[{"data":1,"prerenderedAt":10556},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":374,"course-wordcounts":10424,"ref-card-index":10480},[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":169,"blurb":376,"body":377,"description":10386,"extension":10387,"meta":10388,"module":153,"navigation":10390,"path":170,"practice":10391,"rawbody":10406,"readingTime":10407,"seo":10412,"sources":10413,"status":10420,"stem":10421,"summary":172,"topics":10422,"__hash__":10423},"course\u002F01.algorithms\u002F06.graphs\u002F03.minimum-spanning-trees.md","",{"type":378,"value":379,"toc":10374},"minimark",[380,404,409,412,954,1045,1104,1445,1448,1712,1882,1886,1962,2066,2210,2533,4125,4251,4261,4265,4280,4471,4526,4539,4711,4714,5023,5027,5092,5239,5529,5588,5632,6300,6305,6493,6793,7047,7298,7379,7383,7436,7523,7752,8001,8026,8121,8334,8626,9148,9247,9342,9346,9427,9737,9751,9755,10286,10370],[381,382,383,384,388,389,392,393,396,397,403],"p",{},"Suppose you must lay cable to connect a set of towns, and every possible\nconnection has a known cost. You want all the towns linked — any town reachable\nfrom any other — while spending as little as possible. Laying a redundant link\nwould only waste money, so the cheapest solution can contain no cycle: it is a\n",[385,386,387],"strong",{},"tree"," that ",[385,390,391],{},"spans"," every town. Finding the cheapest such tree is the\n",[385,394,395],{},"minimum spanning tree"," problem, and it is the first place in this course\nwhere a ",[398,399,400],"a",{"href":211},[401,402,207],"em",{}," strategy is provably\noptimal.",[405,406,408],"h2",{"id":407},"the-problem","The problem",[381,410,411],{},"Stated precisely, in the shape the rest of this lesson will use:",[413,414,416,560,582],"callout",{"type":415},"problem",[381,417,418,421,422,425,426,500,501,559],{},[385,419,420],{},"Input."," A weighted, ",[401,423,424],{},"undirected"," graph ",[427,428,431],"span",{"className":429},[430],"katex",[427,432,436,464],{"className":433,"ariaHidden":435},[434],"katex-html","true",[427,437,440,445,451,456,461],{"className":438},[439],"base",[427,441],{"className":442,"style":444},[443],"strut","height:0.6833em;",[427,446,450],{"className":447},[448,449],"mord","mathnormal","G",[427,452],{"className":453,"style":455},[454],"mspace","margin-right:0.2778em;",[427,457,460],{"className":458},[459],"mrel","=",[427,462],{"className":463,"style":455},[454],[427,465,467,471,476,481,486,490,495],{"className":466},[439],[427,468],{"className":469,"style":470},[443],"height:1em;vertical-align:-0.25em;",[427,472,475],{"className":473},[474],"mopen","(",[427,477,480],{"className":478,"style":479},[448,449],"margin-right:0.2222em;","V",[427,482,485],{"className":483},[484],"mpunct",",",[427,487],{"className":488,"style":489},[454],"margin-right:0.1667em;",[427,491,494],{"className":492,"style":493},[448,449],"margin-right:0.0576em;","E",[427,496,499],{"className":497},[498],"mclose",")"," with a weight function\n",[427,502,504],{"className":503},[430],[427,505,507,528,547],{"className":506,"ariaHidden":435},[434],[427,508,510,514,518,521,525],{"className":509},[439],[427,511],{"className":512,"style":513},[443],"height:0.4306em;",[427,515,517],{"className":516},[448,449],"c",[427,519],{"className":520,"style":455},[454],[427,522,524],{"className":523},[459],":",[427,526],{"className":527,"style":455},[454],[427,529,531,534,537,540,544],{"className":530},[439],[427,532],{"className":533,"style":444},[443],[427,535,494],{"className":536,"style":493},[448,449],[427,538],{"className":539,"style":455},[454],[427,541,543],{"className":542},[459],"→",[427,545],{"className":546,"style":455},[454],[427,548,550,554],{"className":549},[439],[427,551],{"className":552,"style":553},[443],"height:0.6889em;",[427,555,558],{"className":556},[448,557],"mathbb","R",". Negative weights are allowed.",[381,561,562,565,566,581],{},[385,563,564],{},"Precondition."," ",[427,567,569],{"className":568},[430],[427,570,572],{"className":571,"ariaHidden":435},[434],[427,573,575,578],{"className":574},[439],[427,576],{"className":577,"style":444},[443],[427,579,450],{"className":580},[448,449]," is connected.",[381,583,584,587,588,590,591,606,607,684,685,765,766,953],{},[385,585,586],{},"Output."," A ",[385,589,395],{}," (MST) of ",[427,592,594],{"className":593},[430],[427,595,597],{"className":596,"ariaHidden":435},[434],[427,598,600,603],{"className":599},[439],[427,601],{"className":602,"style":444},[443],[427,604,450],{"className":605},[448,449],": a subset\n",[427,608,610],{"className":609},[430],[427,611,613,675],{"className":612,"ariaHidden":435},[434],[427,614,616,620,665,668,672],{"className":615},[439],[427,617],{"className":618,"style":619},[443],"height:0.8879em;vertical-align:-0.136em;",[427,621,623,626],{"className":622},[448],[427,624,494],{"className":625,"style":493},[448,449],[427,627,630],{"className":628},[629],"msupsub",[427,631,634],{"className":632},[633],"vlist-t",[427,635,638],{"className":636},[637],"vlist-r",[427,639,643],{"className":640,"style":642},[641],"vlist","height:0.7519em;",[427,644,646,651],{"style":645},"top:-3.063em;margin-right:0.05em;",[427,647],{"className":648,"style":650},[649],"pstrut","height:2.7em;",[427,652,658],{"className":653},[654,655,656,657],"sizing","reset-size6","size3","mtight",[427,659,661],{"className":660},[448,657],[427,662,664],{"className":663},[448,657],"′",[427,666],{"className":667,"style":455},[454],[427,669,671],{"className":670},[459],"⊆",[427,673],{"className":674,"style":455},[454],[427,676,678,681],{"className":677},[439],[427,679],{"className":680,"style":444},[443],[427,682,494],{"className":683,"style":493},[448,449]," such that ",[427,686,688],{"className":687},[430],[427,689,691,711],{"className":690,"ariaHidden":435},[434],[427,692,694,697,702,705,708],{"className":693},[439],[427,695],{"className":696,"style":444},[443],[427,698,701],{"className":699,"style":700},[448,449],"margin-right:0.1389em;","T",[427,703],{"className":704,"style":455},[454],[427,706,460],{"className":707},[459],[427,709],{"className":710,"style":455},[454],[427,712,714,718,721,724,727,730,762],{"className":713},[439],[427,715],{"className":716,"style":717},[443],"height:1.0019em;vertical-align:-0.25em;",[427,719,475],{"className":720},[474],[427,722,480],{"className":723,"style":479},[448,449],[427,725,485],{"className":726},[484],[427,728],{"className":729,"style":489},[454],[427,731,733,736],{"className":732},[448],[427,734,494],{"className":735,"style":493},[448,449],[427,737,739],{"className":738},[629],[427,740,742],{"className":741},[633],[427,743,745],{"className":744},[637],[427,746,748],{"className":747,"style":642},[641],[427,749,750,753],{"style":645},[427,751],{"className":752,"style":650},[649],[427,754,756],{"className":755},[654,655,656,657],[427,757,759],{"className":758},[448,657],[427,760,664],{"className":761},[448,657],[427,763,499],{"className":764},[498]," is a tree and\n",[427,767,769],{"className":768},[430],[427,770,772,804],{"className":771,"ariaHidden":435},[434],[427,773,775,778,786,789,792,795,798,801],{"className":774},[439],[427,776],{"className":777,"style":470},[443],[427,779,782],{"className":780},[448,781],"text",[427,783,785],{"className":784},[448],"cost",[427,787,475],{"className":788},[474],[427,790,701],{"className":791,"style":700},[448,449],[427,793,499],{"className":794},[498],[427,796],{"className":797,"style":455},[454],[427,799,460],{"className":800},[459],[427,802],{"className":803,"style":455},[454],[427,805,807,811,907,910],{"className":806},[439],[427,808],{"className":809,"style":810},[443],"height:1.0771em;vertical-align:-0.3271em;",[427,812,815,822],{"className":813},[814],"mop",[427,816,821],{"className":817,"style":820},[814,818,819],"op-symbol","small-op","position:relative;top:0em;","∑",[427,823,825],{"className":824},[629],[427,826,829,898],{"className":827},[633,828],"vlist-t2",[427,830,832,893],{"className":831},[637],[427,833,836],{"className":834,"style":835},[641],"height:0.1786em;",[427,837,839,842],{"style":838},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[427,840],{"className":841,"style":650},[649],[427,843,845],{"className":844},[654,655,656,657],[427,846,848,852,856],{"className":847},[448,657],[427,849,851],{"className":850},[448,449,657],"e",[427,853,855],{"className":854},[459,657],"∈",[427,857,859,862],{"className":858},[448,657],[427,860,494],{"className":861,"style":493},[448,449,657],[427,863,865],{"className":864},[629],[427,866,868],{"className":867},[633],[427,869,871],{"className":870},[637],[427,872,875],{"className":873,"style":874},[641],"height:0.6828em;",[427,876,878,882],{"style":877},"top:-2.786em;margin-right:0.0714em;",[427,879],{"className":880,"style":881},[649],"height:2.5em;",[427,883,887],{"className":884},[654,885,886,657],"reset-size3","size1",[427,888,890],{"className":889},[448,657],[427,891,664],{"className":892},[448,657],[427,894,897],{"className":895},[896],"vlist-s","​",[427,899,901],{"className":900},[637],[427,902,905],{"className":903,"style":904},[641],"height:0.3271em;",[427,906],{},[427,908],{"className":909,"style":489},[454],[427,911,913,916],{"className":912},[448],[427,914,517],{"className":915},[448,449],[427,917,919],{"className":918},[629],[427,920,922,944],{"className":921},[633,828],[427,923,925,941],{"className":924},[637],[427,926,929],{"className":927,"style":928},[641],"height:0.1514em;",[427,930,932,935],{"style":931},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[427,933],{"className":934,"style":650},[649],[427,936,938],{"className":937},[654,655,656,657],[427,939,851],{"className":940},[448,449,657],[427,942,897],{"className":943},[896],[427,945,947],{"className":946},[637],[427,948,951],{"className":949,"style":950},[641],"height:0.15em;",[427,952],{}," is minimized.",[381,955,956,957,959,960,963,964,1005,1006,1044],{},"Here a ",[385,958,387],{}," means a ",[401,961,962],{},"connected, acyclic"," graph, not to be confused with a\nrooted or rooted-and-ordered tree; an MST has no distinguished root. A spanning\ntree on ",[427,965,967],{"className":966},[430],[427,968,970,989],{"className":969,"ariaHidden":435},[434],[427,971,973,976,980,983,986],{"className":972},[439],[427,974],{"className":975,"style":513},[443],[427,977,979],{"className":978},[448,449],"n",[427,981],{"className":982,"style":455},[454],[427,984,460],{"className":985},[459],[427,987],{"className":988,"style":455},[454],[427,990,992,995,999,1002],{"className":991},[439],[427,993],{"className":994,"style":470},[443],[427,996,998],{"className":997},[448],"∣",[427,1000,480],{"className":1001,"style":479},[448,449],[427,1003,998],{"className":1004},[448]," vertices always has exactly ",[427,1007,1009],{"className":1008},[430],[427,1010,1012,1033],{"className":1011,"ariaHidden":435},[434],[427,1013,1015,1019,1022,1025,1030],{"className":1014},[439],[427,1016],{"className":1017,"style":1018},[443],"height:0.6667em;vertical-align:-0.0833em;",[427,1020,979],{"className":1021},[448,449],[427,1023],{"className":1024,"style":479},[454],[427,1026,1029],{"className":1027},[1028],"mbin","−",[427,1031],{"className":1032,"style":479},[454],[427,1034,1036,1040],{"className":1035},[439],[427,1037],{"className":1038,"style":1039},[443],"height:0.6444em;",[427,1041,1043],{"className":1042},[448],"1"," edges: enough to connect\neverything, one fewer than would create a cycle.",[381,1046,1047,1048,565,1051,1054,1055,1103],{},"It is worth recalling ",[401,1049,1050],{},"why",[1052,1053,387],"q",{}," is exactly the right object. For a graph\n",[427,1056,1058],{"className":1057},[430],[427,1059,1061,1079],{"className":1060,"ariaHidden":435},[434],[427,1062,1064,1067,1070,1073,1076],{"className":1063},[439],[427,1065],{"className":1066,"style":444},[443],[427,1068,450],{"className":1069},[448,449],[427,1071],{"className":1072,"style":455},[454],[427,1074,460],{"className":1075},[459],[427,1077],{"className":1078,"style":455},[454],[427,1080,1082,1085,1088,1091,1094,1097,1100],{"className":1081},[439],[427,1083],{"className":1084,"style":470},[443],[427,1086,475],{"className":1087},[474],[427,1089,480],{"className":1090,"style":479},[448,449],[427,1092,485],{"className":1093},[484],[427,1095],{"className":1096,"style":489},[454],[427,1098,494],{"className":1099,"style":493},[448,449],[427,1101,499],{"className":1102},[498]," the following are equivalent, and any one of them could serve as the\ndefinition.",[413,1105,1107,1112],{"type":1106},"theorem",[381,1108,1109],{},[385,1110,1111],{},"Theorem (Tree characterizations).",[1113,1114,1115,1134,1156,1282,1364],"ol",{},[1116,1117,1118,1133],"li",{},[427,1119,1121],{"className":1120},[430],[427,1122,1124],{"className":1123,"ariaHidden":435},[434],[427,1125,1127,1130],{"className":1126},[439],[427,1128],{"className":1129,"style":444},[443],[427,1131,450],{"className":1132},[448,449]," is a tree (connected and acyclic).",[1116,1135,1136,1151,1152,1155],{},[427,1137,1139],{"className":1138},[430],[427,1140,1142],{"className":1141,"ariaHidden":435},[434],[427,1143,1145,1148],{"className":1144},[439],[427,1146],{"className":1147,"style":444},[443],[427,1149,450],{"className":1150},[448,449]," is connected and every edge is a ",[385,1153,1154],{},"bridge"," (an edge whose removal\nincreases the number of connected 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creates a\ncycle.",[1116,1283,1284,1299,1300,1363],{},[427,1285,1287],{"className":1286},[430],[427,1288,1290],{"className":1289,"ariaHidden":435},[434],[427,1291,1293,1296],{"className":1292},[439],[427,1294],{"className":1295,"style":444},[443],[427,1297,450],{"className":1298},[448,449]," is connected and ",[427,1301,1303],{"className":1302},[430],[427,1304,1306,1330,1354],{"className":1305,"ariaHidden":435},[434],[427,1307,1309,1312,1315,1318,1321,1324,1327],{"className":1308},[439],[427,1310],{"className":1311,"style":470},[443],[427,1313,998],{"className":1314},[448],[427,1316,494],{"className":1317,"style":493},[448,449],[427,1319,998],{"className":1320},[448],[427,1322],{"className":1323,"style":455},[454],[427,1325,460],{"className":1326},[459],[427,1328],{"className":1329,"style":455},[454],[427,1331,1333,1336,1339,1342,1345,1348,1351],{"className":1332},[439],[427,1334],{"className":1335,"style":470},[443],[427,1337,998],{"className":1338},[448],[427,1340,480],{"className":1341,"style":479},[448,449],[427,1343,998],{"className":1344},[448],[427,1346],{"className":1347,"style":479},[454],[427,1349,1029],{"className":1350},[1028],[427,1352],{"className":1353,"style":479},[454],[427,1355,1357,1360],{"className":1356},[439],[427,1358],{"className":1359,"style":1039},[443],[427,1361,1043],{"className":1362},[448],".",[1116,1365,1366,1381,1382,1363],{},[427,1367,1369],{"className":1368},[430],[427,1370,1372],{"className":1371,"ariaHidden":435},[434],[427,1373,1375,1378],{"className":1374},[439],[427,1376],{"className":1377,"style":444},[443],[427,1379,450],{"className":1380},[448,449]," is acyclic and ",[427,1383,1385],{"className":1384},[430],[427,1386,1388,1412,1436],{"className":1387,"ariaHidden":435},[434],[427,1389,1391,1394,1397,1400,1403,1406,1409],{"className":1390},[439],[427,1392],{"className":1393,"style":470},[443],[427,1395,998],{"className":1396},[448],[427,1398,494],{"className":1399,"style":493},[448,449],[427,1401,998],{"className":1402},[448],[427,1404],{"className":1405,"style":455},[454],[427,1407,460],{"className":1408},[459],[427,1410],{"className":1411,"style":455},[454],[427,1413,1415,1418,1421,1424,1427,1430,1433],{"className":1414},[439],[427,1416],{"className":1417,"style":470},[443],[427,1419,998],{"className":1420},[448],[427,1422,480],{"className":1423,"style":479},[448,449],[427,1425,998],{"className":1426},[448],[427,1428],{"className":1429,"style":479},[454],[427,1431,1029],{"className":1432},[1028],[427,1434],{"className":1435,"style":479},[454],[427,1437,1439,1442],{"className":1438},[439],[427,1440],{"className":1441,"style":1039},[443],[427,1443,1043],{"className":1444},[448],[381,1446,1447],{},"Below, 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20.7 31.023 3.448",[1462,1689,1691],{"transform":1690},"translate(69.139 35.776)",[1467,1692],{"d":1693,"fill":1464,"stroke":1464,"className":1694,"style":1570},"M-66.833-0.338L-69.363-0.338L-69.363-0.618Q-68.395-0.618-68.395-0.827L-68.395-4.446Q-68.788-4.258-69.410-4.258L-69.410-4.539Q-68.993-4.539-68.629-4.640Q-68.265-4.740-68.009-4.986L-67.883-4.986Q-67.818-4.969-67.801-4.901L-67.801-0.827Q-67.801-0.618-66.833-0.618",[1478],[1462,1696,1697,1700],{"stroke":1613,"style":1614},[1467,1698],{"fill":1469,"d":1699},"M36.676 24.148 67.699 20.7",[1462,1701,1703],{"transform":1702},"translate(120.354 35.776)",[1467,1704],{"d":1657,"fill":1464,"stroke":1464,"className":1705,"style":1570},[1478],[1707,1708,1711],"figcaption",{"className":1709},[1710],"tikz-cap","A weighted graph with one minimum spanning