.459-9.959-9.959-9.959S106.45-1.97 106.45 3.53s4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958m-9.959 0",[1854,9568,9570],{"transform":9569},"translate(163.033 -55.873)",[1866,9571],{"d":9490,"fill":1856,"stroke":1856,"className":9572,"style":1909},[1870],[1866,9574],{"fill":1860,"d":9575},"M85.11-16.787v-22.132",[1866,9577],{"d":9578},"m85.11-41.425-1.352 3.584 1.351-1.178 1.35 1.178Z",[1854,9580,9581,9584],{"fill":2436},[1866,9582],{"stroke":1860,"d":9583},"M84.91-29.39c0-5.5-4.46-9.96-9.96-9.96s-9.958 4.46-9.958 9.96 4.459 9.958 9.959 9.958 9.958-4.459 9.958-9.959m-9.96 0",[1854,9585,9587],{"transform":9586},"translate(121.576 -88.793)",[1866,9588],{"d":9500,"fill":1856,"stroke":1856,"className":9589,"style":1909},[1870],[1866,9591],{"fill":1860,"d":9592},"m93.324-46.178 43.679 31.766",[1866,9594],{"d":9595,"style":9596},"m139.03-12.938-2.105-3.2.159 1.785-1.748.399Z","stroke-width:.399988",[1854,9598,9599,9602],{"fill":2436},[1866,9600],{"stroke":1860,"d":9601},"M115.286-37.45c0-5.499-4.459-9.958-9.959-9.958s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958m-9.959 0",[1854,9603,9605],{"transform":9604},"translate(151.953 -96.852)",[1866,9606],{"d":9607,"fill":1856,"stroke":1856,"className":9608,"style":1909},"M-48.240 60.581Q-48.240 60.140-47.937 59.819Q-47.635 59.498-47.183 59.306L-47.423 59.166Q-47.693 59.006-47.859 58.748Q-48.024 58.490-48.024 58.192Q-48.024 57.840-47.819 57.568Q-47.614 57.297-47.293 57.153Q-46.972 57.010-46.630 57.010Q-46.308 57.010-45.985 57.126Q-45.662 57.242-45.451 57.483Q-45.239 57.724-45.239 58.059Q-45.239 58.421-45.483 58.684Q-45.727 58.948-46.107 59.125L-45.707 59.361Q-45.512 59.474-45.353 59.643Q-45.194 59.812-45.107 60.021Q-45.020 60.229-45.020 60.462Q-45.020 60.865-45.254 61.169Q-45.488 61.473-45.862 61.636Q-46.237 61.798-46.630 61.798Q-47.016 61.798-47.385 61.661Q-47.754 61.525-47.997 61.248Q-48.240 60.971-48.240 60.581M-47.792 60.581Q-47.792 60.868-47.623 61.091Q-47.453 61.313-47.185 61.429Q-46.917 61.545-46.630 61.545Q-46.192 61.545-45.830 61.328Q-45.468 61.111-45.468 60.704Q-45.468 60.503-45.596 60.325Q-45.724 60.147-45.902 60.048L-46.924 59.453Q-47.163 59.563-47.361 59.729Q-47.559 59.894-47.676 60.110Q-47.792 60.325-47.792 60.581M-47.269 58.452L-46.349 58.985Q-46.042 58.825-45.840 58.592Q-45.639 58.360-45.639 58.059Q-45.639 57.820-45.784 57.630Q-45.929 57.440-46.161 57.341Q-46.394 57.242-46.630 57.242Q-46.852 57.242-47.081 57.312Q-47.310 57.382-47.467 57.539Q-47.624 57.697-47.624 57.926Q-47.624 58.240-47.269 58.452",[1870],[1866,9610],{"fill":1860,"d":9611},"M139.49-46.178 95.81-14.412",[1866,9613],{"d":9614,"style":9596},"m93.785-12.938 3.693-1.016-1.747-.4.158-1.785Z",[1854,9616,9617,9620],{"fill":2436},[1866,9618],{"stroke":1860,"d":9619},"M137.445-37.45c0-5.499-4.458-9.958-9.958-9.958s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.958-4.458 9.958-9.958m-9.958 0",[1854,9621,9623],{"transform":9622},"translate(174.113 -96.852)",[1866,9624],{"d":9625,"fill":1856,"stroke":1856,"className":9626,"style":1909},"M-47.183 61.450Q-47.183 60.944-47.054 60.436Q-46.924 59.929-46.686 59.467Q-46.449 59.006-46.114 58.585L-45.468 57.772L-46.281 57.772Q-46.866 57.772-47.262 57.780Q-47.659 57.789-47.682 57.809Q-47.785 57.926-47.864 58.452L-48.130 58.452L-47.884 56.928L-47.618 56.928L-47.618 56.948Q-47.618 57.016-47.542 57.059Q-47.467 57.102-47.389 57.109Q-47.197 57.133-47.002 57.139Q-46.807 57.146-46.616 57.148Q-46.425 57.150-46.226 57.150L-44.805 57.150L-44.805 57.338Q-44.815 57.386-44.825 57.396L-45.881 58.719Q-46.100 58.992-46.223 59.305Q-46.346 59.617-46.404 59.966Q-46.462 60.315-46.476 60.646Q-46.490 60.978-46.490 61.450Q-46.490 61.600-46.589 61.699Q-46.688 61.798-46.835 61.798Q-46.985 61.798-47.084 61.699Q-47.183 61.600-47.183 61.450",[1870],[1854,9628,9629,9636],{"stroke":1860},[1854,9630,9632],{"transform":9631},"translate(14.18 -19.67)",[1866,9633],{"d":9634,"fill":1856,"stroke":1856,"className":9635,"style":1871},"M-47.115 61.736Q-47.471 61.736-47.740 61.556Q-48.010 61.377-48.150 61.082Q-48.291 60.787-48.291 60.428Q-48.291 60.041-48.129 59.627Q-47.967 59.213-47.683 58.875Q-47.400 58.537-47.031 58.334Q-46.662 58.131-46.260 58.131Q-45.767 58.131-45.490 58.580L-45.053 56.810Q-45.010 56.646-45.010 56.596Q-45.010 56.490-45.506 56.490Q-45.603 56.459-45.603 56.361L-45.580 56.260Q-45.549 56.205-45.490 56.193L-44.389 56.107Q-44.346 56.107-44.306 56.138Q-44.267 56.170-44.267 56.224L-45.420 60.826Q-45.459 61.072-45.459 61.123Q-45.459 61.482-45.213 61.482Q-45.068 61.482-44.967 61.375Q-44.865 61.267-44.801 61.113Q-44.736 60.959-44.687 60.769Q-44.639 60.580-44.619 60.482Q-44.592 60.412-44.529 60.412L-44.428 60.412Q-44.389 60.412-44.363 60.445Q-44.338 60.478-44.338 60.506Q-44.338 60.521-44.346 60.537Q-44.459 61.029-44.658 61.383Q-44.857 61.736-45.228 61.736Q-45.510 61.736-45.736 61.594Q-45.963 61.451-46.045 61.193Q-46.267 61.435-46.541 61.586Q-46.814 61.736-47.115 61.736M-47.099 61.482Q-46.889 61.482-46.680 61.369Q-46.471 61.256-46.301 61.082Q-46.131 60.908-46.002 60.713Q-46.014 60.728-46.023 60.742Q-46.033 60.756-46.045 60.771L-45.611 59.049Q-45.650 58.869-45.736 58.719Q-45.822 58.568-45.959 58.476Q-46.096 58.385-46.275 58.385Q-46.611 58.385-46.883 58.666Q-47.154 58.947-47.314 59.322Q-47.439 59.642-47.537 60.062Q-47.635 60.482-47.635 60.763Q-47.635 61.053-47.502 61.267Q-47.369 61.482-47.099 61.482",[1870],[1854,9637,9638],{"transform":9631},[1866,9639],{"d":9640,"fill":1856,"stroke":1856,"className":9641,"style":9642},"M-42.018 60.323Q-42.385 60.071-42.654 59.751Q-42.924 59.430-43.117 59.028Q-43.310 58.627-43.406 58.195Q-43.501 57.763-43.501 57.335Q-43.501 56.904-43.404 56.472Q-43.307 56.040-43.119 55.643Q-42.930 55.246-42.657 54.919Q-42.385 54.593-42.018 54.347Q-42.010 54.341-41.977 54.335L-41.910 54.335Q-41.840 54.356-41.840 54.420Q-41.840 54.455-41.857 54.473Q-42.273 54.807-42.537 55.268Q-42.801 55.730-42.922 56.252Q-43.044 56.775-43.044 57.335Q-43.044 57.895-42.922 58.418Q-42.801 58.940-42.537 59.402Q-42.273 59.863-41.857 60.197Q-41.840 60.224-41.840 60.250Q-41.840 60.314-41.910 60.335L-41.977 60.335Q-42.010 60.329-42.018 60.323M-39.528 58.961Q-40.091 58.961-40.420 58.674Q-40.750 58.387-40.877 57.937Q-41.005 57.487-41.005 56.922Q-41.005 56.518-40.939 56.153Q-40.873 55.788-40.710 55.492Q-40.548 55.196-40.256 55.021Q-39.965 54.845-39.528 54.845Q-39.092 54.845-38.802 55.021Q-38.512 55.196-38.348 55.491Q-38.183 55.785-38.119 56.143Q-38.055 56.500-38.055 56.922Q-38.055 57.487-38.182 57.937Q-38.309 58.387-38.636 58.674Q-38.963 58.961-39.528 58.961M-39.528 58.744Q-39.124 58.744-38.929 58.434Q-38.734 58.123-38.690 57.731Q-38.646 57.338-38.646 56.825Q-38.646 56.330-38.690 55.971Q-38.734 55.612-38.929 55.337Q-39.124 55.062-39.528 55.062Q-39.932 55.062-40.127 55.337Q-40.322 55.612-40.366 55.971Q-40.410 56.330-40.410 56.825Q-40.410 57.338-40.366 57.731Q-40.322 58.123-40.127 58.434Q-39.932 58.744-39.528 58.744M-37.076 60.335L-37.143 60.335Q-37.220 60.317-37.220 60.250Q-37.220 60.212-37.196 60.197Q-36.786 59.866-36.519 59.402Q-36.253 58.938-36.131 58.407Q-36.010 57.877-36.010 57.335Q-36.010 56.793-36.131 56.263Q-36.253 55.732-36.519 55.268Q-36.786 54.804-37.196 54.473Q-37.220 54.449-37.220 54.420Q-37.220 54.356-37.143 54.335L-37.076 54.335Q-37.047 54.341-37.041 54.347Q-36.681 54.587-36.399 54.924Q-36.118 55.261-35.932 55.650Q-35.746 56.040-35.652 56.472Q-35.558 56.904-35.558 57.335Q-35.558 57.763-35.652 58.195Q-35.746 58.627-35.935 59.021Q-36.124 59.415-36.402 59.748Q-36.681 60.080-37.041 60.323Q-37.047 60.329-37.076 60.335",[1870],"stroke-width:0.180",[1854,9644,9646],{"fill":9645,"stroke":9645},"gray",[1854,9647,9649],{"transform":9648},"translate(-1.701 -10.339)",[1866,9650],{"d":9651,"fill":9645,"stroke":9645,"className":9652,"style":9653},"M-45.838 61.658L-47.879 61.658L-47.879 61.419Q-47.098 61.419-47.098 61.299L-47.098 58.753Q-47.276 58.828-47.480 58.858Q-47.684 58.887-47.913 58.887L-47.913 58.648Q-47.577 58.648-47.293 58.579Q-47.010 58.511-46.800 58.328L-46.695 58.328Q-46.668 58.328-46.644 58.352Q-46.619 58.377-46.619 58.404L-46.619 61.299Q-46.619 61.419-45.838 61.419",[1870],"stroke-width:0.150",[1854,9655,9656],{"fill":9645,"stroke":9645},[1854,9657,9659],{"transform":9658},"translate(12.525 -10.339)",[1866,9660],{"d":9661,"fill":9645,"stroke":9645,"className":9662,"style":9653},"M-45.838 61.658L-48.165 61.658L-48.165 61.477Q-48.162 61.465-48.143 61.438L-47.103 60.562Q-46.795 60.303-46.641 60.159Q-46.488 60.015-46.358 59.798Q-46.229 59.580-46.229 59.339Q-46.229 59.097-46.356 58.921Q-46.483 58.745-46.687 58.656Q-46.890 58.567-47.130 58.567Q-47.337 58.567-47.533 58.655Q-47.728 58.743-47.833 58.909Q-47.713 58.909-47.636 59.001Q-47.559 59.092-47.559 59.207Q-47.559 59.334-47.646 59.423Q-47.733 59.512-47.860 59.512Q-47.989 59.512-48.077 59.422Q-48.165 59.331-48.165 59.207Q-48.165 58.926-47.992 58.727Q-47.818 58.528-47.547 58.428Q-47.276 58.328-47.003 58.328Q-46.678 58.328-46.373 58.435Q-46.068 58.543-45.871 58.770Q-45.675 58.997-45.675 59.334Q-45.675 59.571-45.788 59.763Q-45.902 59.956-46.062 60.094Q-46.222 60.232-46.504 60.420Q-46.786 60.608-46.864 60.667L-47.530 61.158L-47.074 61.158Q-46.641 61.158-46.347 61.151Q-46.053 61.145-46.038 61.133Q-45.960 61.035-45.904 60.674L-45.675 60.674",[1870],[1854,9664,9665],{"fill":9645,"stroke":9645},[1854,9666,9668],{"transform":9667},"translate(26.751 -10.339)",[1866,9669],{"d":9670,"fill":9645,"stroke":9645,"className":9671,"style":9653},"M-47.833 61.319Q-47.579 61.543-46.944 61.543Q-46.722 61.543-46.557 61.444Q-46.392 61.346-46.308 61.170Q-46.224 60.994-46.224 60.779Q-46.224 60.557-46.311 60.385Q-46.397 60.213-46.563 60.113Q-46.729 60.012-46.949 60.012L-47.374 60.012Q-47.428 60-47.440 59.949L-47.440 59.888Q-47.428 59.844-47.374 59.827L-47.015 59.803Q-46.829 59.803-46.673 59.691Q-46.517 59.580-46.430 59.401Q-46.344 59.221-46.344 59.043Q-46.344 58.885-46.424 58.769Q-46.505 58.653-46.639 58.595Q-46.773 58.538-46.939 58.538Q-47.454 58.538-47.689 58.767Q-47.581 58.789-47.515 58.869Q-47.450 58.948-47.450 59.058Q-47.450 59.182-47.535 59.268Q-47.620 59.353-47.750 59.353Q-47.872 59.353-47.957 59.268Q-48.043 59.182-48.043 59.058Q-48.043 58.802-47.871 58.637Q-47.699 58.472-47.445 58.400Q-47.191 58.328-46.939 58.328Q-46.707 58.328-46.438 58.404Q-46.168 58.479-45.981 58.640Q-45.794 58.802-45.794 59.043Q-45.794 59.361-46.013 59.584Q-46.231 59.807-46.558 59.908Q-46.388 59.942-46.225 60.014Q-46.063 60.086-45.924 60.197Q-45.785 60.308-45.702 60.454Q-45.619 60.601-45.619 60.779Q-45.619 61.016-45.737 61.201Q-45.855 61.387-46.058 61.515Q-46.261 61.643-46.493 61.706Q-46.724 61.768-46.939 61.768Q-47.230 61.768-47.520 61.703Q-47.811 61.638-48.015 61.469Q-48.219 61.299-48.219 61.009Q-48.219 60.874-48.128 60.784Q-48.038 60.694-47.899 60.694Q-47.811 60.694-47.739 60.735Q-47.667 60.777-47.625 60.849Q-47.584 60.921-47.584 61.009Q-47.584 61.126-47.652 61.210Q-47.721 61.294-47.833 61.319",[1870],[1854,9673,9674],{"fill":9645,"stroke":9645},[1854,9675,9677],{"transform":9676},"translate(40.978 -10.339)",[1866,9678],{"d":9679,"fill":9645,"stroke":9645,"className":9680,"style":9653},"M-46.649 60.843L-48.304 60.843L-48.304 60.603L-46.439 58.362Q-46.412 58.328-46.358 58.328L-46.229 58.328Q-46.200 58.328-46.174 58.351Q-46.148 58.374-46.148 58.408L-46.148 60.603L-45.533 60.603L-45.533 60.843L-46.148 60.843L-46.148 61.299Q-46.148 61.419-45.538 61.419L-45.538 61.658L-47.259 61.658L-47.259 61.419Q-46.649 61.419-46.649 61.299L-46.649 60.843M-46.610 58.882L-48.038 60.603L-46.610 60.603",[1870],[1854,9682,9683],{"fill":9645,"stroke":9645},[1854,9684,9686],{"transform":9685},"translate(-13.651 1.611)",[1866,9687],{"d":9651,"fill":9645,"stroke":9645,"className":9688,"style":9653},[1870],[1854,9690,9691],{"fill":9645,"stroke":9645},[1854,9692,9694],{"transform":9693},"translate(-13.651 14.415)",[1866,9695],{"d":9661,"fill":9645,"stroke":9645,"className":9696,"style":9653},[1870],[1854,9698,9699],{"fill":9645,"stroke":9645},[1854,9700,9702],{"transform":9701},"translate(-13.651 27.218)",[1866,9703],{"d":9670,"fill":9645,"stroke":9645,"className":9704,"style":9653},[1870],[1854,9706,9707],{"fill":9645,"stroke":9645},[1854,9708,9710],{"transform":9709},"translate(-13.651 40.022)",[1866,9711],{"d":9679,"fill":9645,"stroke":9645,"className":9712,"style":9653},[1870],[1854,9714,9716],{"transform":9715},"translate(-1.993 2.256)",[1866,9717],{"d":9718,"fill":1856,"stroke":1856,"className":9719,"style":1909},"M-46.630 61.798Q-47.265 61.798-47.629 61.453Q-47.994 61.108-48.129 60.583Q-48.264 60.058-48.264 59.433Q-48.264 58.408-47.908 57.709Q-47.553 57.010-46.630 57.010Q-45.703 57.010-45.351 57.709Q-44.999 58.408-44.999 59.433Q-44.999 60.058-45.134 60.583Q-45.269 61.108-45.632 61.453Q-45.994 61.798-46.630 61.798M-46.630 61.573Q-46.192 61.573-45.979 61.198Q-45.765 60.824-45.715 60.357Q-45.666 59.891-45.666 59.313Q-45.666 58.760-45.715 58.332Q-45.765 57.905-45.977 57.570Q-46.189 57.235-46.630 57.235Q-46.972 57.235-47.175 57.442Q-47.378 57.649-47.465 57.961Q-47.553 58.274-47.575 58.590Q-47.597 58.907-47.597 59.313Q-47.597 59.730-47.575 60.072Q-47.553 60.414-47.464 60.762Q-47.375 61.111-47.170 61.342Q-46.965 61.573-46.630 61.573",[1870],[1854,9721,9723],{"transform":9722},"translate(12.233 2.256)",[1866,9724],{"d":9510,"fill":1856,"stroke":1856,"className":9725,"style":1909},[1870],[1854,9727,9729],{"transform":9728},"translate(26.46 2.256)",[1866,9730],{"d":9607,"fill":1856,"stroke":1856,"className":9731,"style":1909},[1870],[1854,9733,9735],{"transform":9734},"translate(38.693 1.507)",[1866,9736],{"d":9737,"fill":1856,"stroke":1856,"className":9738,"style":1909},"M-46.596 61.726Q-47.016 61.726-47.353 61.506Q-47.689 61.285-47.881 60.921Q-48.072 60.557-48.072 60.147Q-48.072 59.843-47.958 59.553Q-47.843 59.262-47.645 59.045Q-47.447 58.828-47.171 58.700Q-46.896 58.572-46.575 58.572Q-46.315 58.572-46.066 58.635Q-45.816 58.698-45.606 58.806Q-45.396 58.913-45.182 59.083Q-44.969 59.252-44.818 59.433L-44.572 59.720Q-44.343 59.399-44.051 59.137Q-43.759 58.876-43.408 58.724Q-43.058 58.572-42.675 58.572Q-42.364 58.572-42.087 58.701Q-41.810 58.831-41.614 59.048Q-41.417 59.265-41.308 59.551Q-41.199 59.836-41.199 60.147Q-41.199 60.462-41.308 60.744Q-41.417 61.026-41.624 61.251Q-41.831 61.477-42.108 61.602Q-42.385 61.726-42.696 61.726Q-43.195 61.726-43.649 61.496Q-44.104 61.265-44.452 60.868L-44.699 60.581Q-44.876 60.831-45.078 61.033Q-45.280 61.234-45.524 61.393Q-45.768 61.552-46.040 61.639Q-46.312 61.726-46.596 61.726M-46.667 61.443Q-46.144 61.443-45.703 61.120Q-45.263 60.797-44.938 60.301L-45.314 59.860Q-45.652 59.460-45.975 59.223Q-46.298 58.985-46.688 58.985Q-47.002 58.985-47.269 59.142Q-47.536 59.300-47.693 59.570Q-47.850 59.840-47.850 60.154Q-47.850 60.482-47.696 60.780Q-47.542 61.077-47.274 61.260Q-47.006 61.443-46.667 61.443M-44.336 60L-43.957 60.441Q-43.721 60.711-43.530 60.891Q-43.338 61.070-43.097 61.193Q-42.856 61.316-42.586 61.316Q-42.275 61.316-42.007 61.157Q-41.739 60.998-41.581 60.727Q-41.424 60.455-41.424 60.147Q-41.424 59.823-41.578 59.522Q-41.732 59.221-42.002 59.040Q-42.272 58.859-42.607 58.859Q-42.955 58.859-43.280 59.014Q-43.605 59.170-43.863 59.424Q-44.121 59.679-44.336 60",[1870],[1854,9740,9742],{"transform":9741},"translate(-3.986 14.31)",[1866,9743],{"d":9737,"fill":1856,"stroke":1856,"className":9744,"style":1909},[1870],[1854,9746,9748],{"transform":9747},"translate(12.233 15.06)",[1866,9749],{"d":9718,"fill":1856,"stroke":1856,"className":9750,"style":1909},[1870],[1854,9752,9754],{"transform":9753},"translate(26.46 15.06)",[1866,9755],{"d":9500,"fill":1856,"stroke":1856,"className":9756,"style":1909},[1870],[1854,9758,9760],{"transform":9759},"translate(40.686 15.06)",[1866,9761],{"d":9625,"fill":1856,"stroke":1856,"className":9762,"style":1909},[1870],[1854,9764,9766],{"transform":9765},"translate(-3.986 27.114)",[1866,9767],{"d":9737,"fill":1856,"stroke":1856,"className":9768,"style":1909},[1870],[1854,9770,9772],{"transform":9771},"translate(10.24 27.114)",[1866,9773],{"d":9737,"fill":1856,"stroke":1856,"className":9774,"style":1909},[1870],[1854,9776,9778],{"transform":9777},"translate(26.46 27.863)",[1866,9779],{"d":9718,"fill":1856,"stroke":1856,"className":9780,"style":1909},[1870],[1854,9782,9784],{"transform":9783},"translate(40.686 27.863)",[1866,9785],{"d":9490,"fill":1856,"stroke":1856,"className":9786,"style":1909},[1870],[1854,9788,9790],{"transform":9789},"translate(-1.993 40.667)",[1866,9791],{"d":9500,"fill":1856,"stroke":1856,"className":9792,"style":1909},[1870],[1854,9794,9796],{"transform":9795},"translate(10.24 39.918)",[1866,9797],{"d":9737,"fill":1856,"stroke":1856,"className":9798,"style":1909},[1870],[1854,9800,9802],{"transform":9801},"translate(24.467 39.918)",[1866,9803],{"d":9737,"fill":1856,"stroke":1856,"className":9804,"style":1909},[1870],[1854,9806,9808],{"transform":9807},"translate(40.686 40.667)",[1866,9809],{"d":9718,"fill":1856,"stroke":1856,"className":9810,"style":1909},[1870],[1854,9812,9813],{"fill":9645,"stroke":9645},[1854,9814,9816,9823],{"fill":9645,"stroke":1860,"fontFamily":9815,"fontSize":1901},"cmr7",[1854,9817,9819],{"transform":9818},"translate(.63 53.534)",[1866,9820],{"d":9821,"fill":9645,"stroke":9645,"className":9822,"style":1909},"M-48.305 60.147Q-48.305 59.809-48.164 59.518Q-48.024 59.228-47.780 59.014Q-47.536 58.801-47.231 58.686Q-46.927 58.572-46.602 58.572Q-46.332 58.572-46.069 58.671Q-45.806 58.770-45.615 58.948L-45.615 57.550Q-45.615 57.280-45.722 57.218Q-45.830 57.157-46.141 57.157L-46.141 56.876L-45.064 56.801L-45.064 60.985Q-45.064 61.173-45.010 61.256Q-44.955 61.340-44.854 61.359Q-44.753 61.378-44.538 61.378L-44.538 61.658L-45.645 61.726L-45.645 61.309Q-46.062 61.726-46.688 61.726Q-47.119 61.726-47.491 61.514Q-47.864 61.303-48.084 60.942Q-48.305 60.581-48.305 60.147M-46.630 61.504Q-46.421 61.504-46.235 61.432Q-46.049 61.361-45.895 61.224Q-45.741 61.087-45.645 60.909L-45.645 59.300Q-45.731 59.153-45.876 59.033Q-46.021 58.913-46.191 58.854Q-46.360 58.794-46.541 58.794Q-47.101 58.794-47.370 59.183Q-47.638 59.573-47.638 60.154Q-47.638 60.725-47.404 61.115Q-47.170 61.504-46.630 61.504M-42.272 61.658L-43.824 61.658L-43.824 61.378Q-43.598 61.378-43.449 61.344Q-43.301 61.309-43.301 61.169L-43.301 59.320Q-43.301 59.132-43.348 59.048Q-43.396 58.965-43.494 58.946Q-43.591 58.927-43.803 58.927L-43.803 58.647L-42.747 58.572L-42.747 61.169Q-42.747 61.309-42.615 61.344Q-42.484 61.378-42.272 61.378L-42.272 61.658M-43.543 57.351Q-43.543 57.180-43.420 57.061Q-43.297 56.941-43.126 56.941Q-42.959 56.941-42.836 57.061Q-42.713 57.180-42.713 57.351Q-42.713 57.526-42.836 57.649Q-42.959 57.772-43.126 57.772Q-43.297 57.772-43.420 57.649Q-43.543 57.526-43.543 57.351M-39.876 61.658L-41.612 61.658L-41.612 61.378Q-41.383 61.378-41.234 61.344Q-41.086 61.309-41.086 61.169L-41.086 59.320Q-41.086 59.050-41.193 58.989Q-41.301 58.927-41.612 58.927L-41.612 58.647L-40.583 58.572L-40.583 59.279Q-40.453 58.971-40.211 58.772Q-39.968 58.572-39.650 58.572Q-39.431 58.572-39.261 58.696Q-39.090 58.821-39.090 59.033Q-39.090 59.170-39.189 59.269Q-39.288 59.368-39.421 59.368Q-39.558 59.368-39.657 59.269Q-39.756 59.170-39.756 59.033Q-39.756 58.893-39.657 58.794Q-39.948 58.794-40.148 58.990Q-40.348 59.187-40.440 59.481Q-40.532 59.775-40.532 60.055L-40.532 61.169Q-40.532 61.378-39.876 61.378L-39.876 61.658M-38.546 60.123Q-38.546 59.802-38.421 59.513Q-38.297 59.224-38.071 59.001Q-37.846 58.777-37.550 58.657Q-37.254 58.537-36.936 58.537Q-36.608 58.537-36.347 58.637Q-36.085 58.736-35.909 58.918Q-35.733 59.101-35.639 59.359Q-35.545 59.617-35.545 59.949Q-35.545 60.041-35.627 60.062L-37.883 60.062L-37.883 60.123Q-37.883 60.711-37.599 61.094Q-37.316 61.477-36.748 61.477Q-36.427 61.477-36.159 61.284Q-35.890 61.091-35.802 60.776Q-35.795 60.735-35.720 60.721L-35.627 60.721Q-35.545 60.745-35.545 60.817Q-35.545 60.824-35.552 60.851Q-35.665 61.248-36.036 61.487Q-36.407 61.726-36.830 61.726Q-37.268 61.726-37.668 61.518Q-38.068 61.309-38.307 60.942Q-38.546 60.575-38.546 60.123M-37.876 59.853L-36.061 59.853Q-36.061 59.576-36.159 59.324Q-36.256 59.071-36.454 58.915Q-36.653 58.760-36.936 58.760Q-37.213 58.760-37.427 58.918Q-37.640 59.077-37.758 59.332Q-37.876 59.587-37.876 59.853M-34.957 60.147Q-34.957 59.819-34.822 59.518Q-34.687 59.218-34.452 58.997Q-34.216 58.777-33.911 58.657Q-33.607 58.537-33.283 58.537Q-32.777 58.537-32.428 58.640Q-32.079 58.742-32.079 59.118Q-32.079 59.265-32.177 59.366Q-32.274 59.467-32.421 59.467Q-32.575 59.467-32.674 59.368Q-32.773 59.269-32.773 59.118Q-32.773 58.930-32.633 58.838Q-32.835 58.787-33.276 58.787Q-33.631 58.787-33.860 58.983Q-34.089 59.180-34.190 59.489Q-34.291 59.799-34.291 60.147Q-34.291 60.496-34.164 60.802Q-34.038 61.108-33.783 61.292Q-33.529 61.477-33.173 61.477Q-32.951 61.477-32.766 61.393Q-32.582 61.309-32.447 61.154Q-32.312 60.998-32.254 60.790Q-32.240 60.735-32.185 60.735L-32.073 60.735Q-32.042 60.735-32.020 60.759Q-31.997 60.783-31.997 60.817L-31.997 60.838Q-32.083 61.125-32.271 61.323Q-32.459 61.521-32.724 61.624Q-32.989 61.726-33.283 61.726Q-33.713 61.726-34.101 61.520Q-34.489 61.313-34.723 60.950Q-34.957 60.588-34.957 60.147M-30.883 60.817L-30.883 58.920L-31.522 58.920L-31.522 58.698Q-31.204 58.698-30.987 58.488Q-30.770 58.278-30.670 57.968Q-30.569 57.659-30.569 57.351L-30.302 57.351L-30.302 58.640L-29.225 58.640L-29.225 58.920L-30.302 58.920L-30.302 60.804Q-30.302 61.080-30.198 61.279Q-30.094 61.477-29.834 61.477Q-29.677 61.477-29.571 61.373Q-29.465 61.268-29.415 61.115Q-29.366 60.961-29.366 60.804L-29.366 60.390L-29.099 60.390L-29.099 60.817Q-29.099 61.043-29.198 61.253Q-29.297 61.463-29.482 61.595Q-29.666 61.726-29.895 61.726Q-30.333 61.726-30.608 61.489Q-30.883 61.251-30.883 60.817",[1870],[1854,9824,9825],{"transform":9818},[1866,9826],{"d":9827,"fill":9645,"stroke":9645,"className":9828,"style":1909},"M-25.610 60.123Q-25.610 59.802-25.485 59.513Q-25.360 59.224-25.134 59.001Q-24.909 58.777-24.613 58.657Q-24.318 58.537-24 58.537Q-23.672 58.537-23.410 58.637Q-23.149 58.736-22.973 58.918Q-22.797 59.101-22.703 59.359Q-22.609 59.617-22.609 59.949Q-22.609 60.041-22.691 60.062L-24.946 60.062L-24.946 60.123Q-24.946 60.711-24.663 61.094Q-24.379 61.477-23.812 61.477Q-23.490 61.477-23.222 61.284Q-22.954 61.091-22.865 60.776Q-22.858 60.735-22.783 60.721L-22.691 60.721Q-22.609 60.745-22.609 60.817Q-22.609 60.824-22.615 60.851Q-22.728 61.248-23.099 61.487Q-23.470 61.726-23.894 61.726Q-24.331 61.726-24.731 61.518Q-25.131 61.309-25.370 60.942Q-25.610 60.575-25.610 60.123M-24.940 59.853L-23.125 59.853Q-23.125 59.576-23.222 59.324Q-23.320 59.071-23.518 58.915Q-23.716 58.760-24 58.760Q-24.277 58.760-24.490 58.918Q-24.704 59.077-24.822 59.332Q-24.940 59.587-24.940 59.853M-22.021 60.147Q-22.021 59.809-21.881 59.518Q-21.740 59.228-21.496 59.014Q-21.252 58.801-20.947 58.686Q-20.643 58.572-20.319 58.572Q-20.049 58.572-19.785 58.671Q-19.522 58.770-19.331 58.948L-19.331 57.550Q-19.331 57.280-19.438 57.218Q-19.546 57.157-19.857 57.157L-19.857 56.876L-18.780 56.801L-18.780 60.985Q-18.780 61.173-18.726 61.256Q-18.671 61.340-18.570 61.359Q-18.469 61.378-18.254 61.378L-18.254 61.658L-19.362 61.726L-19.362 61.309Q-19.779 61.726-20.404 61.726Q-20.835 61.726-21.207 61.514Q-21.580 61.303-21.800 60.942Q-22.021 60.581-22.021 60.147M-20.346 61.504Q-20.137 61.504-19.951 61.432Q-19.765 61.361-19.611 61.224Q-19.457 61.087-19.362 60.909L-19.362 59.300Q-19.447 59.153-19.592 59.033Q-19.737 58.913-19.907 58.854Q-20.076 58.794-20.257 58.794Q-20.818 58.794-21.086 59.183Q-21.354 59.573-21.354 60.154Q-21.354 60.725-21.120 61.115Q-20.886 61.504-20.346 61.504M-17.646 62.191Q-17.646 61.945-17.449 61.761Q-17.253 61.576-16.996 61.497Q-17.133 61.385-17.205 61.224Q-17.277 61.063-17.277 60.882Q-17.277 60.561-17.065 60.315Q-17.400 60.017-17.400 59.607Q-17.400 59.146-17.010 58.859Q-16.620 58.572-16.142 58.572Q-15.670 58.572-15.335 58.818Q-15.161 58.664-14.951 58.582Q-14.740 58.500-14.511 58.500Q-14.347 58.500-14.226 58.607Q-14.105 58.715-14.105 58.879Q-14.105 58.975-14.176 59.047Q-14.248 59.118-14.341 59.118Q-14.440 59.118-14.510 59.045Q-14.580 58.971-14.580 58.872Q-14.580 58.818-14.566 58.787L-14.559 58.773Q-14.552 58.753-14.544 58.742Q-14.535 58.732-14.532 58.725Q-14.887 58.725-15.175 58.948Q-14.887 59.241-14.887 59.607Q-14.887 59.922-15.072 60.154Q-15.257 60.387-15.545 60.515Q-15.834 60.643-16.142 60.643Q-16.343 60.643-16.535 60.593Q-16.726 60.544-16.904 60.434Q-16.996 60.561-16.996 60.704Q-16.996 60.886-16.868 61.021Q-16.740 61.156-16.555 61.156L-15.923 61.156Q-15.475 61.156-15.106 61.227Q-14.737 61.299-14.477 61.528Q-14.217 61.757-14.217 62.191Q-14.217 62.512-14.513 62.714Q-14.809 62.916-15.212 63.005Q-15.615 63.094-15.930 63.094Q-16.248 63.094-16.651 63.005Q-17.054 62.916-17.350 62.714Q-17.646 62.512-17.646 62.191M-17.191 62.191Q-17.191 62.420-16.972 62.569Q-16.754 62.718-16.461 62.786Q-16.169 62.854-15.930 62.854Q-15.766 62.854-15.557 62.818Q-15.349 62.783-15.142 62.702Q-14.935 62.622-14.804 62.494Q-14.672 62.366-14.672 62.191Q-14.672 61.839-15.053 61.745Q-15.434 61.651-15.937 61.651L-16.555 61.651Q-16.795 61.651-16.993 61.802Q-17.191 61.952-17.191 62.191M-16.142 60.404Q-15.475 60.404-15.475 59.607Q-15.475 58.807-16.142 58.807Q-16.812 58.807-16.812 59.607Q-16.812 60.404-16.142 60.404M-13.664 60.123Q-13.664 59.802-13.539 59.513Q-13.414 59.224-13.189 59.001Q-12.963 58.777-12.667 58.657Q-12.372 58.537-12.054 58.537Q-11.726 58.537-11.464 58.637Q-11.203 58.736-11.027 58.918Q-10.851 59.101-10.757 59.359Q-10.663 59.617-10.663 59.949Q-10.663 60.041-10.745 60.062L-13.001 60.062L-13.001 60.123Q-13.001 60.711-12.717 61.094Q-12.433 61.477-11.866 61.477Q-11.545 61.477-11.276 61.284Q-11.008 61.091-10.919 60.776Q-10.912 60.735-10.837 60.721L-10.745 60.721Q-10.663 60.745-10.663 60.817Q-10.663 60.824-10.670 60.851Q-10.782 61.248-11.153 61.487Q-11.524 61.726-11.948 61.726Q-12.385 61.726-12.785 61.518Q-13.185 61.309-13.425 60.942Q-13.664 60.575-13.664 60.123M-12.994 59.853L-11.179 59.853Q-11.179 59.576-11.276 59.324Q-11.374 59.071-11.572 58.915Q-11.770 58.760-12.054 58.760Q-12.331 58.760-12.544 58.918Q-12.758 59.077-12.876 59.332Q-12.994 59.587-12.994 59.853M-10.075 61.651L-10.075 60.588Q-10.075 60.564-10.048 60.537Q-10.020 60.510-9.996 60.510L-9.887 60.510Q-9.822 60.510-9.808 60.568Q-9.713 61.002-9.466 61.253Q-9.220 61.504-8.807 61.504Q-8.465 61.504-8.212 61.371Q-7.959 61.238-7.959 60.930Q-7.959 60.773-8.053 60.658Q-8.147 60.544-8.286 60.475Q-8.424 60.407-8.591 60.369L-9.173 60.270Q-9.528 60.202-9.801 59.981Q-10.075 59.761-10.075 59.419Q-10.075 59.170-9.964 58.995Q-9.853 58.821-9.666 58.722Q-9.480 58.623-9.265 58.580Q-9.050 58.537-8.807 58.537Q-8.393 58.537-8.113 58.719L-7.898 58.544Q-7.887 58.541-7.881 58.539Q-7.874 58.537-7.863 58.537L-7.812 58.537Q-7.785 58.537-7.761 58.561Q-7.737 58.585-7.737 58.613L-7.737 59.460Q-7.737 59.481-7.761 59.508Q-7.785 59.535-7.812 59.535L-7.925 59.535Q-7.952 59.535-7.978 59.510Q-8.004 59.484-8.004 59.460Q-8.004 59.224-8.110 59.060Q-8.216 58.896-8.398 58.814Q-8.581 58.732-8.814 58.732Q-9.142 58.732-9.398 58.835Q-9.654 58.937-9.654 59.214Q-9.654 59.409-9.472 59.518Q-9.289 59.628-9.060 59.669L-8.486 59.775Q-8.239 59.823-8.026 59.951Q-7.812 60.079-7.675 60.282Q-7.539 60.486-7.539 60.735Q-7.539 61.248-7.904 61.487Q-8.270 61.726-8.807 61.726Q-9.302 61.726-9.634 61.432L-9.901 61.706Q-9.921 61.726-9.948 61.726L-9.996 61.726Q-10.020 61.726-10.048 61.699Q-10.075 61.672-10.075 61.651",[1870],[1854,9830,9831,9837],{"stroke":1860},[1854,9832,9834],{"transform":9833},"translate(88.158 -19.67)",[1866,9835],{"d":9634,"fill":1856,"stroke":1856,"className":9836,"style":1871},[1870],[1854,9838,9839],{"transform":9833},[1866,9840],{"d":9841,"fill":1856,"stroke":1856,"className":9842,"style":9642},"M-42.018 60.323Q-42.385 60.071-42.654 59.751Q-42.924 59.430-43.117 59.028Q-43.310 58.627-43.406 58.195Q-43.501 57.763-43.501 57.335Q-43.501 56.904-43.404 56.472Q-43.307 56.040-43.119 55.643Q-42.930 55.246-42.657 54.919Q-42.385 54.593-42.018 54.347Q-42.010 54.341-41.977 54.335L-41.910 54.335Q-41.840 54.356-41.840 54.420Q-41.840 54.455-41.857 54.473Q-42.273 54.807-42.537 55.268Q-42.801 55.730-42.922 56.252Q-43.044 56.775-43.044 57.335Q-43.044 57.895-42.922 58.418Q-42.801 58.940-42.537 59.402Q-42.273 59.863-41.857 60.197Q-41.840 60.224-41.840 60.250Q-41.840 60.314-41.910 60.335L-41.977 60.335Q-42.010 60.329-42.018 60.323M-38.318 58.835L-40.609 58.835L-40.609 58.577Q-39.733 58.577-39.733 58.404L-39.733 55.325Q-39.927 55.413-40.158 55.450Q-40.390 55.486-40.644 55.486L-40.644 55.229Q-40.266 55.229-39.946 55.144Q-39.625 55.059-39.396 54.845L-39.276 54.845Q-39.244 54.845-39.219 54.868Q-39.194 54.892-39.194 54.930L-39.194 58.404Q-39.194 58.577-38.318 58.577L-38.318 58.835M-37.076 60.335L-37.143 60.335Q-37.220 60.317-37.220 60.250Q-37.220 60.212-37.196 60.197Q-36.786 59.866-36.519 59.402Q-36.253 58.938-36.131 58.407Q-36.010 57.877-36.010 57.335Q-36.010 56.793-36.131 56.263Q-36.253 55.732-36.519 55.268Q-36.786 54.804-37.196 54.473Q-37.220 54.449-37.220 54.420Q-37.220 54.356-37.143 54.335L-37.076 54.335Q-37.047 54.341-37.041 54.347Q-36.681 54.587-36.399 54.924Q-36.118 55.261-35.932 55.650Q-35.746 56.040-35.652 56.472Q-35.558 56.904-35.558 57.335Q-35.558 57.763-35.652 58.195Q-35.746 58.627-35.935 59.021Q-36.124 59.415-36.402 59.748Q-36.681 60.080-37.041 60.323Q-37.047 60.329-37.076 60.335",[1870],[1854,9844,9845],{"fill":1989,"stroke":1989},[1854,9846,9848],{"transform":9847},"translate(72.276 -10.339)",[1866,9849],{"d":9651,"fill":1989,"stroke":1989,"className":9850,"style":9653},[1870],[1854,9852,9853],{"fill":9645,"stroke":9645},[1854,9854,9856],{"transform":9855},"translate(86.502 -10.339)",[1866,9857],{"d":9661,"fill":9645,"stroke":9645,"className":9858,"style":9653},[1870],[1854,9860,9861],{"fill":9645,"stroke":9645},[1854,9862,9864],{"transform":9863},"translate(100.728 -10.339)",[1866,9865],{"d":9670,"fill":9645,"stroke":9645,"className":9866,"style":9653},[1870],[1854,9868,9869],{"fill":9645,"stroke":9645},[1854,9870,9872],{"transform":9871},"translate(114.955 -10.339)",[1866,9873],{"d":9679,"fill":9645,"stroke":9645,"className":9874,"style":9653},[1870],[1854,9876,9877],{"fill":1989,"stroke":1989},[1854,9878,9880],{"transform":9879},"translate(60.326 1.611)",[1866,9881],{"d":9651,"fill":1989,"stroke":1989,"className":9882,"style":9653},[1870],[1854,9884,9885],{"fill":9645,"stroke":9645},[1854,9886,9888],{"transform":9887},"translate(60.326 14.415)",[1866,9889],{"d":9661,"fill":9645,"stroke":9645,"className":9890,"style":9653},[1870],[1854,9892,9893],{"fill":9645,"stroke":9645},[1854,9894,9896],{"transform":9895},"translate(60.326 27.218)",[1866,9897],{"d":9670,"fill":9645,"stroke":9645,"className":9898,"style":9653},[1870],[1854,9900,9901],{"fill":9645,"stroke":9645},[1854,9902,9904],{"transform":9903},"translate(60.326 40.022)",[1866,9905],{"d":9679,"fill":9645,"stroke":9645,"className":9906,"style":9653},[1870],[1866,9908],{"fill":1988,"stroke":1860,"d":9909},"M33.04 105.76V94.379H46.13v11.38ZM47.267 105.76V94.379h13.088v11.38Zm13.088-11.381",[1854,9911,9913],{"transform":9912},"translate(71.984 2.256)",[1866,9914],{"d":9718,"fill":1856,"stroke":1856,"className":9915,"style":1909},[1870],[1854,9917,9919],{"transform":9918},"translate(86.21 2.256)",[1866,9920],{"d":9510,"fill":1856,"stroke":1856,"className":9921,"style":1909},[1870],[1854,9923,9925],{"transform":9924},"translate(100.437 2.256)",[1866,9926],{"d":9607,"fill":1856,"stroke":1856,"className":9927,"style":1909},[1870],[1854,9929,9931],{"transform":9930},"translate(112.67 1.507)",[1866,9932],{"d":9737,"fill":1856,"stroke":1856,"className":9933,"style":1909},[1870],[1854,9935,9937],{"transform":9936},"translate(69.991 14.31)",[1866,9938],{"d":9737,"fill":1856,"stroke":1856,"className":9939,"style":1909},[1870],[1854,9941,9943],{"transform":9942},"translate(86.21 15.06)",[1866,9944],{"d":9718,"fill":1856,"stroke":1856,"className":9945,"style":1909},[1870],[1854,9947,9949],{"transform":9948},"translate(100.437 15.06)",[1866,9950],{"d":9500,"fill":1856,"stroke":1856,"className":9951,"style":1909},[1870],[1854,9953,9955],{"transform":9954},"translate(114.663 15.06)",[1866,9956],{"d":9625,"fill":1856,"stroke":1856,"className":9957,"style":1909},[1870],[1854,9959,9961],{"transform":9960},"translate(69.991 27.114)",[1866,9962],{"d":9737,"fill":1856,"stroke":1856,"className":9963,"style":1909},[1870],[1854,9965,9967],{"transform":9966},"translate(84.217 27.114)",[1866,9968],{"d":9737,"fill":1856,"stroke":1856,"className":9969,"style":1909},[1870],[1854,9971,9973],{"transform":9972},"translate(100.437 27.863)",[1866,9974],{"d":9718,"fill":1856,"stroke":1856,"className":9975,"style":1909},[1870],[1854,9977,9979],{"transform":9978},"translate(114.663 27.863)",[1866,9980],{"d":9490,"fill":1856,"stroke":1856,"className":9981,"style":1909},[1870],[1854,9983,9985],{"transform":9984},"translate(71.984 40.667)",[1866,9986],{"d":9500,"fill":1856,"stroke":1856,"className":9987,"style":1909},[1870],[1854,9989,9991],{"transform":9990},"translate(86.21 40.667)",[1866,9992],{"d":9993,"fill":1856,"stroke":1856,"className":9994,"style":1909},"M-47.812 60.896L-47.843 60.896Q-47.706 61.193-47.409 61.369Q-47.112 61.545-46.784 61.545Q-46.421 61.545-46.194 61.367Q-45.967 61.190-45.873 60.901Q-45.779 60.612-45.779 60.250Q-45.779 59.935-45.833 59.650Q-45.888 59.365-46.061 59.159Q-46.233 58.954-46.548 58.954Q-46.821 58.954-47.004 59.021Q-47.187 59.088-47.291 59.177Q-47.395 59.265-47.491 59.375Q-47.587 59.484-47.631 59.494L-47.710 59.494Q-47.782 59.477-47.799 59.406L-47.799 57.088Q-47.799 57.054-47.775 57.032Q-47.751 57.010-47.717 57.010L-47.689 57.010Q-47.402 57.126-47.134 57.180Q-46.866 57.235-46.589 57.235Q-46.312 57.235-46.042 57.180Q-45.772 57.126-45.492 57.010L-45.468 57.010Q-45.433 57.010-45.410 57.033Q-45.386 57.057-45.386 57.088L-45.386 57.157Q-45.386 57.184-45.406 57.204Q-45.680 57.519-46.064 57.695Q-46.449 57.871-46.862 57.871Q-47.201 57.871-47.518 57.785L-47.518 59.067Q-47.122 58.732-46.548 58.732Q-46.144 58.732-45.808 58.942Q-45.471 59.153-45.278 59.505Q-45.085 59.857-45.085 60.257Q-45.085 60.588-45.225 60.874Q-45.365 61.159-45.609 61.369Q-45.854 61.579-46.156 61.689Q-46.459 61.798-46.777 61.798Q-47.136 61.798-47.462 61.634Q-47.788 61.470-47.983 61.178Q-48.178 60.886-48.178 60.523Q-48.178 60.373-48.072 60.267Q-47.966 60.161-47.812 60.161Q-47.659 60.161-47.554 60.265Q-47.450 60.369-47.450 60.523Q-47.450 60.680-47.554 60.788Q-47.659 60.896-47.812 60.896",[1870],[1854,9996,9998],{"transform":9997},"translate(98.444 40.667)",[1866,9999],{"d":10000,"fill":1856,"stroke":1856,"className":10001,"style":1909},"M-45.293 61.658L-47.823 61.658L-47.823 61.378Q-46.855 61.378-46.855 61.169L-46.855 57.550Q-47.248 57.738-47.870 57.738L-47.870 57.457Q-47.453 57.457-47.089 57.356Q-46.725 57.256-46.469 57.010L-46.343 57.010Q-46.278 57.027-46.261 57.095L-46.261 61.169Q-46.261 61.378-45.293 61.378L-45.293 61.658M-42.648 61.798Q-43.284 61.798-43.648 61.453Q-44.012 61.108-44.147 60.583Q-44.282 60.058-44.282 59.433Q-44.282 58.408-43.926 57.709Q-43.571 57.010-42.648 57.010Q-41.722 57.010-41.369 57.709Q-41.017 58.408-41.017 59.433Q-41.017 60.058-41.152 60.583Q-41.287 61.108-41.650 61.453Q-42.012 61.798-42.648 61.798M-42.648 61.573Q-42.210 61.573-41.997 61.198Q-41.783 60.824-41.734 60.357Q-41.684 59.891-41.684 59.313Q-41.684 58.760-41.734 58.332Q-41.783 57.905-41.995 57.570Q-42.207 57.235-42.648 57.235Q-42.990 57.235-43.193 57.442Q-43.396 57.649-43.484 57.961Q-43.571 58.274-43.593 58.590Q-43.615 58.907-43.615 59.313Q-43.615 59.730-43.593 60.072Q-43.571 60.414-43.482 60.762Q-43.393 61.111-43.188 61.342Q-42.983 61.573-42.648 61.573",[1870],[1854,10003,10005],{"transform":10004},"translate(114.663 40.667)",[1866,10006],{"d":9718,"fill":1856,"stroke":1856,"className":10007,"style":1909},[1870],[1854,10009,10010],{"fill":9645,"stroke":9645},[1854,10011,10012,10019,10025,10031],{"fill":9645,"stroke":1860,"fontSize":1901},[1854,10013,10015],{"transform":10014},"translate(82.653 53.534)",[1866,10016],{"d":10017,"fill":9645,"stroke":9645,"className":10018,"style":1909},"M-46.715 61.631L-47.843 59.132Q-47.915 58.985-48.045 58.953Q-48.175 58.920-48.404 58.920L-48.404 58.640L-46.890 58.640L-46.890 58.920Q-47.242 58.920-47.242 59.067Q-47.242 59.112-47.231 59.132L-46.367 61.050L-45.587 59.320Q-45.553 59.252-45.553 59.173Q-45.553 59.060-45.637 58.990Q-45.721 58.920-45.840 58.920L-45.840 58.640L-44.644 58.640L-44.644 58.920Q-44.863 58.920-45.034 59.023Q-45.204 59.125-45.293 59.320L-46.329 61.631Q-46.377 61.726-46.483 61.726L-46.561 61.726Q-46.667 61.726-46.715 61.631M-42.487 61.658L-44.039 61.658L-44.039 61.378Q-43.813 61.378-43.665 61.344Q-43.516 61.309-43.516 61.169L-43.516 59.320Q-43.516 59.132-43.564 59.048Q-43.612 58.965-43.709 58.946Q-43.806 58.927-44.018 58.927L-44.018 58.647L-42.962 58.572L-42.962 61.169Q-42.962 61.309-42.831 61.344Q-42.699 61.378-42.487 61.378L-42.487 61.658M-43.759 57.351Q-43.759 57.180-43.636 57.061Q-43.513 56.941-43.342 56.941Q-43.174 56.941-43.051 57.061Q-42.928 57.180-42.928 57.351Q-42.928 57.526-43.051 57.649Q-43.174 57.772-43.342 57.772Q-43.513 57.772-43.636 57.649Q-43.759 57.526-43.759 57.351M-41.783 60.930Q-41.783 60.598-41.559 60.371Q-41.335 60.144-40.992 60.016Q-40.648 59.887-40.276 59.835Q-39.903 59.782-39.599 59.782L-39.599 59.529Q-39.599 59.324-39.707 59.144Q-39.814 58.965-39.995 58.862Q-40.177 58.760-40.385 58.760Q-40.792 58.760-41.028 58.852Q-40.939 58.889-40.893 58.973Q-40.847 59.057-40.847 59.159Q-40.847 59.255-40.893 59.334Q-40.939 59.412-41.019 59.457Q-41.099 59.501-41.188 59.501Q-41.339 59.501-41.440 59.404Q-41.540 59.306-41.540 59.159Q-41.540 58.537-40.385 58.537Q-40.173 58.537-39.924 58.601Q-39.674 58.664-39.473 58.783Q-39.271 58.903-39.144 59.088Q-39.018 59.272-39.018 59.515L-39.018 61.091Q-39.018 61.207-38.956 61.303Q-38.895 61.398-38.782 61.398Q-38.673 61.398-38.608 61.304Q-38.543 61.210-38.543 61.091L-38.543 60.643L-38.276 60.643L-38.276 61.091Q-38.276 61.361-38.504 61.526Q-38.731 61.692-39.011 61.692Q-39.220 61.692-39.356 61.538Q-39.493 61.385-39.517 61.169Q-39.664 61.436-39.946 61.581Q-40.228 61.726-40.553 61.726Q-40.829 61.726-41.113 61.651Q-41.397 61.576-41.590 61.397Q-41.783 61.217-41.783 60.930M-41.168 60.930Q-41.168 61.104-41.067 61.234Q-40.966 61.364-40.811 61.434Q-40.655 61.504-40.491 61.504Q-40.272 61.504-40.064 61.407Q-39.855 61.309-39.727 61.128Q-39.599 60.947-39.599 60.721L-39.599 59.993Q-39.924 59.993-40.289 60.084Q-40.655 60.175-40.911 60.387Q-41.168 60.598-41.168 60.930",[1870],[1854,10020,10021],{"transform":10014},[1866,10022],{"d":10023,"fill":9645,"stroke":9645,"className":10024,"style":1909},"M-33.709 62.485L-33.709 60.790Q-33.709 60.540-33.868 60.364Q-34.027 60.188-34.272 60.105Q-34.516 60.021-34.752 60.021Q-34.783 60.021-34.806 59.997Q-34.830 59.973-34.830 59.942L-34.830 59.874Q-34.830 59.843-34.806 59.819Q-34.783 59.795-34.752 59.795Q-34.355 59.795-34.032 59.594Q-33.709 59.392-33.709 59.026L-33.709 57.331Q-33.709 57.071-33.561 56.887Q-33.412 56.702-33.176 56.598Q-32.940 56.493-32.686 56.451Q-32.431 56.408-32.175 56.408L-32.106 56.408Q-32.079 56.408-32.053 56.434Q-32.028 56.459-32.028 56.487L-32.028 56.555Q-32.028 56.586-32.052 56.610Q-32.076 56.634-32.106 56.634Q-32.346 56.634-32.583 56.707Q-32.821 56.781-32.981 56.945Q-33.142 57.109-33.142 57.345L-33.142 59.040Q-33.142 59.269-33.262 59.445Q-33.381 59.621-33.574 59.734Q-33.767 59.846-33.990 59.908Q-33.767 59.970-33.574 60.082Q-33.381 60.195-33.262 60.371Q-33.142 60.547-33.142 60.776L-33.142 62.471Q-33.142 62.707-32.981 62.871Q-32.821 63.035-32.583 63.109Q-32.346 63.182-32.106 63.182Q-32.076 63.182-32.052 63.206Q-32.028 63.230-32.028 63.261L-32.028 63.329Q-32.028 63.357-32.053 63.382Q-32.079 63.408-32.106 63.408L-32.175 63.408Q-32.424 63.408-32.686 63.364Q-32.947 63.319-33.181 63.217Q-33.415 63.114-33.562 62.931Q-33.709 62.748-33.709 62.485",[1870],[1854,10026,10027],{"transform":10014},[1866,10028],{"d":10029,"fill":9645,"stroke":9645,"className":10030,"style":1909},"M-28.050 61.658L-30.580 61.658L-30.580 61.378Q-29.612 61.378-29.612 61.169L-29.612 57.550Q-30.005 57.738-30.627 57.738L-30.627 57.457Q-30.210 57.457-29.846 57.356Q-29.482 57.256-29.226 57.010L-29.100 57.010Q-29.035 57.027-29.018 57.095L-29.018 61.169Q-29.018 61.378-28.050 61.378",[1870],[1854,10032,10033],{"transform":10014},[1866,10034],{"d":10035,"fill":9645,"stroke":9645,"className":10036,"style":1909},"M-26.747 63.329L-26.747 63.261Q-26.747 63.230-26.723 63.206Q-26.700 63.182-26.669 63.182Q-26.430 63.182-26.190 63.111Q-25.951 63.039-25.789 62.877Q-25.626 62.714-25.626 62.471L-25.626 60.776Q-25.626 60.434-25.380 60.217Q-25.134 60-24.779 59.908Q-25.001 59.846-25.194 59.734Q-25.387 59.621-25.507 59.445Q-25.626 59.269-25.626 59.040L-25.626 57.345Q-25.626 57.102-25.789 56.939Q-25.951 56.777-26.190 56.705Q-26.430 56.634-26.669 56.634Q-26.700 56.634-26.723 56.610Q-26.747 56.586-26.747 56.555L-26.747 56.487Q-26.747 56.459-26.722 56.434Q-26.696 56.408-26.669 56.408L-26.600 56.408Q-26.265 56.408-25.910 56.488Q-25.555 56.569-25.307 56.777Q-25.059 56.986-25.059 57.331L-25.059 59.026Q-25.059 59.392-24.738 59.594Q-24.416 59.795-24.023 59.795Q-23.993 59.795-23.969 59.819Q-23.945 59.843-23.945 59.874L-23.945 59.942Q-23.945 59.973-23.969 59.997Q-23.993 60.021-24.023 60.021Q-24.256 60.021-24.498 60.105Q-24.741 60.188-24.900 60.364Q-25.059 60.540-25.059 60.790L-25.059 62.485Q-25.059 62.830-25.307 63.039Q-25.555 63.247-25.910 63.328Q-26.265 63.408-26.600 63.408L-26.669 63.408Q-26.696 63.408-26.722 63.382Q-26.747 63.357-26.747 63.329",[1870],[1854,10038,10039,10045],{"stroke":1860},[1854,10040,10042],{"transform":10041},"translate(162.135 -19.67)",[1866,10043],{"d":9634,"fill":1856,"stroke":1856,"className":10044,"style":1871},[1870],[1854,10046,10047],{"transform":10041},[1866,10048],{"d":10049,"fill":1856,"stroke":1856,"className":10050,"style":9642},"M-42.018 60.323Q-42.385 60.071-42.654 59.751Q-42.924 59.430-43.117 59.028Q-43.310 58.627-43.406 58.195Q-43.501 57.763-43.501 57.335Q-43.501 56.904-43.404 56.472Q-43.307 56.040-43.119 55.643Q-42.930 55.246-42.657 54.919Q-42.385 54.593-42.018 54.347Q-42.010 54.341-41.977 54.335L-41.910 54.335Q-41.840 54.356-41.840 54.420Q-41.840 54.455-41.857 54.473Q-42.273 54.807-42.537 55.268Q-42.801 55.730-42.922 56.252Q-43.044 56.775-43.044 57.335Q-43.044 57.895-42.922 58.418Q-42.801 58.940-42.537 59.402Q-42.273 59.863-41.857 60.197Q-41.840 60.224-41.840 60.250Q-41.840 60.314-41.910 60.335L-41.977 60.335Q-42.010 60.329-42.018 60.323M-38.318 58.835L-40.929 58.835L-40.929 58.650Q-40.923 58.627-40.902 58.601L-39.751 57.546Q-39.411 57.235-39.231 57.049Q-39.051 56.863-38.906 56.603Q-38.761 56.342-38.761 56.046Q-38.761 55.773-38.887 55.558Q-39.013 55.343-39.232 55.223Q-39.452 55.103-39.727 55.103Q-39.903 55.103-40.073 55.160Q-40.243 55.217-40.375 55.324Q-40.507 55.431-40.586 55.589Q-40.498 55.589-40.420 55.633Q-40.343 55.677-40.299 55.753Q-40.255 55.829-40.255 55.926Q-40.255 56.066-40.351 56.163Q-40.448 56.260-40.592 56.260Q-40.729 56.260-40.829 56.160Q-40.929 56.061-40.929 55.926Q-40.929 55.601-40.738 55.353Q-40.548 55.106-40.245 54.975Q-39.941 54.845-39.625 54.845Q-39.244 54.845-38.901 54.980Q-38.558 55.114-38.345 55.387Q-38.131 55.659-38.131 56.046Q-38.131 56.321-38.255 56.548Q-38.380 56.775-38.560 56.947Q-38.740 57.118-39.065 57.358Q-39.391 57.599-39.475 57.666L-40.231 58.270L-39.698 58.270Q-39.209 58.270-38.878 58.262Q-38.547 58.255-38.532 58.240Q-38.474 58.170-38.441 58.035Q-38.409 57.900-38.377 57.689L-38.131 57.689L-38.318 58.835M-37.076 60.335L-37.143 60.335Q-37.220 60.317-37.220 60.250Q-37.220 60.212-37.196 60.197Q-36.786 59.866-36.519 59.402Q-36.253 58.938-36.131 58.407Q-36.010 57.877-36.010 57.335Q-36.010 56.793-36.131 56.263Q-36.253 55.732-36.519 55.268Q-36.786 54.804-37.196 54.473Q-37.220 54.449-37.220 54.420Q-37.220 54.356-37.143 54.335L-37.076 54.335Q-37.047 54.341-37.041 54.347Q-36.681 54.587-36.399 54.924Q-36.118 55.261-35.932 55.650Q-35.746 56.040-35.652 56.472Q-35.558 56.904-35.558 57.335Q-35.558 57.763-35.652 58.195Q-35.746 58.627-35.935 59.021Q-36.124 59.415-36.402 59.748Q-36.681 60.080-37.041 60.323Q-37.047 60.329-37.076 60.335",[1870],[1854,10052,10053],{"fill":9645,"stroke":9645},[1854,10054,10056],{"transform":10055},"translate(146.253 -10.339)",[1866,10057],{"d":9651,"fill":9645,"stroke":9645,"className":10058,"style":9653},[1870],[1854,10060,10061],{"fill":1989,"stroke":1989},[1854,10062,10064],{"transform":10063},"translate(160.48 -10.339)",[1866,10065],{"d":9661,"fill":1989,"stroke":1989,"className":10066,"style":9653},[1870],[1854,10068,10069],{"fill":9645,"stroke":9645},[1854,10070,10072],{"transform":10071},"translate(174.706 -10.339)",[1866,10073],{"d":9670,"fill":9645,"stroke":9645,"className":10074,"style":9653},[1870],[1854,10076,10077],{"fill":9645,"stroke":9645},[1854,10078,10080],{"transform":10079},"translate(188.932 -10.339)",[1866,10081],{"d":9679,"fill":9645,"stroke":9645,"className":10082,"style":9653},[1870],[1854,10084,10085],{"fill":9645,"stroke":9645},[1854,10086,10088],{"transform":10087},"translate(134.303 1.611)",[1866,10089],{"d":9651,"fill":9645,"stroke":9645,"className":10090,"style":9653},[1870],[1854,10092,10093],{"fill":1989,"stroke":1989},[1854,10094,10096],{"transform":10095},"translate(134.303 14.415)",[1866,10097],{"d":9661,"fill":1989,"stroke":1989,"className":10098,"style":9653},[1870],[1854,10100,10101],{"fill":9645,"stroke":9645},[1854,10102,10104],{"transform":10103},"translate(134.303 27.218)",[1866,10105],{"d":9670,"fill":9645,"stroke":9645,"className":10106,"style":9653},[1870],[1854,10108,10109],{"fill":9645,"stroke":9645},[1854,10110,10112],{"transform":10111},"translate(134.303 40.022)",[1866,10113],{"d":9679,"fill":9645,"stroke":9645,"className":10114,"style":9653},[1870],[1866,10116],{"fill":1988,"stroke":1860,"d":10117},"M121.244 67.349V55.968h13.088v11.38ZM135.47 67.349V55.968h13.089v11.38ZM121.244 105.76V94.379h13.088v11.38Zm13.088-11.381",[1854,10119,10121],{"transform":10120},"translate(145.961 2.256)",[1866,10122],{"d":9718,"fill":1856,"stroke":1856,"className":10123,"style":1909},[1870],[1854,10125,10127],{"transform":10126},"translate(160.188 2.256)",[1866,10128],{"d":9510,"fill":1856,"stroke":1856,"className":10129,"style":1909},[1870],[1854,10131,10133],{"transform":10132},"translate(174.414 2.256)",[1866,10134],{"d":9993,"fill":1856,"stroke":1856,"className":10135,"style":1909},[1870],[1854,10137,10139],{"transform":10138},"translate(186.647 2.256)",[1866,10140],{"d":10000,"fill":1856,"stroke":1856,"className":10141,"style":1909},[1870],[1854,10143,10145],{"transform":10144},"translate(143.968 14.31)",[1866,10146],{"d":9737,"fill":1856,"stroke":1856,"className":10147,"style":1909},[1870],[1854,10149,10151],{"transform":10150},"translate(160.188 15.06)",[1866,10152],{"d":9718,"fill":1856,"stroke":1856,"className":10153,"style":1909},[1870],[1854,10155,10157],{"transform":10156},"translate(174.414 15.06)",[1866,10158],{"d":9500,"fill":1856,"stroke":1856,"className":10159,"style":1909},[1870],[1854,10161,10163],{"transform":10162},"translate(188.64 15.06)",[1866,10164],{"d":9625,"fill":1856,"stroke":1856,"className":10165,"style":1909},[1870],[1854,10167,10169],{"transform":10168},"translate(143.968 27.114)",[1866,10170],{"d":9737,"fill":1856,"stroke":1856,"className":10171,"style":1909},[1870],[1854,10173,10175],{"transform":10174},"translate(158.195 27.114)",[1866,10176],{"d":9737,"fill":1856,"stroke":1856,"className":10177,"style":1909},[1870],[1854,10179,10181],{"transform":10180},"translate(174.414 27.863)",[1866,10182],{"d":9718,"fill":1856,"stroke":1856,"className":10183,"style":1909},[1870],[1854,10185,10187],{"transform":10186},"translate(188.64 27.863)",[1866,10188],{"d":9490,"fill":1856,"stroke":1856,"className":10189,"style":1909},[1870],[1854,10191,10193],{"transform":10192},"translate(145.961 40.667)",[1866,10194],{"d":9500,"fill":1856,"stroke":1856,"className":10195,"style":1909},[1870],[1854,10197,10199],{"transform":10198},"translate(160.188 40.667)",[1866,10200],{"d":9993,"fill":1856,"stroke":1856,"className":10201,"style":1909},[1870],[1854,10203,10205],{"transform":10204},"translate(174.414 40.667)",[1866,10206],{"d":9625,"fill":1856,"stroke":1856,"className":10207,"style":1909},[1870],[1854,10209,10211],{"transform":10210},"translate(188.64 40.667)",[1866,10212],{"d":9718,"fill":1856,"stroke":1856,"className":10213,"style":1909},[1870],[1854,10215,10216],{"fill":9645,"stroke":9645},[1854,10217,10218,10224,10229,10234,10240,10246],{"fill":9645,"stroke":1860,"fontSize":1901},[1854,10219,10221],{"transform":10220},"translate(152.767 53.534)",[1866,10222],{"d":10017,"fill":9645,"stroke":9645,"className":10223,"style":1909},[1870],[1854,10225,10226],{"transform":10220},[1866,10227],{"d":10023,"fill":9645,"stroke":9645,"className":10228,"style":1909},[1870],[1854,10230,10231],{"transform":10220},[1866,10232],{"d":10029,"fill":9645,"stroke":9645,"className":10233,"style":1909},[1870],[1854,10235,10236],{"transform":10220},[1866,10237],{"d":10238,"fill":9645,"stroke":9645,"className":10239,"style":1909},"M-26.522 62.888Q-26.522 62.854-26.494 62.827Q-26.228 62.605-26.078 62.278Q-25.927 61.952-25.927 61.596L-25.927 61.559Q-26.030 61.658-26.201 61.658Q-26.378 61.658-26.500 61.537Q-26.621 61.415-26.621 61.238Q-26.621 61.067-26.500 60.945Q-26.378 60.824-26.201 60.824Q-25.941 60.824-25.821 61.063Q-25.702 61.303-25.702 61.596Q-25.702 62-25.872 62.369Q-26.043 62.738-26.341 62.994Q-26.371 63.015-26.395 63.015Q-26.440 63.015-26.481 62.974Q-26.522 62.933-26.522 62.888",[1870],[1854,10241,10242],{"transform":10220},[1866,10243],{"d":10244,"fill":9645,"stroke":9645,"className":10245,"style":1909},"M-20.323 61.658L-23.208 61.658L-23.208 61.456Q-23.208 61.426-23.181 61.398L-21.933 60.181Q-21.861 60.106-21.819 60.064Q-21.776 60.021-21.697 59.942Q-21.284 59.529-21.053 59.171Q-20.822 58.814-20.822 58.390Q-20.822 58.158-20.901 57.955Q-20.980 57.751-21.121 57.601Q-21.263 57.450-21.458 57.370Q-21.653 57.290-21.885 57.290Q-22.196 57.290-22.454 57.449Q-22.712 57.608-22.842 57.885L-22.822 57.885Q-22.654 57.885-22.547 57.996Q-22.439 58.107-22.439 58.271Q-22.439 58.428-22.548 58.541Q-22.658 58.654-22.822 58.654Q-22.982 58.654-23.095 58.541Q-23.208 58.428-23.208 58.271Q-23.208 57.895-23 57.608Q-22.791 57.321-22.456 57.165Q-22.121 57.010-21.766 57.010Q-21.342 57.010-20.962 57.168Q-20.583 57.327-20.349 57.644Q-20.115 57.960-20.115 58.390Q-20.115 58.701-20.255 58.970Q-20.395 59.238-20.600 59.443Q-20.805 59.648-21.168 59.930Q-21.530 60.212-21.639 60.308L-22.494 61.036L-21.851 61.036Q-21.588 61.036-21.299 61.034Q-21.010 61.033-20.792 61.024Q-20.573 61.015-20.556 60.998Q-20.494 60.933-20.457 60.766Q-20.419 60.598-20.381 60.356L-20.115 60.356",[1870],[1854,10247,10248],{"transform":10220},[1866,10249],{"d":10250,"fill":9645,"stroke":9645,"className":10251,"style":1909},"M-19.020 63.329L-19.020 63.261Q-19.020 63.230-18.996 63.206Q-18.973 63.182-18.942 63.182Q-18.703 63.182-18.463 63.111Q-18.224 63.039-18.062 62.877Q-17.899 62.714-17.899 62.471L-17.899 60.776Q-17.899 60.434-17.653 60.217Q-17.407 60-17.052 59.908Q-17.274 59.846-17.467 59.734Q-17.660 59.621-17.780 59.445Q-17.899 59.269-17.899 59.040L-17.899 57.345Q-17.899 57.102-18.062 56.939Q-18.224 56.777-18.463 56.705Q-18.703 56.634-18.942 56.634Q-18.973 56.634-18.996 56.610Q-19.020 56.586-19.020 56.555L-19.020 56.487Q-19.020 56.459-18.995 56.434Q-18.969 56.408-18.942 56.408L-18.873 56.408Q-18.538 56.408-18.183 56.488Q-17.828 56.569-17.580 56.777Q-17.332 56.986-17.332 57.331L-17.332 59.026Q-17.332 59.392-17.011 59.594Q-16.689 59.795-16.296 59.795Q-16.266 59.795-16.242 59.819Q-16.218 59.843-16.218 59.874L-16.218 59.942Q-16.218 59.973-16.242 59.997Q-16.266 60.021-16.296 60.021Q-16.529 60.021-16.771 60.105Q-17.014 60.188-17.173 60.364Q-17.332 60.540-17.332 60.790L-17.332 62.485Q-17.332 62.830-17.580 63.039Q-17.828 63.247-18.183 63.328Q-18.538 63.408-18.873 63.408L-18.942 63.408Q-18.969 63.408-18.995 63.382Q-19.020 63.357-19.020 63.329",[1870],[1854,10253,10254,10260],{"stroke":1860},[1854,10255,10257],{"transform":10256},"translate(236.112 -19.67)",[1866,10258],{"d":9634,"fill":1856,"stroke":1856,"className":10259,"style":1871},[1870],[1854,10261,10262],{"transform":10256},[1866,10263],{"d":10264,"fill":1856,"stroke":1856,"className":10265,"style":9642},"M-42.018 60.323Q-42.385 60.071-42.654 59.751Q-42.924 59.430-43.117 59.028Q-43.310 58.627-43.406 58.195Q-43.501 57.763-43.501 57.335Q-43.501 56.904-43.404 56.472Q-43.307 56.040-43.119 55.643Q-42.930 55.246-42.657 54.919Q-42.385 54.593-42.018 54.347Q-42.010 54.341-41.977 54.335L-41.910 54.335Q-41.840 54.356-41.840 54.420Q-41.840 54.455-41.857 54.473Q-42.273 54.807-42.537 55.268Q-42.801 55.730-42.922 56.252Q-43.044 56.775-43.044 57.335Q-43.044 57.895-42.922 58.418Q-42.801 58.940-42.537 59.402Q-42.273 59.863-41.857 60.197Q-41.840 60.224-41.840 60.250Q-41.840 60.314-41.910 60.335L-41.977 60.335Q-42.010 60.329-42.018 60.323M-40.586 58.384Q-40.290 58.721-39.560 58.721Q-39.303 58.721-39.122 58.593Q-38.942 58.466-38.854 58.258Q-38.766 58.050-38.766 57.792Q-38.766 57.397-38.973 57.126Q-39.180 56.855-39.566 56.855L-40.032 56.855Q-40.097 56.840-40.111 56.778L-40.111 56.711Q-40.097 56.655-40.032 56.638L-39.631 56.614Q-39.420 56.614-39.251 56.472Q-39.083 56.330-38.991 56.116Q-38.898 55.902-38.898 55.686Q-38.898 55.398-39.083 55.233Q-39.267 55.067-39.560 55.067Q-39.821 55.067-40.045 55.135Q-40.269 55.202-40.416 55.360Q-40.287 55.378-40.208 55.467Q-40.129 55.557-40.129 55.686Q-40.129 55.823-40.224 55.918Q-40.319 56.014-40.460 56.014Q-40.595 56.014-40.691 55.917Q-40.788 55.820-40.788 55.686Q-40.788 55.398-40.598 55.207Q-40.407 55.015-40.126 54.930Q-39.845 54.845-39.560 54.845Q-39.285 54.845-38.985 54.936Q-38.684 55.026-38.476 55.215Q-38.268 55.404-38.268 55.686Q-38.268 56.055-38.515 56.327Q-38.761 56.600-39.133 56.729Q-38.714 56.822-38.396 57.105Q-38.078 57.388-38.078 57.786Q-38.078 58.149-38.298 58.415Q-38.517 58.680-38.863 58.820Q-39.209 58.961-39.560 58.961Q-39.783 58.961-40.031 58.913Q-40.278 58.864-40.498 58.754Q-40.718 58.645-40.849 58.466Q-40.981 58.287-40.981 58.032Q-40.981 57.883-40.879 57.780Q-40.776 57.678-40.627 57.678Q-40.477 57.678-40.375 57.780Q-40.272 57.883-40.272 58.032Q-40.272 58.164-40.362 58.265Q-40.451 58.366-40.586 58.384M-37.076 60.335L-37.143 60.335Q-37.220 60.317-37.220 60.250Q-37.220 60.212-37.196 60.197Q-36.786 59.866-36.519 59.402Q-36.253 58.938-36.131 58.407Q-36.010 57.877-36.010 57.335Q-36.010 56.793-36.131 56.263Q-36.253 55.732-36.519 55.268Q-36.786 54.804-37.196 54.473Q-37.220 54.449-37.220 54.420Q-37.220 54.356-37.143 54.335L-37.076 54.335Q-37.047 54.341-37.041 54.347Q-36.681 54.587-36.399 54.924Q-36.118 55.261-35.932 55.650Q-35.746 56.040-35.652 56.472Q-35.558 56.904-35.558 57.335Q-35.558 57.763-35.652 58.195Q-35.746 58.627-35.935 59.021Q-36.124 59.415-36.402 59.748Q-36.681 60.080-37.041 60.323Q-37.047 60.329-37.076 60.335",[1870],[1854,10267,10268],{"fill":9645,"stroke":9645},[1854,10269,10271],{"transform":10270},"translate(220.23 -10.339)",[1866,10272],{"d":9651,"fill":9645,"stroke":9645,"className":10273,"style":9653},[1870],[1854,10275,10276],{"fill":9645,"stroke":9645},[1854,10277,10279],{"transform":10278},"translate(234.456 -10.339)",[1866,10280],{"d":9661,"fill":9645,"stroke":9645,"className":10281,"style":9653},[1870],[1854,10283,10284],{"fill":1989,"stroke":1989},[1854,10285,10287],{"transform":10286},"translate(248.683 -10.339)",[1866,10288],{"d":9670,"fill":1989,"stroke":1989,"className":10289,"style":9653},[1870],[1854,10291,10292],{"fill":9645,"stroke":9645},[1854,10293,10295],{"transform":10294},"translate(262.91 -10.339)",[1866,10296],{"d":9679,"fill":9645,"stroke":9645,"className":10297,"style":9653},[1870],[1854,10299,10300],{"fill":9645,"stroke":9645},[1854,10301,10303],{"transform":10302},"translate(208.28 1.611)",[1866,10304],{"d":9651,"fill":9645,"stroke":9645,"className":10305,"style":9653},[1870],[1854,10307,10308],{"fill":9645,"stroke":9645},[1854,10309,10311],{"transform":10310},"translate(208.28 14.415)",[1866,10312],{"d":9661,"fill":9645,"stroke":9645,"className":10313,"style":9653},[1870],[1854,10315,10316],{"fill":1989,"stroke":1989},[1854,10317,10319],{"transform":10318},"translate(208.28 27.218)",[1866,10320],{"d":9670,"fill":1989,"stroke":1989,"className":10321,"style":9653},[1870],[1854,10323,10324],{"fill":9645,"stroke":9645},[1854,10325,10327],{"transform":10326},"translate(208.28 40.022)",[1866,10328],{"d":9679,"fill":9645,"stroke":9645,"className":10329,"style":9653},[1870],[1866,10331],{"fill":1988,"stroke":1860,"d":10332},"M209.448 67.349V55.968h13.088v11.38ZM209.448 80.152v-11.38h13.088v11.38Zm13.088-11.38",[1854,10334,10336],{"transform":10335},"translate(219.938 2.256)",[1866,10337],{"d":9718,"fill":1856,"stroke":1856,"className":10338,"style":1909},[1870],[1854,10340,10342],{"transform":10341},"translate(234.165 2.256)",[1866,10343],{"d":9510,"fill":1856,"stroke":1856,"className":10344,"style":1909},[1870],[1854,10346,10348],{"transform":10347},"translate(248.391 2.256)",[1866,10349],{"d":9993,"fill":1856,"stroke":1856,"className":10350,"style":1909},[1870],[1854,10352,10354],{"transform":10353},"translate(262.618 2.256)",[1866,10355],{"d":10356,"fill":1856,"stroke":1856,"className":10357,"style":1909},"M-46.630 61.798Q-47.088 61.798-47.406 61.583Q-47.723 61.367-47.905 61.015Q-48.086 60.663-48.163 60.243Q-48.240 59.823-48.240 59.395Q-48.240 58.811-47.987 58.255Q-47.734 57.700-47.264 57.355Q-46.794 57.010-46.196 57.010Q-45.786 57.010-45.502 57.208Q-45.218 57.406-45.218 57.809Q-45.218 57.905-45.264 57.984Q-45.310 58.062-45.391 58.107Q-45.471 58.151-45.560 58.151Q-45.707 58.151-45.808 58.054Q-45.909 57.956-45.909 57.809Q-45.909 57.679-45.818 57.572Q-45.727 57.464-45.594 57.464Q-45.782 57.242-46.196 57.242Q-46.510 57.242-46.784 57.406Q-47.057 57.570-47.224 57.844Q-47.412 58.134-47.477 58.500Q-47.542 58.866-47.542 59.320Q-47.392 59.026-47.127 58.848Q-46.862 58.671-46.548 58.671Q-46.117 58.671-45.768 58.877Q-45.420 59.084-45.220 59.440Q-45.020 59.795-45.020 60.222Q-45.020 60.667-45.237 61.027Q-45.454 61.388-45.827 61.593Q-46.199 61.798-46.630 61.798M-46.630 61.545Q-46.254 61.545-46.050 61.362Q-45.847 61.179-45.784 60.896Q-45.721 60.612-45.721 60.222Q-45.721 59.836-45.775 59.556Q-45.830 59.276-46.025 59.084Q-46.220 58.893-46.589 58.893Q-46.879 58.893-47.091 59.069Q-47.303 59.245-47.411 59.518Q-47.518 59.792-47.518 60.075L-47.518 60.216L-47.518 60.257Q-47.518 60.762-47.306 61.154Q-47.095 61.545-46.630 61.545",[1870],[1854,10359,10361],{"transform":10360},"translate(217.945 14.31)",[1866,10362],{"d":9737,"fill":1856,"stroke":1856,"className":10363,"style":1909},[1870],[1854,10365,10367],{"transform":10366},"translate(234.165 15.06)",[1866,10368],{"d":9718,"fill":1856,"stroke":1856,"className":10369,"style":1909},[1870],[1854,10371,10373],{"transform":10372},"translate(248.391 15.06)",[1866,10374],{"d":9500,"fill":1856,"stroke":1856,"className":10375,"style":1909},[1870],[1854,10377,10379],{"transform":10378},"translate(262.618 15.06)",[1866,10380],{"d":9510,"fill":1856,"stroke":1856,"className":10381,"style":1909},[1870],[1854,10383,10385],{"transform":10384},"translate(217.945 27.114)",[1866,10386],{"d":9737,"fill":1856,"stroke":1856,"className":10387,"style":1909},[1870],[1854,10389,10391],{"transform":10390},"translate(232.172 27.114)",[1866,10392],{"d":9737,"fill":1856,"stroke":1856,"className":10393,"style":1909},[1870],[1854,10395,10397],{"transform":10396},"translate(248.391 27.863)",[1866,10398],{"d":9718,"fill":1856,"stroke":1856,"className":10399,"style":1909},[1870],[1854,10401,10403],{"transform":10402},"translate(262.618 27.863)",[1866,10404],{"d":9490,"fill":1856,"stroke":1856,"className":10405,"style":1909},[1870],[1854,10407,10409],{"transform":10408},"translate(219.938 