tree shown in thick edges.",[381,1713,1714,1715,1881],{},"The thick tree spans all nine towns with total weight\n",[427,1716,1718],{"className":1717},[430],[427,1719,1721,1741,1759,1777,1796,1815,1834,1852,1871],{"className":1720,"ariaHidden":435},[434],[427,1722,1724,1728,1731,1734,1738],{"className":1723},[439],[427,1725],{"className":1726,"style":1727},[443],"height:0.7278em;vertical-align:-0.0833em;",[427,1729,1043],{"className":1730},[448],[427,1732],{"className":1733,"style":479},[454],[427,1735,1737],{"className":1736},[1028],"+",[427,1739],{"className":1740,"style":479},[454],[427,1742,1744,1747,1750,1753,1756],{"className":1743},[439],[427,1745],{"className":1746,"style":1727},[443],[427,1748,1228],{"className":1749},[448],[427,1751],{"className":1752,"style":479},[454],[427,1754,1737],{"className":1755},[1028],[427,1757],{"className":1758,"style":479},[454],[427,1760,1762,1765,1768,1771,1774],{"className":1761},[439],[427,1763],{"className":1764,"style":1727},[443],[427,1766,1228],{"className":1767},[448],[427,1769],{"className":1770,"style":479},[454],[427,1772,1737],{"className":1773},[1028],[427,1775],{"className":1776,"style":479},[454],[427,1778,1780,1783,1787,1790,1793],{"className":1779},[439],[427,1781],{"className":1782,"style":1727},[443],[427,1784,1786],{"className":1785},[448],"4",[427,1788],{"className":1789,"style":479},[454],[427,1791,1737],{"className":1792},[1028],[427,1794],{"className":1795,"style":479},[454],[427,1797,1799,1802,1806,1809,1812],{"className":1798},[439],[427,1800],{"className":1801,"style":1727},[443],[427,1803,1805],{"className":1804},[448],"6",[427,1807],{"className":1808,"style":479},[454],[427,1810,1737],{"className":1811},[1028],[427,1813],{"className":1814,"style":479},[454],[427,1816,1818,1821,1825,1828,1831],{"className":1817},[439],[427,1819],{"className":1820,"style":1727},[443],[427,1822,1824],{"className":1823},[448],"7",[427,1826],{"className":1827,"style":479},[454],[427,1829,1737],{"className":1830},[1028],[427,1832],{"className":1833,"style":479},[454],[427,1835,1837,1840,1843,1846,1849],{"className":1836},[439],[427,1838],{"className":1839,"style":1727},[443],[427,1841,1824],{"className":1842},[448],[427,1844],{"className":1845,"style":479},[454],[427,1847,1737],{"className":1848},[1028],[427,1850],{"className":1851,"style":479},[454],[427,1853,1855,1858,1862,1865,1868],{"className":1854},[439],[427,1856],{"className":1857,"style":1039},[443],[427,1859,1861],{"className":1860},[448],"9",[427,1863],{"className":1864,"style":455},[454],[427,1866,460],{"className":1867},[459],[427,1869],{"className":1870,"style":455},[454],[427,1872,1874,1877],{"className":1873},[439],[427,1875],{"className":1876,"style":1039},[443],[427,1878,1880],{"className":1879},[448],"38","; no spanning tree is cheaper.",[405,1883,1885],{"id":1884},"the-cut-property","The cut property",[381,1887,1888,1889,1905,1906,1909,1910,1913,1914,1929,1930,1952,1953],{},"All three algorithms below are instances of a single greedy template: maintain a\nset ",[427,1890,1892],{"className":1891},[430],[427,1893,1895],{"className":1894,"ariaHidden":435},[434],[427,1896,1898,1901],{"className":1897},[439],[427,1899],{"className":1900,"style":444},[443],[427,1902,1904],{"className":1903},[448,449],"A"," of edges that is always a subset of ",[401,1907,1908],{},"some"," MST, and at each step add one\nmore ",[385,1911,1912],{},"safe"," edge, meaning an edge that can be added to ",[427,1915,1917],{"className":1916},[430],[427,1918,1920],{"className":1919,"ariaHidden":435},[434],[427,1921,1923,1926],{"className":1922},[439],[427,1924],{"className":1925,"style":444},[443],[427,1927,1904],{"className":1928},[448,449]," while keeping\n",[427,1931,1933],{"className":1932},[430],[427,1934,1936],{"className":1935,"ariaHidden":435},[434],[427,1937,1939,1943,1946,1949],{"className":1938},[439],[427,1940],{"className":1941,"style":1942},[443],"height:0.8193em;vertical-align:-0.136em;",[427,1944,1904],{"className":1945},[448,449],[427,1947],{"className":1948,"style":455},[454],[427,1950,671],{"className":1951},[459]," (some MST). The entire theory of MSTs reduces to recognizing safe\nedges, and one lemma does that for us.",[1954,1955,1956],"sup",{},[398,1957,1043],{"href":1958,"ariaDescribedBy":1959,"dataFootnoteRef":376,"id":1961},"#user-content-fn-clrs-cut",[1960],"footnote-label","user-content-fnref-clrs-cut",[381,1963,1964,1965,565,1968,2017,2018,2021,2022,2025,2026,2041,2042,2057,2058,2061,2062,2065],{},"First, the vocabulary. A ",[385,1966,1967],{},"cut",[427,1969,1971],{"className":1970},[430],[427,1972,1974,2005],{"className":1973,"ariaHidden":435},[434],[427,1975,1977,1980,1983,1987,1990,1993,1996,1999,2002],{"className":1976},[439],[427,1978],{"className":1979,"style":470},[443],[427,1981,475],{"className":1982},[474],[427,1984,1986],{"className":1985,"style":493},[448,449],"S",[427,1988,485],{"className":1989},[484],[427,1991],{"className":1992,"style":489},[454],[427,1994,480],{"className":1995,"style":479},[448,449],[427,1997],{"className":1998,"style":479},[454],[427,2000,1268],{"className":2001},[1028],[427,2003],{"className":2004,"style":479},[454],[427,2006,2008,2011,2014],{"className":2007},[439],[427,2009],{"className":2010,"style":470},[443],[427,2012,1986],{"className":2013,"style":493},[448,449],[427,2015,499],{"className":2016},[498]," is a partition of the\nvertices into two groups. An edge ",[385,2019,2020],{},"crosses"," the cut if its endpoints lie on\nopposite sides. A cut ",[385,2023,2024],{},"respects"," an edge set ",[427,2027,2029],{"className":2028},[430],[427,2030,2032],{"className":2031,"ariaHidden":435},[434],[427,2033,2035,2038],{"className":2034},[439],[427,2036],{"className":2037,"style":444},[443],[427,2039,1904],{"className":2040},[448,449]," if no edge of ",[427,2043,2045],{"className":2044},[430],[427,2046,2048],{"className":2047,"ariaHidden":435},[434],[427,2049,2051,2054],{"className":2050},[439],[427,2052],{"className":2053,"style":444},[443],[427,2055,1904],{"className":2056},[448,449]," crosses it.\nAn edge crossing the cut is ",[385,2059,2060],{},"light"," (or ",[401,2063,2064],{},"cheapest",") if it has the minimum\nweight of all crossing edges.",[1449,2067,2069,2190],{"className":2068},[1452,1453],[1455,2070,2074],{"xmlns":1457,"width":2071,"height":2072,"viewBox":2073},"293.711","123.704","-75 -75 220.283 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cut splitting vertices into ",[427,2195,2197],{"className":2196},[430],[427,2198,2200],{"className":2199,"ariaHidden":435},[434],[427,2201,2203,2206],{"className":2202},[439],[427,2204],{"className":2205,"style":444},[443],[427,2207,1986],{"className":2208,"style":493},[448,449]," and its complement, with the light crossing edge highlighted.",[413,2211,2213],{"type":2212},"lemma",[381,2214,2215,2218,2219,2234,2235,2250,2251,2299,2300,2315,2316,2367,2368,2402,2403,2454,2455,2470,2471,2486,2487,1363],{},[385,2216,2217],{},"Property (Cut rule)."," Let ",[427,2220,2222],{"className":2221},[430],[427,2223,2225],{"className":2224,"ariaHidden":435},[434],[427,2226,2228,2231],{"className":2227},[439],[427,2229],{"className":2230,"style":444},[443],[427,2232,1904],{"className":2233},[448,449]," be a subset of some MST of ",[427,2236,2238],{"className":2237},[430],[427,2239,2241],{"className":2240,"ariaHidden":435},[434],[427,2242,2244,2247],{"className":2243},[439],[427,2245],{"className":2246,"style":444},[443],[427,2248,450],{"className":2249},[448,449],". Let\n",[427,2252,2254],{"className":2253},[430],[427,2255,2257,2287],{"className":2256,"ariaHidden":435},[434],[427,2258,2260,2263,2266,2269,2272,2275,2278,2281,2284],{"className":2259},[439],[427,2261],{"className":2262,"style":470},[443],[427,2264,475],{"className":2265},[474],[427,2267,1986],{"className":2268,"style":493},[448,449],[427,2270,485],{"className":2271},[484],[427,2273],{"className":2274,"style":489},[454],[427,2276,480],{"className":2277,"style":479},[448,449],[427,2279],{"className":2280,"style":479},[454],[427,2282,1268],{"className":2283},[1028],[427,2285],{"className":2286,"style":479},[454],[427,2288,2290,2293,2296],{"className":2289},[439],[427,2291],{"className":2292,"style":470},[443],[427,2294,1986],{"className":2295,"style":493},[448,449],[427,2297,499],{"className":2298},[498]," be any cut that respects ",[427,2301,2303],{"className":2302},[430],[427,2304,2306],{"className":2305,"ariaHidden":435},[434],[427,2307,2309,2312],{"className":2308},[439],[427,2310],{"className":2311,"style":444},[443],[427,2313,1904],{"className":2314},[448,449],", and let ",[427,2317,2319],{"className":2318},[430],[427,2320,2322,2340],{"className":2321,"ariaHidden":435},[434],[427,2323,2325,2328,2331,2334,2337],{"className":2324},[439],[427,2326],{"className":2327,"style":513},[443],[427,2329,851],{"className":2330},[448,449],[427,2332],{"className":2333,"style":455},[454],[427,2335,460],{"className":2336},[459],[427,2338],{"className":2339,"style":455},[454],[427,2341,2343,2346,2349,2353,2356,2359,2364],{"className":2342},[439],[427,2344],{"className":2345,"style":470},[443],[427,2347,475],{"className":2348},[474],[427,2350,2352],{"className":2351},[448,449],"u",[427,2354,485],{"className":2355},[484],[427,2357],{"className":2358,"style":489},[454],[427,2360,2363],{"className":2361,"style":2362},[448,449],"margin-right:0.0359em;","v",[427,2365,499],{"className":2366},[498]," with\n",[427,2369,2371],{"className":2370},[430],[427,2372,2374,2393],{"className":2373,"ariaHidden":435},[434],[427,2375,2377,2381,2384,2387,2390],{"className":2376},[439],[427,2378],{"className":2379,"style":2380},[443],"height:0.5782em;vertical-align:-0.0391em;",[427,2382,2352],{"className":2383},[448,449],[427,2385],{"className":2386,"style":455},[454],[427,2388,855],{"className":2389},[459],[427,2391],{"className":2392,"style":455},[454],[427,2394,2396,2399],{"className":2395},[439],[427,2397],{"className":2398,"style":444},[443],[427,2400,1986],{"className":2401,"style":493},[448,449],", ",[427,2404,2406],{"className":2405},[430],[427,2407,2409,2427,2445],{"className":2408,"ariaHidden":435},[434],[427,2410,2412,2415,2418,2421,2424],{"className":2411},[439],[427,2413],{"className":2414,"style":2380},[443],[427,2416,2363],{"className":2417,"style":2362},[448,449],[427,2419],{"className":2420,"style":455},[454],[427,2422,855],{"className":2423},[459],[427,2425],{"className":2426,"style":455},[454],[427,2428,2430,2433,2436,2439,2442],{"className":2429},[439],[427,2431],{"className":2432,"style":470},[443],[427,2434,480],{"className":2435,"style":479},[448,449],[427,2437],{"className":2438,"style":479},[454],[427,2440,1268],{"className":2441},[1028],[427,2443],{"className":2444,"style":479},[454],[427,2446,2448,2451],{"className":2447},[439],[427,2449],{"className":2450,"style":444},[443],[427,2452,1986],{"className":2453,"style":493},[448,449]," be a light edge crossing that cut. Then ",[427,2456,2458],{"className":2457},[430],[427,2459,2461],{"className":2460,"ariaHidden":435},[434],[427,2462,2464,2467],{"className":2463},[439],[427,2465],{"className":2466,"style":513},[443],[427,2468,851],{"className":2469},[448,449]," is\nsafe for ",[427,2472,2474],{"className":2473},[430],[427,2475,2477],{"className":2476,"ariaHidden":435},[434],[427,2478,2480,2483],{"className":2479},[439],[427,2481],{"className":2482,"style":444},[443],[427,2484,1904],{"className":2485},[448,449],": there is an MST containing ",[427,2488,2490],{"className":2489},[430],[427,2491,2493,2512],{"className":2492,"ariaHidden":435},[434],[427,2494,2496,2499,2502,2505,2509],{"className":2495},[439],[427,2497],{"className":2498,"style":444},[443],[427,2500,1904],{"className":2501},[448,449],[427,2503],{"className":2504,"style":479},[454],[427,2506,2508],{"className":2507},[1028],"∪",[427,2510],{"className":2511,"style":479},[454],[427,2513,2515,2518],{"className":2514},[439],[427,2516],{"className":2517,"style":470},[443],[427,2519,2522,2526,2529],{"className":2520},[2521],"minner",[427,2523,2525],{"className":2524,"style":1194},[474,1193],"{",[427,2527,851],{"className":2528},[448,449],[427,2530,2532],{"className":2531,"style":1194},[498,1193],"}",[413,2534,2536,2753,2958,3254,3410,3648,3934],{"type":2535},"proof",[381,2537,2538,2218,2541,2584,2585,2600,2601,2604,2605,2620,2621,2662,2663,2704,2705,1363],{},[385,2539,2540],{},"Proof (an exchange argument — done carefully).",[427,2542,2544],{"className":2543},[430],[427,2545,2547],{"className":2546,"ariaHidden":435},[434],[427,2548,2550,2554],{"className":2549},[439],[427,2551],{"className":2552,"style":2553},[443],"height:0.6887em;",[427,2555,2557,2560],{"className":2556},[448],[427,2558,701],{"className":2559,"style":700},[448,449],[427,2561,2563],{"className":2562},[629],[427,2564,2566],{"className":2565},[633],[427,2567,2569],{"className":2568},[637],[427,2570,2572],{"className":2571,"style":2553},[641],[427,2573,2574,2577],{"style":645},[427,2575],{"className":2576,"style":650},[649],[427,2578,2580],{"className":2579},[654,655,656,657],[427,2581,2583],{"className":2582},[1028,657],"∗"," be an MST\ncontaining ",[427,2586,2588],{"className":2587},[430],[427,2589,2591],{"className":2590,"ariaHidden":435},[434],[427,2592,2594,2597],{"className":2593},[439],[427,2595],{"className":2596,"style":444},[443],[427,2598,1904],{"className":2599},[448,449],", and suppose, for contradiction, that ",[401,2602,2603],{},"no"," MST contains ",[427,2606,2608],{"className":2607},[430],[427,2609,2611],{"className":2610,"ariaHidden":435},[434],[427,2612,2614,2617],{"className":2613},[439],[427,2615],{"className":2616,"style":513},[443],[427,2618,851],{"className":2619},[448,449],". Since\n",[427,2622,2624],{"className":2623},[430],[427,2625,2627],{"className":2626,"ariaHidden":435},[434],[427,2628,2630,2633],{"className":2629},[439],[427,2631],{"className":2632,"style":2553},[443],[427,2634,2636,2639],{"className":2635},[448],[427,2637,701],{"className":2638,"style":700},[448,449],[427,2640,2642],{"className":2641},[629],[427,2643,2645],{"className":2644},[633],[427,2646,2648],{"className":2647},[637],[427,2649,2651],{"className":2650,"style":2553},[641],[427,2652,2653,2656],{"style":645},[427,2654],{"className":2655,"style":650},[649],[427,2657,2659],{"className":2658},[654,655,656,657],[427,2660,2583],{"className":2661},[1028,657]," is connected, there is at least one edge of ",[427,2664,2666],{"className":2665},[430],[427,2667,2669],{"className":2668,"ariaHidden":435},[434],[427,2670,2672,2675],{"className":2671},[439],[427,2673],{"className":2674,"style":2553},[443],[427,2676,2678,2681],{"className":2677},[448],[427,2679,701],{"className":2680,"style":700},[448,449],[427,2682,2684],{"className":2683},[629],[427,2685,2687],{"className":2686},[633],[427,2688,2690],{"className":2689},[637],[427,2691,2693],{"className":2692,"style":2553},[641],[427,2694,2695,2698],{"style":645},[427,2696],{"className":2697,"style":650},[649],[427,2699,2701],{"className":2700},[654,655,656,657],[427,2702,2583],{"className":2703},[1028,657]," crossing the cut\n",[427,2706,2708],{"className":2707},[430],[427,2709,2711,2741],{"className":2710,"ariaHidden":435},[434],[427,2712,2714,2717,2720,2723,2726,2729,2732,2735,2738],{"className":2713},[439],[427,2715],{"className":2716,"style":470},[443],[427,2718,475],{"className":2719},[474],[427,2721,1986],{"className":2722,"style":493},[448,449],[427,2724,485],{"className":2725},[484],[427,2727],{"className":2728,"style":489},[454],[427,2730,480],{"className":2731,"style":479},[448,449],[427,2733],{"className":2734,"style":479},[454],[427,2736,1268],{"className":2737},[1028],[427,2739],{"className":2740,"style":479},[454],[427,2742,2744,2747,2750],{"className":2743},[439],[427,2745],{"className":2746,"style":470},[443],[427,2748,1986],{"className":2749,"style":493},[448,449],[427,2751,499],{"className":2752},[498],[381,2754,2755,2756,2759,2760,2822,2823,2838,2839,2917,2918,2921,2922,2937,2938,2953,2954,2957],{},"It is tempting to grab ",[401,2757,2758],{},"any"," such crossing edge ",[427,2761,2763],{"className":2762},[430],[427,2764,2766,2787],{"className":2765,"ariaHidden":435},[434],[427,2767,2769,2773,2778,2781,2784],{"className":2768},[439],[427,2770],{"className":2771,"style":2772},[443],"height:0.8889em;vertical-align:-0.1944em;",[427,2774,2777],{"className":2775,"style":2776},[448,449],"margin-right:0.1076em;","f",[427,2779],{"className":2780,"style":455},[454],[427,2782,855],{"className":2783},[459],[427,2785],{"className":2786,"style":455},[454],[427,2788,2790,2793],{"className":2789},[439],[427,2791],{"className":2792,"style":2553},[443],[427,2794,2796,2799],{"className":2795},[448],[427,2797,701],{"className":2798,"style":700},[448,449],[427,2800,2802],{"className":2801},[629],[427,2803,2805],{"className":2804},[633],[427,2806,2808],{"className":2807},[637],[427,2809,2811],{"className":2810,"style":2553},[641],[427,2812,2813,2816],{"style":645},[427,2814],{"className":2815,"style":650},[649],[427,2817,2819],{"className":2818},[654,655,656,657],[427,2820,2583],{"className":2821},[1028,657],", swap