40.667)",[1866,10410],{"d":9500,"fill":1856,"stroke":1856,"className":10411,"style":1909},[1870],[1854,10413,10415],{"transform":10414},"translate(234.165 40.667)",[1866,10416],{"d":9993,"fill":1856,"stroke":1856,"className":10417,"style":1909},[1870],[1854,10419,10421],{"transform":10420},"translate(248.391 40.667)",[1866,10422],{"d":9625,"fill":1856,"stroke":1856,"className":10423,"style":1909},[1870],[1854,10425,10427],{"transform":10426},"translate(262.618 40.667)",[1866,10428],{"d":9718,"fill":1856,"stroke":1856,"className":10429,"style":1909},[1870],[1854,10431,10432],{"fill":9645,"stroke":9645},[1854,10433,10434,10440,10445,10450,10455,10460,10466,10472],{"fill":9645,"stroke":1860,"fontSize":1901},[1854,10435,10437],{"transform":10436},"translate(222.88 53.534)",[1866,10438],{"d":10017,"fill":9645,"stroke":9645,"className":10439,"style":1909},[1870],[1854,10441,10442],{"transform":10436},[1866,10443],{"d":10023,"fill":9645,"stroke":9645,"className":10444,"style":1909},[1870],[1854,10446,10447],{"transform":10436},[1866,10448],{"d":10029,"fill":9645,"stroke":9645,"className":10449,"style":1909},[1870],[1854,10451,10452],{"transform":10436},[1866,10453],{"d":10238,"fill":9645,"stroke":9645,"className":10454,"style":1909},[1870],[1854,10456,10457],{"transform":10436},[1866,10458],{"d":10244,"fill":9645,"stroke":9645,"className":10459,"style":1909},[1870],[1854,10461,10462],{"transform":10436},[1866,10463],{"d":10464,"fill":9645,"stroke":9645,"className":10465,"style":1909},"M-18.795 62.888Q-18.795 62.854-18.767 62.827Q-18.501 62.605-18.351 62.278Q-18.200 61.952-18.200 61.596L-18.200 61.559Q-18.303 61.658-18.474 61.658Q-18.651 61.658-18.773 61.537Q-18.894 61.415-18.894 61.238Q-18.894 61.067-18.773 60.945Q-18.651 60.824-18.474 60.824Q-18.214 60.824-18.094 61.063Q-17.975 61.303-17.975 61.596Q-17.975 62-18.145 62.369Q-18.316 62.738-18.614 62.994Q-18.644 63.015-18.668 63.015Q-18.713 63.015-18.754 62.974Q-18.795 62.933-18.795 62.888",[1870],[1854,10467,10468],{"transform":10436},[1866,10469],{"d":10470,"fill":9645,"stroke":9645,"className":10471,"style":1909},"M-15.126 61.111Q-15.006 61.268-14.815 61.367Q-14.623 61.467-14.408 61.506Q-14.193 61.545-13.970 61.545Q-13.673 61.545-13.478 61.390Q-13.283 61.234-13.193 60.980Q-13.102 60.725-13.102 60.441Q-13.102 60.147-13.194 59.896Q-13.287 59.645-13.485 59.489Q-13.683 59.334-13.977 59.334L-14.493 59.334Q-14.521 59.334-14.546 59.308Q-14.572 59.283-14.572 59.259L-14.572 59.187Q-14.572 59.156-14.546 59.134Q-14.521 59.112-14.493 59.112L-14.052 59.081Q-13.690 59.081-13.470 58.724Q-13.249 58.366-13.249 57.977Q-13.249 57.649-13.444 57.445Q-13.639 57.242-13.970 57.242Q-14.257 57.242-14.510 57.326Q-14.763 57.409-14.927 57.597Q-14.780 57.597-14.680 57.712Q-14.579 57.826-14.579 57.977Q-14.579 58.127-14.685 58.237Q-14.791 58.346-14.948 58.346Q-15.109 58.346-15.218 58.237Q-15.327 58.127-15.327 57.977Q-15.327 57.652-15.119 57.433Q-14.910 57.215-14.594 57.112Q-14.278 57.010-13.970 57.010Q-13.652 57.010-13.324 57.114Q-12.996 57.218-12.769 57.440Q-12.542 57.662-12.542 57.977Q-12.542 58.411-12.829 58.736Q-13.116 59.060-13.550 59.207Q-13.239 59.272-12.959 59.438Q-12.678 59.604-12.501 59.862Q-12.323 60.120-12.323 60.441Q-12.323 60.851-12.567 61.161Q-12.812 61.470-13.193 61.634Q-13.574 61.798-13.970 61.798Q-14.339 61.798-14.697 61.685Q-15.054 61.573-15.298 61.323Q-15.543 61.074-15.543 60.704Q-15.543 60.533-15.426 60.421Q-15.310 60.308-15.139 60.308Q-15.030 60.308-14.939 60.359Q-14.849 60.410-14.794 60.503Q-14.739 60.595-14.739 60.704Q-14.739 60.872-14.852 60.991Q-14.965 61.111-15.126 61.111",[1870],[1854,10473,10474],{"transform":10436},[1866,10475],{"d":10476,"fill":9645,"stroke":9645,"className":10477,"style":1909},"M-11.293 63.329L-11.293 63.261Q-11.293 63.230-11.269 63.206Q-11.246 63.182-11.215 63.182Q-10.976 63.182-10.736 63.111Q-10.497 63.039-10.335 62.877Q-10.172 62.714-10.172 62.471L-10.172 60.776Q-10.172 60.434-9.926 60.217Q-9.680 60-9.325 59.908Q-9.547 59.846-9.740 59.734Q-9.933 59.621-10.053 59.445Q-10.172 59.269-10.172 59.040L-10.172 57.345Q-10.172 57.102-10.335 56.939Q-10.497 56.777-10.736 56.705Q-10.976 56.634-11.215 56.634Q-11.246 56.634-11.269 56.610Q-11.293 56.586-11.293 56.555L-11.293 56.487Q-11.293 56.459-11.268 56.434Q-11.242 56.408-11.215 56.408L-11.146 56.408Q-10.811 56.408-10.456 56.488Q-10.101 56.569-9.853 56.777Q-9.605 56.986-9.605 57.331L-9.605 59.026Q-9.605 59.392-9.284 59.594Q-8.962 59.795-8.569 59.795Q-8.539 59.795-8.515 59.819Q-8.491 59.843-8.491 59.874L-8.491 59.942Q-8.491 59.973-8.515 59.997Q-8.539 60.021-8.569 60.021Q-8.802 60.021-9.044 60.105Q-9.287 60.188-9.446 60.364Q-9.605 60.540-9.605 60.790L-9.605 62.485Q-9.605 62.830-9.853 63.039Q-10.101 63.247-10.456 63.328Q-10.811 63.408-11.146 63.408L-11.215 63.408Q-11.242 63.408-11.268 63.382Q-11.293 63.357-11.293 63.329",[1870],[1854,10479,10480,10486],{"stroke":1860},[1854,10481,10483],{"transform":10482},"translate(310.09 -19.67)",[1866,10484],{"d":9634,"fill":1856,"stroke":1856,"className":10485,"style":1871},[1870],[1854,10487,10488],{"transform":10482},[1866,10489],{"d":10490,"fill":1856,"stroke":1856,"className":10491,"style":9642},"M-42.018 60.323Q-42.385 60.071-42.654 59.751Q-42.924 59.430-43.117 59.028Q-43.310 58.627-43.406 58.195Q-43.501 57.763-43.501 57.335Q-43.501 56.904-43.404 56.472Q-43.307 56.040-43.119 55.643Q-42.930 55.246-42.657 54.919Q-42.385 54.593-42.018 54.347Q-42.010 54.341-41.977 54.335L-41.910 54.335Q-41.840 54.356-41.840 54.420Q-41.840 54.455-41.857 54.473Q-42.273 54.807-42.537 55.268Q-42.801 55.730-42.922 56.252Q-43.044 56.775-43.044 57.335Q-43.044 57.895-42.922 58.418Q-42.801 58.940-42.537 59.402Q-42.273 59.863-41.857 60.197Q-41.840 60.224-41.840 60.250Q-41.840 60.314-41.910 60.335L-41.977 60.335Q-42.010 60.329-42.018 60.323M-39.229 57.856L-41.084 57.856L-41.084 57.599L-38.989 54.892Q-38.951 54.845-38.892 54.845L-38.761 54.845Q-38.720 54.845-38.692 54.873Q-38.664 54.900-38.664 54.941L-38.664 57.599L-37.975 57.599L-37.975 57.856L-38.664 57.856L-38.664 58.404Q-38.664 58.577-37.987 58.577L-37.987 58.835L-39.906 58.835L-39.906 58.577Q-39.229 58.577-39.229 58.404L-39.229 57.856M-39.188 55.516L-40.794 57.599L-39.188 57.599L-39.188 55.516M-37.076 60.335L-37.143 60.335Q-37.220 60.317-37.220 60.250Q-37.220 60.212-37.196 60.197Q-36.786 59.866-36.519 59.402Q-36.253 58.938-36.131 58.407Q-36.010 57.877-36.010 57.335Q-36.010 56.793-36.131 56.263Q-36.253 55.732-36.519 55.268Q-36.786 54.804-37.196 54.473Q-37.220 54.449-37.220 54.420Q-37.220 54.356-37.143 54.335L-37.076 54.335Q-37.047 54.341-37.041 54.347Q-36.681 54.587-36.399 54.924Q-36.118 55.261-35.932 55.650Q-35.746 56.040-35.652 56.472Q-35.558 56.904-35.558 57.335Q-35.558 57.763-35.652 58.195Q-35.746 58.627-35.935 59.021Q-36.124 59.415-36.402 59.748Q-36.681 60.080-37.041 60.323Q-37.047 60.329-37.076 60.335",[1870],[1854,10493,10494],{"fill":9645,"stroke":9645},[1854,10495,10497],{"transform":10496},"translate(294.207 -10.339)",[1866,10498],{"d":9651,"fill":9645,"stroke":9645,"className":10499,"style":9653},[1870],[1854,10501,10502],{"fill":9645,"stroke":9645},[1854,10503,10505],{"transform":10504},"translate(308.434 -10.339)",[1866,10506],{"d":9661,"fill":9645,"stroke":9645,"className":10507,"style":9653},[1870],[1854,10509,10510],{"fill":9645,"stroke":9645},[1854,10511,10513],{"transform":10512},"translate(322.66 -10.339)",[1866,10514],{"d":9670,"fill":9645,"stroke":9645,"className":10515,"style":9653},[1870],[1854,10517,10518],{"fill":1989,"stroke":1989},[1854,10519,10521],{"transform":10520},"translate(336.886 -10.339)",[1866,10522],{"d":9679,"fill":1989,"stroke":1989,"className":10523,"style":9653},[1870],[1854,10525,10526],{"fill":9645,"stroke":9645},[1854,10527,10529],{"transform":10528},"translate(282.257 1.611)",[1866,10530],{"d":9651,"fill":9645,"stroke":9645,"className":10531,"style":9653},[1870],[1854,10533,10534],{"fill":9645,"stroke":9645},[1854,10535,10537],{"transform":10536},"translate(282.257 14.415)",[1866,10538],{"d":9661,"fill":9645,"stroke":9645,"className":10539,"style":9653},[1870],[1854,10541,10542],{"fill":9645,"stroke":9645},[1854,10543,10545],{"transform":10544},"translate(282.257 27.218)",[1866,10546],{"d":9670,"fill":9645,"stroke":9645,"className":10547,"style":9653},[1870],[1854,10549,10550],{"fill":1989,"stroke":1989},[1854,10551,10553],{"transform":10552},"translate(282.257 40.022)",[1866,10554],{"d":9679,"fill":1989,"stroke":1989,"className":10555,"style":9653},[1870],[1866,10557],{"fill":1988,"stroke":1860,"d":10558},"M240.746 80.152v-11.38h13.088v11.38ZM240.746 92.956V81.575h13.088v11.38ZM254.972 92.956V81.575h13.088v11.38Zm13.088-11.381",[1854,10560,10562],{"transform":10561},"translate(293.916 2.256)",[1866,10563],{"d":9718,"fill":1856,"stroke":1856,"className":10564,"style":1909},[1870],[1854,10566,10568],{"transform":10567},"translate(308.142 2.256)",[1866,10569],{"d":9510,"fill":1856,"stroke":1856,"className":10570,"style":1909},[1870],[1854,10572,10574],{"transform":10573},"translate(322.368 2.256)",[1866,10575],{"d":9993,"fill":1856,"stroke":1856,"className":10576,"style":1909},[1870],[1854,10578,10580],{"transform":10579},"translate(336.595 2.256)",[1866,10581],{"d":10356,"fill":1856,"stroke":1856,"className":10582,"style":1909},[1870],[1854,10584,10586],{"transform":10585},"translate(293.916 15.06)",[1866,10587],{"d":9993,"fill":1856,"stroke":1856,"className":10588,"style":1909},[1870],[1854,10590,10592],{"transform":10591},"translate(308.142 15.06)",[1866,10593],{"d":9718,"fill":1856,"stroke":1856,"className":10594,"style":1909},[1870],[1854,10596,10598],{"transform":10597},"translate(322.368 15.06)",[1866,10599],{"d":9500,"fill":1856,"stroke":1856,"className":10600,"style":1909},[1870],[1854,10602,10604],{"transform":10603},"translate(336.595 15.06)",[1866,10605],{"d":9510,"fill":1856,"stroke":1856,"className":10606,"style":1909},[1870],[1854,10608,10610],{"transform":10609},"translate(293.916 27.863)",[1866,10611],{"d":9510,"fill":1856,"stroke":1856,"className":10612,"style":1909},[1870],[1854,10614,10616],{"transform":10615},"translate(308.142 27.863)",[1866,10617],{"d":10356,"fill":1856,"stroke":1856,"className":10618,"style":1909},[1870],[1854,10620,10622],{"transform":10621},"translate(322.368 27.863)",[1866,10623],{"d":9718,"fill":1856,"stroke":1856,"className":10624,"style":1909},[1870],[1854,10626,10628],{"transform":10627},"translate(336.595 27.863)",[1866,10629],{"d":9490,"fill":1856,"stroke":1856,"className":10630,"style":1909},[1870],[1854,10632,10634],{"transform":10633},"translate(293.916 40.667)",[1866,10635],{"d":9500,"fill":1856,"stroke":1856,"className":10636,"style":1909},[1870],[1854,10638,10640],{"transform":10639},"translate(308.142 40.667)",[1866,10641],{"d":9993,"fill":1856,"stroke":1856,"className":10642,"style":1909},[1870],[1854,10644,10646],{"transform":10645},"translate(322.368 40.667)",[1866,10647],{"d":9625,"fill":1856,"stroke":1856,"className":10648,"style":1909},[1870],[1854,10650,10652],{"transform":10651},"translate(336.595 40.667)",[1866,10653],{"d":9718,"fill":1856,"stroke":1856,"className":10654,"style":1909},[1870],[1854,10656,10657],{"fill":9645,"stroke":9645},[1854,10658,10659,10665,10670,10675,10680,10686,10692,10698,10704,10710],{"fill":9645,"stroke":1860,"fontSize":1901},[1854,10660,10662],{"transform":10661},"translate(293.24 53.534)",[1866,10663],{"d":10017,"fill":9645,"stroke":9645,"className":10664,"style":1909},[1870],[1854,10666,10667],{"transform":10661},[1866,10668],{"d":10023,"fill":9645,"stroke":9645,"className":10669,"style":1909},[1870],[1854,10671,10672],{"transform":10661},[1866,10673],{"d":10029,"fill":9645,"stroke":9645,"className":10674,"style":1909},[1870],[1854,10676,10677],{"transform":10661},[1866,10678],{"d":10238,"fill":9645,"stroke":9645,"className":10679,"style":1909},[1870],[1854,10681,10682],{"transform":10661},[1866,10683],{"d":10684,"fill":9645,"stroke":9645,"className":10685,"style":1909},"M-22.880 61.238Q-22.880 61.070-22.753 60.947Q-22.627 60.824-22.460 60.824Q-22.292 60.824-22.169 60.947Q-22.046 61.070-22.046 61.238Q-22.046 61.412-22.169 61.535Q-22.292 61.658-22.460 61.658Q-22.627 61.658-22.753 61.535Q-22.880 61.412-22.880 61.238",[1870],[1854,10687,10688],{"transform":10661},[1866,10689],{"d":10690,"fill":9645,"stroke":9645,"className":10691,"style":1909},"M-19.139 61.238Q-19.139 61.070-19.012 60.947Q-18.886 60.824-18.719 60.824Q-18.551 60.824-18.428 60.947Q-18.305 61.070-18.305 61.238Q-18.305 61.412-18.428 61.535Q-18.551 61.658-18.719 61.658Q-18.886 61.658-19.012 61.535Q-19.139 61.412-19.139 61.238",[1870],[1854,10693,10694],{"transform":10661},[1866,10695],{"d":10696,"fill":9645,"stroke":9645,"className":10697,"style":1909},"M-15.398 61.238Q-15.398 61.070-15.271 60.947Q-15.145 60.824-14.978 60.824Q-14.810 60.824-14.687 60.947Q-14.564 61.070-14.564 61.238Q-14.564 61.412-14.687 61.535Q-14.810 61.658-14.978 61.658Q-15.145 61.658-15.271 61.535Q-15.398 61.412-15.398 61.238",[1870],[1854,10699,10700],{"transform":10661},[1866,10701],{"d":10702,"fill":9645,"stroke":9645,"className":10703,"style":1909},"M-11.559 62.888Q-11.559 62.854-11.531 62.827Q-11.265 62.605-11.114 62.278Q-10.964 61.952-10.964 61.596L-10.964 61.559Q-11.067 61.658-11.238 61.658Q-11.415 61.658-11.537 61.537Q-11.658 61.415-11.658 61.238Q-11.658 61.067-11.537 60.945Q-11.415 60.824-11.238 60.824Q-10.978 60.824-10.858 61.063Q-10.739 61.303-10.739 61.596Q-10.739 62-10.909 62.369Q-11.080 62.738-11.378 62.994Q-11.408 63.015-11.432 63.015Q-11.477 63.015-11.518 62.974Q-11.559 62.933-11.559 62.888",[1870],[1854,10705,10706],{"transform":10661},[1866,10707],{"d":10708,"fill":9645,"stroke":9645,"className":10709,"style":1909},"M-6.369 60.510L-8.413 60.510L-8.413 60.229L-6.082 57.057Q-6.047 57.010-5.982 57.010L-5.846 57.010Q-5.801 57.010-5.774 57.037Q-5.747 57.064-5.747 57.109L-5.747 60.229L-4.984 60.229L-4.984 60.510L-5.747 60.510L-5.747 61.169Q-5.747 61.378-4.991 61.378L-4.991 61.658L-7.124 61.658L-7.124 61.378Q-6.369 61.378-6.369 61.169L-6.369 60.510M-6.321 57.785L-8.112 60.229L-6.321 60.229",[1870],[1854,10711,10712],{"transform":10661},[1866,10713],{"d":10714,"fill":9645,"stroke":9645,"className":10715,"style":1909},"M-4.057 63.329L-4.057 63.261Q-4.057 63.230-4.033 63.206Q-4.010 63.182-3.979 63.182Q-3.740 63.182-3.500 63.111Q-3.261 63.039-3.099 62.877Q-2.936 62.714-2.936 62.471L-2.936 60.776Q-2.936 60.434-2.690 60.217Q-2.444 60-2.089 59.908Q-2.311 59.846-2.504 59.734Q-2.697 59.621-2.817 59.445Q-2.936 59.269-2.936 59.040L-2.936 57.345Q-2.936 57.102-3.099 56.939Q-3.261 56.777-3.500 56.705Q-3.740 56.634-3.979 56.634Q-4.010 56.634-4.033 56.610Q-4.057 56.586-4.057 56.555L-4.057 56.487Q-4.057 56.459-4.032 56.434Q-4.006 56.408-3.979 56.408L-3.910 56.408Q-3.575 56.408-3.220 56.488Q-2.865 56.569-2.617 56.777Q-2.369 56.986-2.369 57.331L-2.369 59.026Q-2.369 59.392-2.048 59.594Q-1.726 59.795-1.333 59.795Q-1.303 59.795-1.279 59.819Q-1.255 59.843-1.255 59.874L-1.255 59.942Q-1.255 59.973-1.279 59.997Q-1.303 60.021-1.333 60.021Q-1.566 60.021-1.808 60.105Q-2.051 60.188-2.210 60.364Q-2.369 60.540-2.369 60.790L-2.369 62.485Q-2.369 62.830-2.617 63.039Q-2.865 63.247-3.220 63.328Q-3.575 63.408-3.910 63.408L-3.979 63.408Q-4.006 63.408-4.032 63.382Q-4.057 63.357-4.057 63.329",[1870],[1854,10717,10718,10725,10731,10737,10743,10749,10755,10761,10767,10773,10779,10785,10791,10797,10803,10809,10815,10821,10827,10833,10839,10845,10851,10857,10863,10869,10875,10881,10887,10893,10899,10905,10911,10917,10923,10929,10935],{"stroke":1860,"fontSize":1901},[1854,10719,10721],{"transform":10720},"translate(8.157 -38.546)",[1866,10722],{"d":10723,"fill":1856,"stroke":1856,"className":10724,"style":1909},"M-8.388 53.658L-10.592 53.658L-10.592 53.378Q-9.833 53.378-9.833 53.169L-9.833 49.368Q-9.833 49.157-10.592 49.157L-10.592 48.876L-8.388 48.876L-8.388 49.157Q-9.143 49.157-9.143 49.368L-9.143 53.169Q-9.143 53.378-8.388 53.378L-8.388 53.658M-6.053 53.658L-7.687 53.658L-7.687 53.378Q-7.458 53.378-7.309 53.344Q-7.161 53.309-7.161 53.169L-7.161 51.320Q-7.161 51.050-7.268 50.989Q-7.376 50.927-7.687 50.927L-7.687 50.647L-6.627 50.572L-6.627 51.221Q-6.456 50.913-6.152 50.742Q-5.848 50.572-5.503 50.572Q-4.997 50.572-4.713 50.795Q-4.430 51.019-4.430 51.515L-4.430 53.169Q-4.430 53.306-4.281 53.342Q-4.132 53.378-3.907 53.378L-3.907 53.658L-5.537 53.658L-5.537 53.378Q-5.308 53.378-5.159 53.344Q-5.011 53.309-5.011 53.169L-5.011 51.529Q-5.011 51.194-5.130 50.994Q-5.250 50.794-5.564 50.794Q-5.834 50.794-6.069 50.930Q-6.303 51.067-6.441 51.301Q-6.580 51.535-6.580 51.809L-6.580 53.169Q-6.580 53.306-6.429 53.342Q-6.279 53.378-6.053 53.378L-6.053 53.658M-3.319 52.147Q-3.319 51.809-3.179 51.518Q-3.038 51.228-2.794 51.014Q-2.550 50.801-2.246 50.686Q-1.941 50.572-1.617 50.572Q-1.347 50.572-1.083 50.671Q-0.820 50.770-0.629 50.948L-0.629 49.550Q-0.629 49.280-0.736 49.218Q-0.844 49.157-1.155 49.157L-1.155 48.876L-0.079 48.801L-0.079 52.985Q-0.079 53.173-0.024 53.256Q0.031 53.340 0.132 53.359Q0.233 53.378 0.448 53.378L0.448 53.658L-0.660 53.726L-0.660 53.309Q-1.077 53.726-1.702 53.726Q-2.133 53.726-2.505 53.514Q-2.878 53.303-3.098 52.942Q-3.319 52.581-3.319 52.147M-1.644 53.504Q-1.435 53.504-1.249 53.432Q-1.063 53.361-0.909 53.224Q-0.755 53.087-0.660 52.909L-0.660 51.300Q-0.745 51.153-0.890 51.033Q-1.036 50.913-1.205 50.854Q-1.374 50.794-1.555 50.794Q-2.116 50.794-2.384 51.183Q-2.652 51.573-2.652 52.154Q-2.652 52.725-2.418 53.115Q-2.184 53.504-1.644 53.504M2.714 53.658L1.162 53.658L1.162 53.378Q1.388 53.378 1.536 53.344Q1.685 53.309 1.685 53.169L1.685 51.320Q1.685 51.132 1.637 51.048Q1.589 50.965 1.492 50.946Q1.395 50.927 1.183 50.927L1.183 50.647L2.239 50.572L2.239 53.169Q2.239 53.309 2.370 53.344Q2.502 53.378 2.714 53.378L2.714 53.658M1.442 49.351Q1.442 49.180 1.566 49.061Q1.689 48.941 1.859 48.941Q2.027 48.941 2.150 49.061Q2.273 49.180 2.273 49.351Q2.273 49.526 2.150 49.649Q2.027 49.772 1.859 49.772Q1.689 49.772 1.566 49.649Q1.442 49.526 1.442 49.351M3.360 52.147Q3.360 51.819 3.495 51.518Q3.630 51.218 3.866 50.997Q4.102 50.777 4.406 50.657Q4.710 50.537 5.035 50.537Q5.541 50.537 5.889 50.640Q6.238 50.742 6.238 51.118Q6.238 51.265 6.140 51.366Q6.043 51.467 5.896 51.467Q5.742 51.467 5.643 51.368Q5.544 51.269 5.544 51.118Q5.544 50.930 5.684 50.838Q5.483 50.787 5.042 50.787Q4.686 50.787 4.457 50.983Q4.228 51.180 4.127 51.489Q4.026 51.799 4.026 52.147Q4.026 52.496 4.153 52.802Q4.279 53.108 4.534 53.292Q4.789 53.477 5.144 53.477Q5.366 53.477 5.551 53.393Q5.735 53.309 5.870 53.154Q6.005 52.998 6.064 52.790Q6.077 52.735 6.132 52.735L6.245 52.735Q6.275 52.735 6.298 52.759Q6.320 52.783 6.320 52.817L6.320 52.838Q6.234 53.125 6.046 53.323Q5.858 53.521 5.594 53.624Q5.329 53.726 5.035 53.726Q4.604 53.726 4.216 53.520Q3.828 53.313 3.594 52.950Q3.360 52.588 3.360 52.147M6.867 52.123Q6.867 51.802 6.992 51.513Q7.116 51.224 7.342 51.001Q7.567 50.777 7.863 50.657Q8.159 50.537 8.477 50.537Q8.805 50.537 9.066 50.637Q9.328 50.736 9.504 50.918Q9.680 51.101 9.774 51.359Q9.868 51.617 9.868 51.949Q9.868 52.041 9.786 52.062L7.530 52.062L7.530 52.123Q7.530 52.711 7.814 53.094Q8.097 53.477 8.665 53.477Q8.986 53.477 9.254 53.284Q9.523 53.091 9.611 52.776Q9.618 52.735 9.693 52.721L9.786 52.721Q9.868 52.745 9.868 52.817Q9.868 52.824 9.861 52.851Q9.748 53.248 9.377 53.487Q9.006 53.726 8.583 53.726Q8.145 53.726 7.745 53.518Q7.345 53.309 7.106 52.942Q6.867 52.575 6.867 52.123M7.537 51.853L9.352 51.853Q9.352 51.576 9.254 51.324Q9.157 51.071 8.959 50.915Q8.760 50.760 8.477 50.760Q8.200 50.760 7.986 50.918Q7.773 51.077 7.655 51.332Q7.537 51.587 7.537 51.853M10.456 53.651L10.456 52.588Q10.456 52.564 10.483 52.537Q10.510 52.510 10.534 52.510L10.644 52.510Q10.709 52.510 10.722 52.568Q10.818 53.002 11.064 53.253Q11.310 53.504 11.724 53.504Q12.066 53.504 12.318 53.371Q12.571 53.238 12.571 52.930Q12.571 52.773 12.477 52.658Q12.383 52.544 12.245 52.475Q12.107 52.407 11.939 52.369L11.358 52.270Q11.003 52.202 10.729 51.981Q10.456 51.761 10.456 51.419Q10.456 51.170 10.567 50.995Q10.678 50.821 10.864 50.722Q11.050 50.623 11.266 50.580Q11.481 50.537 11.724 50.537Q12.137 50.537 12.418 50.719L12.633 50.544Q12.643 50.541 12.650 50.539Q12.657 50.537 12.667 50.537L12.718 50.537Q12.746 50.537 12.770 50.561Q12.794 50.585 12.794 50.613L12.794 51.460Q12.794 51.481 12.770 51.508Q12.746 51.535 12.718 51.535L12.606 51.535Q12.578 51.535 12.553 51.510Q12.527 51.484 12.527 51.460Q12.527 51.224 12.421 51.060Q12.315 50.896 12.132 50.814Q11.949 50.732 11.717 50.732Q11.389 50.732 11.132 50.835Q10.876 50.937 10.876 51.214Q10.876 51.409 11.059 51.518Q11.242 51.628 11.471 51.669L12.045 51.775Q12.291 51.823 12.505 51.951Q12.718 52.079 12.855 52.282Q12.992 52.486 12.992 52.735Q12.992 53.248 12.626 53.487Q12.260 53.726 11.724 53.726Q11.228 53.726 10.897 53.432L10.630 53.706Q10.609 53.726 10.582 53.726L10.534 53.726Q10.510 53.726 10.483 53.699Q10.456 53.672 10.456 53.651",[1870],[1854,10726,10727],{"transform":10720},[1866,10728],{"d":10729,"fill":1856,"stroke":1856,"className":10730,"style":1909},"M19.352 53.658L16.822 53.658L16.822 53.378Q17.790 53.378 17.790 53.169L17.790 49.550Q17.397 49.738 16.775 49.738L16.775 49.457Q17.192 49.457 17.556 49.356Q17.920 49.256 18.176 49.010L18.302 49.010Q18.367 49.027 18.384 49.095L18.384 53.169Q18.384 53.378 19.352 53.378",[1870],[1854,10732,10733],{"transform":10720},[1866,10734],{"d":10735,"fill":1856,"stroke":1856,"className":10736,"style":1909},"M20.880 54.888Q20.880 54.854 20.908 54.827Q21.174 54.605 21.325 54.278Q21.475 53.952 21.475 53.596L21.475 53.559Q21.372 53.658 21.201 53.658Q21.024 53.658 20.902 53.537Q20.781 53.415 20.781 53.238Q20.781 53.067 20.902 52.945Q21.024 52.824 21.201 52.824Q21.461 52.824 21.581 53.063Q21.700 53.303 21.700 53.596Q21.700 54 21.530 54.369Q21.359 54.738 21.061 54.994Q21.031 55.015 21.007 55.015Q20.962 55.015 20.921 54.974Q20.880 54.933 20.880 54.888",[1870],[1854,10738,10739],{"transform":10720},[1866,10740],{"d":10741,"fill":1856,"stroke":1856,"className":10742,"style":1909},"M27.079 53.658L24.194 53.658L24.194 53.456Q24.194 53.426 24.221 53.398L25.469 52.181Q25.541 52.106 25.583 52.064Q25.626 52.021 25.705 51.942Q26.118 51.529 26.349 51.171Q26.580 50.814 26.580 50.390Q26.580 50.158 26.501 49.955Q26.422 49.751 26.281 49.601Q26.139 49.450 25.944 49.370Q25.749 49.290 25.517 49.290Q25.206 49.290 24.948 49.449Q24.690 49.608 24.560 49.885L24.580 49.885Q24.748 49.885 24.855 49.996Q24.963 50.107 24.963 50.271Q24.963 50.428 24.854 50.541Q24.744 50.654 24.580 50.654Q24.420 50.654 24.307 50.541Q24.194 50.428 24.194 50.271Q24.194 49.895 24.402 49.608Q24.611 49.321 24.946 49.165Q25.281 49.010 25.636 49.010Q26.060 49.010 26.440 49.168Q26.819 49.327 27.053 49.644Q27.287 49.960 27.287 50.390Q27.287 50.701 27.147 50.970Q27.007 51.238 26.802 51.443Q26.597 51.648 26.234 51.930Q25.872 52.212 25.763 52.308L24.908 53.036L25.551 53.036Q25.814 53.036 26.103 53.034Q26.392 53.033 26.610 53.024Q26.829 53.015 26.846 52.998Q26.908 52.933 26.945 52.766Q26.983 52.598 27.021 52.356L27.287 52.356",[1870],[1854,10744,10745],{"transform":10720},[1866,10746],{"d":10747,"fill":1856,"stroke":1856,"className":10748,"style":1909},"M28.607 54.888Q28.607 54.854 28.635 54.827Q28.901 54.605 29.052 54.278Q29.202 53.952 29.202 53.596L29.202 53.559Q29.099 53.658 28.928 53.658Q28.751 53.658 28.629 53.537Q28.508 53.415 28.508 53.238Q28.508 53.067 28.629 52.945Q28.751 52.824 28.928 52.824Q29.188 52.824 29.308 53.063Q29.427 53.303 29.427 53.596Q29.427 54 29.257 54.369Q29.086 54.738 28.788 54.994Q28.758 55.015 28.734 55.015Q28.689 55.015 28.648 54.974Q28.607 54.933 28.607 54.888",[1870],[1854,10750,10751],{"transform":10720},[1866,10752],{"d":10753,"fill":1856,"stroke":1856,"className":10754,"style":1909},"M32.276 53.111Q32.396 53.268 32.587 53.367Q32.779 53.467 32.994 53.506Q33.209 53.545 33.432 53.545Q33.729 53.545 33.924 53.390Q34.119 53.234 34.209 52.980Q34.300 52.725 34.300 52.441Q34.300 52.147 34.208 51.896Q34.115 51.645 33.917 51.489Q33.719 51.334 33.425 51.334L32.909 51.334Q32.881 51.334 32.856 51.308Q32.830 51.283 32.830 51.259L32.830 51.187Q32.830 51.156 32.856 51.134Q32.881 51.112 32.909 51.112L33.350 51.081Q33.712 51.081 33.932 50.724Q34.153 50.366 34.153 49.977Q34.153 49.649 33.958 49.445Q33.763 49.242 33.432 49.242Q33.145 49.242 32.892 49.326Q32.639 49.409 32.475 49.597Q32.622 49.597 32.722 49.712Q32.823 49.826 32.823 49.977Q32.823 50.127 32.717 50.237Q32.611 50.346 32.454 50.346Q32.293 50.346 32.184 50.237Q32.075 50.127 32.075 49.977Q32.075 49.652 32.283 49.433Q32.492 49.215 32.808 49.112Q33.124 49.010 33.432 49.010Q33.750 49.010 34.078 49.114Q34.406 49.218 34.633 49.440Q34.860 49.662 34.860 49.977Q34.860 50.411 34.573 50.736Q34.286 51.060 33.852 51.207Q34.163 51.272 34.443 51.438Q34.724 51.604 34.901 51.862Q35.079 52.120 35.079 52.441Q35.079 52.851 34.835 53.161Q34.590 53.470 34.209 53.634Q33.828 53.798 33.432 53.798Q33.063 53.798 32.705 53.685Q32.348 53.573 32.104 53.323Q31.859 53.074 31.859 52.704Q31.859 52.533 31.976 52.421Q32.092 52.308 32.263 52.308Q32.372 52.308 32.463 52.359Q32.553 52.410 32.608 52.503Q32.663 52.595 32.663 52.704Q32.663 52.872 32.550 52.991Q32.437 53.111 32.276 53.111",[1870],[1854,10756,10757],{"transform":10720},[1866,10758],{"d":10759,"fill":1856,"stroke":1856,"className":10760,"style":1909},"M36.334 54.888Q36.334 54.854 36.362 54.827Q36.628 54.605 36.779 54.278Q36.929 53.952 36.929 53.596L36.929 53.559Q36.826 53.658 36.655 53.658Q36.478 53.658 36.356 53.537Q36.235 53.415 36.235 53.238Q36.235 53.067 36.356 52.945Q36.478 52.824 36.655 52.824Q36.915 52.824 37.035 53.063Q37.154 53.303 37.154 53.596Q37.154 54 36.984 54.369Q36.813 54.738 36.515 54.994Q36.485 55.015 36.461 55.015Q36.416 55.015 36.375 54.974Q36.334 54.933 36.334 54.888",[1870],[1854,10762,10763],{"transform":10720},[1866,10764],{"d":10765,"fill":1856,"stroke":1856,"className":10766,"style":1909},"M41.524 52.510L39.480 52.510L39.480 52.229L41.811 49.057Q41.846 49.010 41.911 49.010L42.047 49.010Q42.092 49.010 42.119 49.037Q42.146 49.064 42.146 49.109L42.146 52.229L42.909 52.229L42.909 52.510L42.146 52.510L42.146 53.169Q42.146 53.378 42.902 53.378L42.902 53.658L40.769 53.658L40.769 53.378Q41.524 53.378 41.524 53.169L41.524 52.510M41.572 49.785L39.781 52.229L41.572 52.229",[1870],[1854,10768,10769],{"transform":10720},[1866,10770],{"d":10771,"fill":1856,"stroke":1856,"className":10772,"style":1909},"M47.870 53.658L46.267 53.658L46.267 53.378Q46.493 53.378 46.642 53.344Q46.790 53.309 46.790 53.169L46.790 49.550Q46.790 49.280 46.683 49.218Q46.575 49.157 46.267 49.157L46.267 48.876L47.344 48.801L47.344 53.169Q47.344 53.306 47.494 53.342Q47.645 53.378 47.870 53.378L47.870 53.658M48.523 52.930Q48.523 52.598 48.747 52.371Q48.971 52.144 49.315 52.016Q49.658 51.887 50.031 51.835Q50.403 51.782 50.707 51.782L50.707 51.529Q50.707 51.324 50.600 51.144Q50.492 50.965 50.311 50.862Q50.130 50.760 49.921 50.760Q49.514 50.760 49.279 50.852Q49.367 50.889 49.414 50.973Q49.460 51.057 49.460 51.159Q49.460 51.255 49.414 51.334Q49.367 51.412 49.287 51.457Q49.207 51.501 49.118 51.501Q48.968 51.501 48.867 51.404Q48.766 51.306 48.766 51.159Q48.766 50.537 49.921 50.537Q50.133 50.537 50.383 50.601Q50.632 50.664 50.834 50.783Q51.035 50.903 51.162 51.088Q51.288 51.272 51.288 51.515L51.288 53.091Q51.288 53.207 51.350 53.303Q51.411 53.398 51.524 53.398Q51.634 53.398 51.699 53.304Q51.763 53.210 51.763 53.091L51.763 52.643L52.030 52.643L52.030 53.091Q52.030 53.361 51.803 53.526Q51.575 53.692 51.295 53.692Q51.087 53.692 50.950 53.538Q50.813 53.385 50.789 53.169Q50.642 53.436 50.360 53.581Q50.078 53.726 49.754 53.726Q49.477 53.726 49.193 53.651Q48.909 53.576 48.716 53.397Q48.523 53.217 48.523 52.930M49.138 52.930Q49.138 53.104 49.239 53.234Q49.340 53.364 49.496 53.434Q49.651 53.504 49.815 53.504Q50.034 53.504 50.242 53.407Q50.451 53.309 50.579 53.128Q50.707 52.947 50.707 52.721L50.707 51.993Q50.383 51.993 50.017 52.084Q49.651 52.175 49.395 52.387Q49.138 52.598 49.138 52.930M53.254 53.658L52.987 53.658L52.987 49.550Q52.987 49.280 52.879 49.218Q52.772 49.157 52.461 49.157L52.461 48.876L53.541 48.801L53.541 50.971Q53.749 50.780 54.035 50.676Q54.320 50.572 54.617 50.572Q54.935 50.572 55.233 50.693Q55.530 50.814 55.752 51.030Q55.974 51.245 56.101 51.530Q56.227 51.816 56.227 52.147Q56.227 52.592 55.988 52.956Q55.749 53.320 55.356 53.523Q54.963 53.726 54.518 53.726Q54.324 53.726 54.134 53.670Q53.944 53.614 53.784 53.509Q53.623 53.405 53.483 53.244L53.254 53.658M53.568 51.313L53.568 52.930Q53.705 53.190 53.946 53.347Q54.187 53.504 54.464 53.504Q54.758 53.504 54.970 53.397Q55.181 53.289 55.315 53.097Q55.448 52.906 55.506 52.667Q55.564 52.428 55.564 52.147Q55.564 51.788 55.470 51.484Q55.376 51.180 55.149 50.987Q54.922 50.794 54.556 50.794Q54.255 50.794 53.989 50.930Q53.722 51.067 53.568 51.313",[1870],[1854,10774,10775],{"transform":10720},[1866,10776],{"d":10777,"fill":1856,"stroke":1856,"className":10778,"style":1909},"M57.043 52.123Q57.043 51.802 57.168 51.513Q57.293 51.224 57.519 51.001Q57.744 50.777 58.040 50.657Q58.335 50.537 58.653 50.537Q58.981 50.537 59.243 50.637Q59.504 50.736 59.680 50.918Q59.856 51.101 59.950 51.359Q60.044 51.617 60.044 51.949Q60.044 52.041 59.962 52.062L57.707 52.062L57.707 52.123Q57.707 52.711 57.990 53.094Q58.274 53.477 58.841 53.477Q59.163 53.477 59.431 53.284Q59.699 53.091 59.788 52.776Q59.795 52.735 59.870 52.721L59.962 52.721Q60.044 52.745 60.044 52.817Q60.044 52.824 60.038 52.851Q59.925 53.248 59.554 53.487Q59.183 53.726 58.759 53.726Q58.322 53.726 57.922 53.518Q57.522 53.309 57.283 52.942Q57.043 52.575 57.043 52.123M57.713 51.853L59.528 51.853Q59.528 51.576 59.431 51.324Q59.333 51.071 59.135 50.915Q58.937 50.760 58.653 50.760Q58.376 50.760 58.163 50.918Q57.949 51.077 57.831 51.332Q57.713 51.587 57.713 51.853M62.300 53.658L60.697 53.658L60.697 53.378Q60.923 53.378 61.072 53.344Q61.220 53.309 61.220 53.169L61.220 49.550Q61.220 49.280 61.113 49.218Q61.005 49.157 60.697 49.157L60.697 48.876L61.774 48.801L61.774 53.169Q61.774 53.306 61.924 53.342Q62.075 53.378 62.300 53.378",[1870],[1854,10780,10781],{"transform":10720},[1866,10782],{"d":10783,"fill":1856,"stroke":1856,"className":10784,"style":1909},"M66.124 52.817L66.124 50.920L65.485 50.920L65.485 50.698Q65.803 50.698 66.020 50.488Q66.237 50.278 66.337 49.968Q66.438 49.659 66.438 49.351L66.705 49.351L66.705 50.640L67.782 50.640L67.782 50.920L66.705 50.920L66.705 52.804Q66.705 53.080 66.809 53.279Q66.913 53.477 67.173 53.477Q67.330 53.477 67.436 53.373Q67.542 53.268 67.592 53.115Q67.641 52.961 67.641 52.804L67.641 52.390L67.908 52.390L67.908 52.817Q67.908 53.043 67.809 53.253Q67.710 53.463 67.525 53.595Q67.341 53.726 67.112 53.726Q66.674 53.726 66.399 53.489Q66.124 53.251 66.124 52.817M70.400 53.658L68.766 53.658L68.766 53.378Q68.995 53.378 69.144 53.344Q69.292 53.309 69.292 53.169L69.292 49.550Q69.292 49.280 69.185 49.218Q69.077 49.157 68.766 49.157L68.766 48.876L69.846 48.801L69.846 51.187Q69.952 51.002 70.130 50.860Q70.307 50.719 70.516 50.645Q70.724 50.572 70.950 50.572Q71.456 50.572 71.740 50.795Q72.023 51.019 72.023 51.515L72.023 53.169Q72.023 53.306 72.172 53.342Q72.321 53.378 72.546 53.378L72.546 53.658L70.916 53.658L70.916 53.378Q71.145 53.378 71.293 53.344Q71.442 53.309 71.442 53.169L71.442 51.529Q71.442 51.194 71.323 50.994Q71.203 50.794 70.888 50.794Q70.618 50.794 70.384 50.930Q70.150 51.067 70.012 51.301Q69.873 51.535 69.873 51.809L69.873 53.169Q69.873 53.306 70.024 53.342Q70.174 53.378 70.400 53.378L70.400 53.658M73.093 52.123Q73.093 51.802 73.218 51.513Q73.343 51.224 73.568 51.001Q73.794 50.777 74.089 50.657Q74.385 50.537 74.703 50.537Q75.031 50.537 75.293 50.637Q75.554 50.736 75.730 50.918Q75.906 51.101 76 51.359Q76.094 51.617 76.094 51.949Q76.094 52.041 76.012 52.062L73.756 52.062L73.756 52.123Q73.756 52.711 74.040 53.094Q74.324 53.477 74.891 53.477Q75.212 53.477 75.481 53.284Q75.749 53.091 75.838 52.776Q75.845 52.735 75.920 52.721L76.012 52.721Q76.094 52.745 76.094 52.817Q76.094 52.824 76.087 52.851Q75.974 53.248 75.604 53.487Q75.233 53.726 74.809 53.726Q74.371 53.726 73.971 53.518Q73.572 53.309 73.332 52.942Q73.093 52.575 73.093 52.123M73.763 51.853L75.578 51.853Q75.578 51.576 75.481 51.324Q75.383 51.071 75.185 50.915Q74.987 50.760 74.703 50.760Q74.426 50.760 74.212 50.918Q73.999 51.077 73.881 51.332Q73.763 51.587 73.763 51.853",[1870],[1854,10786,10787],{"transform":10720},[1866,10788],{"d":10789,"fill":1856,"stroke":1856,"className":10790,"style":1909},"M81.187 53.658L79.454 53.658L79.454 53.378Q79.680 53.378 79.829 53.344Q79.977 53.309 79.977 53.169L79.977 50.920L79.389 50.920L79.389 50.640L79.977 50.640L79.977 49.823Q79.977 49.505 80.155 49.257Q80.333 49.010 80.623 48.869Q80.914 48.729 81.225 48.729Q81.481 48.729 81.685 48.871Q81.888 49.013 81.888 49.256Q81.888 49.392 81.789 49.491Q81.690 49.591 81.553 49.591Q81.416 49.591 81.317 49.491Q81.218 49.392 81.218 49.256Q81.218 49.075 81.358 48.982Q81.280 48.955 81.180 48.955Q80.972 48.955 80.818 49.088Q80.664 49.221 80.584 49.425Q80.504 49.628 80.504 49.837L80.504 50.640L81.392 50.640L81.392 50.920L80.531 50.920L80.531 53.169Q80.531 53.378 81.187 53.378L81.187 53.658M81.826 52.175Q81.826 51.833 81.961 51.534Q82.096 51.235 82.336 51.011Q82.575 50.787 82.893 50.662Q83.211 50.537 83.542 50.537Q83.987 50.537 84.387 50.753Q84.786 50.968 85.021 51.346Q85.255 51.723 85.255 52.175Q85.255 52.516 85.113 52.800Q84.971 53.084 84.727 53.291Q84.482 53.497 84.173 53.612Q83.864 53.726 83.542 53.726Q83.112 53.726 82.710 53.525Q82.308 53.323 82.067 52.971Q81.826 52.619 81.826 52.175M83.542 53.477Q84.144 53.477 84.368 53.099Q84.592 52.721 84.592 52.089Q84.592 51.477 84.357 51.118Q84.123 50.760 83.542 50.760Q82.490 50.760 82.490 52.089Q82.490 52.721 82.715 53.099Q82.941 53.477 83.542 53.477M86.424 52.824L86.424 51.320Q86.424 51.050 86.316 50.989Q86.208 50.927 85.897 50.927L85.897 50.647L87.005 50.572L87.005 52.804L87.005 52.824Q87.005 53.104 87.056 53.248Q87.107 53.391 87.249 53.448Q87.391 53.504 87.678 53.504Q87.931 53.504 88.136 53.364Q88.341 53.224 88.457 52.998Q88.574 52.773 88.574 52.523L88.574 51.320Q88.574 51.050 88.466 50.989Q88.358 50.927 88.047 50.927L88.047 50.647L89.155 50.572L89.155 52.985Q89.155 53.176 89.208 53.258Q89.261 53.340 89.361 53.359Q89.462 53.378 89.678 53.378L89.678 53.658L88.601 53.726L88.601 53.162Q88.492 53.344 88.346 53.467Q88.201 53.590 88.015 53.658Q87.828 53.726 87.627 53.726Q86.424 53.726 86.424 52.824M92.015 53.658L90.279 53.658L90.279 53.378Q90.508 53.378 90.657 53.344Q90.805 53.309 90.805 53.169L90.805 51.320Q90.805 51.050 90.698 50.989Q90.590 50.927 90.279 50.927L90.279 50.647L91.308 50.572L91.308 51.279Q91.438 50.971 91.680 50.772Q91.923 50.572 92.241 50.572Q92.460 50.572 92.631 50.696Q92.802 50.821 92.802 51.033Q92.802 51.170 92.702 51.269Q92.603 51.368 92.470 51.368Q92.333 51.368 92.234 51.269Q92.135 51.170 92.135 51.033Q92.135 50.893 92.234 50.794Q91.944 50.794 91.744 50.990Q91.544 51.187 91.451 51.481Q91.359 51.775 91.359 52.055L91.359 53.169Q91.359 53.378 92.015 53.378",[1870],[1854,10792,10793],{"transform":10720},[1866,10794],{"d":10795,"fill":1856,"stroke":1856,"className":10796,"style":1909},"M97.681 53.631L96.553 51.132Q96.481 50.985 96.351 50.953Q96.221 50.920 95.992 50.920L95.992 50.640L97.506 50.640L97.506 50.920Q97.154 50.920 97.154 51.067Q97.154 51.112 97.165 51.132L98.029 53.050L98.809 51.320Q98.843 51.252 98.843 51.173Q98.843 51.060 98.759 50.990Q98.675 50.920 98.556 50.920L98.556 50.640L99.752 50.640L99.752 50.920Q99.533 50.920 99.362 51.023Q99.192 51.125 99.103 51.320L98.067 53.631Q98.019 53.726 97.913 53.726L97.835 53.726Q97.729 53.726 97.681 53.631",[1870],[1854,10798,10799],{"transform":10720},[1866,10800],{"d":10801,"fill":1856,"stroke":1856,"className":10802,"style":1909},"M100.036 52.123Q100.036 51.802 100.161 51.513Q100.286 51.224 100.512 51.001Q100.737 50.777 101.033 50.657Q101.328 50.537 101.646 50.537Q101.974 50.537 102.236 50.637Q102.497 50.736 102.673 50.918Q102.849 51.101 102.943 51.359Q103.037 51.617 103.037 51.949Q103.037 52.041 102.955 52.062L100.700 52.062L100.700 52.123Q100.700 52.711 100.983 53.094Q101.267 53.477 101.834 53.477Q102.156 53.477 102.424 53.284Q102.692 53.091 102.781 52.776Q102.788 52.735 102.863 52.721L102.955 52.721Q103.037 52.745 103.037 52.817Q103.037 52.824 103.031 52.851Q102.918 53.248 102.547 53.487Q102.176 53.726 101.752 53.726Q101.315 53.726 100.915 53.518Q100.515 53.309 100.276 52.942Q100.036 52.575 100.036 52.123M100.706 51.853L102.521 51.853Q102.521 51.576 102.424 51.324Q102.326 51.071 102.128 50.915Q101.930 50.760 101.646 50.760Q101.369 50.760 101.156 50.918Q100.942 51.077 100.824 51.332Q100.706 51.587 100.706 51.853M105.375 53.658L103.639 53.658L103.639 53.378Q103.868 53.378 104.017 53.344Q104.165 53.309 104.165 53.169L104.165 51.320Q104.165 51.050 104.058 50.989Q103.950 50.927 103.639 50.927L103.639 50.647L104.668 50.572L104.668 51.279Q104.798 50.971 105.040 50.772Q105.283 50.572 105.601 50.572Q105.820 50.572 105.991 50.696Q106.161 50.821 106.161 51.033Q106.161 51.170 106.062 51.269Q105.963 51.368 105.830 51.368Q105.693 51.368 105.594 51.269Q105.495 51.170 105.495 51.033Q105.495 50.893 105.594 50.794Q105.304 50.794 105.104 50.990Q104.904 51.187 104.811 51.481Q104.719 51.775 104.719 52.055L104.719 53.169Q104.719 53.378 105.375 53.378L105.375 53.658M107.272 52.817L107.272 50.920L106.633 50.920L106.633 50.698Q106.951 50.698 107.168 50.488Q107.385 50.278 107.486 49.968Q107.587 49.659 107.587 49.351L107.853 49.351L107.853 50.640L108.930 50.640L108.930 50.920L107.853 50.920L107.853 52.804Q107.853 53.080 107.958 53.279Q108.062 53.477 108.322 53.477Q108.479 53.477 108.585 53.373Q108.691 53.268 108.740 53.115Q108.790 52.961 108.790 52.804L108.790 52.390L109.056 52.390L109.056 52.817Q109.056 53.043 108.957 53.253Q108.858 53.463 108.674 53.595Q108.489 53.726 108.260 53.726Q107.823 53.726 107.547 53.489Q107.272 53.251 107.272 52.817M111.483 53.658L109.931 53.658L109.931 53.378Q110.157 53.378 110.306 53.344Q110.454 53.309 110.454 53.169L110.454 51.320Q110.454 51.132 110.407 51.048Q110.359 50.965 110.261 50.946Q110.164 50.927 109.952 50.927L109.952 50.647L111.008 50.572L111.008 53.169Q111.008 53.309 111.140 53.344Q111.271 53.378 111.483 53.378L111.483 53.658M110.212 49.351Q110.212 49.180 110.335 49.061Q110.458 48.941 110.629 48.941Q110.796 48.941 110.919 49.061Q111.042 49.180 111.042 49.351Q111.042 49.526 110.919 49.649Q110.796 49.772 110.629 49.772Q110.458 49.772 110.335 49.649Q110.212 49.526 110.212 49.351M112.129 52.147Q112.129 51.819 112.264 51.518Q112.399 51.218 112.635 50.997Q112.871 50.777 113.175 50.657Q113.479 50.537 113.804 50.537Q114.310 50.537 114.659 50.640Q115.007 50.742 115.007 51.118Q115.007 51.265 114.910 51.366Q114.812 51.467 114.665 51.467Q114.512 51.467 114.412 51.368Q114.313 51.269 114.313 51.118Q114.313 50.930 114.453 50.838Q114.252 50.787 113.811 50.787Q113.455 50.787 113.226 50.983Q112.997 51.180 112.897 51.489Q112.796 51.799 112.796 52.147Q112.796 52.496 112.922 52.802Q113.049 53.108 113.303 53.292Q113.558 53.477 113.913 53.477Q114.136 53.477 114.320 53.393Q114.505 53.309 114.640 53.154Q114.775 52.998 114.833 52.790Q114.846 52.735 114.901 52.735L115.014 52.735Q115.045 52.735 115.067 52.759Q115.089 52.783 115.089 52.817L115.089 52.838Q115.004 53.125 114.816 53.323Q114.628 53.521 114.363 53.624Q114.098 53.726 113.804 53.726Q113.373 53.726 112.985 53.520Q112.597 53.313 112.363 52.950Q112.129 52.588 112.129 52.147M115.636 52.123Q115.636 51.802 115.761 51.513Q115.886 51.224 116.111 51.001Q116.337 50.777 116.632 50.657Q116.928 50.537 117.246 50.537Q117.574 50.537 117.836 50.637Q118.097 50.736 118.273 50.918Q118.449 51.101 118.543 51.359Q118.637 51.617 118.637 51.949Q118.637 52.041 118.555 52.062L116.299 52.062L116.299 52.123Q116.299 52.711 116.583 53.094Q116.867 53.477 117.434 53.477Q117.755 53.477 118.023 53.284Q118.292 53.091 118.381 52.776Q118.388 52.735 118.463 52.721L118.555 52.721Q118.637 52.745 118.637 52.817Q118.637 52.824 118.630 52.851Q118.517 53.248 118.147 53.487Q117.776 53.726 117.352 53.726Q116.914 53.726 116.514 53.518Q116.115 53.309 115.875 52.942Q115.636 52.575 115.636 52.123M116.306 51.853L118.121 51.853Q118.121 51.576 118.023 51.324Q117.926 51.071 117.728 50.915Q117.530 50.760 117.246 50.760Q116.969 50.760 116.755 50.918Q116.542 51.077 116.424 51.332Q116.306 51.587 116.306 51.853M119.225 53.651L119.225 52.588Q119.225 52.564 119.252 52.537Q119.280 52.510 119.304 52.510L119.413 52.510Q119.478 52.510 119.492 52.568Q119.587 53.002 119.833 53.253Q120.079 53.504 120.493 53.504Q120.835 53.504 121.088 53.371Q121.341 53.238 121.341 52.930Q121.341 52.773 121.247 52.658Q121.153 52.544 121.014 52.475Q120.876 52.407 120.708 52.369L120.127 52.270Q119.772 52.202 119.498 51.981Q119.225 51.761 119.225 51.419Q119.225 51.170 119.336 50.995Q119.447 50.821 119.633 50.722Q119.820 50.623 120.035 50.580Q120.250 50.537 120.493 50.537Q120.907 50.537 121.187 50.719L121.402 50.544Q121.412 50.541 121.419 50.539Q121.426 50.537 121.436 50.537L121.488 50.537Q121.515 50.537 121.539 50.561Q121.563 50.585 121.563 50.613L121.563 51.460Q121.563 51.481 121.539 51.508Q121.515 51.535 121.488 51.535L121.375 51.535Q121.347 51.535 121.322 51.510Q121.296 51.484 121.296 51.460Q121.296 51.224 121.190 51.060Q121.084 50.896 120.901 50.814Q120.719 50.732 120.486 50.732Q120.158 50.732 119.902 50.835Q119.645 50.937 119.645 51.214Q119.645 51.409 119.828 51.518Q120.011 51.628 120.240 51.669L120.814 51.775Q121.060 51.823 121.274 51.951Q121.488 52.079 121.624 52.282Q121.761 52.486 121.761 52.735Q121.761 53.248 121.395 53.487Q121.030 53.726 120.493 53.726Q119.997 53.726 119.666 53.432L119.399 53.706Q119.379 53.726 119.351 53.726L119.304 53.726Q119.280 53.726 119.252 53.699Q119.225 53.672 119.225 53.651M122.889 54.888Q122.889 54.854 122.909 54.834Q123.149 54.595 123.284 54.268Q123.419 53.942 123.419 53.610Q123.333 53.658 123.210 53.658Q123.029 53.658 122.909 53.538Q122.790 53.419 122.790 53.238Q122.790 53.063 122.909 52.944Q123.029 52.824 123.210 52.824Q123.381 52.824 123.479 52.947Q123.576 53.070 123.610 53.246Q123.644 53.422 123.644 53.596Q123.644 53.979 123.497 54.343Q123.350 54.707 123.077 54.988Q123.050 55.015 123.012 55.015Q122.971 55.015 122.930 54.974Q122.889 54.933 122.889 54.888M122.790 51.054Q122.790 50.886 122.913 50.763Q123.036 50.640 123.210 50.640Q123.378 50.640 123.501 50.763Q123.624 50.886 123.624 51.054Q123.624 51.228 123.501 51.351Q123.378 51.474 123.210 51.474Q123.036 51.474 122.913 51.351Q122.790 51.228 122.790 51.054",[1870],[1854,10804,10805],{"transform":10720},[1866,10806],{"d":10807,"fill":1856,"stroke":1856,"className":10808,"style":1909},"M127.911 52.817L127.911 50.920L127.272 50.920L127.272 50.698Q127.590 50.698 127.807 50.488Q128.024 50.278 128.124 49.968Q128.225 49.659 128.225 49.351L128.492 49.351L128.492 50.640L129.569 50.640L129.569 50.920L128.492 50.920L128.492 52.804Q128.492 53.080 128.596 53.279Q128.700 53.477 128.960 53.477Q129.117 53.477 129.223 53.373Q129.329 53.268 129.379 53.115Q129.428 52.961 129.428 52.804L129.428 52.390L129.695 52.390L129.695 52.817Q129.695 53.043 129.596 53.253Q129.497 53.463 129.312 53.595Q129.128 53.726 128.899 53.726Q128.461 53.726 128.186 53.489Q127.911 53.251 127.911 52.817M132.187 53.658L130.553 53.658L130.553 53.378Q130.782 53.378 130.931 53.344Q131.079 53.309 131.079 53.169L131.079 49.550Q131.079 49.280 130.972 49.218Q130.864 49.157 130.553 49.157L130.553 48.876L131.633 48.801L131.633 51.187Q131.739 51.002 131.917 50.860Q132.094 50.719 132.303 50.645Q132.511 50.572 132.737 50.572Q133.243 50.572 133.527 50.795Q133.810 51.019 133.810 51.515L133.810 53.169Q133.810 53.306 133.959 53.342Q134.108 53.378 134.333 53.378L134.333 53.658L132.703 53.658L132.703 53.378Q132.932 53.378 133.080 53.344Q133.229 53.309 133.229 53.169L133.229 51.529Q133.229 51.194 133.110 50.994Q132.990 50.794 132.675 50.794Q132.405 50.794 132.171 50.930Q131.937 51.067 131.799 51.301Q131.660 51.535 131.660 51.809L131.660 53.169Q131.660 53.306 131.811 53.342Q131.961 53.378 132.187 53.378L132.187 53.658M134.880 52.123Q134.880 51.802 135.005 51.513Q135.130 51.224 135.355 51.001Q135.581 50.777 135.876 50.657Q136.172 50.537 136.490 50.537Q136.818 50.537 137.080 50.637Q137.341 50.736 137.517 50.918Q137.693 51.101 137.787 51.359Q137.881 51.617 137.881 51.949Q137.881 52.041 137.799 52.062L135.543 52.062L135.543 52.123Q135.543 52.711 135.827 53.094Q136.111 53.477 136.678 53.477Q136.999 53.477 137.268 53.284Q137.536 53.091 137.625 52.776Q137.632 52.735 137.707 52.721L137.799 52.721Q137.881 52.745 137.881 52.817Q137.881 52.824 137.874 52.851Q137.761 53.248 137.391 53.487Q137.020 53.726 136.596 53.726Q136.158 53.726 135.758 53.518Q135.359 53.309 135.119 52.942Q134.880 52.575 134.880 52.123M135.550 51.853L137.365 51.853Q137.365 51.576 137.268 51.324Q137.170 51.071 136.972 50.915Q136.774 50.760 136.490 50.760Q136.213 50.760 135.999 50.918Q135.786 51.077 135.668 51.332Q135.550 51.587 135.550 51.853",[1870],[1854,10810,10811],{"transform":10720},[1866,10812],{"d":10813,"fill":1989,"stroke":1989,"className":10814,"style":1909},"M141.983 53.658L141.716 53.658L141.716 49.550Q141.716 49.280 141.609 49.218Q141.501 49.157 141.190 49.157L141.190 48.876L142.270 48.801L142.270 50.971Q142.479 50.780 142.764 50.676Q143.049 50.572 143.347 50.572Q143.665 50.572 143.962 50.693Q144.259 50.814 144.482 51.030Q144.704 51.245 144.830 51.530Q144.957 51.816 144.957 52.147Q144.957 52.592 144.717 52.956Q144.478 53.320 144.085 53.523Q143.692 53.726 143.248 53.726Q143.053 53.726 142.863 53.670Q142.674 53.614 142.513 53.509Q142.352 53.405 142.212 53.244L141.983 53.658M142.298 51.313L142.298 52.930Q142.434 53.190 142.675 53.347Q142.916 53.504 143.193 53.504Q143.487 53.504 143.699 53.397Q143.911 53.289 144.044 53.097Q144.177 52.906 144.236 52.667Q144.294 52.428 144.294 52.147Q144.294 51.788 144.200 51.484Q144.106 51.180 143.878 50.987Q143.651 50.794 143.285 50.794Q142.985 50.794 142.718 50.930Q142.451 51.067 142.298 51.313M147.260 53.658L145.657 53.658L145.657 53.378Q145.883 53.378 146.032 53.344Q146.180 53.309 146.180 53.169L146.180 49.550Q146.180 49.280 146.073 49.218Q145.965 49.157 145.657 49.157L145.657 48.876L146.734 48.801L146.734 53.169Q146.734 53.306 146.884 53.342Q147.035 53.378 147.260 53.378L147.260 53.658M148.429 52.824L148.429 51.320Q148.429 51.050 148.322 50.989Q148.214 50.927 147.903 50.927L147.903 50.647L149.010 50.572L149.010 52.804L149.010 52.824Q149.010 53.104 149.062 53.248Q149.113 53.391 149.255 53.448Q149.397 53.504 149.684 53.504Q149.937 53.504 150.142 53.364Q150.347 53.224 150.463 52.998Q150.579 52.773 150.579 52.523L150.579 51.320Q150.579 51.050 150.472 50.989Q150.364 50.927 150.053 50.927L150.053 50.647L151.160 50.572L151.160 52.985Q151.160 53.176 151.213 53.258Q151.266 53.340 151.367 53.359Q151.468 53.378 151.683 53.378L151.683 53.658L150.607 53.726L150.607 53.162Q150.497 53.344 150.352 53.467Q150.207 53.590 150.020 53.658Q149.834 53.726 149.633 53.726Q148.429 53.726 148.429 52.824M152.230 52.123Q152.230 51.802 152.355 51.513Q152.480 51.224 152.705 51.001Q152.931 50.777 153.227 50.657Q153.522 50.537 153.840 50.537Q154.168 50.537 154.430 50.637Q154.691 50.736 154.867 50.918Q155.043 51.101 155.137 51.359Q155.231 51.617 155.231 51.949Q155.231 52.041 155.149 52.062L152.893 52.062L152.893 52.123Q152.893 52.711 153.177 53.094Q153.461 53.477 154.028 53.477Q154.349 53.477 154.618 53.284Q154.886 53.091 154.975 52.776Q154.982 52.735 155.057 52.721L155.149 52.721Q155.231 52.745 155.231 52.817Q155.231 52.824 155.224 52.851Q155.112 53.248 154.741 53.487Q154.370 53.726 153.946 53.726Q153.508 53.726 153.109 53.518Q152.709 53.309 152.469 52.942Q152.230 52.575 152.230 52.123M152.900 51.853L154.715 51.853Q154.715 51.576 154.618 51.324Q154.520 51.071 154.322 50.915Q154.124 50.760 153.840 50.760Q153.563 50.760 153.350 50.918Q153.136 51.077 153.018 51.332Q152.900 51.587 152.900 51.853",[1870],[1854,10816,10817],{"transform":10720},[1866,10818],{"d":10819,"fill":1856,"stroke":1856,"className":10820,"style":1909},"M160.140 53.658L158.588 53.658L158.588 53.378Q158.814 53.378 158.963 53.344Q159.111 53.309 159.111 53.169L159.111 51.320Q159.111 51.132 159.063 51.048Q159.016 50.965 158.918 50.946Q158.821 50.927 158.609 50.927L158.609 50.647L159.665 50.572L159.665 53.169Q159.665 53.309 159.797 53.344Q159.928 53.378 160.140 53.378L160.140 53.658M158.869 49.351Q158.869 49.180 158.992 49.061Q159.115 48.941 159.286 48.941Q159.453 48.941 159.576 49.061Q159.699 49.180 159.699 49.351Q159.699 49.526 159.576 49.649Q159.453 49.772 159.286 49.772Q159.115 49.772 158.992 49.649Q158.869 49.526 158.869 49.351M162.468 53.658L160.834 53.658L160.834 53.378Q161.063 53.378 161.212 53.344Q161.360 53.309 161.360 53.169L161.360 51.320Q161.360 51.050 161.253 50.989Q161.145 50.927 160.834 50.927L160.834 50.647L161.894 50.572L161.894 51.221Q162.064 50.913 162.369 50.742Q162.673 50.572 163.018 50.572Q163.524 50.572 163.808 50.795Q164.091 51.019 164.091 51.515L164.091 53.169Q164.091 53.306 164.240 53.342Q164.389 53.378 164.614 53.378L164.614 53.658L162.984 53.658L162.984 53.378Q163.213 53.378 163.362 53.344Q163.510 53.309 163.510 53.169L163.510 51.529Q163.510 51.194 163.391 50.994Q163.271 50.794 162.957 50.794Q162.687 50.794 162.452 50.930Q162.218 51.067 162.080 51.301Q161.941 51.535 161.941 51.809L161.941 53.169Q161.941 53.306 162.092 53.342Q162.242 53.378 162.468 53.378L162.468 53.658M165.202 52.147Q165.202 51.809 165.342 51.518Q165.482 51.228 165.727 51.014Q165.971 50.801 166.275 50.686Q166.580 50.572 166.904 50.572Q167.174 50.572 167.438 50.671Q167.701 50.770 167.892 50.948L167.892 49.550Q167.892 49.280 167.784 49.218Q167.677 49.157 167.366 49.157L167.366 48.876L168.442 48.801L168.442 52.985Q168.442 53.173 168.497 53.256Q168.552 53.340 168.653 53.359Q168.753 53.378 168.969 53.378L168.969 53.658L167.861 53.726L167.861 53.309Q167.444 53.726 166.819 53.726Q166.388 53.726 166.016 53.514Q165.643 53.303 165.423 52.942Q165.202 52.581 165.202 52.147M166.877 53.504Q167.085 53.504 167.272 53.432Q167.458 53.361 167.612 53.224Q167.766 53.087 167.861 52.909L167.861 51.300Q167.776 51.153 167.631 51.033Q167.485 50.913 167.316 50.854Q167.147 50.794 166.966 50.794Q166.405 50.794 166.137 51.183Q165.869 51.573 165.869 52.154Q165.869 52.725 166.103 53.115Q166.337 53.504 166.877 53.504M169.577 52.123Q169.577 51.802 169.702 51.513Q169.827 51.224 170.052 51.001Q170.278 50.777 170.574 50.657Q170.869 50.537 171.187 50.537Q171.515 50.537 171.777 50.637Q172.038 50.736 172.214 50.918Q172.390 51.101 172.484 51.359Q172.578 51.617 172.578 51.949Q172.578 52.041 172.496 52.062L170.240 52.062L170.240 52.123Q170.240 52.711 170.524 53.094Q170.808 53.477 171.375 53.477Q171.696 53.477 171.965 53.284Q172.233 53.091 172.322 52.776Q172.329 52.735 172.404 52.721L172.496 52.721Q172.578 52.745 172.578 52.817Q172.578 52.824 172.571 52.851Q172.459 53.248 172.088 53.487Q171.717 53.726 171.293 53.726Q170.855 53.726 170.456 53.518Q170.056 53.309 169.816 52.942Q169.577 52.575 169.577 52.123M170.247 51.853L172.062 51.853Q172.062 51.576 171.965 51.324Q171.867 51.071 171.669 50.915Q171.471 50.760 171.187 50.760Q170.910 50.760 170.697 50.918Q170.483 51.077 170.365 51.332Q170.247 51.587 170.247 51.853M174.349 53.658L173.026 53.658L173.026 53.378Q173.586 53.378 173.966 52.978L174.680 52.181L173.768 51.132Q173.631 50.985 173.482 50.953Q173.334 50.920 173.067 50.920L173.067 50.640L174.567 50.640L174.567 50.920Q174.376 50.920 174.376 51.054Q174.376 51.084 174.407 51.132L175.001 51.816L175.442 51.320Q175.555 51.190 175.555 51.074Q175.555 51.012 175.518 50.966Q175.480 50.920 175.422 50.920L175.422 50.640L176.738 50.640L176.738 50.920Q176.177 50.920 175.798 51.320L175.176 52.021L176.170 53.169Q176.270 53.268 176.370 53.313Q176.471 53.357 176.582 53.367Q176.693 53.378 176.871 53.378L176.871 53.658L175.377 53.658L175.377 53.378Q175.442 53.378 175.502 53.344Q175.562 53.309 175.562 53.244Q175.562 53.197 175.531 53.169L174.855 52.383L174.321 52.978Q174.209 53.108 174.209 53.224Q174.209 53.289 174.250 53.333Q174.291 53.378 174.349 53.378",[1870],[1854,10822,10823],{"transform":10720},[1866,10824],{"d":10825,"fill":1856,"stroke":1856,"className":10826,"style":1909},"M181.689 53.658L180.137 53.658L180.137 53.378Q180.363 53.378 180.512 53.344Q180.660 53.309 180.660 53.169L180.660 51.320Q180.660 51.132 180.612 51.048Q180.565 50.965 180.467 50.946Q180.370 50.927 180.158 50.927L180.158 50.647L181.214 50.572L181.214 53.169Q181.214 53.309 181.346 53.344Q181.477 53.378 181.689 53.378L181.689 53.658M180.418 49.351Q180.418 49.180 180.541 49.061Q180.664 48.941 180.835 48.941Q181.002 48.941 181.125 49.061Q181.248 49.180 181.248 49.351Q181.248 49.526 181.125 49.649Q181.002 49.772 180.835 49.772Q180.664 49.772 180.541 49.649Q180.418 49.526 180.418 49.351M182.335 53.651L182.335 52.588Q182.335 52.564 182.362 52.537Q182.390 52.510 182.414 52.510L182.523 52.510Q182.588 52.510 182.602 52.568Q182.697 53.002 182.944 53.253Q183.190 53.504 183.603 53.504Q183.945 53.504 184.198 53.371Q184.451 53.238 184.451 52.930Q184.451 52.773 184.357 52.658Q184.263 52.544 184.124 52.475Q183.986 52.407 183.819 52.369L183.237 52.270Q182.882 52.202 182.609 51.981Q182.335 51.761 182.335 51.419Q182.335 51.170 182.446 50.995Q182.557 50.821 182.744 50.722Q182.930 50.623 183.145 50.580Q183.361 50.537 183.603 50.537Q184.017 50.537 184.297 50.719L184.512 50.544Q184.523 50.541 184.529 50.539Q184.536 50.537 184.547 50.537L184.598 50.537Q184.625 50.537 184.649 50.561Q184.673 50.585 184.673 50.613L184.673 51.460Q184.673 51.481 184.649 51.508Q184.625 51.535 184.598 51.535L184.485 51.535Q184.458 51.535 184.432 51.510Q184.406 51.484 184.406 51.460Q184.406 51.224 184.300 51.060Q184.195 50.896 184.012 50.814Q183.829 50.732 183.596 50.732Q183.268 50.732 183.012 50.835Q182.756 50.937 182.756 51.214Q182.756 51.409 182.938 51.518Q183.121 51.628 183.350 51.669L183.925 51.775Q184.171 51.823 184.384 51.951Q184.598 52.079 184.735 52.282Q184.871 52.486 184.871 52.735Q184.871 53.248 184.506 53.487Q184.140 53.726 183.603 53.726Q183.108 53.726 182.776 53.432L182.509 53.706Q182.489 53.726 182.462 53.726L182.414 53.726Q182.390 53.726 182.362 53.699Q182.335 53.672 182.335 53.651",[1870],[1854,10828,10829],{"transform":10720},[1866,10830],{"d":10831,"fill":1856,"stroke":1856,"className":10832,"style":1909},"M188.725 52.817L188.725 50.920L188.086 50.920L188.086 50.698Q188.404 50.698 188.621 50.488Q188.838 50.278 188.938 49.968Q189.039 49.659 189.039 49.351L189.306 49.351L189.306 50.640L190.383 50.640L190.383 50.920L189.306 50.920L189.306 52.804Q189.306 53.080 189.410 53.279Q189.514 53.477 189.774 53.477Q189.931 53.477 190.037 53.373Q190.143 53.268 190.193 53.115Q190.242 52.961 190.242 52.804L190.242 52.390L190.509 52.390L190.509 52.817Q190.509 53.043 190.410 53.253Q190.311 53.463 190.126 53.595Q189.942 53.726 189.713 53.726Q189.275 53.726 189 53.489Q188.725 53.251 188.725 52.817M193.001 53.658L191.367 53.658L191.367 53.378Q191.596 53.378 191.745 53.344Q191.893 53.309 191.893 53.169L191.893 49.550Q191.893 49.280 191.786 49.218Q191.678 49.157 191.367 49.157L191.367 48.876L192.447 48.801L192.447 51.187Q192.553 51.002 192.731 50.860Q192.908 50.719 193.117 50.645Q193.325 50.572 193.551 50.572Q194.057 50.572 194.341 50.795Q194.624 51.019 194.624 51.515L194.624 53.169Q194.624 53.306 194.773 53.342Q194.922 53.378 195.147 53.378L195.147 53.658L193.517 53.658L193.517 53.378Q193.746 53.378 193.894 53.344Q194.043 53.309 194.043 53.169L194.043 51.529Q194.043 51.194 193.924 50.994Q193.804 50.794 193.489 50.794Q193.219 50.794 192.985 50.930Q192.751 51.067 192.613 51.301Q192.474 51.535 192.474 51.809L192.474 53.169Q192.474 53.306 192.625 53.342Q192.775 53.378 193.001 53.378L193.001 53.658M195.694 52.123Q195.694 51.802 195.819 51.513Q195.944 51.224 196.169 51.001Q196.395 50.777 196.690 50.657Q196.986 50.537 197.304 50.537Q197.632 50.537 197.894 50.637Q198.155 50.736 198.331 50.918Q198.507 51.101 198.601 51.359Q198.695 51.617 198.695 51.949Q198.695 52.041 198.613 52.062L196.357 52.062L196.357 52.123Q196.357 52.711 196.641 53.094Q196.925 53.477 197.492 53.477Q197.813 53.477 198.082 53.284Q198.350 53.091 198.439 52.776Q198.446 52.735 198.521 52.721L198.613 52.721Q198.695 52.745 198.695 52.817Q198.695 52.824 198.688 52.851Q198.575 53.248 198.205 53.487Q197.834 53.726 197.410 53.726Q196.972 53.726 196.572 53.518Q196.173 53.309 195.933 52.942Q195.694 52.575 195.694 52.123M196.364 51.853L198.179 51.853Q198.179 51.576 198.082 51.324Q197.984 51.071 197.786 50.915Q197.588 50.760 197.304 50.760Q197.027 50.760 196.813 50.918Q196.600 51.077 196.482 51.332Q196.364 51.587 196.364 51.853",[1870],[1854,10834,10835],{"transform":10720},[1866,10836],{"d":10837,"fill":1856,"stroke":1856,"className":10838,"style":1909},"M-46.688 61.658L-48.240 61.658L-48.240 61.378Q-48.014 61.378-47.865 61.344Q-47.717 61.309-47.717 61.169L-47.717 59.320Q-47.717 59.132-47.765 59.048Q-47.812 58.965-47.910 58.946Q-48.007 58.927-48.219 58.927L-48.219 58.647L-47.163 58.572L-47.163 61.169Q-47.163 61.309-47.031 61.344Q-46.900 61.378-46.688 61.378L-46.688 61.658M-47.959 57.351Q-47.959 57.180-47.836 57.061Q-47.713 56.941-47.542 56.941Q-47.375 56.941-47.252 57.061Q-47.129 57.180-47.129 57.351Q-47.129 57.526-47.252 57.649Q-47.375 57.772-47.542 57.772Q-47.713 57.772-47.836 57.649Q-47.959 57.526-47.959 57.351M-44.360 61.658L-45.994 61.658L-45.994 61.378Q-45.765 61.378-45.616 61.344Q-45.468 61.309-45.468 61.169L-45.468 59.320Q-45.468 59.050-45.575 58.989Q-45.683 58.927-45.994 58.927L-45.994 58.647L-44.934 58.572L-44.934 59.221Q-44.764 58.913-44.459 58.742Q-44.155 58.572-43.810 58.572Q-43.304 58.572-43.020 58.795Q-42.737 59.019-42.737 59.515L-42.737 61.169Q-42.737 61.306-42.588 61.342Q-42.439 61.378-42.214 61.378L-42.214 61.658L-43.844 61.658L-43.844 61.378Q-43.615 61.378-43.466 61.344Q-43.318 61.309-43.318 61.169L-43.318 59.529Q-43.318 59.194-43.437 58.994Q-43.557 58.794-43.871 58.794Q-44.141 58.794-44.376 58.930Q-44.610 59.067-44.748 59.301Q-44.887 59.535-44.887 59.809L-44.887 61.169Q-44.887 61.306-44.736 61.342Q-44.586 61.378-44.360 61.378",[1870],[1854,10840,10841],{"transform":10720},[1866,10842],{"d":10843,"fill":1856,"stroke":1856,"className":10844,"style":1909},"M-41.313 60.817L-41.313 58.920L-41.952 58.920L-41.952 58.698Q-41.634 58.698-41.417 58.488Q-41.200 58.278-41.100 57.968Q-40.999 57.659-40.999 57.351L-40.732 57.351L-40.732 58.640L-39.655 58.640L-39.655 58.920L-40.732 58.920L-40.732 60.804Q-40.732 61.080-40.628 61.279Q-40.524 61.477-40.264 61.477Q-40.107 61.477-40.001 61.373Q-39.895 61.268-39.845 61.115Q-39.796 60.961-39.796 60.804L-39.796 60.390L-39.529 60.390L-39.529 60.817Q-39.529 61.043-39.628 61.253Q-39.727 61.463-39.912 61.595Q-40.096 61.726-40.325 61.726Q-40.763 61.726-41.038 61.489Q-41.313 61.251-41.313 60.817M-38.760 60.123Q-38.760 59.802-38.635 59.513Q-38.510 59.224-38.285 59.001Q-38.059 58.777-37.764 58.657Q-37.468 58.537-37.150 58.537Q-36.822 58.537-36.560 58.637Q-36.299 58.736-36.123 58.918Q-35.947 59.101-35.853 59.359Q-35.759 59.617-35.759 59.949Q-35.759 60.041-35.841 60.062L-38.097 60.062L-38.097 60.123Q-38.097 60.711-37.813 61.094Q-37.529 61.477-36.962 61.477Q-36.641 61.477-36.373 61.284Q-36.104 61.091-36.015 60.776Q-36.008 60.735-35.933 60.721L-35.841 60.721Q-35.759 60.745-35.759 60.817Q-35.759 60.824-35.766 60.851Q-35.879 61.248-36.249 61.487Q-36.620 61.726-37.044 61.726Q-37.482 61.726-37.882 61.518Q-38.281 61.309-38.521 60.942Q-38.760 60.575-38.760 60.123M-38.090 59.853L-36.275 59.853Q-36.275 59.576-36.373 59.324Q-36.470 59.071-36.668 58.915Q-36.866 58.760-37.150 58.760Q-37.427 58.760-37.641 58.918Q-37.854 59.077-37.972 59.332Q-38.090 59.587-38.090 59.853M-33.421 61.658L-35.157 61.658L-35.157 61.378Q-34.928 61.378-34.780 61.344Q-34.631 61.309-34.631 61.169L-34.631 59.320Q-34.631 59.050-34.739 58.989Q-34.846 58.927-35.157 58.927L-35.157 58.647L-34.129 58.572L-34.129 59.279Q-33.999 58.971-33.756 58.772Q-33.513 58.572-33.196 58.572Q-32.977 58.572-32.806 58.696Q-32.635 58.821-32.635 59.033Q-32.635 59.170-32.734 59.269Q-32.833 59.368-32.967 59.368Q-33.103 59.368-33.202 