it for ",[427,2824,2826],{"className":2825},[430],[427,2827,2829],{"className":2828,"ariaHidden":435},[434],[427,2830,2832,2835],{"className":2831},[439],[427,2833],{"className":2834,"style":513},[443],[427,2836,851],{"className":2837},[448,449],", and\nargue ",[427,2840,2842],{"className":2841},[430],[427,2843,2845,2890,2908],{"className":2844,"ariaHidden":435},[434],[427,2846,2848,2852,2881,2884,2887],{"className":2847},[439],[427,2849],{"className":2850,"style":2851},[443],"height:0.772em;vertical-align:-0.0833em;",[427,2853,2855,2858],{"className":2854},[448],[427,2856,701],{"className":2857,"style":700},[448,449],[427,2859,2861],{"className":2860},[629],[427,2862,2864],{"className":2863},[633],[427,2865,2867],{"className":2866},[637],[427,2868,2870],{"className":2869,"style":2553},[641],[427,2871,2872,2875],{"style":645},[427,2873],{"className":2874,"style":650},[649],[427,2876,2878],{"className":2877},[654,655,656,657],[427,2879,2583],{"className":2880},[1028,657],[427,2882],{"className":2883,"style":479},[454],[427,2885,1029],{"className":2886},[1028],[427,2888],{"className":2889,"style":479},[454],[427,2891,2893,2896,2899,2902,2905],{"className":2892},[439],[427,2894],{"className":2895,"style":2772},[443],[427,2897,2777],{"className":2898,"style":2776},[448,449],[427,2900],{"className":2901,"style":479},[454],[427,2903,1737],{"className":2904},[1028],[427,2906],{"className":2907,"style":479},[454],[427,2909,2911,2914],{"className":2910},[439],[427,2912],{"className":2913,"style":513},[443],[427,2915,851],{"className":2916},[448,449]," is a cheaper tree. ",[385,2919,2920],{},"This is a mistake",": removing an\narbitrary crossing edge ",[427,2923,2925],{"className":2924},[430],[427,2926,2928],{"className":2927,"ariaHidden":435},[434],[427,2929,2931,2934],{"className":2930},[439],[427,2932],{"className":2933,"style":2772},[443],[427,2935,2777],{"className":2936,"style":2776},[448,449]," need not reconnect into a tree once we add ",[427,2939,2941],{"className":2940},[430],[427,2942,2944],{"className":2943,"ariaHidden":435},[434],[427,2945,2947,2950],{"className":2946},[439],[427,2948],{"className":2949,"style":513},[443],[427,2951,851],{"className":2952},[448,449],"; the\nresult can be disconnected or contain a cycle. We must remove the ",[401,2955,2956],{},"right"," edge.",[381,2959,2960,2961,2976,2977,3018,3019,3034,3035,3095,3096,3112,3113,3161,3162,565,3164,3179,3180,3253],{},"So instead: adding ",[427,2962,2964],{"className":2963},[430],[427,2965,2967],{"className":2966,"ariaHidden":435},[434],[427,2968,2970,2973],{"className":2969},[439],[427,2971],{"className":2972,"style":513},[443],[427,2974,851],{"className":2975},[448,449]," to ",[427,2978,2980],{"className":2979},[430],[427,2981,2983],{"className":2982,"ariaHidden":435},[434],[427,2984,2986,2989],{"className":2985},[439],[427,2987],{"className":2988,"style":2553},[443],[427,2990,2992,2995],{"className":2991},[448],[427,2993,701],{"className":2994,"style":700},[448,449],[427,2996,2998],{"className":2997},[629],[427,2999,3001],{"className":3000},[633],[427,3002,3004],{"className":3003},[637],[427,3005,3007],{"className":3006,"style":2553},[641],[427,3008,3009,3012],{"style":645},[427,3010],{"className":3011,"style":650},[649],[427,3013,3015],{"className":3014},[654,655,656,657],[427,3016,2583],{"className":3017},[1028,657]," creates a unique cycle, and because ",[427,3020,3022],{"className":3021},[430],[427,3023,3025],{"className":3024,"ariaHidden":435},[434],[427,3026,3028,3031],{"className":3027},[439],[427,3029],{"className":3030,"style":513},[443],[427,3032,851],{"className":3033},[448,449]," crosses\nthe cut, that cycle must cross back at some edge ",[427,3036,3038],{"className":3037},[430],[427,3039,3041,3060],{"className":3040,"ariaHidden":435},[434],[427,3042,3044,3048,3051,3054,3057],{"className":3043},[439],[427,3045],{"className":3046,"style":3047},[443],"height:0.7335em;vertical-align:-0.1944em;",[427,3049,1462],{"className":3050,"style":2362},[448,449],[427,3052],{"className":3053,"style":455},[454],[427,3055,855],{"className":3056},[459],[427,3058],{"className":3059,"style":455},[454],[427,3061,3063,3066],{"className":3062},[439],[427,3064],{"className":3065,"style":2553},[443],[427,3067,3069,3072],{"className":3068},[448],[427,3070,701],{"className":3071,"style":700},[448,449],[427,3073,3075],{"className":3074},[629],[427,3076,3078],{"className":3077},[633],[427,3079,3081],{"className":3080},[637],[427,3082,3084],{"className":3083,"style":2553},[641],[427,3085,3086,3089],{"style":645},[427,3087],{"className":3088,"style":650},[649],[427,3090,3092],{"className":3091},[654,655,656,657],[427,3093,2583],{"className":3094},[1028,657],", with ",[427,3097,3099],{"className":3098},[430],[427,3100,3102],{"className":3101,"ariaHidden":435},[434],[427,3103,3105,3109],{"className":3104},[439],[427,3106],{"className":3107,"style":3108},[443],"height:0.625em;vertical-align:-0.1944em;",[427,3110,1462],{"className":3111,"style":2362},[448,449]," also\ncrossing ",[427,3114,3116],{"className":3115},[430],[427,3117,3119,3149],{"className":3118,"ariaHidden":435},[434],[427,3120,3122,3125,3128,3131,3134,3137,3140,3143,3146],{"className":3121},[439],[427,3123],{"className":3124,"style":470},[443],[427,3126,475],{"className":3127},[474],[427,3129,1986],{"className":3130,"style":493},[448,449],[427,3132,485],{"className":3133},[484],[427,3135],{"className":3136,"style":489},[454],[427,3138,480],{"className":3139,"style":479},[448,449],[427,3141],{"className":3142,"style":479},[454],[427,3144,1268],{"className":3145},[1028],[427,3147],{"className":3148,"style":479},[454],[427,3150,3152,3155,3158],{"className":3151},[439],[427,3153],{"className":3154,"style":470},[443],[427,3156,1986],{"className":3157,"style":493},[448,449],[427,3159,499],{"className":3160},[498],". Because the cut ",[385,3163,2024],{},[427,3165,3167],{"className":3166},[430],[427,3168,3170],{"className":3169,"ariaHidden":435},[434],[427,3171,3173,3176],{"className":3172},[439],[427,3174],{"className":3175,"style":444},[443],[427,3177,1904],{"className":3178},[448,449],", this ",[427,3181,3183],{"className":3182},[430],[427,3184,3186,3244],{"className":3185,"ariaHidden":435},[434],[427,3187,3189,3192,3195,3198,3241],{"className":3188},[439],[427,3190],{"className":3191,"style":470},[443],[427,3193,1462],{"className":3194,"style":2362},[448,449],[427,3196],{"className":3197,"style":455},[454],[427,3199,3201,3207],{"className":3200},[459],[427,3202,3204],{"className":3203},[448],[427,3205,855],{"className":3206},[459],[427,3208,3211],{"className":3209},[448,3210],"vbox",[427,3212,3215],{"className":3213},[3214],"thinbox",[427,3216,3219,3222,3237],{"className":3217},[3218],"llap",[427,3220],{"className":3221,"style":470},[443],[427,3223,3226],{"className":3224},[3225],"inner",[427,3227,3229,3233],{"className":3228},[448],[427,3230,3232],{"className":3231},[448],"\u002F",[427,3234],{"className":3235,"style":3236},[454],"margin-right:0.0556em;",[427,3238],{"className":3239},[3240],"fix",[427,3242],{"className":3243,"style":455},[454],[427,3245,3247,3250],{"className":3246},[439],[427,3248],{"className":3249,"style":444},[443],[427,3251,1904],{"className":3252},[448,449],".\nForm",[427,3255,3258],{"className":3256},[3257],"katex-display",[427,3259,3261],{"className":3260},[430],[427,3262,3264,3313,3359,3386],{"className":3263,"ariaHidden":435},[434],[427,3265,3267,3271,3304,3307,3310],{"className":3266},[439],[427,3268],{"className":3269,"style":3270},[443],"height:0.8019em;",[427,3272,3274,3277],{"className":3273},[448],[427,3275,701],{"className":3276,"style":700},[448,449],[427,3278,3280],{"className":3279},[629],[427,3281,3283],{"className":3282},[633],[427,3284,3286],{"className":3285},[637],[427,3287,3289],{"className":3288,"style":3270},[641],[427,3290,3292,3295],{"style":3291},"top:-3.113em;margin-right:0.05em;",[427,3293],{"className":3294,"style":650},[649],[427,3296,3298],{"className":3297},[654,655,656,657],[427,3299,3301],{"className":3300},[448,657],[427,3302,664],{"className":3303},[448,657],[427,3305],{"className":3306,"style":455},[454],[427,3308,460],{"className":3309},[459],[427,3311],{"className":3312,"style":455},[454],[427,3314,3316,3320,3350,3353,3356],{"className":3315},[439],[427,3317],{"className":3318,"style":3319},[443],"height:0.822em;vertical-align:-0.0833em;",[427,3321,3323,3326],{"className":3322},[448],[427,3324,701],{"className":3325,"style":700},[448,449],[427,3327,3329],{"className":3328},[629],[427,3330,3332],{"className":3331},[633],[427,3333,3335],{"className":3334},[637],[427,3336,3339],{"className":3337,"style":3338},[641],"height:0.7387em;",[427,3340,3341,3344],{"style":3291},[427,3342],{"className":3343,"style":650},[649],[427,3345,3347],{"className":3346},[654,655,656,657],[427,3348,2583],{"className":3349},[1028,657],[427,3351],{"className":3352,"style":479},[454],[427,3354,1029],{"className":3355},[1028],[427,3357],{"className":3358,"style":479},[454],[427,3360,3362,3365,3377,3380,3383],{"className":3361},[439],[427,3363],{"className":3364,"style":470},[443],[427,3366,3368,3371,3374],{"className":3367},[2521],[427,3369,2525],{"className":3370,"style":1194},[474,1193],[427,3372,1462],{"className":3373,"style":2362},[448,449],[427,3375,2532],{"className":3376,"style":1194},[498,1193],[427,3378],{"className":3379,"style":479},[454],[427,3381,1737],{"className":3382},[1028],[427,3384],{"className":3385,"style":479},[454],[427,3387,3389,3392,3404,3407],{"className":3388},[439],[427,3390],{"className":3391,"style":470},[443],[427,3393,3395,3398,3401],{"className":3394},[2521],[427,3396,2525],{"className":3397,"style":1194},[474,1193],[427,3399,851],{"className":3400},[448,449],[427,3402,2532],{"className":3403,"style":1194},[498,1193],[427,3405],{"className":3406,"style":489},[454],[427,3408,1363],{"className":3409},[448],[381,3411,3412,3413,3428,3429,3473,3474,3516,3517,3532,3533,3535,3536,3647],{},"Deleting ",[427,3414,3416],{"className":3415},[430],[427,3417,3419],{"className":3418,"ariaHidden":435},[434],[427,3420,3422,3425],{"className":3421},[439],[427,3423],{"className":3424,"style":3108},[443],[427,3426,1462],{"className":3427,"style":2362},[448,449]," breaks the unique cycle, so ",[427,3430,3432],{"className":3431},[430],[427,3433,3435],{"className":3434,"ariaHidden":435},[434],[427,3436,3438,3441],{"className":3437},[439],[427,3439],{"className":3440,"style":642},[443],[427,3442,3444,3447],{"className":3443},[448],[427,3445,701],{"className":3446,"style":700},[448,449],[427,3448,3450],{"className":3449},[629],[427,3451,3453],{"className":3452},[633],[427,3454,3456],{"className":3455},[637],[427,3457,3459],{"className":3458,"style":642},[641],[427,3460,3461,3464],{"style":645},[427,3462],{"className":3463,"style":650},[649],[427,3465,3467],{"className":3466},[654,655,656,657],[427,3468,3470],{"className":3469},[448,657],[427,3471,664],{"className":3472},[448,657]," is again a spanning tree, and it\nstill contains ",[427,3475,3477],{"className":3476},[430],[427,3478,3480,3498],{"className":3479,"ariaHidden":435},[434],[427,3481,3483,3486,3489,3492,3495],{"className":3482},[439],[427,3484],{"className":3485,"style":444},[443],[427,3487,1904],{"className":3488},[448,449],[427,3490],{"className":3491,"style":479},[454],[427,3493,2508],{"className":3494},[1028],[427,3496],{"className":3497,"style":479},[454],[427,3499,3501,3504],{"className":3500},[439],[427,3502],{"className":3503,"style":470},[443],[427,3505,3507,3510,3513],{"className":3506},[2521],[427,3508,2525],{"className":3509,"style":1194},[474,1193],[427,3511,851],{"className":3512},[448,449],[427,3514,2532],{"className":3515,"style":1194},[498,1193],". Since ",[427,3518,3520],{"className":3519},[430],[427,3521,3523],{"className":3522,"ariaHidden":435},[434],[427,3524,3526,3529],{"className":3525},[439],[427,3527],{"className":3528,"style":513},[443],[427,3530,851],{"className":3531},[448,449]," is the ",[401,3534,2060],{}," crossing edge,\n",[427,3537,3539],{"className":3538},[430],[427,3540,3542,3599],{"className":3541,"ariaHidden":435},[434],[427,3543,3545,3549,3589,3592,3596],{"className":3544},[439],[427,3546],{"className":3547,"style":3548},[443],"height:0.786em;vertical-align:-0.15em;",[427,3550,3552,3555],{"className":3551},[448],[427,3553,517],{"className":3554},[448,449],[427,3556,3558],{"className":3557},[629],[427,3559,3561,3581],{"className":3560},[633,828],[427,3562,3564,3578],{"className":3563},[637],[427,3565,3567],{"className":3566,"style":928},[641],[427,3568,3569,3572],{"style":931},[427,3570],{"className":3571,"style":650},[649],[427,3573,3575],{"className":3574},[654,655,656,657],[427,3576,851],{"className":3577},[448,449,657],[427,3579,897],{"className":3580},[896],[427,3582,3584],{"className":3583},[637],[427,3585,3587],{"className":3586,"style":950},[641],[427,3588],{},[427,3590],{"className":3591,"style":455},[454],[427,3593,3595],{"className":3594},[459],"≤",[427,3597],{"className":3598,"style":455},[454],[427,3600,3602,3606],{"className":3601},[439],[427,3603],{"className":3604,"style":3605},[443],"height:0.7167em;vertical-align:-0.2861em;",[427,3607,3609,3612],{"className":3608},[448],[427,3610,517],{"className":3611},[448,449],[427,3613,3615],{"className":3614},[629],[427,3616,3618,3638],{"className":3617},[633,828],[427,3619,3621,3635],{"className":3620},[637],[427,3622,3624],{"className":3623,"style":928},[641],[427,3625,3626,3629],{"style":931},[427,3627],{"className":3628,"style":650},[649],[427,3630,3632],{"className":3631},[654,655,656,657],[427,3633,1462],{"className":3634,"style":2362},[448,449,657],[427,3636,897],{"className":3637},[896],[427,3639,3641],{"className":3640},[637],[427,3642,3645],{"className":3643,"style":3644},[641],"height:0.2861em;",[427,3646],{},", hence",[427,3649,3651],{"className":3650},[3257],[427,3652,3654],{"className":3653},[430],[427,3655,3657,3717,3773,3829,3884],{"className":3656,"ariaHidden":435},[434],[427,3658,3660,3664,3670,3673,3705,3708,3711,3714],{"className":3659},[439],[427,3661],{"className":3662,"style":3663},[443],"height:1.0519em;vertical-align:-0.25em;",[427,3665,3667],{"className":3666},[448,781],[427,3668,785],{"className":3669},[448],[427,3671,475],{"className":3672},[474],[427,3674,3676,3679],{"className":3675},[448],[427,3677,701],{"className":3678,"style":700},[448,449],[427,3680,3682],{"className":3681},[629],[427,3683,3685],{"className":3684},[633],[427,3686,3688],{"className":3687},[637],[427,3689,3691],{"className":3690,"style":3270},[641],[427,3692,3693,3696],{"style":3291},[427,3694],{"className":3695,"style":650},[649],[427,3697,3699],{"className":3698},[654,655,656,657],[427,3700,3702],{"className":3701},[448,657],[427,3703,664],{"className":3704},[448,657],[427,3706,499],{"className":3707},[498],[427,3709],{"className":3710,"style":455},[454],[427,3712,460],{"className":3713},[459],[427,3715],{"className":3716,"style":455},[454],[427,3718,3720,3723,3729,3732,3761,3764,3767,3770],{"className":3719},[439],[427,3721],{"className":3722,"style":470},[443],[427,3724,3726],{"className":3725},[448,781],[427,3727,785],{"className":3728},[448],[427,3730,475],{"className":3731},[474],[427,3733,3735,3738],{"className":3734},[448],[427,3736,701],{"className":3737,"style":700},[448,449],[427,3739,3741],{"className":3740},[629],[427,3742,3744],{"className":3743},[633],[427,3745,3747],{"className":3746},[637],[427,3748,3750],{"className":3749,"style":3338},[641],[427,3751,3752,3755],{"style":3291},[427,3753],{"className":3754,"style":650},[649],[427,3756,3758],{"className":3757},[654,655,656,657],[427,3759,2583],{"className":3760},[1028,657],[427,3762,499],{"className":3763},[498],[427,3765],{"className":3766,"style":479},[454],[427,3768,1029],{"className":3769},[1028],[427,3771],{"className":3772,"style":479},[454],[427,3774,3776,3780,3820,3823,3826],{"className":3775},[439],[427,3777],{"className":3778,"style":3779},[443],"height:0.8694em;vertical-align:-0.2861em;",[427,3781,3783,3786],{"className":3782},[448],[427,3784,517],{"className":3785},[448,449],[427,3787,3789],{"className":3788},[629],[427,3790,3792,3812],{"className":3791},[633,828],[427,3793,3795,3809],{"className":3794},[637],[427,3796,3798],{"className":3797,"style":928},[641],[427,3799,3800,3803],{"style":931},[427,3801],{"className":3802,"style":650},[649],[427,3804,3806],{"className":3805},[654,655,656,657],[427,3807,1462],{"className":3808,"style":2362},[448,449,657],[427,3810,897],{"className":3811},[896],[427,3813,3815],{"className":3814},[637],[427,3816,3818],{"className":3817,"style":3644},[641],[427,3819],{},[427,3821],{"className":3822,"style":479},[454],[427,3824,1737],{"className":3825},[1028],[427,3827],{"className":3828,"style":479},[454],[427,3830,3832,3835,3875,3878,3881],{"className":3831},[439],[427,3833],{"className":3834,"style":3548},[443],[427,3836,3838,3841],{"className":3837},[448],[427,3839,517],{"className":3840},[448,449],[427,3842,3844],{"className":3843},[629],[427,3845,3847,3867],{"className":3846},[633,828],[427,3848,3850,3864],{"className":3849},[637],[427,3851,3853],{"className":3852,"style":928},[641],[427,3854,3855,3858],{"style":931},[427,3856],{"className":3857,"style":650},[649],[427,3859,3861],{"className":3860},[654,655,656,657],[427,3862,851],{"className":3863},[448,449,657],[427,3865,897],{"className":3866},[896],[427,3868,3870],{"className":3869},[637],[427,3871,3873],{"className":3872,"style":950},[641],[427,3874],{},[427,3876],{"className":3877,"style":455},[454],[427,3879,3595],{"className":3880},[459],[427,3882],{"className":3883,"style":455},[454],[427,3885,3887,3890,3896,3899,3928,3931],{"className":3886},[439],[427,3888],{"className":3889,"style":470},[443],[427,3891,3893],{"className":3892},[448,781],[427,3894,785],{"className":3895},[448],[427,3897,475],{"className":3898},[474],[427,3900,3902,3905],{"className":3901},[448],[427,3903,701],{"className":3904,"style":700},[448,449],[427,3906,3908],{"className":3907},[629],[427,3909,3911],{"className":3910},[633],[427,3912,3914],{"className":3913},[637],[427,3915,3917],{"className":3916,"style":3338},[641],[427,3918,3919,3922],{"style":3291},[427,3920],{"className":3921,"style":650},[649],[427,3923,3925],{"className":3924},[654,655,656,657],[427,3926,2583],{"className":3927},[1028,657],[427,3929,499],{"className":3930},[498],[427,3932,1363],{"className":3933},[448],[381,3935,3936,3937,3978,3979,4023,4024,4027,4028,4043,4044,4086,4087,4102,4103],{},"But ",[427,3938,3940],{"className":3939},[430],[427,3941,3943],{"className":3942,"ariaHidden":435},[434],[427,3944,3946,3949],{"className":3945},[439],[427,3947],{"className":3948,"style":2553},[443],[427,3950,3952,3955],{"className":3951},[448],[427,3953,701],{"className":3954,"style":700},[448,449],[427,3956,3958],{"className":3957},[629],[427,3959,3961],{"className":3960},[633],[427,3962,3964],{"className":3963},[637],[427,3965,3967],{"className":3966,"style":2553},[641],[427,3968,3969,3972],{"style":645},[427,3970],{"className":3971,"style":650},[649],[427,3973,3975],{"className":3974},[654,655,656,657],[427,3976,2583],{"className":3977},[1028,657]," was minimum, so equality holds: ",[427,3980,3982],{"className":3981},[430],[427,3983,3985],{"className":3984,"ariaHidden":435},[434],[427,3986,3988,3991],{"className":3987},[439],[427,3989],{"className":3990,"style":642},[443],[427,3992,3994,3997],{"className":3993},[448],[427,3995,701],{"className":3996,"style":700},[448,449],[427,3998,4000],{"className":3999},[629],[427,4001,4003],{"className":4002},[633],[427,4004,4006],{"className":4005},[637],[427,4007,4009],{"className":4008,"style":642},[641],[427,4010,4011,4014],{"style":645},[427,4012],{"className":4013,"style":650},[649],[427,4015,4017],{"className":4016},[654,655,656,657],[427,4018,4020],{"className":4019},[448,657],[427,4021,664],{"className":4022},[448,657]," is ",[401,4025,4026],{},"also"," an MST, and it contains\n",[427,4029,4031],{"className":4030},[430],[427,4032,4034],{"className":4033,"ariaHidden":435},[434],[427,4035,4037,4040],{"className":4036},[439],[427,4038],{"className":4039,"style":513},[443],[427,4041,851],{"className":4042},[448,449],". Contradiction. Therefore some MST contains ",[427,4045,4047],{"className":4046},[430],[427,4048,4050,4068],{"className":4049,"ariaHidden":435},[434],[427,4051,4053,4056,4059,4062,4065],{"className":4052},[439],[427,4054],{"className":4055,"style":444},[443],[427,4057,1904],{"className":4058},[448,449],[427,4060],{"className":4061,"style":479},[454],[427,4063,2508],{"className":4064},[1028],[427,4066],{"className":4067,"style":479},[454],[427,4069,4071,4074],{"className":4070},[439],[427,4072],{"className":4073,"style":470},[443],[427,4075,4077,4080,4083],{"className":4076},[2521],[427,4078,2525],{"className":4079,"style":1194},[474,1193],[427,4081,851],{"className":4082},[448,449],[427,4084,2532],{"className":4085,"style":1194},[498,1193],", i.e. ",[427,4088,4090],{"className":4089},[430],[427,4091,4093],{"className":4092,"ariaHidden":435},[434],[427,4094,4096,4099],{"className":4095},[439],[427,4097],{"className":4098,"style":513},[443],[427,4100,851],{"className":4101},[448,449]," is\nsafe. 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exchange argument: adding edge ",[427,4220,4222],{"className":4221},[430],[427,4223,4225],{"className":4224,"ariaHidden":435},[434],[427,4226,4228,4231],{"className":4227},[439],[427,4229],{"className":4230,"style":513},[443],[427,4232,851],{"className":4233},[448,449]," creates a cycle whose crossing edge ",[427,4236,4238],{"className":4237},[430],[427,4239,4241],{"className":4240,"ariaHidden":435},[434],[427,4242,4244,4247],{"className":4243},[439],[427,4245],{"className":4246,"style":3108},[443],[427,4248,1462],{"className":4249,"style":2362},[448,449]," is swapped out.",[381,4252,4253,4254,4257,4258],{},"The cut property is the engine. Every correct MST algorithm, whether Borůvka's,\nPrim's, or Kruskal's, is just a strategy for ",[401,4255,4256],{},"choosing which cut to apply it\nto",", and ",[385,4259,4260],{},"all three only ever add edges that obey the cut rule.",[405,4262,4264],{"id":4263},"borůvkas-algorithm","Borůvka's algorithm",[381,4266,4267,4268,4275,4276,4279],{},"The oldest MST algorithm (Otakar Borůvka, 1926) is also the most directly cut-rule\ndriven, and it parallelises well.",[1954,4269,4270],{},[398,4271,1228],{"href":4272,"ariaDescribedBy":4273,"dataFootnoteRef":376,"id":4274},"#user-content-fn-erickson-mst",[1960],"user-content-fnref-erickson-mst"," The idea: every component, in every\nround, ",[401,4277,4278],{},"simultaneously"," reaches out along its own cheapest outgoing edge.",[381,4281,4282,4283,4298,4299,4314,4315,4318,4319,4336,4337,4385,4386,4401,4402,4431,4432,4470],{},"Maintain a forest ",[427,4284,4286],{"className":4285},[430],[427,4287,4289],{"className":4288,"ariaHidden":435},[434],[427,4290,4292,4295],{"className":4291},[439],[427,4293],{"className":4294,"style":444},[443],[427,4296,1904],{"className":4297},[448,449],", initially the ",[427,4300,4302],{"className":4301},[430],[427,4303,4305],{"className":4304,"ariaHidden":435},[434],[427,4306,4308,4311],{"className":4307},[439],[427,4309],{"className":4310,"style":513},[443],[427,4312,979],{"className":4313},[448,449]," isolated vertices. In each ",[385,4316,4317],{},"round",",\neach current component ",[427,4320,4322],{"className":4321},[430],[427,4323,4325],{"className":4324,"ariaHidden":435},[434],[427,4326,4328,4331],{"className":4327},[439],[427,4329],{"className":4330,"style":444},[443],[427,4332,4335],{"className":4333,"style":4334},[448,449],"margin-right:0.0715em;","C"," looks at the cut ",[427,4338,4340],{"className":4339},[430],[427,4341,4343,4373],{"className":4342,"ariaHidden":435},[434],[427,4344,4346,4349,4352,4355,4358,4361,4364,4367,4370],{"className":4345},[439],[427,4347],{"className":4348,"style":470},[443],[427,4350,475],{"className":4351},[474],[427,4353,4335],{"className":4354,"style":4334},[448,449],[427,4356,485],{"className":4357},[484],[427,4359],{"className":4360,"style":489},[454],[427,4362,480],{"className":4363,"style":479},[448,449],[427,4365],{"className":4366,"style":479},[454],[427,4368,1268],{"className":4369},[1028],[427,4371],{"className":4372,"style":479},[454],[427,4374,4376,4379,4382],{"className":4375},[439],[427,4377],{"className":4378,"style":470},[443],[427,4380,4335],{"className":4381,"style":4334},[448,449],[427,4383,499],{"className":4384},[498],", which respects\n",[427,4387,4389],{"className":4388},[430],[427,4390,4392],{"className":4391,"ariaHidden":435},[434],[427,4393,4395,4398],{"className":4394},[439],[427,4396],{"className":4397,"style":444},[443],[427,4399,1904],{"className":4400},[448,449],", and selects its lightest crossing edge. By the cut property every such edge\nis safe, so we add them all at once and merge the components they join. Because\nevery component at least halves in count each round (each one merges with at\nleast one neighbor), the number of components drops by a factor ",[427,4403,4405],{"className":4404},[430],[427,4406,4408,4422],{"className":4407,"ariaHidden":435},[434],[427,4409,4411,4415,4419],{"className":4410},[439],[427,4412],{"className":4413,"style":4414},[443],"height:0.7719em;vertical-align:-0.136em;",[427,4416,4418],{"className":4417},[459],"≥",[427,4420],{"className":4421,"style":455},[454],[427,4423,4425,4428],{"className":4424},[439],[427,4426],{"className":4427,"style":1039},[443],[427,4429,1228],{"className":4430},[448]," per\nround, so only ",[427,4433,4435],{"className":4434},[430],[427,4436,4438],{"className":4437,"ariaHidden":435},[434],[427,4439,4441,4444,4449,4452,4461,4464,4467],{"className":4440},[439],[427,4442],{"className":4443,"style":470},[443],[427,4445,4448],{"className":4446,"style":4447},[448,449],"margin-right:0.0278em;","O",[427,4450,475],{"className":4451},[474],[427,4453,4455],{"className":4454},[814],[427,4456,4460],{"className":4457,"style":4459},[448,4458],"mathrm","margin-right:0.0139em;","log",[427,4462],{"className":4463,"style":489},[454],[427,4465,480],{"className":4466,"style":479},[448,449],[427,4468,499],{"className":4469},[498]," rounds are needed.",[4472,4473,4477],"pre",{"className":4474,"code":4475,"language":4476,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Borůvka}(G, c)$ — every component grabs its cheapest exit edge\nnumber: 1\n$A \\gets \\emptyset$ \u002F\u002F $n$ singleton components\nwhile $A$ has more than one component do\n  foreach component $C$ of $(V, A)$ do\n    $e_C \\gets$ the lightest edge crossing $(C,\\, V \\setminus C)$ \u002F\u002F safe by cut rule\n  foreach distinct edge $e_C$ chosen do\n    $A \\gets A \\cup \\set{e_C}$ \u002F\u002F add all exit edges at once\nreturn $A$\n","algorithm",[4478,4479,4480,4486,4491,4496,4501,4506,4511,4516,4521],"code",{"__ignoreMap":376},[427,4481,4483],{"class":4482,"line":6},"line",[427,4484,4485],{},"caption: $\\textsc{Borůvka}(G, c)$ — every component grabs its cheapest exit edge\n",[427,4487,4488],{"class":4482,"line":18},[427,4489,4490],{},"number: 1\n",[427,4492,4493],{"class":4482,"line":24},[427,4494,4495],{},"$A \\gets \\emptyset$ \u002F\u002F $n$ singleton components\n",[427,4497,4498],{"class":4482,"line":73},[427,4499,4500],{},"while $A$ has more than one component do\n",[427,4502,4503],{"class":4482,"line":102},[427,4504,4505],{},"  foreach component $C$ of $(V, A)$ do\n",[427,4507,4508],{"class":4482,"line":108},[427,4509,4510],{},"    $e_C \\gets$ the lightest edge crossing $(C,\\, V \\setminus C)$ \u002F\u002F safe by cut rule\n",[427,4512,4513],{"class":4482,"line":116},[427,4514,4515],{},"  foreach distinct edge $e_C$ chosen do\n",[427,4517,4518],{"class":4482,"line":196},[427,4519,4520],{},"    $A \\gets A \\cup \\set{e_C}$ \u002F\u002F add all exit edges at once\n",[427,4522,4523],{"class":4482,"line":202},[427,4524,4525],{},"return $A$\n",[413,4527,4529],{"type":4528},"remark",[381,4530,4531,4534,4535,4538],{},[385,4532,4533],{},"Remark (A subtlety)."," To keep the result acyclic when weights tie, break ties by a\nfixed total order on edges (e.g. by index). Otherwise two components could pick\n",[401,4536,4537],{},"each other's"," edge as different edges of equal weight and create a cycle.",[381,4540,4541,4544,4545,4569,4570,4603,4604,4670,4671,4710],{},[385,4542,4543],{},"Running time."," Each round scans all edges to find component-minimum exits in\n",[427,4546,4548],{"className":4547},[430],[427,4549,4551],{"className":4550,"ariaHidden":435},[434],[427,4552,4554,4557,4560,4563,4566],{"className":4553},[439],[427,4555],{"className":4556,"style":470},[443],[427,4558,4448],{"className":4559,"style":4447},[448,449],[427,4561,475],{"className":4562},[474],[427,4564,494],{"className":4565,"style":493},[448,449],[427,4567,499],{"className":4568},[498]," time and there are ",[427,4571,4573],{"className":4572},[430],[427,4574,4576],{"className":4575,"ariaHidden":435},[434],[427,4577,4579,4582,4585,4588,4594,4597,4600],{"className":4578},[439],[427,4580],{"className":4581,"style":470},[443],[427,4583,4448],{"className":4584,"style":4447},[448,449],[427,4586,475],{"className":4587},[474],[427,4589,4591],{"className":4590},[814],[427,4592,4460],{"className":4593,"style":4459},[448,4458],[427,4595],{"className":4596,"style":489},[454],[427,4598,480],{"className":4599,"style":479},[448,449],[427,4601,499],{"className":4602},[498]," rounds, so ",[427,4605,4607],{"className":4606},[430],[427,4608,4610],{"className":4609,"ariaHidden":435},[434],[427,4611,4613,4617],{"className":4612},[439],[427,4614],{"className":4615,"style":4616},[443],"height:0.6944em;",[427,4618,4622],{"className":4619},[4620,4621],"enclosing","textsc",[427,4623,4625,4629,4666],{"className":4624},[448,781],[427,4626,4628],{"className":4627},[448],"Bor",[427,4630,4633],{"className":4631},[448,4632],"accent",[427,4634,4636],{"className":4635},[633],[427,4637,4639],{"className":4638},[637],[427,4640,4642,4652],{"className":4641,"style":4616},[641],[427,4643,4645,4649],{"style":4644},"top:-3em;",[427,4646],{"className":4647,"style":4648},[649],"height:3em;",[427,4650,2352],{"className":4651},[448],[427,4653,4654,4657],{"style":4644},[427,4655],{"className":4656,"style":4648},[649],[427,4658,4662],{"className":4659,"style":4661},[4660],"accent-body","left:-0.375em;",[427,4663,4665],{"className":4664},[448],"˚",[427,4667,4669],{"className":4668},[448],"vka"," runs in\n",[427,4672,4674],{"className":4673},[430],[427,4675,4677],{"className":4676,"ariaHidden":435},[434],[427,4678,4680,4683,4686,4689,4692,4695,4701,4704,4707],{"className":4679},[439],[427,4681],{"className":4682,"style":470},[443],[427,4684,4448],{"className":4685,"style":4447},[448,449],[427,4687,475],{"className":4688},[474],[427,4690,494],{"className":4691,"style":493},[448,449],[427,4693],{"className":4694,"style":489},[454],[427,4696,4698],{"className":4697},[814],[427,4699,4460],{"className":4700,"style":4459},[448,4458],[427,4702],{"className":4703,"style":489},[454],[427,4705,480],{"className":4706,"style":479},[448,449],[427,4708,499],{"className":4709},[498],", the same headline bound as Prim and Kruskal, but with the\nuseful property that the per-round work is fully parallel.",[381,4712,4713],{},"One round on six isolated vertices shows the parallel grab. Each singleton (a\ncomponent of one) points an arrow along its own cheapest incident edge; the chosen\nedges, taken together, merge the six components down to two in a single sweep:",[1449,4715,4717,5019],{"className":4716},[1452,1453],[1455,4718,4722],{"xmlns":1457,"width":4719,"height":4720,"viewBox":4721},"312.871","91.994","-75 -75 234.654 68.996",[1462,4723,4724,4727,4734,4737,4744,4747,4754,4757,4764,4767,4774,4777,4784,4802,4819,4836,4853,4870,4887,4904,4914,4922,4930,4938,4946,4954],{"stroke":1464,"style":1465},[1467,4725],{"fill":1469,"d":4726},"M-42.641-60.689c0-6.286-5.096-11.381-11.381-11.381s-11.381 5.095-11.381 11.381 5.095 11.381 11.38 11.381 11.382-5.095 11.382-11.38Zm-11.381 0",[1462,4728,4730],{"transform":4729},"translate(-2.45 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Borůvka round: every component (here singletons) selects its cheapest exit edge (red arrows). The five chosen edges merge six components into two.",[405,5024,5026],{"id":5025},"kruskals-algorithm","Kruskal's algorithm",[381,5028,5029,5051,5052,5055,5056,5071,5072,5075,5076,5091],{},[427,5030,5032],{"className":5031},[430],[427,5033,5035],{"className":5034,"ariaHidden":435},[434],[427,5036,5038,5041],{"className":5037},[439],[427,5039],{"className":5040,"style":4616},[443],[427,5042,5044],{"className":5043},[4620,4621],[427,5045,5047],{"className":5046},[448,781],[427,5048,5050],{"className":5049},[448],"Kruskal","'s strategy is global and edge-centric: consider the edges in order\nof increasing weight, and add each one ",[401,5053,5054],{},"unless"," it would form a cycle. The set\n",[427,5057,5059],{"className":5058},[430],[427,5060,5062],{"className":5061,"ariaHidden":435},[434],[427,5063,5065,5068],{"className":5064},[439],[427,5066],{"className":5067,"style":444},[443],[427,5069,1904],{"className":5070},[448,449]," is a ",[385,5073,5074],{},"forest"," (an acyclic graph) that gradually coalesces into a single\ntree once ",[427,5077,5079],{"className":5078},[430],[427,5080,5082],{"className":5081,"ariaHidden":435},[434],[427,5083,5085,5088],{"className":5084},[439],[427,5086],{"className":5087,"style":444},[443],[427,5089,450],{"className":5090},[448,449]," is fully connected.",[381,5093,5094,5095,5125,5126,5141,5142,5157,5158,5173,5174,3532,5204,5207,5208,5238],{},"Why is each added edge safe? When Kruskal accepts the cheapest remaining edge\n",[427,5096,5098],{"className":5097},[430],[427,5099,5101],{"className":5100,"ariaHidden":435},[434],[427,5102,5104,5107,5110,5113,5116,5119,5122],{"className":5103},[439],[427,5105],{"className":5106,"style":470},[443],[427,5108,475],{"className":5109},[474],[427,5111,2352],{"className":5112},[448,449],[427,5114,485],{"className":5115},[484],[427,5117],{"className":5118,"style":489},[454],[427,5120,2363],{"className":5121,"style":2362},[448,449],[427,5123,499],{"className":5124},[498],", it connects two different trees of the current forest. Take ",[427,5127,5129],{"className":5128},[430],[427,5130,5132],{"className":5131,"ariaHidden":435},[434],[427,5133,5135,5138],{"className":5134},[439],[427,5136],{"className":5137,"style":444},[443],[427,5139,1986],{"className":5140,"style":493},[448,449]," to be\nthe vertices of ",[427,5143,5145],{"className":5144},[430],[427,5146,5148],{"className":5147,"ariaHidden":435},[434],[427,5149,5151,5154],{"className":5150},[439],[427,5152],{"className":5153,"style":513},[443],[427,5155,2352],{"className":5156},[448,449],"'s tree. This cut respects ",[427,5159,5161],{"className":5160},[430],[427,5162,5164],{"className":5163,"ariaHidden":435},[434],[427,5165,5167,5170],{"className":5166},[439],[427,5168],{"className":5169,"style":444},[443],[427,5171,1904],{"className":5172},[448,449]," (no forest edge crosses out of\na tree), and ",[427,5175,5177],{"className":5176},[430],[427,5178,5180],{"className":5179,"ariaHidden":435},[434],[427,5181,5183,5186,5189,5192,5195,5198,5201],{"className":5182},[439],[427,5184],{"className":5185,"style":470},[443],[427,5187,475],{"className":5188},[474],[427,5190,2352],{"className":5191},[448,449],[427,5193,485],{"className":5194},[484],[427,5196],{"className":5197,"style":489},[454],[427,5199,2363],{"className":5200,"style":2362},[448,449],[427,5202,499],{"className":5203},[498],[401,5205,5206],{},"lightest"," edge crossing it; any lighter crossing\nedge would have been considered earlier and would have joined the trees already.