59.269Q-33.301 59.170-33.301 59.033Q-33.301 58.893-33.202 58.794Q-33.493 58.794-33.693 58.990Q-33.893 59.187-33.985 59.481Q-34.077 59.775-34.077 60.055L-34.077 61.169Q-34.077 61.378-33.421 61.378L-33.421 61.658M-30.369 61.658L-32.003 61.658L-32.003 61.378Q-31.774 61.378-31.625 61.344Q-31.476 61.309-31.476 61.169L-31.476 59.320Q-31.476 59.050-31.584 58.989Q-31.692 58.927-32.003 58.927L-32.003 58.647L-30.943 58.572L-30.943 59.221Q-30.772 58.913-30.468 58.742Q-30.164 58.572-29.819 58.572Q-29.419 58.572-29.142 58.712Q-28.865 58.852-28.779 59.200Q-28.612 58.907-28.313 58.739Q-28.014 58.572-27.669 58.572Q-27.163 58.572-26.879 58.795Q-26.595 59.019-26.595 59.515L-26.595 61.169Q-26.595 61.306-26.447 61.342Q-26.298 61.378-26.072 61.378L-26.072 61.658L-27.703 61.658L-27.703 61.378Q-27.477 61.378-27.327 61.342Q-27.176 61.306-27.176 61.169L-27.176 59.529Q-27.176 59.194-27.296 58.994Q-27.416 58.794-27.730 58.794Q-28 58.794-28.234 58.930Q-28.468 59.067-28.607 59.301Q-28.745 59.535-28.745 59.809L-28.745 61.169Q-28.745 61.306-28.597 61.342Q-28.448 61.378-28.222 61.378L-28.222 61.658L-29.853 61.658L-29.853 61.378Q-29.624 61.378-29.475 61.344Q-29.326 61.309-29.326 61.169L-29.326 59.529Q-29.326 59.194-29.446 58.994Q-29.566 58.794-29.880 58.794Q-30.150 58.794-30.384 58.930Q-30.618 59.067-30.757 59.301Q-30.895 59.535-30.895 59.809L-30.895 61.169Q-30.895 61.306-30.745 61.342Q-30.594 61.378-30.369 61.378L-30.369 61.658M-25.526 60.123Q-25.526 59.802-25.401 59.513Q-25.276 59.224-25.050 59.001Q-24.825 58.777-24.529 58.657Q-24.234 58.537-23.916 58.537Q-23.588 58.537-23.326 58.637Q-23.065 58.736-22.889 58.918Q-22.713 59.101-22.619 59.359Q-22.525 59.617-22.525 59.949Q-22.525 60.041-22.607 60.062L-24.862 60.062L-24.862 60.123Q-24.862 60.711-24.579 61.094Q-24.295 61.477-23.728 61.477Q-23.406 61.477-23.138 61.284Q-22.870 61.091-22.781 60.776Q-22.774 60.735-22.699 60.721L-22.607 60.721Q-22.525 60.745-22.525 60.817Q-22.525 60.824-22.531 60.851Q-22.644 61.248-23.015 61.487Q-23.386 61.726-23.810 61.726Q-24.247 61.726-24.647 61.518Q-25.047 61.309-25.286 60.942Q-25.526 60.575-25.526 60.123M-24.856 59.853L-23.041 59.853Q-23.041 59.576-23.138 59.324Q-23.236 59.071-23.434 58.915Q-23.632 58.760-23.916 58.760Q-24.193 58.760-24.406 58.918Q-24.620 59.077-24.738 59.332Q-24.856 59.587-24.856 59.853M-21.937 60.147Q-21.937 59.809-21.797 59.518Q-21.656 59.228-21.412 59.014Q-21.168 58.801-20.863 58.686Q-20.559 58.572-20.235 58.572Q-19.965 58.572-19.701 58.671Q-19.438 58.770-19.247 58.948L-19.247 57.550Q-19.247 57.280-19.354 57.218Q-19.462 57.157-19.773 57.157L-19.773 56.876L-18.696 56.801L-18.696 60.985Q-18.696 61.173-18.642 61.256Q-18.587 61.340-18.486 61.359Q-18.385 61.378-18.170 61.378L-18.170 61.658L-19.278 61.726L-19.278 61.309Q-19.695 61.726-20.320 61.726Q-20.751 61.726-21.123 61.514Q-21.496 61.303-21.716 60.942Q-21.937 60.581-21.937 60.147M-20.262 61.504Q-20.053 61.504-19.867 61.432Q-19.681 61.361-19.527 61.224Q-19.373 61.087-19.278 60.909L-19.278 59.300Q-19.363 59.153-19.508 59.033Q-19.654 58.913-19.823 58.854Q-19.992 58.794-20.173 58.794Q-20.734 58.794-21.002 59.183Q-21.270 59.573-21.270 60.154Q-21.270 60.725-21.036 61.115Q-20.802 61.504-20.262 61.504M-15.904 61.658L-17.456 61.658L-17.456 61.378Q-17.230 61.378-17.081 61.344Q-16.933 61.309-16.933 61.169L-16.933 59.320Q-16.933 59.132-16.981 59.048Q-17.029 58.965-17.126 58.946Q-17.223 58.927-17.435 58.927L-17.435 58.647L-16.379 58.572L-16.379 61.169Q-16.379 61.309-16.248 61.344Q-16.116 61.378-15.904 61.378L-15.904 61.658M-17.175 57.351Q-17.175 57.180-17.052 57.061Q-16.929 56.941-16.758 56.941Q-16.591 56.941-16.468 57.061Q-16.345 57.180-16.345 57.351Q-16.345 57.526-16.468 57.649Q-16.591 57.772-16.758 57.772Q-16.929 57.772-17.052 57.649Q-17.175 57.526-17.175 57.351M-15.200 60.930Q-15.200 60.598-14.976 60.371Q-14.752 60.144-14.409 60.016Q-14.065 59.887-13.693 59.835Q-13.320 59.782-13.016 59.782L-13.016 59.529Q-13.016 59.324-13.123 59.144Q-13.231 58.965-13.412 58.862Q-13.593 58.760-13.802 58.760Q-14.209 58.760-14.445 58.852Q-14.356 58.889-14.310 58.973Q-14.263 59.057-14.263 59.159Q-14.263 59.255-14.310 59.334Q-14.356 59.412-14.436 59.457Q-14.516 59.501-14.605 59.501Q-14.756 59.501-14.856 59.404Q-14.957 59.306-14.957 59.159Q-14.957 58.537-13.802 58.537Q-13.590 58.537-13.341 58.601Q-13.091 58.664-12.889 58.783Q-12.688 58.903-12.561 59.088Q-12.435 59.272-12.435 59.515L-12.435 61.091Q-12.435 61.207-12.373 61.303Q-12.312 61.398-12.199 61.398Q-12.090 61.398-12.025 61.304Q-11.960 61.210-11.960 61.091L-11.960 60.643L-11.693 60.643L-11.693 61.091Q-11.693 61.361-11.920 61.526Q-12.148 61.692-12.428 61.692Q-12.636 61.692-12.773 61.538Q-12.910 61.385-12.934 61.169Q-13.081 61.436-13.363 61.581Q-13.645 61.726-13.969 61.726Q-14.246 61.726-14.530 61.651Q-14.814 61.576-15.007 61.397Q-15.200 61.217-15.200 60.930M-14.585 60.930Q-14.585 61.104-14.484 61.234Q-14.383 61.364-14.227 61.434Q-14.072 61.504-13.908 61.504Q-13.689 61.504-13.481 61.407Q-13.272 61.309-13.144 61.128Q-13.016 60.947-13.016 60.721L-13.016 59.993Q-13.341 59.993-13.706 60.084Q-14.072 60.175-14.328 60.387Q-14.585 60.598-14.585 60.930M-10.750 60.817L-10.750 58.920L-11.389 58.920L-11.389 58.698Q-11.071 58.698-10.854 58.488Q-10.637 58.278-10.536 57.968Q-10.435 57.659-10.435 57.351L-10.169 57.351L-10.169 58.640L-9.092 58.640L-9.092 58.920L-10.169 58.920L-10.169 60.804Q-10.169 61.080-10.064 61.279Q-9.960 61.477-9.700 61.477Q-9.543 61.477-9.437 61.373Q-9.331 61.268-9.282 61.115Q-9.232 60.961-9.232 60.804L-9.232 60.390L-8.966 60.390L-8.966 60.817Q-8.966 61.043-9.065 61.253Q-9.164 61.463-9.348 61.595Q-9.533 61.726-9.762 61.726Q-10.199 61.726-10.475 61.489Q-10.750 61.251-10.750 60.817M-8.196 60.123Q-8.196 59.802-8.072 59.513Q-7.947 59.224-7.721 59.001Q-7.496 58.777-7.200 58.657Q-6.904 58.537-6.587 58.537Q-6.258 58.537-5.997 58.637Q-5.736 58.736-5.560 58.918Q-5.383 59.101-5.289 59.359Q-5.196 59.617-5.196 59.949Q-5.196 60.041-5.278 60.062L-7.533 60.062L-7.533 60.123Q-7.533 60.711-7.250 61.094Q-6.966 61.477-6.399 61.477Q-6.077 61.477-5.809 61.284Q-5.541 61.091-5.452 60.776Q-5.445 60.735-5.370 60.721L-5.278 60.721Q-5.196 60.745-5.196 60.817Q-5.196 60.824-5.202 60.851Q-5.315 61.248-5.686 61.487Q-6.057 61.726-6.481 61.726Q-6.918 61.726-7.318 61.518Q-7.718 61.309-7.957 60.942Q-8.196 60.575-8.196 60.123M-7.527 59.853L-5.712 59.853Q-5.712 59.576-5.809 59.324Q-5.906 59.071-6.105 58.915Q-6.303 58.760-6.587 58.760Q-6.863 58.760-7.077 58.918Q-7.291 59.077-7.409 59.332Q-7.527 59.587-7.527 59.853",[1870],[1854,10846,10847],{"transform":10720},[1866,10848],{"d":10849,"fill":1856,"stroke":1856,"className":10850,"style":1909},"M-0.277 61.631L-1.405 59.132Q-1.477 58.985-1.607 58.953Q-1.737 58.920-1.966 58.920L-1.966 58.640L-0.452 58.640L-0.452 58.920Q-0.804 58.920-0.804 59.067Q-0.804 59.112-0.793 59.132L0.071 61.050L0.851 59.320Q0.885 59.252 0.885 59.173Q0.885 59.060 0.801 58.990Q0.717 58.920 0.598 58.920L0.598 58.640L1.794 58.640L1.794 58.920Q1.575 58.920 1.404 59.023Q1.234 59.125 1.145 59.320L0.109 61.631Q0.061 61.726-0.045 61.726L-0.123 61.726Q-0.229 61.726-0.277 61.631",[1870],[1854,10852,10853],{"transform":10720},[1866,10854],{"d":10855,"fill":1856,"stroke":1856,"className":10856,"style":1909},"M2.078 60.123Q2.078 59.802 2.203 59.513Q2.328 59.224 2.554 59.001Q2.779 58.777 3.075 58.657Q3.370 58.537 3.688 58.537Q4.016 58.537 4.278 58.637Q4.539 58.736 4.715 58.918Q4.891 59.101 4.985 59.359Q5.079 59.617 5.079 59.949Q5.079 60.041 4.997 60.062L2.742 60.062L2.742 60.123Q2.742 60.711 3.025 61.094Q3.309 61.477 3.876 61.477Q4.198 61.477 4.466 61.284Q4.734 61.091 4.823 60.776Q4.830 60.735 4.905 60.721L4.997 60.721Q5.079 60.745 5.079 60.817Q5.079 60.824 5.073 60.851Q4.960 61.248 4.589 61.487Q4.218 61.726 3.794 61.726Q3.357 61.726 2.957 61.518Q2.557 61.309 2.318 60.942Q2.078 60.575 2.078 60.123M2.748 59.853L4.563 59.853Q4.563 59.576 4.466 59.324Q4.368 59.071 4.170 58.915Q3.972 58.760 3.688 58.760Q3.411 58.760 3.198 58.918Q2.984 59.077 2.866 59.332Q2.748 59.587 2.748 59.853M7.417 61.658L5.681 61.658L5.681 61.378Q5.910 61.378 6.059 61.344Q6.207 61.309 6.207 61.169L6.207 59.320Q6.207 59.050 6.100 58.989Q5.992 58.927 5.681 58.927L5.681 58.647L6.710 58.572L6.710 59.279Q6.840 58.971 7.082 58.772Q7.325 58.572 7.643 58.572Q7.862 58.572 8.033 58.696Q8.203 58.821 8.203 59.033Q8.203 59.170 8.104 59.269Q8.005 59.368 7.872 59.368Q7.735 59.368 7.636 59.269Q7.537 59.170 7.537 59.033Q7.537 58.893 7.636 58.794Q7.346 58.794 7.146 58.990Q6.946 59.187 6.853 59.481Q6.761 59.775 6.761 60.055L6.761 61.169Q6.761 61.378 7.417 61.378L7.417 61.658M9.314 60.817L9.314 58.920L8.675 58.920L8.675 58.698Q8.993 58.698 9.210 58.488Q9.427 58.278 9.528 57.968Q9.629 57.659 9.629 57.351L9.895 57.351L9.895 58.640L10.972 58.640L10.972 58.920L9.895 58.920L9.895 60.804Q9.895 61.080 10 61.279Q10.104 61.477 10.364 61.477Q10.521 61.477 10.627 61.373Q10.733 61.268 10.782 61.115Q10.832 60.961 10.832 60.804L10.832 60.390L11.098 60.390L11.098 60.817Q11.098 61.043 10.999 61.253Q10.900 61.463 10.716 61.595Q10.531 61.726 10.302 61.726Q9.865 61.726 9.589 61.489Q9.314 61.251 9.314 60.817M11.867 60.123Q11.867 59.802 11.992 59.513Q12.117 59.224 12.343 59.001Q12.568 58.777 12.864 58.657Q13.159 58.537 13.477 58.537Q13.805 58.537 14.067 58.637Q14.328 58.736 14.504 58.918Q14.680 59.101 14.774 59.359Q14.868 59.617 14.868 59.949Q14.868 60.041 14.786 60.062L12.531 60.062L12.531 60.123Q12.531 60.711 12.814 61.094Q13.098 61.477 13.665 61.477Q13.987 61.477 14.255 61.284Q14.523 61.091 14.612 60.776Q14.619 60.735 14.694 60.721L14.786 60.721Q14.868 60.745 14.868 60.817Q14.868 60.824 14.862 60.851Q14.749 61.248 14.378 61.487Q14.007 61.726 13.583 61.726Q13.146 61.726 12.746 61.518Q12.346 61.309 12.107 60.942Q11.867 60.575 11.867 60.123M12.537 59.853L14.352 59.853Q14.352 59.576 14.255 59.324Q14.158 59.071 13.959 58.915Q13.761 58.760 13.477 58.760Q13.201 58.760 12.987 58.918Q12.773 59.077 12.655 59.332Q12.537 59.587 12.537 59.853M16.639 61.658L15.316 61.658L15.316 61.378Q15.877 61.378 16.256 60.978L16.971 60.181L16.058 59.132Q15.921 58.985 15.773 58.953Q15.624 58.920 15.357 58.920L15.357 58.640L16.858 58.640L16.858 58.920Q16.666 58.920 16.666 59.054Q16.666 59.084 16.697 59.132L17.292 59.816L17.733 59.320Q17.846 59.190 17.846 59.074Q17.846 59.012 17.808 58.966Q17.770 58.920 17.712 58.920L17.712 58.640L19.028 58.640L19.028 58.920Q18.468 58.920 18.088 59.320L17.466 60.021L18.461 61.169Q18.560 61.268 18.661 61.313Q18.762 61.357 18.873 61.367Q18.984 61.378 19.161 61.378L19.161 61.658L17.668 61.658L17.668 61.378Q17.733 61.378 17.793 61.344Q17.852 61.309 17.852 61.244Q17.852 61.197 17.822 61.169L17.145 60.383L16.612 60.978Q16.499 61.108 16.499 61.224Q16.499 61.289 16.540 61.333Q16.581 61.378 16.639 61.378",[1870],[1854,10858,10859],{"transform":10720},[1866,10860],{"d":10861,"fill":1856,"stroke":1856,"className":10862,"style":1909},"M22.588 61.497Q22.588 61.456 22.595 61.432L23.586 57.437Q23.624 57.341 23.624 57.249Q23.624 57.157 23.190 57.157Q23.104 57.129 23.104 57.044L23.132 56.934Q23.139 56.893 23.210 56.876L24.191 56.801Q24.236 56.801 24.265 56.827Q24.294 56.852 24.294 56.904L23.573 59.782Q23.785 59.693 23.933 59.568Q24.082 59.443 24.420 59.139Q24.759 58.835 25 58.703Q25.241 58.572 25.500 58.572Q25.719 58.572 25.868 58.717Q26.017 58.862 26.017 59.081Q26.017 59.276 25.900 59.421Q25.784 59.566 25.596 59.566Q25.480 59.566 25.398 59.493Q25.316 59.419 25.316 59.300Q25.316 59.146 25.429 59.016Q25.541 58.886 25.695 58.886Q25.613 58.794 25.483 58.794Q25.224 58.794 24.974 58.963Q24.725 59.132 24.373 59.453Q24.020 59.775 23.846 59.881Q24.947 60 24.947 60.629Q24.947 60.728 24.911 60.892Q24.875 61.056 24.875 61.145Q24.875 61.504 25.114 61.504Q25.357 61.504 25.504 61.232Q25.651 60.961 25.729 60.629Q25.753 60.568 25.808 60.568L25.917 60.568Q25.952 60.568 25.974 60.593Q25.996 60.619 25.996 60.650Q25.996 60.663 25.989 60.677Q25.890 61.077 25.668 61.402Q25.446 61.726 25.101 61.726Q24.772 61.726 24.552 61.528Q24.332 61.330 24.332 61.009Q24.332 60.950 24.352 60.831Q24.373 60.711 24.373 60.650Q24.373 60.383 24.082 60.236Q23.791 60.089 23.498 60.089L23.152 61.484Q23.122 61.586 23.028 61.656Q22.934 61.726 22.831 61.726Q22.732 61.726 22.660 61.663Q22.588 61.600 22.588 61.497",[1870],[1854,10864,10865],{"transform":10720},[1866,10866],{"d":10867,"fill":1856,"stroke":1856,"className":10868,"style":1909},"M29.550 60.930Q29.550 60.598 29.773 60.371Q29.997 60.144 30.341 60.016Q30.684 59.887 31.057 59.835Q31.429 59.782 31.734 59.782L31.734 59.529Q31.734 59.324 31.626 59.144Q31.518 58.965 31.337 58.862Q31.156 58.760 30.948 58.760Q30.541 58.760 30.305 58.852Q30.394 58.889 30.440 58.973Q30.486 59.057 30.486 59.159Q30.486 59.255 30.440 59.334Q30.394 59.412 30.313 59.457Q30.233 59.501 30.144 59.501Q29.994 59.501 29.893 59.404Q29.792 59.306 29.792 59.159Q29.792 58.537 30.948 58.537Q31.159 58.537 31.409 58.601Q31.658 58.664 31.860 58.783Q32.062 58.903 32.188 59.088Q32.315 59.272 32.315 59.515L32.315 61.091Q32.315 61.207 32.376 61.303Q32.438 61.398 32.551 61.398Q32.660 61.398 32.725 61.304Q32.790 61.210 32.790 61.091L32.790 60.643L33.056 60.643L33.056 61.091Q33.056 61.361 32.829 61.526Q32.602 61.692 32.322 61.692Q32.113 61.692 31.976 61.538Q31.840 61.385 31.816 61.169Q31.669 61.436 31.387 61.581Q31.105 61.726 30.780 61.726Q30.503 61.726 30.219 61.651Q29.936 61.576 29.743 61.397Q29.550 61.217 29.550 60.930M30.165 60.930Q30.165 61.104 30.266 61.234Q30.366 61.364 30.522 61.434Q30.677 61.504 30.842 61.504Q31.060 61.504 31.269 61.407Q31.477 61.309 31.605 61.128Q31.734 60.947 31.734 60.721L31.734 59.993Q31.409 59.993 31.043 60.084Q30.677 60.175 30.421 60.387Q30.165 60.598 30.165 60.930M33.473 60.147Q33.473 59.809 33.614 59.518Q33.754 59.228 33.998 59.014Q34.242 58.801 34.547 58.686Q34.851 58.572 35.176 58.572Q35.446 58.572 35.709 58.671Q35.972 58.770 36.163 58.948L36.163 57.550Q36.163 57.280 36.056 57.218Q35.948 57.157 35.637 57.157L35.637 56.876L36.714 56.801L36.714 60.985Q36.714 61.173 36.768 61.256Q36.823 61.340 36.924 61.359Q37.025 61.378 37.240 61.378L37.240 61.658L36.133 61.726L36.133 61.309Q35.716 61.726 35.090 61.726Q34.659 61.726 34.287 61.514Q33.914 61.303 33.694 60.942Q33.473 60.581 33.473 60.147M35.148 61.504Q35.357 61.504 35.543 61.432Q35.729 61.361 35.883 61.224Q36.037 61.087 36.133 60.909L36.133 59.300Q36.047 59.153 35.902 59.033Q35.757 58.913 35.587 58.854Q35.418 58.794 35.237 58.794Q34.677 58.794 34.408 59.183Q34.140 59.573 34.140 60.154Q34.140 60.725 34.374 61.115Q34.608 61.504 35.148 61.504M39.571 61.658L37.937 61.658L37.937 61.378Q38.166 61.378 38.315 61.344Q38.464 61.309 38.464 61.169L38.464 59.320Q38.464 59.050 38.356 58.989Q38.248 58.927 37.937 58.927L37.937 58.647L38.997 58.572L38.997 59.221Q39.168 58.913 39.472 58.742Q39.776 58.572 40.121 58.572Q40.521 58.572 40.798 58.712Q41.075 58.852 41.160 59.200Q41.328 58.907 41.627 58.739Q41.926 58.572 42.271 58.572Q42.777 58.572 43.061 58.795Q43.344 59.019 43.344 59.515L43.344 61.169Q43.344 61.306 43.493 61.342Q43.642 61.378 43.867 61.378L43.867 61.658L42.237 61.658L42.237 61.378Q42.463 61.378 42.613 61.342Q42.763 61.306 42.763 61.169L42.763 59.529Q42.763 59.194 42.644 58.994Q42.524 58.794 42.210 58.794Q41.940 58.794 41.706 58.930Q41.471 59.067 41.333 59.301Q41.195 59.535 41.195 59.809L41.195 61.169Q41.195 61.306 41.343 61.342Q41.492 61.378 41.718 61.378L41.718 61.658L40.087 61.658L40.087 61.378Q40.316 61.378 40.465 61.344Q40.614 61.309 40.614 61.169L40.614 59.529Q40.614 59.194 40.494 58.994Q40.374 58.794 40.060 58.794Q39.790 58.794 39.556 58.930Q39.322 59.067 39.183 59.301Q39.045 59.535 39.045 59.809L39.045 61.169Q39.045 61.306 39.195 61.342Q39.345 61.378 39.571 61.378L39.571 61.658M46.072 61.658L44.520 61.658L44.520 61.378Q44.746 61.378 44.895 61.344Q45.043 61.309 45.043 61.169L45.043 59.320Q45.043 59.132 44.995 59.048Q44.948 58.965 44.850 58.946Q44.753 58.927 44.541 58.927L44.541 58.647L45.597 58.572L45.597 61.169Q45.597 61.309 45.729 61.344Q45.860 61.378 46.072 61.378L46.072 61.658M44.801 57.351Q44.801 57.180 44.924 57.061Q45.047 56.941 45.218 56.941Q45.385 56.941 45.508 57.061Q45.631 57.180 45.631 57.351Q45.631 57.526 45.508 57.649Q45.385 57.772 45.218 57.772Q45.047 57.772 44.924 57.649Q44.801 57.526 44.801 57.351M47.244 60.817L47.244 58.920L46.605 58.920L46.605 58.698Q46.923 58.698 47.140 58.488Q47.357 58.278 47.458 57.968Q47.559 57.659 47.559 57.351L47.825 57.351L47.825 58.640L48.902 58.640L48.902 58.920L47.825 58.920L47.825 60.804Q47.825 61.080 47.930 61.279Q48.034 61.477 48.294 61.477Q48.451 61.477 48.557 61.373Q48.663 61.268 48.712 61.115Q48.762 60.961 48.762 60.804L48.762 60.390L49.029 60.390L49.029 60.817Q49.029 61.043 48.929 61.253Q48.830 61.463 48.646 61.595Q48.461 61.726 48.232 61.726Q47.795 61.726 47.520 61.489Q47.244 61.251 47.244 60.817M50.365 60.817L50.365 58.920L49.726 58.920L49.726 58.698Q50.044 58.698 50.261 58.488Q50.478 58.278 50.579 57.968Q50.679 57.659 50.679 57.351L50.946 57.351L50.946 58.640L52.023 58.640L52.023 58.920L50.946 58.920L50.946 60.804Q50.946 61.080 51.050 61.279Q51.155 61.477 51.414 61.477Q51.572 61.477 51.677 61.373Q51.783 61.268 51.833 61.115Q51.883 60.961 51.883 60.804L51.883 60.390L52.149 60.390L52.149 60.817Q52.149 61.043 52.050 61.253Q51.951 61.463 51.766 61.595Q51.582 61.726 51.353 61.726Q50.915 61.726 50.640 61.489Q50.365 61.251 50.365 60.817M52.918 60.123Q52.918 59.802 53.043 59.513Q53.168 59.224 53.393 59.001Q53.619 58.777 53.915 58.657Q54.210 58.537 54.528 58.537Q54.856 58.537 55.118 58.637Q55.379 58.736 55.555 58.918Q55.731 59.101 55.825 59.359Q55.919 59.617 55.919 59.949Q55.919 60.041 55.837 60.062L53.581 60.062L53.581 60.123Q53.581 60.711 53.865 61.094Q54.149 61.477 54.716 61.477Q55.037 61.477 55.306 61.284Q55.574 61.091 55.663 60.776Q55.670 60.735 55.745 60.721L55.837 60.721Q55.919 60.745 55.919 60.817Q55.919 60.824 55.912 60.851Q55.800 61.248 55.429 61.487Q55.058 61.726 54.634 61.726Q54.197 61.726 53.797 61.518Q53.397 61.309 53.157 60.942Q52.918 60.575 52.918 60.123M53.588 59.853L55.403 59.853Q55.403 59.576 55.306 59.324Q55.208 59.071 55.010 58.915Q54.812 58.760 54.528 58.760Q54.251 58.760 54.038 58.918Q53.824 59.077 53.706 59.332Q53.588 59.587 53.588 59.853M56.507 60.147Q56.507 59.809 56.647 59.518Q56.787 59.228 57.032 59.014Q57.276 58.801 57.580 58.686Q57.885 58.572 58.209 58.572Q58.479 58.572 58.742 58.671Q59.006 58.770 59.197 58.948L59.197 57.550Q59.197 57.280 59.089 57.218Q58.982 57.157 58.671 57.157L58.671 56.876L59.747 56.801L59.747 60.985Q59.747 61.173 59.802 61.256Q59.857 61.340 59.958 61.359Q60.058 61.378 60.274 61.378L60.274 61.658L59.166 61.726L59.166 61.309Q58.749 61.726 58.124 61.726Q57.693 61.726 57.321 61.514Q56.948 61.303 56.728 60.942Q56.507 60.581 56.507 60.147M58.182 61.504Q58.390 61.504 58.577 61.432Q58.763 61.361 58.917 61.224Q59.071 61.087 59.166 60.909L59.166 59.300Q59.081 59.153 58.936 59.033Q58.790 58.913 58.621 58.854Q58.452 58.794 58.271 58.794Q57.710 58.794 57.442 59.183Q57.174 59.573 57.174 60.154Q57.174 60.725 57.408 61.115Q57.642 61.504 58.182 61.504",[1870],[1854,10870,10871],{"transform":10720},[1866,10872],{"d":10873,"fill":1856,"stroke":1856,"className":10874,"style":1909},"M64.171 60.817L64.171 58.920L63.532 58.920L63.532 58.698Q63.850 58.698 64.067 58.488Q64.284 58.278 64.384 57.968Q64.485 57.659 64.485 57.351L64.752 57.351L64.752 58.640L65.829 58.640L65.829 58.920L64.752 58.920L64.752 60.804Q64.752 61.080 64.856 61.279Q64.960 61.477 65.220 61.477Q65.377 61.477 65.483 61.373Q65.589 61.268 65.639 61.115Q65.688 60.961 65.688 60.804L65.688 60.390L65.955 60.390L65.955 60.817Q65.955 61.043 65.856 61.253Q65.757 61.463 65.572 61.595Q65.388 61.726 65.159 61.726Q64.721 61.726 64.446 61.489Q64.171 61.251 64.171 60.817M68.447 61.658L66.813 61.658L66.813 61.378Q67.042 61.378 67.191 61.344Q67.339 61.309 67.339 61.169L67.339 57.550Q67.339 57.280 67.232 57.218Q67.124 57.157 66.813 57.157L66.813 56.876L67.893 56.801L67.893 59.187Q67.999 59.002 68.177 58.860Q68.354 58.719 68.563 58.645Q68.771 58.572 68.997 58.572Q69.503 58.572 69.787 58.795Q70.070 59.019 70.070 59.515L70.070 61.169Q70.070 61.306 70.219 61.342Q70.368 61.378 70.593 61.378L70.593 61.658L68.963 61.658L68.963 61.378Q69.192 61.378 69.340 61.344Q69.489 61.309 69.489 61.169L69.489 59.529Q69.489 59.194 69.370 58.994Q69.250 58.794 68.935 58.794Q68.665 58.794 68.431 58.930Q68.197 59.067 68.059 59.301Q67.920 59.535 67.920 59.809L67.920 61.169Q67.920 61.306 68.071 61.342Q68.221 61.378 68.447 61.378L68.447 61.658M71.239 60.930Q71.239 60.598 71.463 60.371Q71.687 60.144 72.030 60.016Q72.374 59.887 72.747 59.835Q73.119 59.782 73.423 59.782L73.423 59.529Q73.423 59.324 73.316 59.144Q73.208 58.965 73.027 58.862Q72.846 58.760 72.637 58.760Q72.230 58.760 71.995 58.852Q72.083 58.889 72.130 58.973Q72.176 59.057 72.176 59.159Q72.176 59.255 72.130 59.334Q72.083 59.412 72.003 59.457Q71.923 59.501 71.834 59.501Q71.684 59.501 71.583 59.404Q71.482 59.306 71.482 59.159Q71.482 58.537 72.637 58.537Q72.849 58.537 73.099 58.601Q73.348 58.664 73.550 58.783Q73.751 58.903 73.878 59.088Q74.004 59.272 74.004 59.515L74.004 61.091Q74.004 61.207 74.066 61.303Q74.127 61.398 74.240 61.398Q74.350 61.398 74.414 61.304Q74.479 61.210 74.479 61.091L74.479 60.643L74.746 60.643L74.746 61.091Q74.746 61.361 74.519 61.526Q74.291 61.692 74.011 61.692Q73.803 61.692 73.666 61.538Q73.529 61.385 73.505 61.169Q73.358 61.436 73.076 61.581Q72.794 61.726 72.470 61.726Q72.193 61.726 71.909 61.651Q71.625 61.576 71.432 61.397Q71.239 61.217 71.239 60.930M71.854 60.930Q71.854 61.104 71.955 61.234Q72.056 61.364 72.212 61.434Q72.367 61.504 72.531 61.504Q72.750 61.504 72.958 61.407Q73.167 61.309 73.295 61.128Q73.423 60.947 73.423 60.721L73.423 59.993Q73.099 59.993 72.733 60.084Q72.367 60.175 72.111 60.387Q71.854 60.598 71.854 60.930M75.689 60.817L75.689 58.920L75.050 58.920L75.050 58.698Q75.368 58.698 75.585 58.488Q75.802 58.278 75.903 57.968Q76.004 57.659 76.004 57.351L76.270 57.351L76.270 58.640L77.347 58.640L77.347 58.920L76.270 58.920L76.270 60.804Q76.270 61.080 76.375 61.279Q76.479 61.477 76.739 61.477Q76.896 61.477 77.002 61.373Q77.108 61.268 77.157 61.115Q77.207 60.961 77.207 60.804L77.207 60.390L77.474 60.390L77.474 60.817Q77.474 61.043 77.374 61.253Q77.275 61.463 77.091 61.595Q76.906 61.726 76.677 61.726Q76.240 61.726 75.965 61.489Q75.689 61.251 75.689 60.817",[1870],[1854,10876,10877],{"transform":10720},[1866,10878],{"d":10879,"fill":1856,"stroke":1856,"className":10880,"style":1909},"M82.741 61.658L81.005 61.658L81.005 61.378Q81.234 61.378 81.383 61.344Q81.531 61.309 81.531 61.169L81.531 59.320Q81.531 59.050 81.424 58.989Q81.316 58.927 81.005 58.927L81.005 58.647L82.034 58.572L82.034 59.279Q82.164 58.971 82.406 58.772Q82.649 58.572 82.967 58.572Q83.186 58.572 83.357 58.696Q83.528 58.821 83.528 59.033Q83.528 59.170 83.428 59.269Q83.329 59.368 83.196 59.368Q83.059 59.368 82.960 59.269Q82.861 59.170 82.861 59.033Q82.861 58.893 82.960 58.794Q82.670 58.794 82.470 58.990Q82.270 59.187 82.177 59.481Q82.085 59.775 82.085 60.055L82.085 61.169Q82.085 61.378 82.741 61.378L82.741 61.658M84.071 60.175Q84.071 59.833 84.206 59.534Q84.341 59.235 84.580 59.011Q84.820 58.787 85.137 58.662Q85.455 58.537 85.787 58.537Q86.231 58.537 86.631 58.753Q87.031 58.968 87.265 59.346Q87.499 59.723 87.499 60.175Q87.499 60.516 87.357 60.800Q87.216 61.084 86.971 61.291Q86.727 61.497 86.417 61.612Q86.108 61.726 85.787 61.726Q85.356 61.726 84.955 61.525Q84.553 61.323 84.312 60.971Q84.071 60.619 84.071 60.175M85.787 61.477Q86.388 61.477 86.612 61.099Q86.836 60.721 86.836 60.089Q86.836 59.477 86.602 59.118Q86.368 58.760 85.787 58.760Q84.734 58.760 84.734 60.089Q84.734 60.721 84.960 61.099Q85.185 61.477 85.787 61.477M88.668 60.824L88.668 59.320Q88.668 59.050 88.561 58.989Q88.453 58.927 88.142 58.927L88.142 58.647L89.249 58.572L89.249 60.804L89.249 60.824Q89.249 61.104 89.301 61.248Q89.352 61.391 89.494 61.448Q89.635 61.504 89.923 61.504Q90.176 61.504 90.381 61.364Q90.586 61.224 90.702 60.998Q90.818 60.773 90.818 60.523L90.818 59.320Q90.818 59.050 90.710 58.989Q90.603 58.927 90.292 58.927L90.292 58.647L91.399 58.572L91.399 60.985Q91.399 61.176 91.452 61.258Q91.505 61.340 91.606 61.359Q91.707 61.378 91.922 61.378L91.922 61.658L90.845 61.726L90.845 61.162Q90.736 61.344 90.591 61.467Q90.446 61.590 90.259 61.658Q90.073 61.726 89.871 61.726Q88.668 61.726 88.668 60.824M94.192 61.658L92.558 61.658L92.558 61.378Q92.787 61.378 92.936 61.344Q93.084 61.309 93.084 61.169L93.084 59.320Q93.084 59.050 92.977 58.989Q92.869 58.927 92.558 58.927L92.558 58.647L93.617 58.572L93.617 59.221Q93.788 58.913 94.093 58.742Q94.397 58.572 94.742 58.572Q95.248 58.572 95.531 58.795Q95.815 59.019 95.815 59.515L95.815 61.169Q95.815 61.306 95.964 61.342Q96.113 61.378 96.338 61.378L96.338 61.658L94.708 61.658L94.708 61.378Q94.937 61.378 95.085 61.344Q95.234 61.309 95.234 61.169L95.234 59.529Q95.234 59.194 95.115 58.994Q94.995 58.794 94.680 58.794Q94.410 58.794 94.176 58.930Q93.942 59.067 93.804 59.301Q93.665 59.535 93.665 59.809L93.665 61.169Q93.665 61.306 93.816 61.342Q93.966 61.378 94.192 61.378L94.192 61.658M96.926 60.147Q96.926 59.809 97.066 59.518Q97.206 59.228 97.451 59.014Q97.695 58.801 97.999 58.686Q98.303 58.572 98.628 58.572Q98.898 58.572 99.161 58.671Q99.425 58.770 99.616 58.948L99.616 57.550Q99.616 57.280 99.508 57.218Q99.401 57.157 99.090 57.157L99.090 56.876L100.166 56.801L100.166 60.985Q100.166 61.173 100.221 61.256Q100.276 61.340 100.376 61.359Q100.477 61.378 100.693 61.378L100.693 61.658L99.585 61.726L99.585 61.309Q99.168 61.726 98.543 61.726Q98.112 61.726 97.740 61.514Q97.367 61.303 97.146 60.942Q96.926 60.581 96.926 60.147M98.601 61.504Q98.809 61.504 98.996 61.432Q99.182 61.361 99.336 61.224Q99.490 61.087 99.585 60.909L99.585 59.300Q99.500 59.153 99.354 59.033Q99.209 58.913 99.040 58.854Q98.871 58.794 98.690 58.794Q98.129 58.794 97.861 59.183Q97.593 59.573 97.593 60.154Q97.593 60.725 97.827 61.115Q98.061 61.504 98.601 61.504",[1870],[1854,10882,10883],{"transform":10720},[1866,10884],{"d":10885,"fill":1856,"stroke":1856,"className":10886,"style":1909},"M106.177 63.408Q105.627 63.008 105.256 62.453Q104.885 61.897 104.704 61.251Q104.523 60.605 104.523 59.908Q104.523 59.395 104.623 58.900Q104.724 58.404 104.929 57.953Q105.134 57.502 105.447 57.110Q105.760 56.719 106.177 56.415Q106.187 56.411 106.194 56.410Q106.201 56.408 106.211 56.408L106.279 56.408Q106.314 56.408 106.336 56.432Q106.358 56.456 106.358 56.493Q106.358 56.538 106.331 56.555Q105.982 56.856 105.729 57.240Q105.476 57.625 105.324 58.066Q105.172 58.507 105.100 58.963Q105.028 59.419 105.028 59.908Q105.028 60.909 105.338 61.796Q105.647 62.683 106.331 63.268Q106.358 63.285 106.358 63.329Q106.358 63.367 106.336 63.391Q106.314 63.415 106.279 63.415L106.211 63.415Q106.204 63.411 106.196 63.410Q106.187 63.408 106.177 63.408M108.785 61.658L107.233 61.658L107.233 61.378Q107.459 61.378 107.607 61.344Q107.756 61.309 107.756 61.169L107.756 59.320Q107.756 59.132 107.708 59.048Q107.660 58.965 107.563 58.946Q107.465 58.927 107.254 58.927L107.254 58.647L108.310 58.572L108.310 61.169Q108.310 61.309 108.441 61.344Q108.573 61.378 108.785 61.378L108.785 61.658M107.513 57.351Q107.513 57.180 107.636 57.061Q107.759 56.941 107.930 56.941Q108.098 56.941 108.221 57.061Q108.344 57.180 108.344 57.351Q108.344 57.526 108.221 57.649Q108.098 57.772 107.930 57.772Q107.759 57.772 107.636 57.649Q107.513 57.526 107.513 57.351M109.957 60.817L109.957 58.920L109.318 58.920L109.318 58.698Q109.636 58.698 109.853 58.488Q110.070 58.278 110.171 57.968Q110.272 57.659 110.272 57.351L110.538 57.351L110.538 58.640L111.615 58.640L111.615 58.920L110.538 58.920L110.538 60.804Q110.538 61.080 110.642 61.279Q110.747 61.477 111.006 61.477Q111.164 61.477 111.270 61.373Q111.376 61.268 111.425 61.115Q111.475 60.961 111.475 60.804L111.475 60.390L111.741 60.390L111.741 60.817Q111.741 61.043 111.642 61.253Q111.543 61.463 111.358 61.595Q111.174 61.726 110.945 61.726Q110.507 61.726 110.232 61.489Q109.957 61.251 109.957 60.817M112.551 61.651L112.551 60.588Q112.551 60.564 112.579 60.537Q112.606 60.510 112.630 60.510L112.739 60.510Q112.804 60.510 112.818 60.568Q112.914 61.002 113.160 61.253Q113.406 61.504 113.819 61.504Q114.161 61.504 114.414 61.371Q114.667 61.238 114.667 60.930Q114.667 60.773 114.573 60.658Q114.479 60.544 114.341 60.475Q114.202 60.407 114.035 60.369L113.454 60.270Q113.098 60.202 112.825 59.981Q112.551 59.761 112.551 59.419Q112.551 59.170 112.662 58.995Q112.774 58.821 112.960 58.722Q113.146 58.623 113.361 58.580Q113.577 58.537 113.819 58.537Q114.233 58.537 114.513 58.719L114.729 58.544Q114.739 58.541 114.746 58.539Q114.753 58.537 114.763 58.537L114.814 58.537Q114.841 58.537 114.865 58.561Q114.889 58.585 114.889 58.613L114.889 59.460Q114.889 59.481 114.865 59.508Q114.841 59.535 114.814 59.535L114.701 59.535Q114.674 59.535 114.648 59.510Q114.623 59.484 114.623 59.460Q114.623 59.224 114.517 59.060Q114.411 58.896 114.228 58.814Q114.045 58.732 113.813 58.732Q113.484 58.732 113.228 58.835Q112.972 58.937 112.972 59.214Q112.972 59.409 113.155 59.518Q113.337 59.628 113.566 59.669L114.141 59.775Q114.387 59.823 114.600 59.951Q114.814 60.079 114.951 60.282Q115.087 60.486 115.087 60.735Q115.087 61.248 114.722 61.487Q114.356 61.726 113.819 61.726Q113.324 61.726 112.992 61.432L112.726 61.706Q112.705 61.726 112.678 61.726L112.630 61.726Q112.606 61.726 112.579 61.699Q112.551 61.672 112.551 61.651",[1870],[1854,10888,10889],{"transform":10720},[1866,10890],{"d":10891,"fill":1856,"stroke":1856,"className":10892,"style":1909},"M120.173 61.658L118.437 61.658L118.437 61.378Q118.666 61.378 118.815 61.344Q118.963 61.309 118.963 61.169L118.963 59.320Q118.963 59.050 118.856 58.989Q118.748 58.927 118.437 58.927L118.437 58.647L119.466 58.572L119.466 59.279Q119.596 58.971 119.838 58.772Q120.081 58.572 120.399 58.572Q120.618 58.572 120.789 58.696Q120.960 58.821 120.960 59.033Q120.960 59.170 120.860 59.269Q120.761 59.368 120.628 59.368Q120.491 59.368 120.392 59.269Q120.293 59.170 120.293 59.033Q120.293 58.893 120.392 58.794Q120.102 58.794 119.902 58.990Q119.702 59.187 119.609 59.481Q119.517 59.775 119.517 60.055L119.517 61.169Q119.517 61.378 120.173 61.378L120.173 61.658M121.503 60.175Q121.503 59.833 121.638 59.534Q121.773 59.235 122.012 59.011Q122.252 58.787 122.569 58.662Q122.887 58.537 123.219 58.537Q123.663 58.537 124.063 58.753Q124.463 58.968 124.697 59.346Q124.931 59.723 124.931 60.175Q124.931 60.516 124.789 60.800Q124.648 61.084 124.403 61.291Q124.159 61.497 123.849 61.612Q123.540 61.726 123.219 61.726Q122.788 61.726 122.387 61.525Q121.985 61.323 121.744 60.971Q121.503 60.619 121.503 60.175M123.219 61.477Q123.820 61.477 124.044 61.099Q124.268 60.721 124.268 60.089Q124.268 59.477 124.034 59.118Q123.800 58.760 123.219 58.760Q122.166 58.760 122.166 60.089Q122.166 60.721 122.392 61.099Q122.617 61.477 123.219 61.477",[1870],[1854,10894,10895],{"transform":10720},[1866,10896],{"d":10897,"fill":1856,"stroke":1856,"className":10898,"style":1909},"M126.707 61.631L125.726 59.132Q125.665 58.989 125.547 58.954Q125.429 58.920 125.213 58.920L125.213 58.640L126.693 58.640L126.693 58.920Q126.314 58.920 126.314 59.081Q126.314 59.091 126.328 59.132L127.042 60.964L127.715 59.259Q127.685 59.187 127.685 59.159Q127.685 59.132 127.657 59.132Q127.596 58.985 127.478 58.953Q127.360 58.920 127.148 58.920L127.148 58.640L128.546 58.640L128.546 58.920Q128.170 58.920 128.170 59.081Q128.170 59.112 128.177 59.132L128.932 61.070L129.619 59.320Q129.640 59.269 129.640 59.214Q129.640 59.074 129.527 58.997Q129.414 58.920 129.274 58.920L129.274 58.640L130.494 58.640L130.494 58.920Q130.289 58.920 130.134 59.026Q129.978 59.132 129.906 59.320L129.001 61.631Q128.966 61.726 128.854 61.726L128.785 61.726Q128.676 61.726 128.638 61.631L127.856 59.628L127.069 61.631Q127.035 61.726 126.922 61.726L126.854 61.726Q126.745 61.726 126.707 61.631",[1870],[1854,10900,10901],{"transform":10720},[1866,10902],{"d":10903,"fill":1856,"stroke":1856,"className":10904,"style":1909},"M133.787 60.721Q133.787 60.346 134.109 60.034L134.997 59.221Q134.703 58.520 134.703 57.816Q134.703 57.519 134.849 57.251Q134.994 56.982 135.252 56.822Q135.510 56.661 135.807 56.661Q136.040 56.661 136.202 56.804Q136.365 56.948 136.443 57.170Q136.522 57.392 136.522 57.618Q136.522 57.939 136.243 58.286Q135.965 58.633 135.544 59.026Q135.729 59.450 136.052 59.898Q136.375 60.346 136.775 60.790Q137.017 60.551 137.240 60.262Q137.462 59.973 137.721 59.580L137.896 59.320Q137.944 59.248 137.944 59.180Q137.944 59.036 137.809 58.978Q137.674 58.920 137.516 58.920L137.516 58.640L139.099 58.640L139.099 58.920Q138.463 58.920 138.203 59.320L137.937 59.713Q137.650 60.140 137.419 60.441Q137.188 60.742 136.949 60.971Q137.192 61.214 137.450 61.366Q137.708 61.518 137.978 61.518Q138.179 61.518 138.366 61.429Q138.552 61.340 138.668 61.178Q138.784 61.015 138.784 60.810L139.051 60.810Q139.051 61.091 138.901 61.316Q138.750 61.542 138.501 61.670Q138.251 61.798 137.971 61.798Q137.253 61.798 136.587 61.296Q135.924 61.798 135.185 61.798Q134.850 61.798 134.529 61.677Q134.208 61.555 133.998 61.309Q133.787 61.063 133.787 60.721M135.250 61.518Q135.831 61.518 136.382 61.125Q136.112 60.892 135.879 60.621Q135.647 60.349 135.442 60.040Q135.237 59.730 135.096 59.440L134.785 59.720Q134.450 60.034 134.450 60.523Q134.450 60.899 134.669 61.209Q134.888 61.518 135.250 61.518M135.452 58.801Q135.807 58.452 136.033 58.168Q136.259 57.885 136.259 57.618Q136.259 57.454 136.214 57.288Q136.170 57.122 136.065 57.003Q135.961 56.883 135.801 56.883Q135.619 56.883 135.493 56.996Q135.366 57.109 135.302 57.280Q135.237 57.450 135.237 57.625Q135.237 58.223 135.452 58.801",[1870],[1854,10906,10907],{"transform":10720},[1866,10908],{"d":10909,"fill":1856,"stroke":1856,"className":10910,"style":1909},"M142.555 60.147Q142.555 59.819 142.690 59.518Q142.825 59.218 143.061 58.997Q143.297 58.777 143.601 58.657Q143.906 58.537 144.230 58.537Q144.736 58.537 145.085 58.640Q145.433 58.742 145.433 59.118Q145.433 59.265 145.336 59.366Q145.239 59.467 145.092 59.467Q144.938 59.467 144.839 59.368Q144.740 59.269 144.740 59.118Q144.740 58.930 144.880 58.838Q144.678 58.787 144.237 58.787Q143.882 58.787 143.653 58.983Q143.424 59.180 143.323 59.489Q143.222 59.799 143.222 60.147Q143.222 60.496 143.348 60.802Q143.475 61.108 143.730 61.292Q143.984 61.477 144.340 61.477Q144.562 61.477 144.746 61.393Q144.931 61.309 145.066 61.154Q145.201 60.998 145.259 60.790Q145.273 60.735 145.327 60.735L145.440 60.735Q145.471 60.735 145.493 60.759Q145.515 60.783 145.515 60.817L145.515 60.838Q145.430 61.125 145.242 61.323Q145.054 61.521 144.789 61.624Q144.524 61.726 144.230 61.726Q143.800 61.726 143.412 61.520Q143.024 61.313 142.790 60.950Q142.555 60.588 142.555 60.147M146.062 60.175Q146.062 59.833 146.197 59.534Q146.332 59.235 146.572 59.011Q146.811 58.787 147.129 58.662Q147.447 58.537 147.778 58.537Q148.222 58.537 148.622 58.753Q149.022 58.968 149.256 59.346Q149.491 59.723 149.491 60.175Q149.491 60.516 149.349 60.800Q149.207 61.084 148.962 61.291Q148.718 61.497 148.409 61.612Q148.099 61.726 147.778 61.726Q147.347 61.726 146.946 61.525Q146.544 61.323 146.303 60.971Q146.062 60.619 146.062 60.175M147.778 61.477Q148.380 61.477 148.604 61.099Q148.827 60.721 148.827 60.089Q148.827 59.477 148.593 59.118Q148.359 58.760 147.778 58.760Q146.725 58.760 146.725 60.089Q146.725 60.721 146.951 61.099Q147.177 61.477 147.778 61.477M151.753 61.658L150.150 61.658L150.150 61.378Q150.376 61.378 150.524 61.344Q150.673 61.309 150.673 61.169L150.673 57.550Q150.673 57.280 150.565 57.218Q150.458 57.157 150.150 57.157L150.150 56.876L151.227 56.801L151.227 61.169Q151.227 61.306 151.377 61.342Q151.528 61.378 151.753 61.378L151.753 61.658M152.922 60.824L152.922 59.320Q152.922 59.050 152.814 58.989Q152.707 58.927 152.396 58.927L152.396 58.647L153.503 58.572L153.503 60.804L153.503 60.824Q153.503 61.104 153.554 61.248Q153.606 61.391 153.748 61.448Q153.889 61.504 154.177 61.504Q154.429 61.504 154.635 61.364Q154.840 61.224 154.956 60.998Q155.072 60.773 155.072 60.523L155.072 59.320Q155.072 59.050 154.964 58.989Q154.857 58.927 154.546 58.927L154.546 58.647L155.653 58.572L155.653 60.985Q155.653 61.176 155.706 61.258Q155.759 61.340 155.860 61.359Q155.961 61.378 156.176 61.378L156.176 61.658L155.099 61.726L155.099 61.162Q154.990 61.344 154.845 61.467Q154.699 61.590 154.513 61.658Q154.327 61.726 154.125 61.726Q152.922 61.726 152.922 60.824M158.446 61.658L156.812 61.658L156.812 61.378Q157.041 61.378 157.189 61.344Q157.338 61.309 157.338 61.169L157.338 59.320Q157.338 59.050 157.231 58.989Q157.123 58.927 156.812 58.927L156.812 58.647L157.871 58.572L157.871 59.221Q158.042 58.913 158.346 58.742Q158.651 58.572 158.996 58.572Q159.396 58.572 159.673 58.712Q159.949 58.852 160.035 59.200Q160.202 58.907 160.501 58.739Q160.801 58.572 161.146 58.572Q161.652 58.572 161.935 58.795Q162.219 59.019 162.219 59.515L162.219 61.169Q162.219 61.306 162.368 61.342Q162.516 61.378 162.742 61.378L162.742 61.658L161.112 61.658L161.112 61.378Q161.337 61.378 161.488 61.342Q161.638 61.306 161.638 61.169L161.638 59.529Q161.638 59.194 161.518 58.994Q161.399 58.794 161.084 58.794Q160.814 58.794 160.580 58.930Q160.346 59.067 160.208 59.301Q160.069 59.535 160.069 59.809L160.069 61.169Q160.069 61.306 160.218 61.342Q160.366 61.378 160.592 61.378L160.592 61.658L158.962 61.658L158.962 61.378Q159.191 61.378 159.339 61.344Q159.488 61.309 159.488 61.169L159.488 59.529Q159.488 59.194 159.368 58.994Q159.249 58.794 158.934 58.794Q158.664 58.794 158.430 58.930Q158.196 59.067 158.058 59.301Q157.919 59.535 157.919 59.809L157.919 61.169Q157.919 61.306 158.070 61.342Q158.220 61.378 158.446 61.378L158.446 61.658M165.012 61.658L163.378 61.658L163.378 61.378Q163.607 61.378 163.755 61.344Q163.904 61.309 163.904 61.169L163.904 59.320Q163.904 59.050 163.796 58.989Q163.689 58.927 163.378 58.927L163.378 58.647L164.437 58.572L164.437 59.221Q164.608 58.913 164.912 58.742Q165.217 58.572 165.562 58.572Q166.068 58.572 166.351 58.795Q166.635 59.019 166.635 59.515L166.635 61.169Q166.635 61.306 166.784 61.342Q166.932 61.378 167.158 61.378L167.158 61.658L165.528 61.658L165.528 61.378Q165.757 61.378 165.905 61.344Q166.054 61.309 166.054 61.169L166.054 59.529Q166.054 59.194 165.934 58.994Q165.815 58.794 165.500 58.794Q165.230 58.794 164.996 58.930Q164.762 59.067 164.624 59.301Q164.485 59.535 164.485 59.809L164.485 61.169Q164.485 61.306 164.636 61.342Q164.786 61.378 165.012 61.378",[1870],[1854,10912,10913],{"transform":10720},[1866,10914],{"d":10915,"fill":1856,"stroke":1856,"className":10916,"style":1909},"M170.458 60.147Q170.458 59.809 170.599 59.518Q170.739 59.228 170.983 59.014Q171.227 58.801 171.532 58.686Q171.836 58.572 172.161 58.572Q172.431 58.572 172.694 58.671Q172.957 58.770 173.148 58.948L173.148 57.550Q173.148 57.280 173.041 57.218Q172.933 57.157 172.622 57.157L172.622 56.876L173.699 56.801L173.699 60.985Q173.699 61.173 173.753 61.256Q173.808 61.340 173.909 61.359Q174.010 61.378 174.225 61.378L174.225 61.658L173.118 61.726L173.118 61.309Q172.701 61.726 172.075 61.726Q171.644 61.726 171.272 61.514Q170.899 61.303 170.679 60.942Q170.458 60.581 170.458 60.147M172.133 61.504Q172.342 61.504 172.528 61.432Q172.714 61.361 172.868 61.224Q173.022 61.087 173.118 60.909L173.118 59.300Q173.032 59.153 172.887 59.033Q172.742 58.913 172.572 58.854Q172.403 58.794 172.222 58.794Q171.662 58.794 171.393 59.183Q171.125 59.573 171.125 60.154Q171.125 60.725 171.359 61.115Q171.593 61.504 172.133 61.504M176.624 61.658L174.888 61.658L174.888 61.378Q175.117 61.378 175.266 61.344Q175.415 61.309 175.415 61.169L175.415 59.320Q175.415 59.050 175.307 58.989Q175.199 58.927 174.888 58.927L174.888 58.647L175.917 58.572L175.917 59.279Q176.047 58.971 176.290 58.772Q176.532 58.572 176.850 58.572Q177.069 58.572 177.240 58.696Q177.411 58.821 177.411 59.033Q177.411 59.170 177.311 59.269Q177.212 59.368 177.079 59.368Q176.942 59.368 176.843 59.269Q176.744 59.170 176.744 59.033Q176.744 58.893 176.843 58.794Q176.553 58.794 176.353 58.990Q176.153 59.187 176.061 59.481Q175.968 59.775 175.968 60.055L175.968 61.169Q175.968 61.378 176.624 61.378L176.624 61.658M179.612 61.658L178.060 61.658L178.060 61.378Q178.286 61.378 178.434 61.344Q178.583 61.309 178.583 61.169L178.583 59.320Q178.583 59.132 178.535 59.048Q178.487 58.965 178.390 58.946Q178.292 58.927 178.081 58.927L178.081 58.647L179.137 58.572L179.137 61.169Q179.137 61.309 179.268 61.344Q179.400 61.378 179.612 61.378L179.612 61.658M178.340 57.351Q178.340 57.180 178.463 57.061Q178.586 56.941 178.757 56.941Q178.925 56.941 179.048 57.061Q179.171 57.180 179.171 57.351Q179.171 57.526 179.048 57.649Q178.925 57.772 178.757 57.772Q178.586 57.772 178.463 57.649Q178.340 57.526 178.340 57.351M181.847 61.631L180.719 59.132Q180.647 58.985 180.518 58.953Q180.388 58.920 180.159 58.920L180.159 58.640L181.673 58.640L181.673 58.920Q181.321 58.920 181.321 59.067Q181.321 59.112 181.331 59.132L182.196 61.050L182.975 59.320Q183.009 59.252 183.009 59.173Q183.009 59.060 182.925 58.990Q182.842 58.920 182.722 58.920L182.722 58.640L183.918 58.640L183.918 58.920Q183.700 58.920 183.529 59.023Q183.358 59.125 183.269 59.320L182.233 61.631Q182.186 61.726 182.080 61.726L182.001 61.726Q181.895 61.726 181.847 61.631",[1870],[1854,10918,10919],{"transform":10720},[1866,10920],{"d":10921,"fill":1856,"stroke":1856,"className":10922,"style":1909},"M184.209 60.123Q184.209 59.802 184.334 59.513Q184.459 59.224 184.685 59.001Q184.910 58.777 185.206 58.657Q185.501 58.537 185.819 58.537Q186.147 58.537 186.409 58.637Q186.670 58.736 186.846 58.918Q187.022 59.101 187.116 59.359Q187.210 59.617 187.210 59.949Q187.210 60.041 187.128 60.062L184.873 60.062L184.873 60.123Q184.873 60.711 185.156 61.094Q185.440 61.477 186.007 61.477Q186.329 61.477 186.597 61.284Q186.865 61.091 186.954 60.776Q186.961 60.735 187.036 60.721L187.128 60.721Q187.210 60.745 187.210 60.817Q187.210 60.824 187.204 60.851Q187.091 61.248 186.720 61.487Q186.349 61.726 185.925 61.726Q185.488 61.726 185.088 61.518Q184.688 61.309 184.449 60.942Q184.209 60.575 184.209 60.123M184.879 59.853L186.694 59.853Q186.694 59.576 186.597 59.324Q186.499 59.071 186.301 58.915Q186.103 58.760 185.819 58.760Q185.542 58.760 185.329 58.918Q185.115 59.077 184.997 59.332Q184.879 59.587 184.879 59.853",[1870],[1854,10924,10925],{"transform":10720},[1866,10926],{"d":10927,"fill":1856,"stroke":1856,"className":10928,"style":1909},"M191.027 60.817L191.027 58.920L190.388 58.920L190.388 58.698Q190.706 58.698 190.923 58.488Q191.140 58.278 191.240 57.968Q191.341 57.659 191.341 57.351L191.608 57.351L191.608 58.640L192.685 58.640L192.685 58.920L191.608 58.920L191.608 60.804Q191.608 61.080 191.712 61.279Q191.816 61.477 192.076 61.477Q192.233 61.477 192.339 61.373Q192.445 61.268 192.495 61.115Q192.544 60.961 192.544 60.804L192.544 60.390L192.811 60.390L192.811 60.817Q192.811 61.043 192.712 61.253Q192.613 61.463 192.428 61.595Q192.244 61.726 192.015 61.726Q191.577 61.726 191.302 61.489Q191.027 61.251 191.027 60.817M195.303 61.658L193.669 61.658L193.669 61.378Q193.898 61.378 194.047 61.344Q194.195 61.309 194.195 61.169L194.195 57.550Q194.195 57.280 194.088 57.218Q193.980 57.157 193.669 57.157L193.669 56.876L194.749 56.801L194.749 59.187Q194.855 59.002 195.033 58.860Q195.210 58.719 195.419 58.645Q195.627 58.572 195.853 58.572Q196.359 58.572 196.643 58.795Q196.926 59.019 196.926 59.515L196.926 61.169Q196.926 61.306 197.075 61.342Q197.224 61.378 197.449 61.378L197.449 61.658L195.819 61.658L195.819 61.378Q196.048 61.378 196.196 61.344Q196.345 61.309 196.345 61.169L196.345 59.529Q196.345 59.194 196.226 58.994Q196.106 58.794 195.791 58.794Q195.521 58.794 195.287 58.930Q195.053 59.067 194.915 59.301Q194.776 59.535 194.776 59.809L194.776 61.169Q194.776 61.306 194.927 61.342Q195.077 61.378 195.303 61.378L195.303 61.658M197.996 60.123Q197.996 59.802 198.121 59.513Q198.246 59.224 198.471 59.001Q198.697 58.777 198.992 58.657Q199.288 58.537 199.606 58.537Q199.934 58.537 200.196 58.637Q200.457 58.736 200.633 58.918Q200.809 59.101 200.903 59.359Q200.997 59.617 200.997 59.949Q200.997 60.041 200.915 60.062L198.659 60.062L198.659 60.123Q198.659 60.711 198.943 61.094Q199.227 61.477 199.794 61.477Q200.115 61.477 200.384 61.284Q200.652 61.091 200.741 60.776Q200.748 60.735 200.823 60.721L200.915 60.721Q200.997 60.745 200.997 60.817Q200.997 60.824 200.990 60.851Q200.877 61.248 200.507 61.487Q200.136 61.726 199.712 61.726Q199.274 61.726 198.874 61.518Q198.475 61.309 198.235 60.942Q197.996 60.575 197.996 60.123M198.666 59.853L200.481 59.853Q200.481 59.576 200.384 59.324Q200.286 59.071 200.088 58.915Q199.890 58.760 199.606 58.760Q199.329 58.760 199.115 58.918Q198.902 59.077 198.784 59.332Q198.666 59.587 198.666 59.853",[1870],[1854,10930,10931],{"transform":10720},[1866,10932],{"d":10933,"fill":1856,"stroke":1856,"className":10934,"style":1909},"M204.867 60.824L204.867 59.320Q204.867 59.050 204.759 58.989Q204.651 58.927 204.340 58.927L204.340 58.647L205.448 58.572L205.448 60.804L205.448 60.824Q205.448 61.104 205.499 61.248Q205.550 61.391 205.692 61.448Q205.834 61.504 206.121 61.504Q206.374 61.504 206.579 61.364Q206.784 61.224 206.900 60.998Q207.017 60.773 207.017 60.523L207.017 59.320Q207.017 59.050 206.909 58.989Q206.801 58.927 206.490 58.927L206.490 58.647L207.598 58.572L207.598 60.985Q207.598 61.176 207.651 61.258Q207.704 61.340 207.804 61.359Q207.905 61.378 208.121 61.378L208.121 61.658L207.044 61.726L207.044 61.162Q206.935 61.344 206.789 61.467Q206.644 61.590 206.458 61.658Q206.271 61.726 206.070 61.726Q204.867 61.726 204.867 60.824M210.353 63.015L208.722 63.015L208.722 62.735Q208.951 62.735 209.100 62.700Q209.249 62.666 209.249 62.526L209.249 59.180Q209.249 59.009 209.112 58.968Q208.975 58.927 208.722 58.927L208.722 58.647L209.802 58.572L209.802 58.978Q210.024 58.777 210.311 58.674Q210.599 58.572 210.906 58.572Q211.333 58.572 211.697 58.785Q212.061 58.999 212.275 59.363Q212.489 59.727 212.489 60.147Q212.489 60.592 212.249 60.956Q212.010 61.320 211.617 61.523Q211.224 61.726 210.780 61.726Q210.513 61.726 210.265 61.626Q210.018 61.525 209.830 61.344L209.830 62.526Q209.830 62.663 209.978 62.699Q210.127 62.735 210.353 62.735L210.353 63.015M209.830 59.327L209.830 60.937Q209.963 61.190 210.206 61.347Q210.448 61.504 210.725 61.504Q211.053 61.504 211.306 61.303Q211.559 61.101 211.692 60.783Q211.826 60.465 211.826 60.147Q211.826 59.918 211.761 59.689Q211.696 59.460 211.568 59.262Q211.439 59.064 211.245 58.944Q211.050 58.825 210.817 58.825Q210.523 58.825 210.255 58.954Q209.987 59.084 209.830 59.327",[1870],[1854,10936,10937],{"transform":10720},[1866,10938],{"d":10939,"fill":1856,"stroke":1856,"className":10940,"style":1909},"M213.340 60.147Q213.340 59.809 213.481 59.518Q213.621 59.228 213.865 59.014Q214.109 58.801 214.414 58.686Q214.718 58.572 215.043 58.572Q215.313 58.572 215.576 58.671Q215.839 58.770 216.030 58.948L216.030 57.550Q216.030 57.280 215.923 57.218Q215.815 57.157 215.504 57.157L215.504 56.876L216.581 56.801L216.581 60.985Q216.581 61.173 216.635 61.256Q216.690 61.340 216.791 61.359Q216.892 61.378 217.107 61.378L217.107 61.658L216 61.726L216 61.309Q215.583 61.726 214.957 61.726Q214.526 61.726 214.154 61.514Q213.781 61.303 213.561 60.942Q213.340 60.581 213.340 60.147M215.015 61.504Q215.224 61.504 215.410 61.432Q215.596 61.361 215.750 61.224Q215.904 61.087 216 60.909L216 59.300Q215.914 59.153 215.769 59.033Q215.624 58.913 215.454 58.854Q215.285 58.794 215.104 58.794Q214.544 58.794 214.275 59.183Q214.007 59.573 214.007 60.154Q214.007 60.725 214.241 61.115Q214.475 61.504 215.015 61.504M217.815 60.930Q217.815 60.598 218.038 60.371Q218.262 60.144 218.606 60.016Q218.949 59.887 219.322 59.835Q219.694 59.782 219.999 59.782L219.999 59.529Q219.999 59.324 219.891 59.144Q219.783 58.965 219.602 58.862Q219.421 58.760 219.213 58.760Q218.806 58.760 218.570 58.852Q218.659 58.889 218.705 58.973Q218.751 59.057 218.751 59.159Q218.751 59.255 218.705 59.334Q218.659 59.412 218.578 59.457Q218.498 59.501 218.409 59.501Q218.259 59.501 218.158 59.404Q218.057 59.306 218.057 59.159Q218.057 58.537 219.213 58.537Q219.424 58.537 219.674 58.601Q219.923 58.664 220.125 58.783Q220.327 58.903 220.453 59.088Q220.580 59.272 220.580 59.515L220.580 61.091Q220.580 61.207 220.641 61.303Q220.703 61.398 220.816 61.398Q220.925 61.398 220.990 61.304Q221.055 61.210 221.055 61.091L221.055 60.643L221.321 60.643L221.321 61.091Q221.321 61.361 221.094 61.526Q220.867 61.692 220.587 61.692Q220.378 61.692 220.241 61.538Q220.105 61.385 220.081 61.169Q219.934 61.436 219.652 61.581Q219.370 61.726 219.045 61.726Q218.768 61.726 218.484 61.651Q218.201 61.576 218.008 61.397Q217.815 61.217 217.815 60.930M218.430 60.930Q218.430 61.104 218.531 61.234Q218.631 61.364 218.787 61.434Q218.943 61.504 219.107 61.504Q219.325 61.504 219.534 61.407Q219.742 61.309 219.870 61.128Q219.999 60.947 219.999 60.721L219.999 59.993Q219.674 59.993 219.308 60.084Q218.943 60.175 218.686 60.387Q218.430 60.598 218.430 60.930M222.265 60.817L222.265 58.920L221.626 58.920L221.626 58.698Q221.943 58.698 222.161 58.488Q222.378 58.278 222.478 57.968Q222.579 57.659 222.579 57.351L222.846 57.351L222.846 58.640L223.922 58.640L223.922 58.920L222.846 58.920L222.846 60.804Q222.846 61.080 222.950 61.279Q223.054 61.477 223.314 61.477Q223.471 61.477 223.577 61.373Q223.683 61.268 223.733 61.115Q223.782 60.961 223.782 60.804L223.782 60.390L224.049 60.390L224.049 60.817Q224.049 61.043 223.950 61.253Q223.851 61.463 223.666 61.595Q223.482 61.726 223.253 61.726Q222.815 61.726 222.540 61.489Q222.265 61.251 222.265 60.817M224.818 60.123Q224.818 59.802 224.943 59.513Q225.068 59.224 225.293 59.001Q225.519 58.777 225.814 58.657Q226.110 58.537 226.428 58.537Q226.756 58.537 227.017 58.637Q227.279 58.736 227.455 58.918Q227.631 59.101 227.725 59.359Q227.819 59.617 227.819 59.949Q227.819 60.041 227.737 60.062L225.481 60.062L225.481 60.123Q225.481 60.711 225.765 61.094Q226.048 61.477 226.616 61.477Q226.937 61.477 227.205 61.284Q227.474 61.091 227.563 60.776Q227.569 60.735 227.645 60.721L227.737 60.721Q227.819 60.745 227.819 60.817Q227.819 60.824 227.812 60.851Q227.699 61.248 227.328 61.487Q226.958 61.726 226.534 61.726Q226.096 61.726 225.696 61.518Q225.297 61.309 225.057 60.942Q224.818 60.575 224.818 60.123M225.488 59.853L227.303 59.853Q227.303 59.576 227.205 59.324Q227.108 59.071 226.910 58.915Q226.712 58.760 226.428 58.760Q226.151 58.760 225.937 58.918Q225.724 59.077 225.606 59.332Q225.488 59.587 225.488 59.853M228.407 61.651L228.407 60.588Q228.407 60.564 228.434 60.537Q228.462 60.510 228.485 60.510L228.595 60.510Q228.660 60.510 228.673 60.568Q228.769 61.002 229.015 61.253Q229.261 61.504 229.675 61.504Q230.017 61.504 230.270 61.371Q230.523 61.238 230.523 60.930Q230.523 60.773 230.429 60.658Q230.335 60.544 230.196 60.475Q230.058 60.407 229.890 60.369L229.309 60.270Q228.954 60.202 228.680 59.981Q228.407 59.761 228.407 59.419Q228.407 59.170 228.518 58.995Q228.629 58.821 228.815 58.722Q229.002 58.623 229.217 58.580Q229.432 58.537 229.675 58.537Q230.089 58.537 230.369 58.719L230.584 58.544Q230.594 58.541 230.601 58.539Q230.608 58.537 230.618 58.537L230.670 58.537Q230.697 58.537 230.721 58.561Q230.745 58.585 230.745 58.613L230.745 59.460Q230.745 59.481 230.721 59.508Q230.697 59.535 230.670 59.535L230.557 59.535Q230.529 59.535 230.504 59.510Q230.478 59.484 230.478 59.460Q230.478 59.224 230.372 59.060Q230.266 58.896 230.083 58.814Q229.901 58.732 229.668 58.732Q229.340 58.732 229.084 58.835Q228.827 58.937 228.827 59.214Q228.827 59.409 229.010 59.518Q229.193 59.628 229.422 59.669L229.996 59.775Q230.242 59.823 230.456 59.951Q230.670 60.079 230.806 60.282Q230.943 60.486 230.943 60.735Q230.943 61.248 230.577 61.487Q230.212 61.726 229.675 61.726Q229.179 61.726 228.848 61.432L228.581 61.706Q228.561 61.726 228.533 61.726L228.485 61.726Q228.462 61.726 228.434 61.699Q228.407 61.672 228.407 61.651M231.893 63.415L231.825 63.415Q231.791 63.415 231.768 63.389Q231.746 63.364 231.746 63.329Q231.746 63.285 231.777 63.268Q232.132 62.964 232.382 62.574Q232.631 62.184 232.784 61.752Q232.936 61.320 233.006 60.851Q233.076 60.383 233.076 59.908Q233.076 59.429 233.006 58.963Q232.936 58.496 232.782 58.061Q232.628 57.625 232.377 57.237Q232.126 56.849 231.777 56.555Q231.746 56.538 231.746 56.493Q231.746 56.459 231.768 56.434Q231.791 56.408 231.825 56.408L231.893 56.408Q231.903 56.408 231.912 56.410Q231.921 56.411 231.931 56.415Q232.474 56.815 232.847 57.368Q233.219 57.922 233.401 58.568Q233.582 59.214 233.582 59.908Q233.582 60.609 233.401 61.256Q233.219 61.904 232.845 62.458Q232.471 63.012 231.931 63.408Q231.921 63.408 231.912 63.410Q231.903 63.411 231.893 63.415M235.092 61.238Q235.092 61.070 235.215 60.947Q235.339 60.824 235.513 60.824Q235.680 60.824 235.803 60.947Q235.926 61.070 235.926 61.238Q235.926 61.412 235.803 61.535Q235.680 61.658 235.513 61.658Q235.339 61.658 235.215 61.535Q235.092 61.412 235.092 61.238",[1870],[2089,10942,10944,10945,10995,10996,11038,11039,11054,11055,11070,11071,11121],{"className":10943},[2092],"Floyd-Warshall, one matrix per round. ",[415,10946,10948],{"className":10947},[418],[415,10949,10951],{"className":10950,"ariaHidden":423},[422],[415,10952,10954,10957],{"className":10953},[427],[415,10955],{"className":10956,"style":8959},[431],[415,10958,10960,10963],{"className":10959},[436],[415,10961,1173],{"className":10962},[436,437],[415,10964,10966],{"className":10965},[582],[415,10967,10969],{"className":10968},[586],[415,10970,10972],{"className":10971},[591],[415,10973,10975],{"className":10974,"style":8959},[595],[415,10976,10977,10980],{"style":8154},[415,10978],{"className":10979,"style":604},[603],[415,10981,10983],{"className":10982},[608,609,610,611],[415,10984,10986,10989,10992],{"className":10985},[436,611],[415,10987,463],{"className":10988},[462,611],[415,10990,727],{"className":10991,"style":726},[436,437,611],[415,10993,487],{"className":10994},[486,611]," uses only vertices ",[415,10997,10999],{"className":10998},[418],[415,11000,11002],{"className":11001,"ariaHidden":423},[422],[415,11003,11005,11008,11011,11014,11017,11020,11023,11026,11029,11032,11035],{"className":11004},[427],[415,11006],{"className":11007,"style":458},[431],[415,11009,2971],{"className":11010},[462],[415,11012,665],{"className":11013},[436],[415,11015,473],{"className":11016},[472],[415,11018],{"className":11019,"style":477},[442],[415,11021,686],{"className":11022},[566],[415,11024],{"className":11025,"style":477},[442],[415,11027,473],{"className":11028},[472],[415,11030],{"className":11031,"style":477},[442],[415,11033,727],{"className":11034,"style":726},[436,437],[415,11036,2978],{"className":11037},[486]," as intermediates; each round's pivot vertex ",[415,11040,11042],{"className":11041},[418],[415,11043,11045],{"className":11044,"ariaHidden":423},[422],[415,11046,11048,11051],{"className":11047},[427],[415,11049],{"className":11050,"style":1163},[431],[415,11052,727],{"className":11053,"style":726},[436,437]," is marked in blue on the indices (its row and column drive that round's updates), and the entries that improved when ",[415,11056,11058],{"className":11057},[418],[415,11059,11061],{"className":11060,"ariaHidden":423},[422],[415,11062,11064,11067],{"className":11063},[427],[415,11065],{"className":11066,"style":1163},[431],[415,11068,727],{"className":11069,"style":726},[436,437]," was admitted are shaded. ",[415,11072,11074],{"className":11073},[418],[415,11075,11077],{"className":11076,"ariaHidden":423},[422],[415,11078,11080,11083],{"className":11079},[427],[415,11081],{"className":11082,"style":8959},[431],[415,11084,11086,11089],{"className":11085},[436],[415,11087,1173],{"className":11088},[436,437],[415,11090,11092],{"className":11091},[582],[415,11093,11095],{"className":11094},[586],[415,11096,11098],{"className":11097},[591],[415,11099,11101],{"className":11100,"style":8959},[595],[415,11102,11103,11106],{"style":8154},[415,11104],{"className":11105,"style":604},[603],[415,11107,11109],{"className":11108},[608,609,610,611],[415,11110,11112,11115,11118],{"className":11111},[436,611],[415,11113,463],{"className":11114},[462,611],[415,11116,4252],{"className":11117},[436,611],[415,11119,487],{"className":11120},[486,611]," holds the all-pairs shortest distances.",[381,11123,11124,11125,3455,11140,3455,11155,11170,11171,11186],{},"Three nested loops over ",[415,11126,11128],{"className":11127},[418],[415,11129,11131],{"className":11130,"ariaHidden":423},[422],[415,11132,11134,11137],{"className":11133},[427],[415,11135],{"className":11136,"style":1163},[431],[415,11138,727],{"className":11139,"style":726},[436,437],[415,11141,11143],{"className":11142},[418],[415,11144,11146],{"className":11145,"ariaHidden":423},[422],[415,11147,11149,11152],{"className":11148},[427],[415,11150],{"className":11151,"style":8333},[431],[415,11153,823],{"className":11154},[436,437],[415,11156,11158],{"className":11157},[418],[415,11159,11161],{"className":11160,"ariaHidden":423},[422],[415,11162,11164,11167],{"className":11163},[427],[415,11165],{"className":11166,"style":8349},[431],[415,11168,8353],{"className":11169,"style":8285},[436,437]," evaluate this recurrence directly: the outer\nloop admits one intermediate vertex ",[415,11172,11174],{"className":11173},[418],[415,11175,11177],{"className":11176,"ariaHidden":423},[422],[415,11178,11180,11183],{"className":11179},[427],[415,11181],{"className":11182,"style":1163},[431],[415,11184,727],{"className":11185,"style":726},[436,437]," per round (exactly the five matrices above),\nand the inner two relax every pair against a route through it.",[1381,11188,11190],{"className":1383,"code":11189,"language":1385,"meta":376,"style":376},"caption: $\\textsc{Floyd-Warshall}(W)$ — all-pairs shortest paths in $O(V^3)$\n$d \\gets W$ \u002F\u002F $d_{ij}$: edge weight; $0$ on the diagonal, else $\\infty$\nfor $k \\gets 1$ to $n$ do \u002F\u002F admit vertex $k$ as an intermediate\n  for $i \\gets 1$ to $n$ do\n    for $j \\gets 1$ to $n$ do\n      if $d_{ik} + d_{kj} \u003C d_{ij}$ then \u002F\u002F routing through $k$ is shorter\n        $d_{ij} \\gets d_{ik} + d_{kj}$\nreturn $d$\n",[1387,11191,11192,11197,11202,11207,11212,11217,11222,11227],{"__ignoreMap":376},[415,11193,11194],{"class":1391,"line":6},[415,11195,11196],{},"caption: $\\textsc{Floyd-Warshall}(W)$ — all-pairs shortest paths in $O(V^3)$\n",[415,11198,11199],{"class":1391,"line":18},[415,11200,11201],{},"$d \\gets W$ \u002F\u002F $d_{ij}$: edge weight; $0$ on the diagonal, else $\\infty$\n",[415,11203,11204],{"class":1391,"line":24},[415,11205,11206],{},"for $k \\gets 1$ to $n$ do \u002F\u002F admit vertex $k$ as an intermediate\n",[415,11208,11209],{"class":1391,"line":73},[415,11210,11211],{},"  for $i \\gets 1$ to $n$ do\n",[415,11213,11214],{"class":1391,"line":102},[415,11215,11216],{},"    for $j \\gets 1$ to $n$ do\n",[415,11218,11219],{"class":1391,"line":108},[415,11220,11221],{},"      if $d_{ik} + d_{kj} \u003C d_{ij}$ then \u002F\u002F routing through $k$ is shorter\n",[415,11223,11224],{"class":1391,"line":116},[415,11225,11226],{},"        $d_{ij} \\gets d_{ik} + d_{kj}$\n",[415,11228,11229],{"class":1391,"line":196},[415,11230,11231],{},"return $d$\n",[381,11233,11234,11235,11285,11286,11336,11337,11344,11345,11419],{},"This is ",[415,11236,11238],{"className":11237},[418],[415,11239,11241],{"className":11240,"ariaHidden":423},[422],[415,11242,11244,11247,11250,11253,11282],{"className":11243},[427],[415,11245],{"className":11246,"style":8126},[431],[415,11248,6799],{"className":11249},[436],[415,11251,463],{"className":11252},[462],[415,11254,11256,11259],{"className":11255},[436],[415,11257,468],{"className":11258,"style":467},[436,437],[415,11260,11262],{"className":11261},[582],[415,11263,11265],{"className":11264},[586],[415,11266,11268],{"className":11267},[591],[415,11269,11271],{"className":11270,"style":8151},[595],[415,11272,11273,11276],{"style":8154},[415,11274],{"className":11275,"style":604},[603],[415,11277,11279],{"className":11278},[608,609,610,611],[415,11280,1633],{"className":11281},[436,611],[415,11283,487],{"className":11284},[486]," time