\nBy the cut property, ",[427,5209,5211],{"className":5210},[430],[427,5212,5214],{"className":5213,"ariaHidden":435},[434],[427,5215,5217,5220,5223,5226,5229,5232,5235],{"className":5216},[439],[427,5218],{"className":5219,"style":470},[443],[427,5221,475],{"className":5222},[474],[427,5224,2352],{"className":5225},[448,449],[427,5227,485],{"className":5228},[484],[427,5230],{"className":5231,"style":489},[454],[427,5233,2363],{"className":5234,"style":2362},[448,449],[427,5236,499],{"className":5237},[498]," is safe.",[381,5240,5241,5242,5277,5278,5377,5378,5408,5409,5415,5416,2402,5438,5460,5461,5489,5490,5505,5506,5528],{},"The cycle test, ",[1052,5243,5244,5245,5260,5261,5276],{},"are ",[427,5246,5248],{"className":5247},[430],[427,5249,5251],{"className":5250,"ariaHidden":435},[434],[427,5252,5254,5257],{"className":5253},[439],[427,5255],{"className":5256,"style":513},[443],[427,5258,2352],{"className":5259},[448,449]," and ",[427,5262,5264],{"className":5263},[430],[427,5265,5267],{"className":5266,"ariaHidden":435},[434],[427,5268,5270,5273],{"className":5269},[439],[427,5271],{"className":5272,"style":513},[443],[427,5274,2363],{"className":5275,"style":2362},[448,449]," already in the same tree?",", is exactly the\nquestion ",[427,5279,5281],{"className":5280},[430],[427,5282,5284,5356],{"className":5283,"ariaHidden":435},[434],[427,5285,5287,5291,5298,5301,5304,5307,5310,5353],{"className":5286},[439],[427,5288],{"className":5289,"style":5290},[443],"height:1.403em;vertical-align:-0.25em;",[427,5292,5294],{"className":5293},[448,781],[427,5295,5297],{"className":5296},[448],"comp",[427,5299,475],{"className":5300},[474],[427,5302,2352],{"className":5303},[448,449],[427,5305,499],{"className":5306},[498],[427,5308],{"className":5309,"style":455},[454],[427,5311,5313],{"className":5312},[459],[427,5314,5317],{"className":5315},[814,5316],"op-limits",[427,5318,5320],{"className":5319},[633],[427,5321,5323],{"className":5322},[637],[427,5324,5327,5337],{"className":5325,"style":5326},[641],"height:1.153em;",[427,5328,5329,5332],{"style":4644},[427,5330],{"className":5331,"style":4648},[649],[427,5333,5334],{},[427,5335,460],{"className":5336},[814],[427,5338,5340,5343],{"style":5339},"top:-3.5669em;margin-left:0em;",[427,5341],{"className":5342,"style":4648},[649],[427,5344,5346],{"className":5345},[654,655,656,657],[427,5347,5349],{"className":5348},[448,657],[427,5350,5352],{"className":5351},[498,657],"?",[427,5354],{"className":5355,"style":455},[454],[427,5357,5359,5362,5368,5371,5374],{"className":5358},[439],[427,5360],{"className":5361,"style":470},[443],[427,5363,5365],{"className":5364},[448,781],[427,5366,5297],{"className":5367},[448],[427,5369,475],{"className":5370},[474],[427,5372,2363],{"className":5373,"style":2362},[448,449],[427,5375,499],{"className":5376},[498],": does adding ",[427,5379,5381],{"className":5380},[430],[427,5382,5384],{"className":5383,"ariaHidden":435},[434],[427,5385,5387,5390,5393,5396,5399,5402,5405],{"className":5386},[439],[427,5388],{"className":5389,"style":470},[443],[427,5391,475],{"className":5392},[474],[427,5394,2352],{"className":5395},[448,449],[427,5397,485],{"className":5398},[484],[427,5400],{"className":5401,"style":489},[454],[427,5403,2363],{"className":5404,"style":2362},[448,449],[427,5406,499],{"className":5407},[498],"\nkeep the graph acyclic? This is what the\n",[398,5410,5411,5414],{"href":107},[385,5412,5413],{},"disjoint-set"," (union-find)"," data\nstructure answers efficiently. It supports ",[427,5417,5419],{"className":5418},[430],[427,5420,5422],{"className":5421,"ariaHidden":435},[434],[427,5423,5425,5428],{"className":5424},[439],[427,5426],{"className":5427,"style":4616},[443],[427,5429,5431],{"className":5430},[4620,4621],[427,5432,5434],{"className":5433},[448,781],[427,5435,5437],{"className":5436},[448],"Make-Set",[427,5439,5441],{"className":5440},[430],[427,5442,5444],{"className":5443,"ariaHidden":435},[434],[427,5445,5447,5450],{"className":5446},[439],[427,5448],{"className":5449,"style":4616},[443],[427,5451,5453],{"className":5452},[4620,4621],[427,5454,5456],{"className":5455},[448,781],[427,5457,5459],{"className":5458},[448],"Find","\n(",[427,5462,5464],{"className":5463},[430],[427,5465,5467],{"className":5466,"ariaHidden":435},[434],[427,5468,5470,5473,5479,5482,5486],{"className":5469},[439],[427,5471],{"className":5472,"style":470},[443],[427,5474,5476],{"className":5475},[448,781],[427,5477,5297],{"className":5478},[448],[427,5480,475],{"className":5481},[474],[427,5483,5485],{"className":5484},[448,449],"x",[427,5487,499],{"className":5488},[498],", which component is ",[427,5491,5493],{"className":5492},[430],[427,5494,5496],{"className":5495,"ariaHidden":435},[434],[427,5497,5499,5502],{"className":5498},[439],[427,5500],{"className":5501,"style":513},[443],[427,5503,5485],{"className":5504},[448,449]," in?), and ",[427,5507,5509],{"className":5508},[430],[427,5510,5512],{"className":5511,"ariaHidden":435},[434],[427,5513,5515,5518],{"className":5514},[439],[427,5516],{"className":5517,"style":444},[443],[427,5519,5521],{"className":5520},[4620,4621],[427,5522,5524],{"className":5523},[448,781],[427,5525,5527],{"className":5526},[448],"Union"," (merge two\ncomponents).",[4472,5530,5532],{"className":4474,"code":5531,"language":4476,"meta":376,"style":376},"caption: $\\textsc{Kruskal}(G, c)$ — grow a forest, cheapest safe edge first\nnumber: 2\n$A \\gets \\emptyset$\nforeach vertex $v \\in V$ do\n  call $\\textsc{Make-Set}(v)$\nsort the edges of $E$ into nondecreasing order by weight $c$\nforeach edge $(u, v) \\in E$ in sorted order do\n  if $\\textsc{Find}(u) \\neq \\textsc{Find}(v)$ then \u002F\u002F stays acyclic\n    $A \\gets A \\cup \\set{(u, v)}$ \u002F\u002F safe edge\n    call $\\textsc{Union}(u, v)$\nreturn $A$\n",[4478,5533,5534,5539,5544,5549,5554,5559,5564,5569,5574,5579,5584],{"__ignoreMap":376},[427,5535,5536],{"class":4482,"line":6},[427,5537,5538],{},"caption: $\\textsc{Kruskal}(G, c)$ — grow a forest, cheapest safe edge first\n",[427,5540,5541],{"class":4482,"line":18},[427,5542,5543],{},"number: 2\n",[427,5545,5546],{"class":4482,"line":24},[427,5547,5548],{},"$A \\gets \\emptyset$\n",[427,5550,5551],{"class":4482,"line":73},[427,5552,5553],{},"foreach vertex $v \\in V$ do\n",[427,5555,5556],{"class":4482,"line":102},[427,5557,5558],{},"  call $\\textsc{Make-Set}(v)$\n",[427,5560,5561],{"class":4482,"line":108},[427,5562,5563],{},"sort the edges of $E$ into nondecreasing order by weight $c$\n",[427,5565,5566],{"class":4482,"line":116},[427,5567,5568],{},"foreach edge $(u, v) \\in E$ in sorted order do\n",[427,5570,5571],{"class":4482,"line":196},[427,5572,5573],{},"  if $\\textsc{Find}(u) \\neq \\textsc{Find}(v)$ then \u002F\u002F stays acyclic\n",[427,5575,5576],{"class":4482,"line":202},[427,5577,5578],{},"    $A \\gets A \\cup \\set{(u, v)}$ \u002F\u002F safe edge\n",[427,5580,5581],{"class":4482,"line":283},[427,5582,5583],{},"    call $\\textsc{Union}(u, v)$\n",[427,5585,5586],{"class":4482,"line":333},[427,5587,4525],{},[381,5589,5590,5591,5612,5613,5616,5617,524],{},"Running ",[427,5592,5594],{"className":5593},[430],[427,5595,5597],{"className":5596,"ariaHidden":435},[434],[427,5598,5600,5603],{"className":5599},[439],[427,5601],{"className":5602,"style":4616},[443],[427,5604,5606],{"className":5605},[4620,4621],[427,5607,5609],{"className":5608},[448,781],[427,5610,5050],{"className":5611},[448]," on the nine-town graph above makes the accept\u002Freject\nrhythm visible. We scan the thirteen edges in nondecreasing weight; the first\neight that join ",[401,5614,5615],{},"distinct"," trees are accepted, and the five that would close a\ncycle are skipped. The eight accepted edges are exactly the MST of weight ",[427,5618,5620],{"className":5619},[430],[427,5621,5623],{"className":5622,"ariaHidden":435},[434],[427,5624,5626,5629],{"className":5625},[439],[427,5627],{"className":5628,"style":1039},[443],[427,5630,1880],{"className":5631},[448],[1449,5633,5635,6281],{"className":5634},[1452,1453],[1455,5636,5640],{"xmlns":1457,"width":5637,"height":5638,"viewBox":5639},"199.535","354.145","-75 -75 149.651 265.609",[1462,5641,5642,5666,5673,5680,5683,5690,5711,5728,5735,5756,5771,5777,5798,5813,5820,5841,5856,5863,5884,5899,5906,5927,5942,5948,5967,5982,5989,6010,6025,6031,6050,6067,6074,6093,6108,6115,6136,6151,6158,6177,6192,6199,6218,6233],{"stroke":1464,"style":1465},[1462,5643,5646,5654,5660],{"stroke":1469,"fontFamily":5644,"fontSize":5645},"cmbx8","8",[1462,5647,5649],{"transform":5648},"translate(-13.95 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16.562-48.632Q16.680-48.755 16.732-48.935Q16.785-49.115 16.785-49.287L16.785-49.759L17.066-49.759L17.066-49.271Q17.066-49.017 16.961-48.777Q16.855-48.537 16.658-48.384Q16.461-48.232 16.203-48.232Q15.887-48.232 15.635-48.355Q15.383-48.478 15.244-48.712Q15.105-48.947 15.105-49.271",[1478],[1462,6270,6271],{"transform":6240},[1467,6272],{"d":6273,"fill":1613,"stroke":1613,"className":6274,"style":5653},"M26.342-49.287L21.029-49.287Q20.951-49.294 20.902-49.343Q20.854-49.392 20.854-49.470Q20.854-49.540 20.901-49.591Q20.947-49.642 21.029-49.654L26.342-49.654Q26.416-49.642 26.463-49.591Q26.510-49.540 26.510-49.470Q26.510-49.392 26.461-49.343Q26.412-49.294 26.342-49.287M26.342-50.974L21.029-50.974Q20.951-50.982 20.902-51.031Q20.854-51.080 20.854-51.158Q20.854-51.228 20.901-51.279Q20.947-51.330 21.029-51.341L26.342-51.341Q26.416-51.330 26.463-51.279Q26.510-51.228 26.510-51.158Q26.510-51.080 26.461-51.031Q26.412-50.982 26.342-50.974",[1478],[1462,6276,6277],{"transform":6240},[1467,6278],{"d":6279,"fill":1613,"stroke":1613,"className":6280,"style":5653},"M30.146-48.943Q30.337-48.669 30.693-48.542Q31.048-48.415 31.431-48.415Q31.767-48.415 31.976-48.601Q32.185-48.787 32.281-49.080Q32.376-49.373 32.376-49.685Q32.376-50.009 32.279-50.304Q32.181-50.599 31.968-50.783Q31.755-50.966 31.423-50.966L30.857-50.966Q30.826-50.966 30.796-50.996Q30.767-51.025 30.767-51.052L30.767-51.134Q30.767-51.169 30.796-51.195Q30.826-51.220 30.857-51.220L31.337-51.255Q31.623-51.255 31.820-51.460Q32.017-51.665 32.113-51.960Q32.208-52.255 32.208-52.533Q32.208-52.912 32.009-53.150Q31.810-53.388 31.431-53.388Q31.111-53.388 30.822-53.281Q30.533-53.173 30.369-52.951Q30.548-52.951 30.671-52.824Q30.794-52.697 30.794-52.525Q30.794-52.353 30.669-52.228Q30.544-52.103 30.369-52.103Q30.197-52.103 30.072-52.228Q29.947-52.353 29.947-52.525Q29.947-52.892 30.171-53.140Q30.396-53.388 30.736-53.509Q31.076-53.630 31.431-53.630Q31.779-53.630 32.142-53.509Q32.505-53.388 32.753-53.138Q33.001-52.888 33.001-52.533Q33.001-52.048 32.683-51.665Q32.365-51.283 31.888-51.111Q32.439-51.001 32.839-50.615Q33.240-50.228 33.240-49.693Q33.240-49.236 32.976-48.880Q32.712-48.525 32.291-48.333Q31.869-48.142 31.431-48.142Q31.021-48.142 30.628-48.277Q30.236-48.412 29.970-48.697Q29.705-48.982 29.705-49.400Q29.705-49.595 29.837-49.724Q29.970-49.853 30.162-49.853Q30.287-49.853 30.390-49.794Q30.494-49.736 30.556-49.630Q30.619-49.525 30.619-49.400Q30.619-49.205 30.484-49.074Q30.349-48.943 30.146-48.943M33.951-49.533Q33.951-50.029 34.277-50.394Q34.603-50.759 35.126-51.005L34.857-51.165Q34.560-51.349 34.376-51.644Q34.193-51.939 34.193-52.279Q34.193-52.673 34.412-52.984Q34.630-53.294 34.984-53.462Q35.337-53.630 35.720-53.630Q35.994-53.630 36.267-53.552Q36.541-53.474 36.757-53.326Q36.974-53.177 37.111-52.951Q37.248-52.724 37.248-52.431Q37.248-52.025 36.978-51.718Q36.708-51.412 36.287-51.197L36.736-50.927Q36.955-50.790 37.123-50.597Q37.291-50.404 37.388-50.165Q37.486-49.927 37.486-49.669Q37.486-49.330 37.335-49.042Q37.185-48.755 36.939-48.558Q36.693-48.361 36.369-48.251Q36.044-48.142 35.720-48.142Q35.291-48.142 34.884-48.304Q34.478-48.466 34.214-48.785Q33.951-49.103 33.951-49.533M34.439-49.533Q34.439-49.048 34.830-48.732Q35.220-48.415 35.720-48.415Q36.013-48.415 36.312-48.527Q36.611-48.638 36.804-48.857Q36.998-49.076 36.998-49.388Q36.998-49.619 36.857-49.830Q36.716-50.040 36.509-50.158L35.400-50.837Q34.982-50.634 34.710-50.298Q34.439-49.962 34.439-49.533M35.025-51.966L36.013-51.365Q36.361-51.548 36.587-51.818Q36.814-52.087 36.814-52.431Q36.814-52.650 36.722-52.826Q36.630-53.001 36.476-53.124Q36.322-53.248 36.121-53.318Q35.919-53.388 35.720-53.388Q35.314-53.388 34.968-53.177Q34.623-52.966 34.623-52.583Q34.623-52.400 34.734-52.238Q34.845-52.076 35.025-51.966",[1478],[1707,6282,6284,6285,1363],{"className":6283},[1710],"Kruskal's edge-by-edge trace on the nine-town graph: accept an edge unless both ends already share a tree. Eight accepts total ",[427,6286,6288],{"className":6287},[430],[427,6289,6291],{"className":6290,"ariaHidden":435},[434],[427,6292,6294,6297],{"className":6293},[439],[427,6295],{"className":6296,"style":1039},[443],[427,6298,1880],{"className":6299},[448],[6301,6302,6304],"h3",{"id":6303},"making-it-fast-union-by-size","Making it fast: union by size",[381,6306,6307,6308,6350,6351,6378,6379,6394,6395,6416,6417,6441,6442,6463,6464,6488,6489,6492],{},"The simplest implementation keeps an array ",[427,6309,6311],{"className":6310},[430],[427,6312,6314],{"className":6313,"ariaHidden":435},[434],[427,6315,6317,6320,6326,6330,6333,6336,6340,6343,6346],{"className":6316},[439],[427,6318],{"className":6319,"style":470},[443],[427,6321,6323],{"className":6322},[448,781],[427,6324,5297],{"className":6325},[448],[427,6327,6329],{"className":6328},[474],"[",[427,6331,1043],{"className":6332},[448],[427,6334],{"className":6335,"style":489},[454],[427,6337,6339],{"className":6338},[2521],"…",[427,6341],{"className":6342,"style":489},[454],[427,6344,979],{"className":6345},[448,449],[427,6347,6349],{"className":6348},[498],"]","\nwhere ",[427,6352,6354],{"className":6353},[430],[427,6355,6357],{"className":6356,"ariaHidden":435},[434],[427,6358,6360,6363,6369,6372,6375],{"className":6359},[439],[427,6361],{"className":6362,"style":470},[443],[427,6364,6366],{"className":6365},[448,781],[427,6367,5297],{"className":6368},[448],[427,6370,6329],{"className":6371},[474],[427,6373,2363],{"className":6374,"style":2362},[448,449],[427,6376,6349],{"className":6377},[498]," names ",[427,6380,6382],{"className":6381},[430],[427,6383,6385],{"className":6384,"ariaHidden":435},[434],[427,6386,6388,6391],{"className":6387},[439],[427,6389],{"className":6390,"style":513},[443],[427,6392,2363],{"className":6393,"style":2362},[448,449],"'s current component. Then ",[427,6396,6398],{"className":6397},[430],[427,6399,6401],{"className":6400,"ariaHidden":435},[434],[427,6402,6404,6407],{"className":6403},[439],[427,6405],{"className":6406,"style":4616},[443],[427,6408,6410],{"className":6409},[4620,4621],[427,6411,6413],{"className":6412},[448,781],[427,6414,5459],{"className":6415},[448]," is\n",[427,6418,6420],{"className":6419},[430],[427,6421,6423],{"className":6422,"ariaHidden":435},[434],[427,6424,6426,6429,6432,6435,6438],{"className":6425},[439],[427,6427],{"className":6428,"style":470},[443],[427,6430,4448],{"className":6431,"style":4447},[448,449],[427,6433,475],{"className":6434},[474],[427,6436,1043],{"className":6437},[448],[427,6439,499],{"className":6440},[498],", but