and ",[415,11287,11289],{"className":11288},[418],[415,11290,11292],{"className":11291,"ariaHidden":423},[422],[415,11293,11295,11298,11301,11304,11333],{"className":11294},[427],[415,11296],{"className":11297,"style":8126},[431],[415,11299,6799],{"className":11300},[436],[415,11302,463],{"className":11303},[462],[415,11305,11307,11310],{"className":11306},[436],[415,11308,468],{"className":11309,"style":467},[436,437],[415,11311,11313],{"className":11312},[582],[415,11314,11316],{"className":11315},[586],[415,11317,11319],{"className":11318},[591],[415,11320,11322],{"className":11321,"style":8151},[595],[415,11323,11324,11327],{"style":8154},[415,11325],{"className":11326,"style":604},[603],[415,11328,11330],{"className":11329},[608,609,610,611],[415,11331,2361],{"className":11332},[436,611],[415,11334,487],{"className":11335},[486]," space: compact, cache-friendly, and a\nclean win on dense graphs.",[1542,11338,11339],{},[385,11340,1633],{"href":11341,"ariaDescribedBy":11342,"dataFootnoteRef":376,"id":11343},"#user-content-fn-skiena-sp",[1548],"user-content-fnref-skiena-sp"," It handles negative edges, and a negative entry on\nthe diagonal (",[415,11346,11348],{"className":11347},[418],[415,11349,11351,11410],{"className":11350,"ariaHidden":423},[422],[415,11352,11354,11357,11401,11404,11407],{"className":11353},[427],[415,11355],{"className":11356,"style":9096},[431],[415,11358,11360,11363],{"className":11359},[436],[415,11361,1173],{"className":11362},[436,437],[415,11364,11366],{"className":11365},[582],[415,11367,11369,11393],{"className":11368},[586,587],[415,11370,11372,11390],{"className":11371},[591],[415,11373,11375],{"className":11374,"style":884},[595],[415,11376,11377,11380],{"style":9117},[415,11378],{"className":11379,"style":604},[603],[415,11381,11383],{"className":11382},[608,609,610,611],[415,11384,11386],{"className":11385},[436,611],[415,11387,11389],{"className":11388},[436,437,611],"ii",[415,11391,620],{"className":11392},[619],[415,11394,11396],{"className":11395},[591],[415,11397,11399],{"className":11398,"style":627},[595],[415,11400],{},[415,11402],{"className":11403,"style":443},[442],[415,11405,1752],{"className":11406},[447],[415,11408],{"className":11409,"style":443},[442],[415,11411,11413,11416],{"className":11412},[427],[415,11414],{"className":11415,"style":1294},[431],[415,11417,615],{"className":11418},[436],") flags a negative cycle.",[400,11421,11423],{"id":11422},"choosing-an-algorithm","Choosing an algorithm",[11425,11426,11427,11446],"table",{},[11428,11429,11430],"thead",{},[11431,11432,11433,11437,11440,11443],"tr",{},[11434,11435,11436],"th",{},"Algorithm",[11434,11438,11439],{},"Solves",[11434,11441,11442],{},"Negative edges?",[11434,11444,11445],{},"Time",[11447,11448,11449,11507,11562,11654,11728],"tbody",{},[11431,11450,11451,11454,11460,11463],{},[11452,11453,387],"td",{},[11452,11455,11456,11457],{},"SSSP, ",[390,11458,11459],{},"unweighted",[11452,11461,11462],{},"n\u002Fa",[11452,11464,11465],{},[415,11466,11468],{"className":11467},[418],[415,11469,11471,11495],{"className":11470,"ariaHidden":423},[422],[415,11472,11474,11477,11480,11483,11486,11489,11492],{"className":11473},[427],[415,11475],{"className":11476,"style":458},[431],[415,11478,4879],{"className":11479,"style":4878},[436,437],[415,11481,463],{"className":11482},[462],[415,11484,468],{"className":11485,"style":467},[436,437],[415,11487],{"className":11488,"style":467},[442],[415,11490,1505],{"className":11491},[902],[415,11493],{"className":11494,"style":467},[442],[415,11496,11498,11501,11504],{"className":11497},[427],[415,11499],{"className":11500,"style":458},[431],[415,11502,482],{"className":11503,"style":481},[436,437],[415,11505,487],{"className":11506},[486],[11431,11508,11509,11512,11515,11518],{},[11452,11510,11511],{},"DAG relaxation",[11452,11513,11514],{},"SSSP on a DAG",[11452,11516,11517],{},"yes",[11452,11519,11520],{},[415,11521,11523],{"className":11522},[418],[415,11524,11526,11550],{"className":11525,"ariaHidden":423},[422],[415,11527,11529,11532,11535,11538,11541,11544,11547],{"className":11528},[427],[415,11530],{"className":11531,"style":458},[431],[415,11533,4879],{"className":11534,"style":4878},[436,437],[415,11536,463],{"className":11537},[462],[415,11539,468],{"className":11540,"style":467},[436,437],[415,11542],{"className":11543,"style":467},[442],[415,11545,1505],{"className":11546},[902],[415,11548],{"className":11549,"style":467},[442],[415,11551,11553,11556,11559],{"className":11552},[427],[415,11554],{"className":11555,"style":458},[431],[415,11557,482],{"className":11558,"style":481},[436,437],[415,11560,487],{"className":11561},[486],[11431,11563,11564,11587,11590,11595],{},[11452,11565,11566],{},[415,11567,11569],{"className":11568},[418],[415,11570,11572],{"className":11571,"ariaHidden":423},[422],[415,11573,11575,11578],{"className":11574},[427],[415,11576],{"className":11577,"style":2641},[431],[415,11579,11581],{"className":11580},[2109,2110],[415,11582,11584],{"className":11583},[436,2114],[415,11585,2651],{"className":11586},[436],[11452,11588,11589],{},"SSSP",[11452,11591,11592],{},[390,11593,11594],{},"no",[11452,11596,11597],{},[415,11598,11600],{"className":11599},[418],[415,11601,11603,11627],{"className":11602,"ariaHidden":423},[422],[415,11604,11606,11609,11612,11615,11618,11621,11624],{"className":11605},[427],[415,11607],{"className":11608,"style":458},[431],[415,11610,4879],{"className":11611,"style":4878},[436,437],[415,11613,463],{"className":11614},[462],[415,11616,482],{"className":11617,"style":481},[436,437],[415,11619],{"className":11620,"style":467},[442],[415,11622,1505],{"className":11623},[902],[415,11625],{"className":11626,"style":467},[442],[415,11628,11630,11633,11636,11639,11645,11648,11651],{"className":11629},[427],[415,11631],{"className":11632,"style":458},[431],[415,11634,468],{"className":11635,"style":467},[436,437],[415,11637],{"className":11638,"style":477},[442],[415,11640,11642],{"className":11641},[787],[415,11643,4919],{"className":11644,"style":4918},[436,4917],[415,11646],{"className":11647,"style":477},[442],[415,11649,468],{"className":11650,"style":467},[436,437],[415,11652,487],{"className":11653},[486],[11431,11655,11656,11679,11682,11684],{},[11452,11657,11658],{},[415,11659,11661],{"className":11660},[418],[415,11662,11664],{"className":11663,"ariaHidden":423},[422],[415,11665,11667,11670],{"className":11666},[427],[415,11668],{"className":11669,"style":1163},[431],[415,11671,11673],{"className":11672},[2109,2110],[415,11674,11676],{"className":11675},[436,2114],[415,11677,5061],{"className":11678},[436],[11452,11680,11681],{},"SSSP, + cycle detection",[11452,11683,11517],{},[11452,11685,11686],{},[415,11687,11689],{"className":11688},[418],[415,11690,11692,11716],{"className":11691,"ariaHidden":423},[422],[415,11693,11695,11698,11701,11704,11707,11710,11713],{"className":11694},[427],[415,11696],{"className":11697,"style":458},[431],[415,11699,4879],{"className":11700,"style":4878},[436,437],[415,11702,463],{"className":11703},[462],[415,11705,468],{"className":11706,"style":467},[436,437],[415,11708],{"className":11709,"style":467},[442],[415,11711,6581],{"className":11712},[902],[415,11714],{"className":11715,"style":467},[442],[415,11717,11719,11722,11725],{"className":11718},[427],[415,11720],{"className":11721,"style":458},[431],[415,11723,482],{"className":11724,"style":481},[436,437],[415,11726,487],{"className":11727},[486],[11431,11729,11730,11753,11758,11760],{},[11452,11731,11732],{},[415,11733,11735],{"className":11734},[418],[415,11736,11738],{"className":11737,"ariaHidden":423},[422],[415,11739,11741,11744],{"className":11740},[427],[415,11742],{"className":11743,"style":2641},[431],[415,11745,11747],{"className":11746},[2109,2110],[415,11748,11750],{"className":11749},[436,2114],[415,11751,8192],{"className":11752},[436],[11452,11754,11755],{},[390,11756,11757],{},"all-pairs",[11452,11759,11517],{},[11452,11761,11762],{},[415,11763,11765],{"className":11764},[418],[415,11766,11768],{"className":11767,"ariaHidden":423},[422],[415,11769,11771,11774,11777,11780,11809],{"className":11770},[427],[415,11772],{"className":11773,"style":8126},[431],[415,11775,4879],{"className":11776,"style":4878},[436,437],[415,11778,463],{"className":11779},[462],[415,11781,11783,11786],{"className":11782},[436],[415,11784,468],{"className":11785,"style":467},[436,437],[415,11787,11789],{"className":11788},[582],[415,11790,11792],{"className":11791},[586],[415,11793,11795],{"className":11794},[591],[415,11796,11798],{"className":11797,"style":8151},[595],[415,11799,11800,11803],{"style":8154},[415,11801],{"className":11802,"style":604},[603],[415,11804,11806],{"className":11805},[608,609,610,611],[415,11807,1633],{"className":11808},[436,611],[415,11810,487],{"className":11811},[486],[381,11813,11814,11815,11836],{},"The decision tree is simple: unweighted, use BFS; a DAG, relax in topological\norder; non-negative weights, use Dijkstra (the fastest for one source); negative\nedges possible, use Bellman-Ford and let it police negative cycles; every pair\nat once, use Floyd-Warshall. All five are, at heart, disciplined applications of\nthe same ",[415,11816,11818],{"className":11817},[418],[415,11819,11821],{"className":11820,"ariaHidden":423},[422],[415,11822,11824,11827],{"className":11823},[427],[415,11825],{"className":11826,"style":1163},[431],[415,11828,11830],{"className":11829},[2109,2110],[415,11831,11833],{"className":11832},[436,2114],[415,11834,2118],{"className":11835},[436]," primitive.",[400,11838,11840],{"id":11839},"takeaways","Takeaways",[3102,11842,11843,11853,11939,12208,12256],{},[3105,11844,11845,11846,11848,11849,2979,11851,1131],{},"Shortest paths rest on ",[390,11847,1344],{}," plus two structural facts: the\n",[390,11850,1428],{},[390,11852,1539],{},[3105,11854,11855,11876,11877,11880,11881,11938],{},[415,11856,11858],{"className":11857},[418],[415,11859,11861],{"className":11860,"ariaHidden":423},[422],[415,11862,11864,11867],{"className":11863},[427],[415,11865],{"className":11866,"style":2641},[431],[415,11868,11870],{"className":11869},[2109,2110],[415,11871,11873],{"className":11872},[436,2114],[415,11874,2651],{"className":11875},[436]," greedily finalizes the closest frontier vertex; correct ",[395,11878,11879],{},"only","\nfor non-negative weights, in ",[415,11882,11884],{"className":11883},[418],[415,11885,11887,11911],{"className":11886,"ariaHidden":423},[422],[415,11888,11890,11893,11896,11899,11902,11905,11908],{"className":11889},[427],[415,11891],{"className":11892,"style":458},[431],[415,11894,4879],{"className":11895,"style":4878},[436,437],[415,11897,463],{"className":11898},[462],[415,11900,482],{"className":11901,"style":481},[436,437],[415,11903],{"className":11904,"style":467},[442],[415,11906,1505],{"className":11907},[902],[415,11909],{"className":11910,"style":467},[442],[415,11912,11914,11917,11920,11923,11929,11932,11935],{"className":11913},[427],[415,11915],{"className":11916,"style":458},[431],[415,11918,468],{"className":11919,"style":467},[436,437],[415,11921],{"className":11922,"style":477},[442],[415,11924,11926],{"className":11925},[787],[415,11927,4919],{"className":11928,"style":4918},[436,4917],[415,11930],{"className":11931,"style":477},[442],[415,11933,468],{"className":11934,"style":467},[436,437],[415,11936,487],{"className":11937},[486]," with a Fibonacci heap.",[3105,11940,11941,531,11962,6827,11965,12001,12002,7950,12035,12062,12063,12117,12118,12160,12161,12164,12165,12207],{},[415,11942,11944],{"className":11943},[418],[415,11945,11947],{"className":11946,"ariaHidden":423},[422],[415,11948,11950,11953],{"className":11949},[427],[415,11951],{"className":11952,"style":1163},[431],[415,11954,11956],{"className":11955},[2109,2110],[415,11957,11959],{"className":11958},[436,2114],[415,11960,5061],{"className":11961},[436],[395,11963,11964],{},"is a dynamic program",[415,11966,11968],{"className":11967},[418],[415,11969,11971],{"className":11970,"ariaHidden":423},[422],[415,11972,11974,11977,11983,11986,11989,11992,11995,11998],{"className":11973},[427],[415,11975],{"className":11976,"style":458},[431],[415,11978,11980],{"className":11979},[436,2114],[415,11981,5181],{"className":11982},[436],[415,11984,463],{"className":11985},[462],[415,11987,727],{"className":11988,"style":726},[436,437],[415,11990,473],{"className":11991},[472],[415,11993],{"className":11994,"style":477},[442],[415,11996,523],{"className":11997,"style":522},[436,437],[415,11999,487],{"className":12000},[486]," = cheapest\n",[415,12003,12005],{"className":12004},[418],[415,12006,12008,12026],{"className":12007,"ariaHidden":423},[422],[415,12009,12011,12014,12017,12020,12023],{"className":12010},[427],[415,12012],{"className":12013,"style":1020},[431],[415,12015,1083],{"className":12016},[436,437],[415,12018],{"className":12019,"style":443},[442],[415,12021,3213],{"className":12022},[447,3212],[415,12024],{"className":12025,"style":443},[442],[415,12027,12029,12032],{"className":12028},[427],[415,12030],{"className":12031,"style":1020},[431],[415,12033,523],{"className":12034,"style":522},[436,437],[415,12036,12038],{"className":12037},[418],[415,12039,12041,12053],{"className":12040,"ariaHidden":423},[422],[415,12042,12044,12047,12050],{"className":12043},[427],[415,12045],{"className":12046,"style":1225},[431],[415,12048,1468],{"className":12049},[447],[415,12051],{"className":12052,"style":443},[442],[415,12054,12056,12059],{"className":12055},[427],[415,12057],{"className":12058,"style":1163},[431],[415,12060,727],{"className":12061,"style":726},[436,437]," edges, and one DP row = one pass of\nrelaxing all edges. Shortest paths need ",[415,12064,12066],{"className":12065},[418],[415,12067,12069,12081,12108],{"className":12068,"ariaHidden":423},[422],[415,12070,12072,12075,12078],{"className":12071},[427],[415,12073],{"className":12074,"style":1225},[431],[415,12076,1468],{"className":12077},[447],[415,12079],{"className":12080,"style":443},[442],[415,12082,12084,12087,12099,12102,12105],{"className":12083},[427],[415,12085],{"className":12086,"style":458},[431],[415,12088,12090,12093,12096],{"className":12089},[566],[415,12091,5095],{"className":12092,"style":571},[462,570],[415,12094,468],{"className":12095,"style":467},[436,437],[415,12097,5095],{"className":12098,"style":571},[486,570],[415,12100],{"className":12101,"style":467},[442],[415,12103,903],{"className":12104},[902],[415,12106],{"className":12107,"style":467},[442],[415,12109,12111,12114],{"className":12110},[427],[415,12112],{"className":12113,"style":1294},[431],[415,12115,665],{"className":12116},[436]," edges (short-circuit\nany cycle), so ",[415,12119,12121],{"className":12120},[418],[415,12122,12124,12151],{"className":12123,"ariaHidden":423},[422],[415,12125,12127,12130,12142,12145,12148],{"className":12126},[427],[415,12128],{"className":12129,"style":458},[431],[415,12131,12133,12136,12139],{"className":12132},[566],[415,12134,5095],{"className":12135,"style":571},[462,570],[415,12137,468],{"className":12138,"style":467},[436,437],[415,12140,5095],{"className":12141,"style":571},[486,570],[415,12143],{"className":12144,"style":467},[442],[415,12146,903],{"className":12147},[902],[415,12149],{"className":12150,"style":467},[442],[415,12152,12154,12157],{"className":12153},[427],[415,12155],{"className":12156,"style":1294},[431],[415,12158,665],{"className":12159},[436]," rounds suffice; one extra round ",[390,12162,12163],{},"detects negative\ncycles",". Slower at ",[415,12166,12168],{"className":12167},[418],[415,12169,12171,12195],{"className":12170,"ariaHidden":423},[422],[415,12172,12174,12177,12180,12183,12186,12189,12192],{"className":12173},[427],[415,12175],{"className":12176,"style":458},[431],[415,12178,6799],{"className":12179},[436],[415,12181,463],{"className":12182},[462],[415,12184,468],{"className":12185,"style":467},[436,437],[415,12187],{"className":12188,"style":467},[442],[415,12190,6581],{"className":12191},[902],[415,12193],{"className":12194,"style":467},[442],[415,12196,12198,12201,12204],{"className":12197},[427],[415,12199],{"className":12200,"style":458},[431],[415,12202,482],{"className":12203,"style":481},[436,437],[415,12205,487],{"className":12206},[486]," but handles negative edges.",[3105,12209,12210,12211,12213,12214,1131],{},"On a ",[390,12212,8022],{},", relaxing in topological order solves SSSP in ",[415,12215,12217],{"className":12216},[418],[415,12218,12220,12244],{"className":12219,"ariaHidden":423},[422],[415,12221,12223,12226,12229,12232,12235,12238,12241],{"className":12222},[427],[415,12224],{"className":12225,"style":458},[431],[415,12227,6799],{"className":12228},[436],[415,12230,463],{"className":12231},[462],[415,12233,468],{"className":12234,"style":467},[436,437],[415,12236],{"className":12237,"style":467},[442],[415,12239,1505],{"className":12240},[902],[415,12242],{"className":12243,"style":467},[442],[415,12245,12247,12250,12253],{"className":12246},[427],[415,12248],{"className":12249,"style":458},[431],[415,12251,482],{"className":12252,"style":481},[436,437],[415,12254,487],{"className":12255},[486],[3105,12257,12258,12279,12280,12282,12283,12333],{},[415,12259,12261],{"className":12260},[418],[415,12262,12264],{"className":12263,"ariaHidden":423},[422],[415,12265,12267,12270],{"className":12266},[427],[415,12268],{"className":12269,"style":2641},[431],[415,12271,12273],{"className":12272},[2109,2110],[415,12274,12276],{"className":12275},[436,2114],[415,12277,8192],{"className":12278},[436]," computes ",[390,12281,11757],{}," shortest paths in ",[415,12284,12286],{"className":12285},[418],[415,12287,12289],{"className":12288,"ariaHidden":423},[422],[415,12290,12292,12295,12298,12301,12330],{"className":12291},[427],[415,12293],{"className":12294,"style":8126},[431],[415,12296,6799],{"className":12297},[436],[415,12299,463],{"className":12300},[462],[415,12302,12304,12307],{"className":12303},[436],[415,12305,468],{"className":12306,"style":467},[436,437],[415,12308,12310],{"className":12309},[582],[415,12311,12313],{"className":12312},[586],[415,12314,12316],{"className":12315},[591],[415,12317,12319],{"className":12318,"style":8151},[595],[415,12320,12321,12324],{"style":8154},[415,12322],{"className":12323,"style":604},[603],[415,12325,12327],{"className":12326},[608,609,610,611],[415,12328,1633],{"className":12329},[436,611],[415,12331,487],{"className":12332},[486]," via\na dynamic program over intermediate vertices.",[12335,12336,12339,12344],"section",{"className":12337,"dataFootnotes":376},[12338],"footnotes",[400,12340,12343],{"className":12341,"id":1548},[12342],"sr-only","Footnotes",[12345,12346,12347,12361,12373],"ol",{},[3105,12348,12350,12353,12354],{"id":12349},"user-content-fn-clrs-relax",[390,12351,12352],{},"CLRS",", Ch. 24 & 25 — Single-Source and All-Pairs Shortest Paths — relaxation, the triangle inequality, and optimal substructure. ",[385,12355,12360],{"href":12356,"ariaLabel":12357,"className":12358,"dataFootnoteBackref":376},"#user-content-fnref-clrs-relax","Back to reference 1",[12359],"data-footnote-backref","↩",[3105,12362,12364,12367,12368],{"id":12363},"user-content-fn-erickson-sp",[390,12365,12366],{},"Erickson",", Ch. 8 & 9 — Shortest Paths — Bellman-Ford's extra relaxation pass detecting a negative cycle. ",[385,12369,12360],{"href":12370,"ariaLabel":12371,"className":12372,"dataFootnoteBackref":376},"#user-content-fnref-erickson-sp","Back to reference 2",[12359],[3105,12374,12376,12379,12380,12430,12431],{"id":12375},"user-content-fn-skiena-sp",[390,12377,12378],{},"Skiena",", §6 — Weighted Graph Algorithms — Floyd-Warshall's ",[415,12381,12383],{"className":12382},[418],[415,12384,12386],{"className":12385,"ariaHidden":423},[422],[415,12387,12389,12392,12395,12398,12427],{"className":12388},[427],[415,12390],{"className":12391,"style":8126},[431],[415,12393,6799],{"className":12394},[436],[415,12396,463],{"className":12397},[462],[415,12399,12401,12404],{"className":12400},[436],[415,12402,468],{"className":12403,"style":467},[436,437],[415,12405,12407],{"className":12406},[582],[415,12408,12410],{"className":12409},[586],[415,12411,12413],{"className":12412},[591],[415,12414,12416],{"className":12415,"style":8151},[595],[415,12417,12418,12421],{"style":8154},[415,12419],{"className":12420,"style":604},[603],[415,12422,12424],{"className":12423},[608,609,610,611],[415,12425,1633],{"className":12426},[436,611],[415,12428,487],{"className":12429},[486]," all-pairs dynamic program. ",[385,12432,12360],{"href":12433,"ariaLabel":12434,"className":12435,"dataFootnoteBackref":376},"#user-content-fnref-skiena-sp","Back to reference 3",[12359],[12437,12438,12439],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}html.dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}",{"title":376,"searchDepth":18,"depth":18,"links":12441},[12442,12443,12444,12445,12446,12447,12448],{"id":402,"depth":18,"text":403},{"id":2620,"depth":18,"text":2621},{"id":5031,"depth":18,"text":5032},{"id":8073,"depth":18,"text":8074},{"id":11422,"depth":18,"text":11423},{"id":11839,"depth":18,"text":11840},{"id":1548,"depth":18,"text":12343},"Every navigation app, every network router, every game pathfinder is solving the\nsame problem: given a weighted graph, find the cheapest route from one place to\nanother. BFS already solved\nthis when every edge counts as one step. Now the edges carry weights (distances,\ntimes, costs), and we want to minimize the total weight along a path. This lesson\nbuilds the shortest-path toolkit from a single primitive shared by every algorithm\nin it.","md",{"moduleNumber":102,"lessonNumber":73,"order":12452},504,true,[12455,12459,12462,12465],{"title":12456,"slug":12457,"difficulty":12458},"Network Delay Time","network-delay-time","Medium",{"title":12460,"slug":12461,"difficulty":12458},"Cheapest Flights Within K Stops","cheapest-flights-within-k-stops",{"title":12463,"slug":12464,"difficulty":12458},"Path with Maximum Probability","path-with-maximum-probability",{"title":12466,"slug":12467,"difficulty":12458},"Find the City With the Smallest Number of Neighbors","find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance","---\ntitle: Shortest Paths\nmodule: Graphs\nmoduleNumber: 5\nlessonNumber: 4\norder: 504\nsummary: >-\n  Finding the cheapest route through a weighted network is one of the most-used\n  algorithms in computing. A single operation — _relaxation_ — underlies them\n  all. Dijkstra's algorithm solves the non-negative case greedily; Bellman-Ford\n  handles negative edges and detects negative cycles; and Floyd-Warshall finds\n  the shortest path between _every_ pair of vertices.\ntopics: [Shortest Paths]\nsources:\n  - book: CLRS\n    ref: \"Ch. 24 & 25 — Single-Source and All-Pairs Shortest Paths\"\n  - book: Skiena\n    ref: \"§6 — Weighted Graph Algorithms\"\n  - book: Erickson\n    ref: \"Ch. 8 & 9 — Shortest Paths\"\npractice:\n  - title: 'Network Delay Time'\n    slug: network-delay-time\n    difficulty: Medium\n  - title: 'Cheapest Flights Within K Stops'\n    slug: cheapest-flights-within-k-stops\n    difficulty: Medium\n  - title: 'Path with Maximum Probability'\n    slug: path-with-maximum-probability\n    difficulty: Medium\n  - title: 'Find the City With the Smallest Number of Neighbors'\n    slug: find-the-city-with-the-smallest-number-of-neighbors-at-a-threshold-distance\n    difficulty: Medium\n---\n\nEvery navigation app, every network router, every game pathfinder is solving the\nsame problem: given a weighted graph, find the cheapest route from one place to\nanother. [BFS](\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal) already solved\nthis when every edge counts as one step. Now the edges carry **weights** (distances,\ntimes, costs), and we want to minimize the _total_ weight along a path. This lesson\nbuilds the shortest-path toolkit from a single primitive shared by every algorithm\nin it.\n\n## The problem and its primitive\n\n> **Definition (Shortest-path distance).** Given a weighted directed graph $G = (V, E)$ with weight $w(u, v)$ on each\n> edge, the **weight of a path** $p = \\vector{v_0, v_1, \\dots, v_k}$ is\n> $w(p) = \\sum_{i=1}^{k} w(v_{i-1}, v_i)$. The **shortest-path distance**\n> $\\delta(u, v)$ is the minimum weight over all paths from $u$ to $v$ (or\n> $\\infty$ if none exists). The **single-source shortest paths** (SSSP) problem\n> asks for $\\delta(s, v)$ from a fixed source $s$ to _every_ vertex $v$.\n\nEvery algorithm maintains two arrays. For each vertex $v$, an estimate $v.d$ is\nan _upper bound_ on $\\delta(s, v)$, always $\\ge$ the true distance, shrinking\ntoward it. A predecessor $v.\\pi$ records the previous vertex on the best path\nfound so far, forming a **shortest-path tree**. We initialize $s.d = 0$ and\n$v.d = \\infty$ for every other vertex.\n\nThe one operation that updates these estimates is **relaxation**: testing\nwhether going _through_ $u$ improves our route to $v$.\n\n```algorithm\ncaption: $\\textsc{Relax}(u, v, w)$ — try the edge $(u,v)$ as a shortcut to $v$\nnumber: 1\nif $u.d + w(u, v) \u003C v.d$ then\n  $v.d \\gets u.d + w(u, v)$ \u002F\u002F cheaper route to v via u\n  $v.\\pi \\gets u$\n```\n\nRelaxation never produces an estimate below the true distance, and it can only\never _lower_ an estimate. Every shortest-path algorithm below is a different\ndiscipline for _deciding which edges to relax, and in what order_. Two facts\nmake relaxation work: the **triangle inequality**\n$\\delta(s, v) \\le \\delta(s, u) + w(u, v)$, and **optimal substructure**: any\nsubpath of a shortest path is itself a shortest path.[^clrs-relax] The latter is what makes\ngreedy and dynamic-programming approaches both viable.\n\nOne relaxation step looks like this. Before, $v$ believes its best route costs\n$9$; we test the edge $(u, v)$ of weight $3$ against $u$'s settled estimate\n$u.d = 5$. Since $5 + 3 = 8 \u003C 9$, the edge is a shortcut: $v.d$ drops to $8$ and\n$v.\\pi$ is rewired to point back through $u$.\n\n$$\n% caption: One $\\textsc{Relax}(u, v, w)$ step. The edge $u \\to v$ beats $v$'s current estimate ($5 + 3 \u003C 9$), so $v.d$ falls to $8$ and its predecessor is rewired to $u$.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n  V\u002F.style={circle, draw, minimum size=11mm, font=\\small},\n  old\u002F.style={font=\\scriptsize},\n  rel\u002F.style={->, line width=1.2pt, draw=red!75!black}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % before\n  \\node[font=\\footnotesize] at (1.4,1.7) {before};\n  \\node[V] (u1) at (0,0) {$5$};\n  \\node[V] (v1) at (2.8,0) {$9$};\n  \\node[old] at (0,-1.0) {$u.d=5$};\n  \\node[old] at (2.8,-1.0) {$v.d=9$};\n  \\draw[->] (u1) -- node[font=\\scriptsize, above]{$3$} (v1);\n  % after\n  \\begin{scope}[xshift=58mm]\n    \\node[font=\\footnotesize] at (1.4,1.7) {after relax};\n    \\node[V] (u2) at (0,0) {$5$};\n    \\node[V, draw=acc, very thick, fill=acc!15] (v2) at (2.8,0) {$8$};\n    \\node[old] at (0,-1.0) {$u.d=5$};\n    \\node[old, acc] at (2.8,-1.0) {$v.d=8$};\n    \\draw[rel] (u2) -- node[font=\\scriptsize, above]{$3$} (v2);\n    \\node[font=\\scriptsize, acc] at (1.4,-1.7) {$v.\\pi \\gets u$};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nWe will trace the algorithms on this small weighted digraph. Note the\n**negative edge** $a \\to b$ of weight $-2$: it is harmless here (there is no\nnegative _cycle_), but it is exactly the kind of edge that will break Dijkstra\nand force us toward a dynamic program.\n\n$$\n% caption: A small weighted digraph with one negative edge $a$ to $b$ of weight $-2$, used to trace the algorithms.\n\\begin{tikzpicture}[\n  vtx\u002F.style={circle, draw, minimum size=8mm, font=\\small},\n  wt\u002F.style={font=\\scriptsize, fill=white, inner sep=1.5pt},\n  >={Stealth[round]}, node distance=22mm]\n  \\node[vtx] (s) {$s$};\n  \\node[vtx] (a) [above right=8mm and 22mm of s] {$a$};\n  \\node[vtx] (b) [below right=8mm and 22mm of s] {$b$};\n  \\node[vtx] (c) [right=22mm of a] {$c$};\n  \\node[vtx] (t) [below right=8mm and 22mm of c] {$t$};\n  \\draw[->] (s) -- node[wt,above left]{$10$} (a);\n  \\draw[->] (s) -- node[wt,below left]{$5$} (b);\n  \\draw[->] (a) -- node[wt,left]{$-2$} (b);\n  \\draw[->] (a) -- node[wt,above]{$1$} (c);\n  \\draw[->] (b) -- node[wt,below,pos=0.4]{$9$} (c);\n  \\draw[->] (c) -- node[wt,above right]{$4$} (t);\n  \\draw[->] (b) -- node[wt,below]{$2$} (t);\n\\end{tikzpicture}\n$$\n\n## Dijkstra's algorithm\n\nWhen _all edge weights are non-negative_, we can be greedy. $\\textsc{Dijkstra}$'s\nalgorithm grows a set $S$ of vertices whose shortest distances are _finalized_.\nAt each step it picks the non-finalized vertex $u$ with the smallest estimate\n$u.d$, finalizes it, and relaxes its outgoing edges. A\n[min-priority queue](\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort) keyed\nby $d$ supplies the next vertex.\n\n```algorithm\ncaption: $\\textsc{Dijkstra}(G, w, s)$ — SSSP for non-negative weights\nnumber: 2\nforeach vertex $v \\in V$ do\n  $v.d \\gets \\infty$\n  $v.\\pi \\gets \\text{nil}$\n$s.d \\gets 0$\n$S \\gets \\emptyset$\n$Q \\gets V$ \u002F\u002F min-PQ keyed by d\nwhile $Q \\neq \\emptyset$ do\n  $u \\gets$ $\\textsc{Extract-Min}(Q)$ \u002F\u002F closest unfinalized\n  $S \\gets S \\cup \\set{u}$ \u002F\u002F u.d now final\n  foreach $v$ adjacent to $u$ do\n    call $\\textsc{Relax}(u, v, w)$ \u002F\u002F Decrease-Key updates Q\nreturn $d$ and $\\pi$\n```\n\n**Why the greedy choice is correct.** Correctness is framed around a\nsingle **loop invariant**: at the start of each iteration of the `while` loop,\nevery _finalized_ vertex already holds its true distance,\n$$\\forall x \\in S:\\quad x.d = \\delta(s, x).