a naive ",[427,6443,6445],{"className":6444},[430],[427,6446,6448],{"className":6447,"ariaHidden":435},[434],[427,6449,6451,6454],{"className":6450},[439],[427,6452],{"className":6453,"style":444},[443],[427,6455,6457],{"className":6456},[4620,4621],[427,6458,6460],{"className":6459},[448,781],[427,6461,5527],{"className":6462},[448]," that relabels one whole side costs ",[427,6465,6467],{"className":6466},[430],[427,6468,6470],{"className":6469,"ariaHidden":435},[434],[427,6471,6473,6476,6479,6482,6485],{"className":6472},[439],[427,6474],{"className":6475,"style":470},[443],[427,6477,4448],{"className":6478,"style":4447},[448,449],[427,6480,475],{"className":6481},[474],[427,6483,979],{"className":6484},[448,449],[427,6486,499],{"className":6487},[498]," in\nthe worst case. The fix is the classic ",[385,6490,6491],{},"union-by-size"," argument:",[6494,6495,6496,6573],"ul",{},[1116,6497,6498,6499,6527,6528,6556,6557,6572],{},"Alongside ",[427,6500,6502],{"className":6501},[430],[427,6503,6505],{"className":6504,"ariaHidden":435},[434],[427,6506,6508,6511,6517,6520,6524],{"className":6507},[439],[427,6509],{"className":6510,"style":470},[443],[427,6512,6514],{"className":6513},[448,781],[427,6515,5297],{"className":6516},[448],[427,6518,6329],{"className":6519},[474],[427,6521,6523],{"className":6522},[448],"⋅",[427,6525,6349],{"className":6526},[498],", keep ",[427,6529,6531],{"className":6530},[430],[427,6532,6534],{"className":6533,"ariaHidden":435},[434],[427,6535,6537,6540,6547,6550,6553],{"className":6536},[439],[427,6538],{"className":6539,"style":470},[443],[427,6541,6543],{"className":6542},[448,781],[427,6544,6546],{"className":6545},[448],"members",[427,6548,6329],{"className":6549},[474],[427,6551,517],{"className":6552},[448,449],[427,6554,6349],{"className":6555},[498],", a linked list of the\nvertices currently in component ",[427,6558,6560],{"className":6559},[430],[427,6561,6563],{"className":6562,"ariaHidden":435},[434],[427,6564,6566,6569],{"className":6565},[439],[427,6567],{"className":6568,"style":513},[443],[427,6570,517],{"className":6571},[448,449],", so we can enumerate a component cheaply.",[1116,6574,6575,6576,6615,6616,6619,6620,6748,6749,6776,6777,6792],{},"On ",[427,6577,6579],{"className":6578},[430],[427,6580,6582],{"className":6581,"ariaHidden":435},[434],[427,6583,6585,6588,6597,6600,6603,6606,6609,6612],{"className":6584},[439],[427,6586],{"className":6587,"style":470},[443],[427,6589,6591],{"className":6590},[4620,4621],[427,6592,6594],{"className":6593},[448,781],[427,6595,5527],{"className":6596},[448],[427,6598,475],{"className":6599},[474],[427,6601,2352],{"className":6602},[448,449],[427,6604,485],{"className":6605},[484],[427,6607],{"className":6608,"style":489},[454],[427,6610,2363],{"className":6611,"style":2362},[448,449],[427,6613,499],{"className":6614},[498],", relabel the ",[385,6617,6618],{},"smaller"," component: choose the side\nwith ",[427,6621,6623],{"className":6622},[430],[427,6624,6626,6693],{"className":6625,"ariaHidden":435},[434],[427,6627,6629,6632,6635,6681,6684,6687,6690],{"className":6628},[439],[427,6630],{"className":6631,"style":470},[443],[427,6633,998],{"className":6634},[448],[427,6636,6638,6644],{"className":6637},[448],[427,6639,6641],{"className":6640},[448,781],[427,6642,5297],{"className":6643},[448],[427,6645,6647],{"className":6646},[629],[427,6648,6650,6672],{"className":6649},[633,828],[427,6651,6653,6669],{"className":6652},[637],[427,6654,6657],{"className":6655,"style":6656},[641],"height:0.0573em;",[427,6658,6660,6663],{"style":6659},"top:-2.4559em;margin-right:0.05em;",[427,6661],{"className":6662,"style":650},[649],[427,6664,6666],{"className":6665},[654,655,656,657],[427,6667,2352],{"className":6668},[448,449,657],[427,6670,897],{"className":6671},[896],[427,6673,6675],{"className":6674},[637],[427,6676,6679],{"className":6677,"style":6678},[641],"height:0.2441em;",[427,6680],{},[427,6682,998],{"className":6683},[448],[427,6685],{"className":6686,"style":455},[454],[427,6688,3595],{"className":6689},[459],[427,6691],{"className":6692,"style":455},[454],[427,6694,6696,6699,6702,6745],{"className":6695},[439],[427,6697],{"className":6698,"style":470},[443],[427,6700,998],{"className":6701},[448],[427,6703,6705,6711],{"className":6704},[448],[427,6706,6708],{"className":6707},[448,781],[427,6709,5297],{"className":6710},[448],[427,6712,6714],{"className":6713},[629],[427,6715,6717,6737],{"className":6716},[633,828],[427,6718,6720,6734],{"className":6719},[637],[427,6721,6723],{"className":6722,"style":6656},[641],[427,6724,6725,6728],{"style":6659},[427,6726],{"className":6727,"style":650},[649],[427,6729,6731],{"className":6730},[654,655,656,657],[427,6732,2363],{"className":6733,"style":2362},[448,449,657],[427,6735,897],{"className":6736},[896],[427,6738,6740],{"className":6739},[637],[427,6741,6743],{"className":6742,"style":6678},[641],[427,6744],{},[427,6746,998],{"className":6747},[448]," and rewrite ",[427,6750,6752],{"className":6751},[430],[427,6753,6755],{"className":6754,"ariaHidden":435},[434],[427,6756,6758,6761,6767,6770,6773],{"className":6757},[439],[427,6759],{"className":6760,"style":470},[443],[427,6762,6764],{"className":6763},[448,781],[427,6765,5297],{"className":6766},[448],[427,6768,6329],{"className":6769},[474],[427,6771,5485],{"className":6772},[448,449],[427,6774,6349],{"className":6775},[498]," for the\nvertices ",[427,6778,6780],{"className":6779},[430],[427,6781,6783],{"className":6782,"ariaHidden":435},[434],[427,6784,6786,6789],{"className":6785},[439],[427,6787],{"className":6788,"style":513},[443],[427,6790,5485],{"className":6791},[448,449]," in that smaller side.",[381,6794,6795,6796,6811,6812,6839,6840,6842,6843,565,6858,6861,6862,6925,6926,6941,6942,6975,6976,7003,7004,7007,7008,1363],{},"Why this wins: whenever a vertex ",[427,6797,6799],{"className":6798},[430],[427,6800,6802],{"className":6801,"ariaHidden":435},[434],[427,6803,6805,6808],{"className":6804},[439],[427,6806],{"className":6807,"style":513},[443],[427,6809,5485],{"className":6810},[448,449]," has its label ",[427,6813,6815],{"className":6814},[430],[427,6816,6818],{"className":6817,"ariaHidden":435},[434],[427,6819,6821,6824,6830,6833,6836],{"className":6820},[439],[427,6822],{"className":6823,"style":470},[443],[427,6825,6827],{"className":6826},[448,781],[427,6828,5297],{"className":6829},[448],[427,6831,6329],{"className":6832},[474],[427,6834,5485],{"className":6835},[448,449],[427,6837,6349],{"className":6838},[498]," rewritten, it\nsat in the ",[401,6841,6618],{}," of the two merged components, so the component containing\n",[427,6844,6846],{"className":6845},[430],[427,6847,6849],{"className":6848,"ariaHidden":435},[434],[427,6850,6852,6855],{"className":6851},[439],[427,6853],{"className":6854,"style":513},[443],[427,6856,5485],{"className":6857},[448,449],[385,6859,6860],{},"at least doubles"," in size. A vertex can be doubled at most ",[427,6863,6865],{"className":6864},[430],[427,6866,6868],{"className":6867,"ariaHidden":435},[434],[427,6869,6871,6875,6919,6922],{"className":6870},[439],[427,6872],{"className":6873,"style":6874},[443],"height:0.9386em;vertical-align:-0.2441em;",[427,6876,6878,6884],{"className":6877},[814],[427,6879,6881],{"className":6880},[814],[427,6882,4460],{"className":6883,"style":4459},[448,4458],[427,6885,6887],{"className":6886},[629],[427,6888,6890,6911],{"className":6889},[633,828],[427,6891,6893,6908],{"className":6892},[637],[427,6894,6897],{"className":6895,"style":6896},[641],"height:0.207em;",[427,6898,6899,6902],{"style":6659},[427,6900],{"className":6901,"style":650},[649],[427,6903,6905],{"className":6904},[654,655,656,657],[427,6906,1228],{"className":6907},[448,657],[427,6909,897],{"className":6910},[896],[427,6912,6914],{"className":6913},[637],[427,6915,6917],{"className":6916,"style":6678},[641],[427,6918],{},[427,6920],{"className":6921,"style":489},[454],[427,6923,979],{"className":6924},[448,449],"\ntimes before it is in a component of all ",[427,6927,6929],{"className":6928},[430],[427,6930,6932],{"className":6931,"ariaHidden":435},[434],[427,6933,6935,6938],{"className":6934},[439],[427,6936],{"className":6937,"style":513},[443],[427,6939,979],{"className":6940},[448,449]," vertices. Hence each vertex is\nrelabelled ",[427,6943,6945],{"className":6944},[430],[427,6946,6948],{"className":6947,"ariaHidden":435},[434],[427,6949,6951,6954,6957,6960,6966,6969,6972],{"className":6950},[439],[427,6952],{"className":6953,"style":470},[443],[427,6955,4448],{"className":6956,"style":4447},[448,449],[427,6958,475],{"className":6959},[474],[427,6961,6963],{"className":6962},[814],[427,6964,4460],{"className":6965,"style":4459},[448,4458],[427,6967],{"className":6968,"style":489},[454],[427,6970,979],{"className":6971},[448,449],[427,6973,499],{"className":6974},[498]," times across the whole run, and the total work spent\nupdating ",[427,6977,6979],{"className":6978},[430],[427,6980,6982],{"className":6981,"ariaHidden":435},[434],[427,6983,6985,6988,6994,6997,7000],{"className":6984},[439],[427,6986],{"className":6987,"style":470},[443],[427,6989,6991],{"className":6990},[448,781],[427,6992,5297],{"className":6993},[448],[427,6995,6329],{"className":6996},[474],[427,6998,6523],{"className":6999},[448],[427,7001,6349],{"className":7002},[498]," over ",[401,7005,7006],{},"all"," unions is ",[427,7009,7011],{"className":7010},[430],[427,7012,7014],{"className":7013,"ariaHidden":435},[434],[427,7015,7017,7020,7023,7026,7029,7032,7038,7041,7044],{"className":7016},[439],[427,7018],{"className":7019,"style":470},[443],[427,7021,4448],{"className":7022,"style":4447},[448,449],[427,7024,475],{"className":7025},[474],[427,7027,979],{"className":7028},[448,449],[427,7030],{"className":7031,"style":489},[454],[427,7033,7035],{"className":7034},[814],[427,7036,4460],{"className":7037,"style":4459},[448,4458],[427,7039],{"className":7040,"style":489},[454],[427,7042,979],{"className":7043},[448,449],[427,7045,499],{"className":7046},[498],[381,7048,7049,7051,7052,7134,7135,7195,7196,7235,7236,7275,7276,7297],{},[385,7050,4543],{}," Sorting the edges dominates: ",[427,7053,7055],{"className":7054},[430],[427,7056,7058,7101],{"className":7057,"ariaHidden":435},[434],[427,7059,7061,7064,7067,7070,7074,7077,7083,7086,7089,7092,7095,7098],{"className":7060},[439],[427,7062],{"className":7063,"style":470},[443],[427,7065,4448],{"className":7066,"style":4447},[448,449],[427,7068,475],{"className":7069},[474],[427,7071,7073],{"className":7072},[448,449],"m",[427,7075],{"className":7076,"style":489},[454],[427,7078,7080],{"className":7079},[814],[427,7081,4460],{"className":7082,"style":4459},[448,4458],[427,7084],{"className":7085,"style":489},[454],[427,7087,7073],{"className":7088},[448,449],[427,7090,499],{"className":7091},[498],[427,7093],{"className":7094,"style":455},[454],[427,7096,460],{"className":7097},[459],[427,7099],{"className":7100,"style":455},[454],[427,7102,7104,7107,7110,7113,7116,7119,7125,7128,7131],{"className":7103},[439],[427,7105],{"className":7106,"style":470},[443],[427,7108,4448],{"className":7109,"style":4447},[448,449],[427,7111,475],{"className":7112},[474],[427,7114,7073],{"className":7115},[448,449],[427,7117],{"className":7118,"style":489},[454],[427,7120,7122],{"className":7121},[814],[427,7123,4460],{"className":7124,"style":4459},[448,4458],[427,7126],{"className":7127,"style":489},[454],[427,7129,979],{"className":7130},[448,449],[427,7132,499],{"className":7133},[498]," since\n",[427,7136,7138],{"className":7137},[430],[427,7139,7141,7159],{"className":7140,"ariaHidden":435},[434],[427,7142,7144,7147,7150,7153,7156],{"className":7143},[439],[427,7145],{"className":7146,"style":4414},[443],[427,7148,7073],{"className":7149},[448,449],[427,7151],{"className":7152,"style":455},[454],[427,7154,3595],{"className":7155},[459],[427,7157],{"className":7158,"style":455},[454],[427,7160,7162,7166],{"className":7161},[439],[427,7163],{"className":7164,"style":7165},[443],"height:0.8141em;",[427,7167,7169,7172],{"className":7168},[448],[427,7170,979],{"className":7171},[448,449],[427,7173,7175],{"className":7174},[629],[427,7176,7178],{"className":7177},[633],[427,7179,7181],{"className":7180},[637],[427,7182,7184],{"className":7183,"style":7165},[641],[427,7185,7186,7189],{"style":645},[427,7187],{"className":7188,"style":650},[649],[427,7190,7192],{"className":7191},[654,655,656,657],[427,7193,1228],{"className":7194},[448,657],". The union-find work adds only ",[427,7197,7199],{"className":7198},[430],[427,7200,7202],{"className":7201,"ariaHidden":435},[434],[427,7203,7205,7208,7211,7214,7217,7220,7226,7229,7232],{"className":7204},[439],[427,7206],{"className":7207,"style":470},[443],[427,7209,4448],{"className":7210,"style":4447},[448,449],[427,7212,475],{"className":7213},[474],[427,7215,979],{"className":7216},[448,449],[427,7218],{"className":7219,"style":489},[454],[427,7221,7223],{"className":7222},[814],[427,7224,4460],{"className":7225,"style":4459},[448,4458],[427,7227],{"className":7228,"style":489},[454],[427,7230,979],{"className":7231},[448,449],[427,7233,499],{"className":7234},[498]," with union by size (and\nis effectively linear, ",[427,7237,7239],{"className":7238},[430],[427,7240,7242],{"className":7241,"ariaHidden":435},[434],[427,7243,7245,7248,7251,7254,7257,7260,7265,7268,7271],{"className":7244},[439],[427,7246],{"className":7247,"style":470},[443],[427,7249,4448],{"className":7250,"style":4447},[448,449],[427,7252,475],{"className":7253},[474],[427,7255,7073],{"className":7256},[448,449],[427,7258],{"className":7259,"style":489},[454],[427,7261,7264],{"className":7262,"style":7263},[448,449],"margin-right:0.0037em;","α",[427,7266,475],{"className":7267},[474],[427,7269,979],{"className":7270},[448,449],[427,7272,7274],{"className":7273},[498],"))",", with the inverse-Ackermann tricks of\nunion-by-rank plus path compression). So overall ",[427,7277,7279],{"className":7278},[430],[427,7280,7282],{"className":7281,"ariaHidden":435},[434],[427,7283,7285,7288],{"className":7284},[439],[427,7286],{"className":7287,"style":4616},[443],[427,7289,7291],{"className":7290},[4620,4621],[427,7292,7294],{"className":7293},[448,781],[427,7295,5050],{"className":7296},[448]," runs in",[427,7299,7301],{"className":7300},[3257],[427,7302,7304],{"className":7303},[430],[427,7305,7307,7343],{"className":7306,"ariaHidden":435},[434],[427,7308,7310,7313,7316,7319,7322,7325,7328,7331,7334,7337,7340],{"className":7309},[439],[427,7311],{"className":7312,"style":470},[443],[427,7314,701],{"className":7315,"style":700},[448,449],[427,7317,475],{"className":7318},[474],[427,7320,979],{"className":7321},[448,449],[427,7323,485],{"className":7324},[484],[427,7326],{"className":7327,"style":489},[454],[427,7329,7073],{"className":7330},[448,449],[427,7332,499],{"className":7333},[498],[427,7335],{"className":7336,"style":455},[454],[427,7338,460],{"className":7339},[459],[427,7341],{"className":7342,"style":455},[454],[427,7344,7346,7349,7352,7355,7358,7361,7367,7370,7373,7376],{"className":7345},[439],[427,7347],{"className":7348,"style":470},[443],[427,7350,4448],{"className":7351,"style":4447},[448,449],[427,7353,475],{"className":7354},[474],[427,7356,7073],{"className":7357},[448,449],[427,7359],{"className":7360,"style":489},[454],[427,7362,7364],{"className":7363},[814],[427,7365,4460],{"className":7366,"style":4459},[448,4458],[427,7368],{"className":7369,"style":489},[454],[427,7371,979],{"className":7372},[448,449],[427,7374,499],{"className":7375},[498],[427,7377,1363],{"className":7378},[448],[405,7380,7382],{"id":7381},"prims-algorithm","Prim's algorithm",[381,7384,7385,7407,7408,7411,7412,7427,7428],{},[427,7386,7388],{"className":7387},[430],[427,7389,7391],{"className":7390,"ariaHidden":435},[434],[427,7392,7394,7397],{"className":7393},[439],[427,7395],{"className":7396,"style":444},[443],[427,7398,7400],{"className":7399},[4620,4621],[427,7401,7403],{"className":7402},[448,781],[427,7404,7406],{"className":7405},[448],"Prim","'s strategy is local and vertex-centric: grow a ",[401,7409,7410],{},"single"," tree outward\nfrom an arbitrary root, repeatedly attaching the cheapest edge that links a tree\nvertex to a non-tree vertex. Here the set ",[427,7413,7415],{"className":7414},[430],[427,7416,7418],{"className":7417,"ariaHidden":435},[434],[427,7419,7421,7424],{"className":7420},[439],[427,7422],{"className":7423,"style":444},[443],[427,7425,1904],{"className":7426},[448,449]," is always one connected tree.",[1954,7429,7430],{},[398,7431,7435],{"href":7432,"ariaDescribedBy":7433,"dataFootnoteRef":376,"id":7434},"#user-content-fn-skiena-mst",[1960],"user-content-fnref-skiena-mst","3",[381,7437,7438,7439,7454,7455,7503,7504,7519,7520,7522],{},"Safety is again the cut property. Let ",[427,7440,7442],{"className":7441},[430],[427,7443,7445],{"className":7444,"ariaHidden":435},[434],[427,7446,7448,7451],{"className":7447},[439],[427,7449],{"className":7450,"style":444},[443],[427,7452,1986],{"className":7453,"style":493},[448,449]," be the vertices currently in the tree.\nThe cut ",[427,7456,7458],{"className":7457},[430],[427,7459,7461,7491],{"className":7460,"ariaHidden":435},[434],[427,7462,7464,7467,7470,7473,7476,7479,7482,7485,7488],{"className":7463},[439],[427,7465],{"className":7466,"style":470},[443],[427,7468,475],{"className":7469},[474],[427,7471,1986],{"className":7472,"style":493},[448,449],[427,7474,485],{"className":7475},[484],[427,7477],{"className":7478,"style":489},[454],[427,7480,480],{"className":7481,"style":479},[448,449],[427,7483],{"className":7484,"style":479},[454],[427,7486,1268],{"className":7487},[1028],[427,7489],{"className":7490,"style":479},[454],[427,7492,7494,7497,7500],{"className":7493},[439],[427,7495],{"className":7496,"style":470},[443],[427,7498,1986],{"className":7499,"style":493},[448,449],[427,7501,499],{"className":7502},[498]," respects ",[427,7505,7507],{"className":7506},[430],[427,7508,7510],{"className":7509,"ariaHidden":435},[434],[427,7511,7513,7516],{"className":7512},[439],[427,7514],{"className":7515,"style":444},[443],[427,7517,1904],{"className":7518},[448,449],", and Prim deliberately picks the\n",[401,7521,5206],{}," edge crossing it, a light edge, so the chosen edge is safe.",[381,7524,7525,7526,7554,7555,7640,7641,7696,7697,7700,7701,7716,7717,7722,7723,7751],{},"This is ",[385,7527,7528,7529,7553],{},"exactly ",[398,7530,7531],{"href":175},[427,7532,7534],{"className":7533},[430],[427,7535,7537],{"className":7536,"ariaHidden":435},[434],[427,7538,7540,7543],{"className":7539},[439],[427,7541],{"className":7542,"style":2772},[443],[427,7544,7546],{"className":7545},[4620,4621],[427,7547,7549],{"className":7548},[448,781],[427,7550,7552],{"className":7551},[448],"Dijkstra"," with\nthe relaxation rule changed",". Where\nDijkstra keys a frontier vertex by ",[427,7556,7558],{"className":7557},[430],[427,7559,7561,7589],{"className":7560,"ariaHidden":435},[434],[427,7562,7564,7567,7571,7574,7577,7580,7583,7586],{"className":7563},[439],[427,7565],{"className":7566,"style":470},[443],[427,7568,7570],{"className":7569},[448,449],"d",[427,7572,6329],{"className":7573},[474],[427,7575,5485],{"className":7576},[448,449],[427,7578,6349],{"className":7579},[498],[427,7581],{"className":7582,"style":479},[454],[427,7584,1737],{"className":7585},[1028],[427,7587],{"className":7588,"style":479},[454],[427,7590,7592,7596],{"className":7591},[439],[427,7593],{"className":7594,"style":7595},[443],"height:0.5806em;vertical-align:-0.15em;",[427,7597,7599,7602],{"className":7598},[448],[427,7600,517],{"className":7601},[448,449],[427,7603,7605],{"className":7604},[629],[427,7606,7608,7632],{"className":7607},[633,828],[427,7609,7611,7629],{"className":7610},[637],[427,7612,7614],{"className":7613,"style":928},[641],[427,7615,7616,7619],{"style":931},[427,7617],{"className":7618,"style":650},[649],[427,7620,7622],{"className":7621},[654,655,656,657],[427,7623,7625],{"className":7624},[448,657],[427,7626,7628],{"className":7627,"style":2362},[448,449,657],"xv",[427,7630,897],{"className":7631},[896],[427,7633,7635],{"className":7634},[637],[427,7636,7638],{"className":7637,"style":950},[641],[427,7639],{}," (distance from the source),\nPrim keys it by ",[427,7642,7644],{"className":7643},[430],[427,7645,7647],{"className":7646,"ariaHidden":435},[434],[427,7648,7650,7653],{"className":7649},[439],[427,7651],{"className":7652,"style":7595},[443],[427,7654,7656,7659],{"className":7655},[448],[427,7657,517],{"className":7658},[448,449],[427,7660,7662],{"className":7661},[629],[427,7663,7665,7688],{"className":7664},[633,828],[427,7666,7668,7685],{"className":7667},[637],[427,7669,7671],{"className":7670,"style":928},[641],[427,7672,7673,7676],{"style":931},[427,7674],{"className":7675,"style":650},[649],[427,7677,7679],{"className":7678},[654,655,656,657],[427,7680,7682],{"className":7681},[448,657],[427,7683,7628],{"className":7684,"style":2362},[448,449,657],[427,7686,897],{"className":7687},[896],[427,7689,7691],{"className":7690},[637],[427,7692,7694],{"className":7693,"style":950},[641],[427,7695],{}," alone (cost to attach to the ",[401,7698,7699],{},"finished"," set). Prim keeps\nevery non-tree vertex ",[427,7702,7704],{"className":7703},[430],[427,7705,7707],{"className":7706,"ariaHidden":435},[434],[427,7708,7710,7713],{"className":7709},[439],[427,7711],{"className":7712,"style":513},[443],[427,7714,2363],{"className":7715,"style":2362},[448,449]," in a\n",[398,7718,7719],{"href":56},[385,7720,7721],{},"min-priority queue"," keyed by\n",[427,7724,7726],{"className":7725},[430],[427,7727,7729],{"className":7728,"ariaHidden":435},[434],[427,7730,7732,7735,7742,7745,7748],{"className":7731},[439],[427,7733],{"className":7734,"style":470},[443],[427,7736,7738],{"className":7737},[448,781],[427,7739,7741],{"className":7740},[448],"key",[427,7743,6329],{"className":7744},[474],[427,7746,2363],{"className":7747,"style":2362},[448,449],[427,7749,6349],{"className":7750},[498],",\nmaintaining the invariant",[427,7753,7755],{"className":7754},[3257],[427,7756,7758],{"className":7757},[430],[427,7759,7761,7784,7807,7837],{"className":7760,"ariaHidden":435},[434],[427,7762,7764,7768,7772,7775,7778,7781],{"className":7763},[439],[427,7765],{"className":7766,"style":7767},[443],"height:0.7335em;vertical-align:-0.0391em;",[427,7769,7771],{"className":7770},[448],"∀",[427,7773,2363],{"className":7774,"style":2362},[448,449],[427,7776],{"className":7777,"style":455},[454],[427,7779,855],{"className":7780},[459],[427,7782],{"className":7783,"style":455},[454],[427,7785,7787,7791,7795,7798,7801,7804],{"className":7786},[439],[427,7788],{"className":7789,"style":7790},[443],"height:0.8778em;vertical-align:-0.1944em;",[427,7792,7794],{"className":7793},[448,449],"Q",[427,7796],{"className":7797,"style":455},[454],[427,7799,524],{"className":7800},[459],[427,7802],{"className":7803,"style":4119},[454],[427,7805],{"className":7806,"style":455},[454],[427,7808,7810,7813,7819,7822,7825,7828,7831,7834],{"className":7809},[439],[427,7811],{"className":7812,"style":470},[443],[427,7814,7816],{"className":7815},[448,781],[427,7817,7741],{"className":7818},[448],[427,7820,6329],{"className":7821},[474],[427,7823,2363],{"className":7824,"style":2362},[448,449],[427,7826,6349],{"className":7827},[498],[427,7829],{"className":7830,"style":455},[454],[427,7832,460],{"className":7833},[459],[427,7835],{"className":7836,"style":455},[454],[427,7838,7840,7844,7934,7937,7940,7983,7987],{"className":7839},[439],[427,7841],{"className":7842,"style":7843},[443],"height:1.1052em;vertical-align:-0.3552em;",[427,7845,7847,7854],{"className":7846},[814],[427,7848,7850],{"className":7849},[814],[427,7851,7853],{"className":7852},[448,4458],"min",[427,7855,7857],{"className":7856},[629],[427,7858,7860,7925],{"className":7859},[633,828],[427,7861,7863,7922],{"className":7862},[637],[427,7864,7867],{"className":7865,"style":7866},[641],"height:0.3448em;",[427,7868,7870,7873],{"style":7869},"top:-2.5198em;margin-right:0.05em;",[427,7871],{"className":7872,"style":650},[649],[427,7874,7876],{"className":7875},[654,655,656,657],[427,7877,7879,7882,7919],{"className":7878},[448,657],[427,7880,5485],{"className":7881},[448,449,657],[427,7883,7885,7891],{"className":7884},[459,657],[427,7886,7888],{"className":7887},[448,657],[427,7889,855],{"className":7890},[459,657],[427,7892,7894],{"className":7893},[448,3210,657],[427,7895,7897],{"className":7896},[3214,657],[427,7898,7900,7903,7916],{"className":7899},[3218,657],[427,7901],{"className":7902,"style":470},[443],[427,7904,7906],{"className":7905},[3225],[427,7907,7909,7912],{"className":7908},[448,657],[427,7910,3232],{"className":7911},[448,657],[427,7913],{"className":7914,"style":7915},[454,657],"margin-right:0.0651em;",[427,7917],{"className":7918},[3240],[427,7920,7794],{"className":7921},[448,449,657],[427,7923,897],{"className":7924},[896],[427,7926,7928],{"className":7927},[637],[427,7929,7932],{"className":7930,"style":7931},[641],"height:0.3552em;",[427,7933],{},[427,7935],{"className":7936,"style":455},[454],[427,7938],{"className":7939,"style":489},[454],[427,7941,7943,7946],{"className":7942},[448],[427,7944,517],{"className":7945},[448,449],[427,7947,7949],{"className":7948},[629],[427,7950,7952,7975],{"className":7951},[633,828],[427,7953,7955,7972],{"className":7954},[637],[427,7956,7958],{"className":7957,"style":928},[641],[427,7959,7960,7963],{"style":931},[427,7961],{"className":7962,"style":650},[649],[427,7964,7966],{"className":7965},[654,655,656,657],[427,7967,7969],{"className":7968},[448,657],[427,7970,7628],{"className":7971,"style":2362},[448,449,657],[427,7973,897],{"className":7974},[896],[427,7976,7978],{"className":7977},[637],[427,7979,7981],{"className":7980,"style":950},[641],[427,7982],{},[427,7984],{"className":7985,"style":7986},[454],"margin-right:2em;",[427,7988,7990,7994,7997],{"className":7989},[448,781],[427,7991,7993],{"className":7992},[448],"(the cheapest way to connect ",[427,7995,2363],{"className":7996,"style":2362},[448,449],[427,7998,8000],{"className":7999},[448]," to a finished vertex).",[381,8002,8003,8004,1363],{},"Extracting the minimum yields the next vertex to absorb; absorbing it may make\nits neighbors cheaper to reach, so we relax their keys with ",[427,8005,8007],{"className":8006},[430],[427,8008,8010],{"className":8009,"ariaHidden":435},[434],[427,8011,8013,8016],{"className":8012},[439],[427,8014],{"className":8015,"style":7790},[443],[427,8017,8019],{"className":8018},[4620,4621],[427,8020,8022],{"className":8021},[448,781],[427,8023,8025],{"className":8024},[448],"Decrease-Key",[4472,8027,8029],{"className":4474,"code":8028,"language":4476,"meta":376,"style":376},"caption: $\\textsc{Prim}(G, c, s)$ — grow one tree from an arbitrary start $s$\nnumber: 3\nforeach vertex $u \\in V$ do\n  $\\text{key}[u] \\gets \\infty$\n  $\\pi[u] \\gets \\text{nil}$\n$\\text{key}[s] \\gets 0$ \u002F\u002F start tree at $s$\n$Q \\gets V$ \u002F\u002F min-PQ keyed by $\\text{key}$\n$E' \\gets \\emptyset$\nwhile $Q \\neq \\emptyset$ do\n  $u \\gets \\textsc{Extract-Min}(Q)$ \u002F\u002F cheapest to attach\n  if $\\pi[u] \\neq \\text{nil}$ then\n    $E' \\gets E' \\cup \\set{(\\pi[u], u)}$ \u002F\u002F commit safe edge\n  foreach $v$ adjacent to $u$ with $v \\in Q$ do\n    if $c_{uv} \u003C \\text{key}[v]$ then\n      $\\pi[v] \\gets u$ \u002F\u002F $v$'s best link to tree\n      $\\text{key}[v] \\gets c_{uv}$ \u002F\u002F Decrease-Key\nreturn $E'$\n",[4478,8030,8031,8036,8041,8046,8051,8056,8061,8066,8071,8076,8081,8086,8091,8097,8103,8109,8115],{"__ignoreMap":376},[427,8032,8033],{"class":4482,"line":6},[427,8034,8035],{},"caption: $\\textsc{Prim}(G, c, s)$ — grow one tree from an arbitrary start $s$\n",[427,8037,8038],{"class":4482,"line":18},[427,8039,8040],{},"number: 3\n",[427,8042,8043],{"class":4482,"line":24},[427,8044,8045],{},"foreach vertex $u \\in V$ do\n",[427,8047,8048],{"class":4482,"line":73},[427,8049,8050],{},"  $\\text{key}[u] \\gets \\infty$\n",[427,8052,8053],{"class":4482,"line":102},[427,8054,8055],{},"  $\\pi[u] \\gets \\text{nil}$\n",[427,8057,8058],{"class":4482,"line":108},[427,8059,8060],{},"$\\text{key}[s] \\gets 0$ \u002F\u002F start tree at $s$\n",[427,8062,8063],{"class":4482,"line":116},[427,8064,8065],{},"$Q \\gets V$ \u002F\u002F min-PQ keyed by $\\text{key}$\n",[427,8067,8068],{"class":4482,"line":196},[427,8069,8070],{},"$E' \\gets \\emptyset$\n",[427,8072,8073],{"class":4482,"line":202},[427,8074,8075],{},"while $Q \\neq \\emptyset$ do\n",[427,8077,8078],{"class":4482,"line":283},[427,8079,8080],{},"  $u \\gets \\textsc{Extract-Min}(Q)$ \u002F\u002F cheapest to attach\n",[427,8082,8083],{"class":4482,"line":333},[427,8084,8085],{},"  if $\\pi[u] \\neq \\text{nil}$ then\n",[427,8087,8088],{"class":4482,"line":354},[427,8089,8090],{},"    $E' \\gets E' \\cup \\set{(\\pi[u], u)}$ \u002F\u002F commit safe edge\n",[427,8092,8094],{"class":4482,"line":8093},13,[427,8095,8096],{},"  foreach $v$ adjacent to $u$ with $v \\in Q$ do\n",[427,8098,8100],{"class":4482,"line":8099},14,[427,8101,8102],{},"    if $c_{uv} \u003C \\text{key}[v]$ then\n",[427,8104,8106],{"class":4482,"line":8105},15,[427,8107,8108],{},"      $\\pi[v] \\gets u$ \u002F\u002F $v$'s best link to tree\n",[427,8110,8112],{"class":4482,"line":8111},16,[427,8113,8114],{},"      $\\text{key}[v] \\gets c_{uv}$ \u002F\u002F Decrease-Key\n",[427,8116,8118],{"class":4482,"line":8117},17,[427,8119,8120],{},"return $E'$\n",[381,8122,8123,8124,8185,8186,8201,8202,8217,8218,8266,8267,8289,8290,8317,8318,8333],{},"A single Prim step looks like this. The tree ",[427,8125,8127],{"className":8126},[430],[427,8128,8130,8148],{"className":8129,"ariaHidden":435},[434],[427,8131,8133,8136,8139,8142,8145],{"className":8132},[439],[427,8134],{"className":8135,"style":444},[443],[427,8137,1986],{"className":8138,"style":493},[448,449],[427,8140],{"className":8141,"style":455},[454],[427,8143,460],{"className":8144},[459],[427,8146],{"className":8147,"style":455},[454],[427,8149,8151,8154],{"className":8150},[439],[427,8152],{"className":8153,"style":470},[443],[427,8155,8157,8160,8163,8166,8169,8173,8176,8179,8182],{"className":8156},[2521],[427,8158,2525],{"className":8159,"style":1194},[474,1193],[427,8161,398],{"className":8162},[448,449],[427,8164,485],{"className":8165},[484],[427,8167],{"className":8168,"style":489},[454],[427,8170,8172],{"className":8171},[448,449],"b",[427,8174,485],{"className":8175},[484],[427,8177],{"className":8178,"style":489},[454],[427,8180,517],{"className":8181},[448,449],[427,8183,2532],{"className":8184,"style":1194},[498,1193]," has been grown\nfrom root ",[427,8187,8189],{"className":8188},[430],[427,8190,8192],{"className":8191,"ariaHidden":435},[434],[427,8193,8195,8198],{"className":8194},[439],[427,8196],{"className":8197,"style":513},[443],[427,8199,398],{"className":8200},[448,449],"; every frontier vertex carries a key equal to its cheapest edge\ninto ",[427,8203,8205],{"className":8204},[430],[427,8206,8208],{"className":8207,"ariaHidden":435},[434],[427,8209,8211,8214],{"className":8210},[439],[427,8212],{"className":8213,"style":444},[443],[427,8215,1986],{"className":8216,"style":493},[448,449],". The cut ",[427,8219,8221],{"className":8220},[430],[427,8222,8224,8254],{"className":8223,"ariaHidden":435},[434],[427,8225,8227,8230,8233,8236,8239,8242,8245,8248,8251],{"className":8226},[439],[427,8228],{"className":8229,"style":470},[443],[427,8231,475],{"className":8232},[474],[427,8234,1986],{"className":8235,"style":493},[448,449],[427,8237,485],{"className":8238},[484],[427,8240],{"className":8241,"style":489},[454],[427,8243,480],{"className":8244,"style":479},[448,449],[427,8246],{"className":8247,"style":479},[454],[427,8249,1268],{"className":8250},[1028],[427,8252],{"className":8253,"style":479},[454],[427,8255,8257,8260,8263],{"className":8256},[439],[427,8258],{"className":8259,"style":470},[443],[427,8261,1986],{"className":8262,"style":493},[448,449],[427,8264,499],{"className":8265},[498]," respects the tree, and ",[427,8268,8270],{"className":8269},[430],[427,8271,8273],{"className":8272,"ariaHidden":435},[434],[427,8274,8276,8279],{"className":8275},[439],[427,8277],{"className":8278,"style":444},[443],[427,8280,8282],{"className":8281},[4620,4621],[427,8283,8285],{"className":8284},[448,781],[427,8286,8288],{"className":8287},[448],"Extract-Min","\nreturns the endpoint of the lightest crossing edge — here ",[427,8291,8293],{"className":8292},[430],[427,8294,8296],{"className":8295,"ariaHidden":435},[434],[427,8297,8299,8303,8306,8313],{"className":8298},[439],[427,8300],{"className":8301,"style":8302},[443],"height:0.6595em;",[427,8304,517],{"className":8305},[448,449],[427,8307,8309],{"className":8308},[448,781],[427,8310,8312],{"className":8311},[448],"–",[427,8314,8316],{"className":8315},[448,449],"i"," at weight\n",[427,8319,8321],{"className":8320},[430],[427,8322,8324],{"className":8323,"ariaHidden":435},[434],[427,8325,8327,8330],{"className":8326},[439],[427,8328],{"className":8329,"style":1039},[443],[427,8331,1228],{"className":8332},[448]," — which the cut property certifies as safe:",[1449,8335,8337,8543],{"className":8336},[1452,1453],[1455,8338,8342],{"xmlns":1457,"width":8339,"height":8340,"viewBox":8341},"286.775","156.488","-75 -75 215.081 