$$\nIt holds initially ($S = \\set{s}$ and $s.d = 0 = \\delta(s, s)$). The maintenance\nstep is the crux of the argument: suppose finalizing $u$ were the first move to\nbreak the invariant. Then either $u.d$ is set too _low_ or too _high_.\n\n- _Too low_ ($u.d \u003C \\delta(s, u)$) is impossible, because relaxation never drops\n  an estimate below the true distance — every value of $u.d$ is the cost of some\n  real $s \\rightsquigarrow u$ walk.\n- _Too high_ ($u.d > \\delta(s, u)$): let $\\pi$ be a genuine shortest path from\n  $s$ to $u$. Walking it from $s$ (which is in $S$), let $(x, y)$ be the **first\n  edge that leaves $S$** — so $x \\in S$, $y \\notin S$. By the invariant\n  $x.d = \\delta(s, x)$, and $(x, y)$ was relaxed when $x$ was finalized, so\n  $y.d = \\delta(s, x) + w(x, y) = \\delta(s, y)$. Now $y$ sits on a shortest path\n  to $u$, and **because every weight is $\\ge 0$**, the rest of $\\pi$ only adds\n  cost: $\\delta(s, y) \\le \\delta(s, u) \\le u.d$. Hence $y.d \\le u.d$, so\n  $\\textsc{Extract-Min}$ would have returned $y$, not $u$ — contradiction.\n\nThe non-negativity is doing all the work in that last inequality: it guarantees\nextending a path never _decreases_ its cost, so the closest frontier vertex can\nbe safely frozen. A single negative edge breaks $\\delta(s, y) \\le \\delta(s, u)$,\nso the greedy commitment becomes unsound.\n\nOn a small non-negative digraph the finalization order is exactly nondecreasing\nin distance, which is the visible signature of the greedy invariant. Notice that\n$a$ is finalized at distance $3$ via the two-hop route $s \\to b \\to a$, _beating_\nthe direct edge $s \\to a$ of weight $4$ — the relaxation through $b$ fired before\n$a$ was ever extracted:\n\n$$\n% caption: Dijkstra finalizes vertices in nondecreasing distance: $s(0), b(1), a(3), c(4), t(7)$. Vertex $a$ settles at $3$ via $s \\to b \\to a$, beating the direct edge of weight $4$.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n  V\u002F.style={circle, draw, minimum size=8mm, font=\\small},\n  wt\u002F.style={font=\\scriptsize, inner sep=1pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[V] (s) at (0,0) {$s$};\n  \\node[V] (b) at (1.8,-1.0) {$b$};\n  \\node[V] (a) at (1.8,1.0) {$a$};\n  \\node[V] (c) at (3.8,0) {$c$};\n  \\node[V] (t) at (5.6,0) {$t$};\n  \\draw[->] (s) -- node[wt, above left]{$4$} (a);\n  \\draw[->] (s) -- node[wt, below left]{$1$} (b);\n  \\draw[->] (b) -- node[wt, left]{$2$} (a);\n  \\draw[->] (b) -- node[wt, below]{$5$} (c);\n  \\draw[->] (a) -- node[wt, above]{$1$} (c);\n  \\draw[->] (a) to[bend left=30] node[wt, above]{$6$} (t);\n  \\draw[->] (c) -- node[wt, below]{$3$} (t);\n  \\node[font=\\footnotesize] at (0.3,-2.3) {\\textbf{finalized:}};\n  \\foreach \\v\u002F\\dd\u002F\\x in {s\u002F0\u002F2.0, b\u002F1\u002F2.9, a\u002F3\u002F3.8, c\u002F4\u002F4.7, t\u002F7\u002F5.6}\n    \\node[draw, rounded corners, fill=acc!12, minimum size=6mm, font=\\scriptsize] at (\\x,-2.3) {$\\v{:}\\dd$};\n\\end{tikzpicture}\n$$\n\n**Running time.** Like [Prim](\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees), Dijkstra\ndoes $V$ $\\textsc{Extract-Min}$ and up to $E$\n$\\textsc{Decrease-Key}$ operations. A binary heap gives $O((V + E)\\log V) = O(E\\log V)$;\na Fibonacci heap improves this to $O(E + V\\log V)$.\n\n## Bellman-Ford as a dynamic program\n\nDijkstra breaks when edges can be **negative** — its greedy finalization assumes\na finalized estimate can never improve, which negative edges violate.\n$\\textsc{Bellman-Ford}$ trades speed for generality. It is best derived\nnot as \"relax everything repeatedly\" but as a genuine\n[**dynamic program**](\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs). That framing is\nworth keeping, because it explains _where the $\\abs{V}-1$ comes from_.\n\n**The subproblem.** Bound the number of edges a walk may use. For each vertex\n$v$ and each budget $k$, define\n$$\n\\text{OPT}(k, v) = \\text{cost of a cheapest } s \\rightsquigarrow v \\text{ walk\nusing at most } k \\text{ edges.}\n$$\nThe full answer we want is $\\text{OPT}(n-1, v)$ for $n = \\abs{V}$, since (as we\nprove below) shortest paths never need more than $n-1$ edges.\n\n**The recurrence.** A cheapest walk to $v$ using at most $k$ edges either uses at\nmost $k-1$ edges already, or it takes one final edge $(u, v)$ after a cheapest\n$\\le (k-1)$-edge walk to $u$. Minimizing over both:\n$$\n\\text{OPT}(k, v) = \\min\\Bigl(\\, \\text{OPT}(k-1, v),\\;\n\\min_{(u,v)\\,\\in\\,E}\\;\\text{OPT}(k-1, u) + w(u, v) \\,\\Bigr),\n$$\nwith base cases $\\text{OPT}(0, s) = 0$ and $\\text{OPT}(0, v) = \\infty$ for\n$v \\neq s$. This is exactly relaxation: filling row $k$ of the table from row\n$k-1$ is one full pass that relaxes every edge. The tabular form\nmakes the layering explicit, with row $k$ holding the best walk of length $\\le k$:\n\n```algorithm\ncaption: $\\textsc{Bellman-Ford-DP}(G, w, s)$ — the explicit DP table\nnumber: 3\n$\\text{OPT}[0, s] \\gets 0$; $\\text{OPT}[0, v] \\gets \\infty$ for $v \\neq s$\nfor $k \\gets 1$ to $n - 1$ do \u002F\u002F one pass over all edges per row\n  foreach vertex $v \\in V$ do\n    $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, v]$ \u002F\u002F inherit: no extra edge\n  foreach edge $(u, v) \\in E$ do\n    if $\\text{OPT}[k-1, u] + w(u, v) \u003C \\text{OPT}[k, v]$ then\n      $\\text{OPT}[k, v] \\gets \\text{OPT}[k-1, u] + w(u, v)$\n      $v.\\pi \\gets u$\nreturn $\\text{OPT}[n-1, \\cdot]$ and $\\pi$\n```\n\n**Why stop after $n-1$ rounds?** This is the theorem that licenses the whole\nalgorithm. _If $G$ has no negative-cost cycle, then for all $s, t$ the distance\n$\\delta(s, t)$ is achieved by a path of length $\\le n - 1$._ The proof is a\nshort-circuiting argument: any $s \\rightsquigarrow t$ **walk** of length $> n-1$\nvisits some vertex twice, so it contains a cycle. Excising that cycle yields a\nshorter walk of cost no greater than the original (the removed cycle has\nnon-negative cost). Repeating until no cycle remains leaves a _simple_ path, with\nat most $n-1$ edges, that is no more expensive. So the rows of the DP stop\nchanging by row $n-1$.\n\n**Saving space.** The recurrence only ever reads row $k-1$, so we collapse the\ntable to a single 1-D array $d[\\cdot]$ updated in place, recovering the familiar\nform of Bellman-Ford as $n-1$ rounds of relaxing every edge. Note **negative\nedges are fine**; only a negative _cycle_ breaks the theorem.\n\n```algorithm\ncaption: $\\textsc{Bellman-Ford}(G, w, s)$ — space-saved; with cycle check\nnumber: 4\nforeach vertex $v \\in V$ do\n  $v.d \\gets \\infty$\n  $v.\\pi \\gets \\text{nil}$\n$s.d \\gets 0$\nfor $i \\gets 1$ to $n - 1$ do \u002F\u002F V-1 full-relaxation rounds\n  foreach edge $(u, v) \\in E$ do\n    call $\\textsc{Relax}(u, v, w)$\nforeach edge $(u, v) \\in E$ do \u002F\u002F extra pass: detect neg cycle\n  if $u.d + w(u, v) \u003C v.d$ then\n    return false \u002F\u002F neg cycle reachable from s\nreturn true\n```\n\n**Detecting negative cycles.** After $n - 1$ rounds, if any edge can _still_ be\nrelaxed, some walk was improved using $n$ edges — only possible if a **negative\ncycle** lets the cost descend without bound. So one extra pass is exactly the\ntest: relax everything once more, and any successful relaxation certifies a\nnegative cycle reachable from $s$[^erickson-sp] (where \"cheapest cost to reach a vertex\" is no\nlonger even well-defined). Dijkstra entirely lacks that capability. The\ncost is $\\Theta(V \\cdot E)$: $\\abs{V} - 1$ passes, each relaxing all $\\abs{E}$\nedges.\n\n> **Example (Tracing it).** On our digraph from $s$, the DP rows settle quickly. Row 1\n> relaxes the source edges: $a.d = 10$, $b.d = 5$. Row 2 propagates one edge\n> further, setting $c.d = a.d + 1 = 11$ and $t.d = b.d + 2 = 7$, and tests the\n> negative edge $a \\to b$, but $10 + (-2) = 8 > 5$, so $b$ keeps its cheaper\n> direct route. Row 3 settles $t.d = c.d + 4 = 15$ vs. the existing $7$, so no\n> change. The table has converged in fewer than $n-1 = 4$ rounds; a final pass\n> changes nothing, certifying no reachable negative cycle.\n\nLaid out as the DP table $\\text{OPT}[k, v]$, the layering is plain: row $k$ holds\nthe cheapest walk of at most $k$ edges, each row computed from the one above by a\nsingle full pass of relaxation. The entries that improved on each row are shaded;\nthe table stops changing after row $2$, well within the $n-1 = 4$ bound:\n\n$$\n% caption: The Bellman-Ford DP table $\\text{OPT}[k, v]$ for the trace digraph. Row $k$ = cheapest $s \\rightsquigarrow v$ walk of $\\le k$ edges; shaded cells improved that round. Rows freeze after $k=2$.\n\\begin{tikzpicture}[\n  cell\u002F.style={draw, minimum size=9mm, inner sep=1pt, font=\\small},\n  hdr\u002F.style={draw=none, font=\\small},\n  imp\u002F.style={cell, fill=acc!18}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % corner + column headers (vertices) + row headers (round k)\n  \\node[hdr] at (0,0) {$k$};\n  \\node[hdr] at (1,0) {$s$}; \\node[hdr] at (2,0) {$a$}; \\node[hdr] at (3,0) {$b$};\n  \\node[hdr] at (4,0) {$c$}; \\node[hdr] at (5,0) {$t$};\n  \\node[hdr] at (0,-1) {$0$}; \\node[hdr] at (0,-2) {$1$};\n  \\node[hdr] at (0,-3) {$2$}; \\node[hdr] at (0,-4) {$3$};\n  % k=0: only s reachable; shaded cells improved this round\n  \\node[imp]  at (1,-1) {$0$}; \\node[cell] at (2,-1) {$\\infty$}; \\node[cell] at (3,-1) {$\\infty$}; \\node[cell] at (4,-1) {$\\infty$}; \\node[cell] at (5,-1) {$\\infty$};\n  % k=1\n  \\node[cell] at (1,-2) {$0$}; \\node[imp]  at (2,-2) {$10$}; \\node[imp]  at (3,-2) {$5$}; \\node[cell] at (4,-2) {$\\infty$}; \\node[cell] at (5,-2) {$\\infty$};\n  % k=2\n  \\node[cell] at (1,-3) {$0$}; \\node[cell] at (2,-3) {$10$}; \\node[cell] at (3,-3) {$5$}; \\node[imp]  at (4,-3) {$11$}; \\node[imp]  at (5,-3) {$7$};\n  % k=3 (frozen — no improvements)\n  \\node[cell] at (1,-4) {$0$}; \\node[cell] at (2,-4) {$10$}; \\node[cell] at (3,-4) {$5$}; \\node[cell] at (4,-4) {$11$}; \\node[cell] at (5,-4) {$7$};\n\\end{tikzpicture}\n$$\n\n> **Remark (A special case worth knowing).** On a **DAG** there are no cycles at all, so\n> we can relax edges in _topological order_ (from the previous lesson) in a\n> single sweep, since every vertex's predecessors are finalized before it is reached.\n> This solves DAG shortest paths in $\\Theta(V + E)$, and it works with negative\n> weights too.\n\n## Floyd-Warshall: all pairs at once\n\nSometimes we want $\\delta(u, v)$ for _every_ pair of vertices, a full distance\nmatrix. Running Bellman-Ford from each source costs $O(V^2 E)$, but\n$\\textsc{Floyd-Warshall}$ does better with a slick dynamic program over _intermediate\nvertices_.\n\nNumber the vertices $1, \\dots, n$. Let $d_{ij}^{(k)}$ be the weight of the\nshortest path from $i$ to $j$ whose intermediate vertices all lie in\n$\\set{1, \\dots, k}$. Either the shortest such path avoids vertex $k$ (then it\nis $d_{ij}^{(k-1)}$) or it routes _through_ $k$, splitting into an $i \\to k$\npiece and a $k \\to j$ piece, each using only earlier intermediates:\n\n$$\nd_{ij}^{(k)} = \\min\\!\\left( d_{ij}^{(k-1)},\\;\nd_{ik}^{(k-1)} + d_{kj}^{(k-1)} \\right).\n$$\n\nOne round of the recurrence makes the mechanism concrete. Starting from\n$d^{(0)}$ — direct edges only — and admitting intermediates from $\\set{1, 2}$\nproduces $d^{(2)}$: the entry $d_{13}$ drops from $8$ to $5$ via $1 \\to 2 \\to 3$,\nand the once-unreachable $d_{42}, d_{43}$ become finite by routing through vertex\n$1$ then $2$. The five matrices $d^{(0)},\\dots,d^{(4)}$ below trace one round per\npermitted intermediate; the entries that improved in each round are shaded:\n\n$$\n% caption: Floyd-Warshall, one matrix per round. $d^{(k)}$ uses only vertices $\\{1,\\dots,k\\}$ as intermediates; each round's pivot vertex $k$ is marked in blue on the indices (its row and column drive that round's updates), and the entries that improved when $k$ was admitted are shaded. $d^{(4)}$ holds the all-pairs shortest distances.\n\\begin{tikzpicture}[font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  % --- graph (top): the two diagonals are straight, forming an X; each weight\n  % label hugs its own edge near the source node, so 8 and 7 can't be confused ---\n  \\begin{scope}[xshift=47mm, yshift=24mm, >=Stealth,\n      every node\u002F.style={circle, draw, minimum size=7mm, font=\\scriptsize},\n      wt\u002F.style={draw=none, font=\\scriptsize, fill=white, inner sep=1pt}]\n    \\node (1) at (0,1.6) {$1$};\n    \\node (2) at (2.2,1.6) {$2$};\n    \\node (3) at (2.2,0) {$3$};\n    \\node (4) at (0,0) {$4$};\n    \\draw[->] (1) -- node[wt, above]{$3$} (2);\n    \\draw[->] (2) -- node[wt, right]{$2$} (3);\n    \\draw[->] (3) -- node[wt, below]{$1$} (4);\n    \\draw[->] (4) -- node[wt, left]{$2$} (1);\n    \\draw[->] (1) -- node[wt, pos=0.26]{$8$} (3);\n    \\draw[->] (2) -- node[wt, pos=0.26]{$7$} (4);\n  \\end{scope}\n  % --- five matrices in a row; \\hi marks the cells that improved this round ---\n  \\tikzset{\n    mlabel\u002F.style={font=\\footnotesize}, idx\u002F.style={font=\\tiny, gray},\n    sub\u002F.style={font=\\scriptsize, gray}, hi\u002F.style={fill=acc!18}}\n  \\newcommand{\\fwmat}[5]{% #1=xshift #2=k #3=highlight #4=rows #5=subtitle\n    \\begin{scope}[xshift=#1 mm]\n      \\node[mlabel] at (0.75,0.82) {$d^{(#2)}$};\n      \\foreach \\j\u002F\\x in {1\u002F0,2\u002F0.5,3\u002F1.0,4\u002F1.5} {\n        \\ifnum\\j=#2 \\node[idx, acc] at (\\x,0.42) {$\\j$};\\else \\node[idx] at (\\x,0.42) {$\\j$};\\fi}\n      \\foreach \\i\u002F\\y in {1\u002F0,2\u002F-0.45,3\u002F-0.9,4\u002F-1.35} {\n        \\ifnum\\i=#2 \\node[idx, acc] at (-0.42,\\y) {$\\i$};\\else \\node[idx] at (-0.42,\\y) {$\\i$};\\fi}\n      \\foreach \\hx\u002F\\hy in {#3} \\fill[hi] (\\hx-0.23,\\hy-0.2) rectangle (\\hx+0.23,\\hy+0.2);\n      \\foreach \\row\u002F\\y in {#4}\n        \\foreach \\val [count=\\ci from 0] in \\row \\node[font=\\scriptsize] at (\\ci*0.5,\\y) {$\\val$};\n      \\node[sub] at (0.75,-1.82) {#5};\n    \\end{scope}}\n  \\fwmat{0}{0}{}{{0,3,8,\\infty}\u002F0,{\\infty,0,2,7}\u002F-0.45,{\\infty,\\infty,0,1}\u002F-0.9,{2,\\infty,\\infty,0}\u002F-1.35}{direct edges}\n  \\fwmat{26}{1}{0.5\u002F-1.35,1.0\u002F-1.35}{{0,3,8,\\infty}\u002F0,{\\infty,0,2,7}\u002F-0.45,{\\infty,\\infty,0,1}\u002F-0.9,{2,5,10,0}\u002F-1.35}{via $\\{1\\}$}\n  \\fwmat{52}{2}{1.0\u002F0,1.5\u002F0,1.0\u002F-1.35}{{0,3,5,10}\u002F0,{\\infty,0,2,7}\u002F-0.45,{\\infty,\\infty,0,1}\u002F-0.9,{2,5,7,0}\u002F-1.35}{via $\\{1,2\\}$}\n  \\fwmat{78}{3}{1.5\u002F0,1.5\u002F-0.45}{{0,3,5,6}\u002F0,{\\infty,0,2,3}\u002F-0.45,{\\infty,\\infty,0,1}\u002F-0.9,{2,5,7,0}\u002F-1.35}{via $\\{1,2,3\\}$}\n  \\fwmat{104}{4}{0\u002F-0.45,0\u002F-0.9,0.5\u002F-0.9}{{0,3,5,6}\u002F0,{5,0,2,3}\u002F-0.45,{3,6,0,1}\u002F-0.9,{2,5,7,0}\u002F-1.35}{via $\\{1,\\dots,4\\}$}\n  % legend tying the matrix indices to the graph vertices (clears the \"what is edge 1?\" confusion)\n  \\node[font=\\scriptsize, align=center] at (5.3,1.55)\n    {Indices $1,2,3,4$ label the four vertices; the {\\color{acc}blue} index is the\\\\\n     intermediate vertex $k$ admitted that round (its row \\& column drive the updates).};\n\\end{tikzpicture}\n$$\n\nThree nested loops over $k$, $i$, $j$ evaluate this recurrence directly: the outer\nloop admits one intermediate vertex $k$ per round (exactly the five matrices above),\nand the inner two relax every pair against a route through it.\n\n```algorithm\ncaption: $\\textsc{Floyd-Warshall}(W)$ — all-pairs shortest paths in $O(V^3)$\n$d \\gets W$ \u002F\u002F $d_{ij}$: edge weight; $0$ on the diagonal, else $\\infty$\nfor $k \\gets 1$ to $n$ do \u002F\u002F admit vertex $k$ as an intermediate\n  for $i \\gets 1$ to $n$ do\n    for $j \\gets 1$ to $n$ do\n      if $d_{ik} + d_{kj} \u003C d_{ij}$ then \u002F\u002F routing through $k$ is shorter\n        $d_{ij} \\gets d_{ik} + d_{kj}$\nreturn $d$\n```\n\nThis is $\\Theta(V^3)$ time and $\\Theta(V^2)$ space: compact, cache-friendly, and a\nclean win on dense graphs.[^skiena-sp] It handles negative edges, and a negative entry on\nthe diagonal ($d_{ii} \u003C 0$) flags a negative cycle.\n\n## Choosing an algorithm\n\n| Algorithm | Solves | Negative edges? | Time |\n| --- | --- | --- | --- |\n| BFS | SSSP, **unweighted** | n\u002Fa | $O(V + E)$ |\n| DAG relaxation | SSSP on a DAG | yes | $O(V + E)$ |\n| $\\textsc{Dijkstra}$ | SSSP | **no** | $O(E + V\\log V)$ |\n| $\\textsc{Bellman-Ford}$ | SSSP, + cycle detection | yes | $O(V \\cdot E)$ |\n| $\\textsc{Floyd-Warshall}$ | **all-pairs** | yes | $O(V^3)$ |\n\nThe decision tree is simple: unweighted, use BFS; a DAG, relax in topological\norder; non-negative weights, use Dijkstra (the fastest for one source); negative\nedges possible, use Bellman-Ford and let it police negative cycles; every pair\nat once, use Floyd-Warshall. All five are, at heart, disciplined applications of\nthe same $\\textsc{Relax}$ primitive.\n\n## Takeaways\n\n- Shortest paths rest on **relaxation** plus two structural facts: the\n  **triangle inequality** and **optimal substructure**.\n- $\\textsc{Dijkstra}$ greedily finalizes the closest frontier vertex; correct _only_\n  for non-negative weights, in $O(E + V\\log V)$ with a Fibonacci heap.\n- $\\textsc{Bellman-Ford}$ _is a dynamic program_: $\\text{OPT}(k, v)$ = cheapest\n  $s \\rightsquigarrow v$ walk of $\\le k$ edges, and one DP row = one pass of\n  relaxing all edges. Shortest paths need $\\le \\abs{V}-1$ edges (short-circuit\n  any cycle), so $\\abs{V}-1$ rounds suffice; one extra round **detects negative\n  cycles**. Slower at $\\Theta(V \\cdot E)$ but handles negative edges.\n- On a **DAG**, relaxing in topological order solves SSSP in $\\Theta(V + E)$.\n- $\\textsc{Floyd-Warshall}$ computes **all-pairs** shortest paths in $\\Theta(V^3)$ via\n  a dynamic program over intermediate vertices.\n\n[^clrs-relax]: **CLRS**, Ch. 24 & 25 — Single-Source and All-Pairs Shortest Paths — relaxation, the triangle inequality, and optimal substructure.\n[^erickson-sp]: **Erickson**, Ch. 8 & 9 — Shortest Paths — Bellman-Ford's extra relaxation pass detecting a negative cycle.\n[^skiena-sp]: **Skiena**, §6 — Weighted Graph Algorithms — Floyd-Warshall's $\\Theta(V^3)$ all-pairs dynamic program.\n",{"text":12470,"minutes":12471,"time":12472,"words":12473},"10 min read",9.89,593400,1978,{"title":174,"description":12449},[12476,12478,12480],{"book":12352,"ref":12477},"Ch. 24 & 25 — Single-Source and All-Pairs Shortest Paths",{"book":12378,"ref":12479},"§6 — Weighted Graph Algorithms",{"book":12366,"ref":12481},"Ch. 8 & 9 — Shortest Paths","available","01.algorithms\u002F06.graphs\u002F04.shortest-paths",[174],"Mz7m1xlyB_hUtarNdB4lvcTd9MHJPIsSdmT9ca37EgA",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":12487,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":12488,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":12489,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":12490,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":12491,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":12492,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":12493,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":12494,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":12495,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":12496,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":12497,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":12498,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":12499,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":12500,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":12501,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":12502,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":12503,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":12504,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":12505,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":12506,"\u002Falgorithms\u002Fsequences\u002Ftries":12507,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":12508,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":12509,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":12510,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":12473,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":12511,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":12512,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":12513,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":12514,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":12515,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":12516,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":12517,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":12518,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":12519,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":12520,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":12521,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":12522,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":12523,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":12524,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":12525,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":12526,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":12527,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":12528,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":12529,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":12530,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":12531,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":12532,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":12503,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":12533,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":12534,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":12535,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":12536,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":12518,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":12537,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":12538,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":12499,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":12539,"\u002Falgorithms":12540,"\u002Ftheory-of-computation":12541,"\u002Fcomputer-architecture":12541,"\u002Fphysical-computing":12541,"\u002Fdatabases":12541,"\u002Fdeep-learning":12541},1763,2107,1738,2628,1723,2048,1697,1044,1542,1565,1679,1586,1388,1465,1971,1455,1533,1483,1578,1791,1481,2704,1658,2070,2080,1568,1451,1291,1543,1883,1443,1599,2038,2241,1744,1678,2288,1929,1657,1412,1554,1418,1713,1798,1694,1762,1534,1595,1262,1495,1630,2306,2142,107,0,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":12543,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":12544,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":12545,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":12546,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":12547,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":12548,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":12549,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":12550,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":12551,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":12552,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":12553,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":12554,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":12555,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":12556,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":12557,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":12558,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":12559,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":12560,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":12561,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":12562,"\u002Falgorithms\u002Fsequences\u002Ftries":12563,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":12564,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":12565,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":12566,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":12567,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":12568,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":12569,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":12570,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":12571,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":12572,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":12573,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":12574,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":12575,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":12576,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":12577,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":12578,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":12579,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":12580,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":12581,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":12582,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":12583,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":12584,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":12585,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":12586,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":12587,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":12588,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":12589,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":12590,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":12591,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":12592,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":12593,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":12594,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":12595,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":12596,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":12597,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":12598,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":12599,"\u002Falgorithms":12600,"\u002Ftheory-of-computation":12603,"\u002Fcomputer-architecture":12606,"\u002Fphysical-computing":12609,"\u002Fdatabases":12612,"\u002Fdeep-learning":12615},{"path":11,"title":10,"module":5,"summary":14},{"path":17,"title":16,"module":5,"summary":20},{"path":23,"title":22,"module":5,"summary":27},{"path":34,"title":33,"module":29,"summary":37},{"path":40,"title":39,"module":29,"summary":43},{"path":46,"title":45,"module":29,"summary":49},{"path":56,"title":55,"module":51,"summary":59},{"path":62,"title":61,"module":51,"summary":64},{"path":67,"title":66,"module":51,"summary":70},{"path":78,"title":77,"module":72,"summary":81},{"path":84,"title":83,"module":72,"summary":87},{"path":90,"title":89,"module":72,"summary":92},{"path":95,"title":94,"module":72,"summary":98},{"path":101,"title":100,"module":72,"summary":104},{"path":107,"title":106,"module":72,"summary":112},{"path":115,"title":114,"module":72,"summary":119},{"path":126,"title":125,"module":121,"summary":129},{"path":132,"title":131,"module":121,"summary":134},{"path":137,"title":136,"module":121,"summary":140},{"path":143,"title":142,"module":121,"summary":146},{"path":149,"title":148,"module":121,"summary":151},{"path":158,"title":157,"module":153,"summary":162},{"path":165,"title":164,"module":153,"summary":167},{"path":170,"title":169,"module":153,"summary":172},{"path":175,"title":174,"module":153,"summary":177},{"path":180,"title":179,"module":153,"summary":182},{"path":185,"title":184,"module":153,"summary":187},{"path":190,"title":189,"module":153,"summary":192},{"path":195,"title":194,"module":153,"summary":198},{"path":201,"title":200,"module":153,"summary":204},{"path":211,"title":210,"module":206,"summary":213},{"path":216,"title":215,"module":206,"summary":219},{"path":222,"title":221,"module":206,"summary":224},{"path":227,"title":226,"module":206,"summary":229},{"path":236,"title":235,"module":231,"summary":238},{"path":241,"title":240,"module":231,"summary":244},{"path":247,"title":246,"module":231,"summary":249},{"path":252,"title":251,"module":231,"summary":254},{"path":257,"title":256,"module":231,"summary":259},{"path":262,"title":261,"module":231,"summary":264},{"path":267,"title":266,"module":231,"summary":269},{"path":272,"title":271,"module":231,"summary":274},{"path":277,"title":276,"module":231,"summary":279},{"path":282,"title":281,"module":231,"summary":285},{"path":292,"title":291,"module":287,"summary":295},{"path":298,"title":297,"module":287,"summary":300},{"path":303,"title":302,"module":287,"summary":305},{"path":312,"title":311,"module":307,"summary":315},{"path":318,"title":317,"module":307,"summary":320},{"path":323,"title":322,"module":307,"summary":325},{"path":328,"title":327,"module":307,"summary":330},{"path":338,"title":337,"module":332,"summary":341},{"path":344,"title":343,"module":332,"summary":346},{"path":349,"title":348,"module":332,"summary":351},{"path":359,"title":358,"module":353,"summary":362},{"path":364,"title":361,"module":353,"summary":366},{"path":369,"title":368,"module":353,"summary":373},{"path":12601,"title":12602,"module":376,"summary":376},"\u002Falgorithms","Algorithms",{"path":12604,"title":12605,"module":376,"summary":376},"\u002Ftheory-of-computation","Theory of Computation",{"path":12607,"title":12608,"module":376,"summary":376},"\u002Fcomputer-architecture","Computer Architecture",{"path":12610,"title":12611,"module":376,"summary":376},"\u002Fphysical-computing","Physical Computing",{"path":12613,"title":12614,"module":376,"summary":376},"\u002Fdatabases","Databases",{"path":12616,"title":12617,"module":376,"summary":376},"\u002Fdeep-learning","Deep Learning",1781526656638]