117.366",[1462,8343,8344,8348,8357,8370,8382,8394,8397,8404,8407,8414,8417,8424,8436,8448,8465,8468,8474,8477,8484],{"stroke":1464,"style":1465},[1467,8345],{"fill":8346,"stroke":2078,"d":8347,"style":4138},"var(--tk-soft-neutral)","M-65.403-7.784V-68.07a4 4 0 0 1 4-4H72.86a4 4 0 0 1 4 4v60.286a4 4 0 0 1-4 4H-61.403a4 4 0 0 1-4-4ZM76.86-72.07",[1462,8349,8350],{"fill":2078,"stroke":2078},[1462,8351,8353],{"transform":8352},"translate(-9.76 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26.523-24.324Q26.812-24.325 27.030-24.334Q27.249-24.343 27.266-24.360Q27.328-24.425 27.365-24.592Q27.403-24.760 27.441-25.002L27.707-25.002",[1478],[1707,8544,8546,8547,8562,8563,8584,8585,8609,8610,8625],{"className":8545},[1710],"A Prim snapshot: the grown tree ",[427,8548,8550],{"className":8549},[430],[427,8551,8553],{"className":8552,"ariaHidden":435},[434],[427,8554,8556,8559],{"className":8555},[439],[427,8557],{"className":8558,"style":444},[443],[427,8560,1986],{"className":8561,"style":493},[448,449]," (shaded, dashed box) and the frontier; ",[427,8564,8566],{"className":8565},[430],[427,8567,8569],{"className":8568,"ariaHidden":435},[434],[427,8570,8572,8575],{"className":8571},[439],[427,8573],{"className":8574,"style":444},[443],[427,8576,8578],{"className":8577},[4620,4621],[427,8579,8581],{"className":8580},[448,781],[427,8582,8288],{"className":8583},[448]," pulls the lightest crossing edge (",[427,8586,8588],{"className":8587},[430],[427,8589,8591],{"className":8590,"ariaHidden":435},[434],[427,8592,8594,8597,8600,8606],{"className":8593},[439],[427,8595],{"className":8596,"style":8302},[443],[427,8598,517],{"className":8599},[448,449],[427,8601,8603],{"className":8602},[448,781],[427,8604,8312],{"className":8605},[448],[427,8607,8316],{"className":8608},[448,449],", weight ",[427,8611,8613],{"className":8612},[430],[427,8614,8616],{"className":8615,"ariaHidden":435},[434],[427,8617,8619,8622],{"className":8618},[439],[427,8620],{"className":8621,"style":1039},[443],[427,8623,1228],{"className":8624},[448],").",[381,8627,8628,8630,8631,565,8646,8667,8668,565,8683,8704,8705,8962,8963,8966,8967,9000,9001,9040,9041,9044,9045,9066,9067,9091,9092,9113,9114,9147],{},[385,8629,4543],{}," Prim performs ",[427,8632,8634],{"className":8633},[430],[427,8635,8637],{"className":8636,"ariaHidden":435},[434],[427,8638,8640,8643],{"className":8639},[439],[427,8641],{"className":8642,"style":513},[443],[427,8644,979],{"className":8645},[448,449],[427,8647,8649],{"className":8648},[430],[427,8650,8652],{"className":8651,"ariaHidden":435},[434],[427,8653,8655,8658],{"className":8654},[439],[427,8656],{"className":8657,"style":444},[443],[427,8659,8661],{"className":8660},[4620,4621],[427,8662,8664],{"className":8663},[448,781],[427,8665,8288],{"className":8666},[448]," operations and up to\n",[427,8669,8671],{"className":8670},[430],[427,8672,8674],{"className":8673,"ariaHidden":435},[434],[427,8675,8677,8680],{"className":8676},[439],[427,8678],{"className":8679,"style":513},[443],[427,8681,7073],{"className":8682},[448,449],[427,8684,8686],{"className":8685},[430],[427,8687,8689],{"className":8688,"ariaHidden":435},[434],[427,8690,8692,8695],{"className":8691},[439],[427,8693],{"className":8694,"style":7790},[443],[427,8696,8698],{"className":8697},[4620,4621],[427,8699,8701],{"className":8700},[448,781],[427,8702,8025],{"className":8703},[448]," operations, so in general\n",[427,8706,8708],{"className":8707},[430],[427,8709,8711,8747,8830,8896],{"className":8710,"ariaHidden":435},[434],[427,8712,8714,8717,8720,8723,8726,8729,8732,8735,8738,8741,8744],{"className":8713},[439],[427,8715],{"className":8716,"style":470},[443],[427,8718,701],{"className":8719,"style":700},[448,449],[427,8721,475],{"className":8722},[474],[427,8724,979],{"className":8725},[448,449],[427,8727,485],{"className":8728},[484],[427,8730],{"className":8731,"style":489},[454],[427,8733,7073],{"className":8734},[448,449],[427,8736,499],{"className":8737},[498],[427,8739],{"className":8740,"style":455},[454],[427,8742,460],{"className":8743},[459],[427,8745],{"className":8746,"style":455},[454],[427,8748,8750,8754,8757,8763,8766,8769,8772,8821,8824,8827],{"className":8749},[439],[427,8751],{"className":8752,"style":8753},[443],"height:1.2em;vertical-align:-0.35em;",[427,8755,4448],{"className":8756,"style":4447},[448,449],[427,8758,8760],{"className":8759},[448],[427,8761,475],{"className":8762},[1198,886],[427,8764,979],{"className":8765},[448,449],[427,8767],{"className":8768,"style":489},[454],[427,8770,475],{"className":8771},[474],[427,8773,8775,8778],{"className":8774},[448],[427,8776,701],{"className":8777,"style":700},[448,449],[427,8779,8781],{"className":8780},[629],[427,8782,8784,8813],{"className":8783},[633,828],[427,8785,8787,8810],{"className":8786},[637],[427,8788,8791],{"className":8789,"style":8790},[641],"height:0.3175em;",[427,8792,8794,8797],{"style":8793},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[427,8795],{"className":8796,"style":650},[649],[427,8798,8800],{"className":8799},[654,655,656,657],[427,8801,8803],{"className":8802},[448,657],[427,8804,8806],{"className":8805},[448,781,657],[427,8807,8809],{"className":8808},[448,657],"ins",[427,8811,897],{"className":8812},[896],[427,8814,8816],{"className":8815},[637],[427,8817,8819],{"className":8818,"style":950},[641],[427,8820],{},[427,8822],{"className":8823,"style":479},[454],[427,8825,1737],{"className":8826},[1028],[427,8828],{"className":8829,"style":479},[454],[427,8831,8833,8836,8884,8887,8890,8893],{"className":8832},[439],[427,8834],{"className":8835,"style":470},[443],[427,8837,8839,8842],{"className":8838},[448],[427,8840,701],{"className":8841,"style":700},[448,449],[427,8843,8845],{"className":8844},[629],[427,8846,8848,8876],{"className":8847},[633,828],[427,8849,8851,8873],{"className":8850},[637],[427,8852,8855],{"className":8853,"style":8854},[641],"height:0.2806em;",[427,8856,8857,8860],{"style":8793},[427,8858],{"className":8859,"style":650},[649],[427,8861,8863],{"className":8862},[654,655,656,657],[427,8864,8866],{"className":8865},[448,657],[427,8867,8869],{"className":8868},[448,781,657],[427,8870,8872],{"className":8871},[448,657],"ext",[427,8874,897],{"className":8875},[896],[427,8877,8879],{"className":8878},[637],[427,8880,8882],{"className":8881,"style":950},[641],[427,8883],{},[427,8885,499],{"className":8886},[498],[427,8888],{"className":8889,"style":479},[454],[427,8891,1737],{"className":8892},[1028],[427,8894],{"className":8895,"style":479},[454],[427,8897,8899,8902,8905,8908,8956],{"className":8898},[439],[427,8900],{"className":8901,"style":8753},[443],[427,8903,7073],{"className":8904},[448,449],[427,8906],{"className":8907,"style":489},[454],[427,8909,8911,8914],{"className":8910},[448],[427,8912,701],{"className":8913,"style":700},[448,449],[427,8915,8917],{"className":8916},[629],[427,8918,8920,8948],{"className":8919},[633,828],[427,8921,8923,8945],{"className":8922},[637],[427,8924,8927],{"className":8925,"style":8926},[641],"height:0.3361em;",[427,8928,8929,8932],{"style":8793},[427,8930],{"className":8931,"style":650},[649],[427,8933,8935],{"className":8934},[654,655,656,657],[427,8936,8938],{"className":8937},[448,657],[427,8939,8941],{"className":8940},[448,781,657],[427,8942,8944],{"className":8943},[448,657],"dec",[427,8946,897],{"className":8947},[896],[427,8949,8951],{"className":8950},[637],[427,8952,8954],{"className":8953,"style":950},[641],[427,8955],{},[427,8957,8959],{"className":8958},[448],[427,8960,499],{"className":8961},[1198,886],".\nWith a ",[385,8964,8965],{},"binary heap"," all three operations cost ",[427,8968,8970],{"className":8969},[430],[427,8971,8973],{"className":8972,"ariaHidden":435},[434],[427,8974,8976,8979,8982,8985,8991,8994,8997],{"className":8975},[439],[427,8977],{"className":8978,"style":470},[443],[427,8980,4448],{"className":8981,"style":4447},[448,449],[427,8983,475],{"className":8984},[474],[427,8986,8988],{"className":8987},[814],[427,8989,4460],{"className":8990,"style":4459},[448,4458],[427,8992],{"className":8993,"style":489},[454],[427,8995,979],{"className":8996},[448,449],[427,8998,499],{"className":8999},[498],", giving\n",[427,9002,9004],{"className":9003},[430],[427,9005,9007],{"className":9006,"ariaHidden":435},[434],[427,9008,9010,9013,9016,9019,9022,9025,9031,9034,9037],{"className":9009},[439],[427,9011],{"className":9012,"style":470},[443],[427,9014,4448],{"className":9015,"style":4447},[448,449],[427,9017,475],{"className":9018},[474],[427,9020,7073],{"className":9021},[448,449],[427,9023],{"className":9024,"style":489},[454],[427,9026,9028],{"className":9027},[814],[427,9029,4460],{"className":9030,"style":4459},[448,4458],[427,9032],{"className":9033,"style":489},[454],[427,9035,979],{"className":9036},[448,449],[427,9038,499],{"className":9039},[498],", the same as Kruskal. With a ",[385,9042,9043],{},"Fibonacci heap",",\n",[427,9046,9048],{"className":9047},[430],[427,9049,9051],{"className":9050,"ariaHidden":435},[434],[427,9052,9054,9057],{"className":9053},[439],[427,9055],{"className":9056,"style":7790},[443],[427,9058,9060],{"className":9059},[4620,4621],[427,9061,9063],{"className":9062},[448,781],[427,9064,8025],{"className":9065},[448]," drops to ",[427,9068,9070],{"className":9069},[430],[427,9071,9073],{"className":9072,"ariaHidden":435},[434],[427,9074,9076,9079,9082,9085,9088],{"className":9075},[439],[427,9077],{"className":9078,"style":470},[443],[427,9080,4448],{"className":9081,"style":4447},[448,449],[427,9083,475],{"className":9084},[474],[427,9086,1043],{"className":9087},[448],[427,9089,499],{"className":9090},[498]," amortized while ",[427,9093,9095],{"className":9094},[430],[427,9096,9098],{"className":9097,"ariaHidden":435},[434],[427,9099,9101,9104],{"className":9100},[439],[427,9102],{"className":9103,"style":444},[443],[427,9105,9107],{"className":9106},[4620,4621],[427,9108,9110],{"className":9109},[448,781],[427,9111,8288],{"className":9112},[448],"\nstays ",[427,9115,9117],{"className":9116},[430],[427,9118,9120],{"className":9119,"ariaHidden":435},[434],[427,9121,9123,9126,9129,9132,9138,9141,9144],{"className":9122},[439],[427,9124],{"className":9125,"style":470},[443],[427,9127,4448],{"className":9128,"style":4447},[448,449],[427,9130,475],{"className":9131},[474],[427,9133,9135],{"className":9134},[814],[427,9136,4460],{"className":9137,"style":4459},[448,4458],[427,9139],{"className":9140,"style":489},[454],[427,9142,979],{"className":9143},[448,449],[427,9145,499],{"className":9146},[498],", improving the total to",[427,9149,9151],{"className":9150},[3257],[427,9152,9154],{"className":9153},[430],[427,9155,9157,9193,9217],{"className":9156,"ariaHidden":435},[434],[427,9158,9160,9163,9166,9169,9172,9175,9178,9181,9184,9187,9190],{"className":9159},[439],[427,9161],{"className":9162,"style":470},[443],[427,9164,701],{"className":9165,"style":700},[448,449],[427,9167,475],{"className":9168},[474],[427,9170,979],{"className":9171},[448,449],[427,9173,485],{"className":9174},[484],[427,9176],{"className":9177,"style":489},[454],[427,9179,7073],{"className":9180},[448,449],[427,9182,499],{"className":9183},[498],[427,9185],{"className":9186,"style":455},[454],[427,9188,460],{"className":9189},[459],[427,9191],{"className":9192,"style":455},[454],[427,9194,9196,9199,9202,9205,9208,9211,9214],{"className":9195},[439],[427,9197],{"className":9198,"style":470},[443],[427,9200,4448],{"className":9201,"style":4447},[448,449],[427,9203,475],{"className":9204},[474],[427,9206,7073],{"className":9207},[448,449],[427,9209],{"className":9210,"style":479},[454],[427,9212,1737],{"className":9213},[1028],[427,9215],{"className":9216,"style":479},[454],[427,9218,9220,9223,9226,9229,9235,9238,9241,9244],{"className":9219},[439],[427,9221],{"className":9222,"style":470},[443],[427,9224,979],{"className":9225},[448,449],[427,9227],{"className":9228,"style":489},[454],[427,9230,9232],{"className":9231},[814],[427,9233,4460],{"className":9234,"style":4459},[448,4458],[427,9236],{"className":9237,"style":489},[454],[427,9239,979],{"className":9240},[448,449],[427,9242,499],{"className":9243},[498],[427,9245,1363],{"className":9246},[448],[381,9248,9249,9250,9308,9309,9333,9334,9337,9338,9341],{},"Notice that when the graph is dense, ",[427,9251,9253],{"className":9252},[430],[427,9254,9256,9274],{"className":9255,"ariaHidden":435},[434],[427,9257,9259,9262,9265,9268,9271],{"className":9258},[439],[427,9260],{"className":9261,"style":513},[443],[427,9263,7073],{"className":9264},[448,449],[427,9266],{"className":9267,"style":455},[454],[427,9269,460],{"className":9270},[459],[427,9272],{"className":9273,"style":455},[454],[427,9275,9277,9280,9284,9287,9290,9293,9299,9302,9305],{"className":9276},[439],[427,9278],{"className":9279,"style":470},[443],[427,9281,9283],{"className":9282},[448],"Ω",[427,9285,475],{"className":9286},[474],[427,9288,979],{"className":9289},[448,449],[427,9291],{"className":9292,"style":489},[454],[427,9294,9296],{"className":9295},[814],[427,9297,4460],{"className":9298,"style":4459},[448,4458],[427,9300],{"className":9301,"style":489},[454],[427,9303,979],{"className":9304},[448,449],[427,9306,499],{"className":9307},[498],", this is\n",[427,9310,9312],{"className":9311},[430],[427,9313,9315],{"className":9314,"ariaHidden":435},[434],[427,9316,9318,9321,9324,9327,9330],{"className":9317},[439],[427,9319],{"className":9320,"style":470},[443],[427,9322,4448],{"className":9323,"style":4447},[448,449],[427,9325,475],{"className":9326},[474],[427,9328,7073],{"className":9329},[448,449],[427,9331,499],{"className":9332},[498],", that is, ",[385,9335,9336],{},"linear",", and\n",[398,9339,9340],{"href":17},"asymptotically"," the best known.",[405,9343,9345],{"id":9344},"which-to-use","Which to use?",[381,9347,9348,9349,5260,9388,1363],{},"Throughout, ",[427,9350,9352],{"className":9351},[430],[427,9353,9355,9373],{"className":9354,"ariaHidden":435},[434],[427,9356,9358,9361,9364,9367,9370],{"className":9357},[439],[427,9359],{"className":9360,"style":513},[443],[427,9362,979],{"className":9363},[448,449],[427,9365],{"className":9366,"style":455},[454],[427,9368,460],{"className":9369},[459],[427,9371],{"className":9372,"style":455},[454],[427,9374,9376,9379,9382,9385],{"className":9375},[439],[427,9377],{"className":9378,"style":470},[443],[427,9380,998],{"className":9381},[448],[427,9383,480],{"className":9384,"style":479},[448,449],[427,9386,998],{"className":9387},[448],[427,9389,9391],{"className":9390},[430],[427,9392,9394,9412],{"className":9393,"ariaHidden":435},[434],[427,9395,9397,9400,9403,9406,9409],{"className":9396},[439],[427,9398],{"className":9399,"style":513},[443],[427,9401,7073],{"className":9402},[448,449],[427,9404],{"className":9405,"style":455},[454],[427,9407,460],{"className":9408},[459],[427,9410],{"className":9411,"style":455},[454],[427,9413,9415,9418,9421,9424],{"className":9414},[439],[427,9416],{"className":9417,"style":470},[443],[427,9419,998],{"className":9420},[448],[427,9422,494],{"className":9423,"style":493},[448,449],[427,9425,998],{"className":9426},[448],[9428,9429,9430,9446],"table",{},[9431,9432,9433],"thead",{},[9434,9435,9436,9439,9442,9444],"tr",{},[9437,9438],"th",{},[9437,9440,9441],{},"Borůvka",[9437,9443,5050],{},[9437,9445,7406],{},[9447,9448,9449,9474,9488,9616,9723],"tbody",{},[9434,9450,9451,9455,9462,9468],{},[9452,9453,9454],"td",{},"Grows",[9452,9456,9457,9458,9461],{},"every ",[385,9459,9460],{},"component"," at once",[9452,9463,9464,9465,9467],{},"a ",[385,9466,5074],{},", globally",[9452,9469,9470,9471,9473],{},"a single ",[385,9472,387],{},", from a root",[9434,9475,9476,9479,9482,9485],{},[9452,9477,9478],{},"Core structure",[9452,9480,9481],{},"component scan \u002F union-find",[9452,9483,9484],{},"union-find (union by size)",[9452,9486,9487],{},"priority queue",[9434,9489,9490,9493,9534,9575],{},[9452,9491,9492],{},"Headline time",[9452,9494,9495],{},[427,9496,9498],{"className":9497},[430],[427,9499,9501],{"className":9500,"ariaHidden":435},[434],[427,9502,9504,9507,9510,9513,9516,9519,9525,9528,9531],{"className":9503},[439],[427,9505],{"className":9506,"style":470},[443],[427,9508,4448],{"className":9509,"style":4447},[448,449],[427,9511,475],{"className":9512},[474],[427,9514,7073],{"className":9515},[448,449],[427,9517],{"className":9518,"style":489},[454],[427,9520,9522],{"className":9521},[814],[427,9523,4460],{"className":9524,"style":4459},[448,4458],[427,9526],{"className":9527,"style":489},[454],[427,9529,979],{"className":9530},[448,449],[427,9532,499],{"className":9533},[498],[9452,9535,9536],{},[427,9537,9539],{"className":9538},[430],[427,9540,9542],{"className":9541,"ariaHidden":435},[434],[427,9543,9545,9548,9551,9554,9557,9560,9566,9569,9572],{"className":9544},[439],[427,9546],{"className":9547,"style":470},[443],[427,9549,4448],{"className":9550,"style":4447},[448,449],[427,9552,475],{"className":9553},[474],[427,9555,7073],{"className":9556},[448,449],[427,9558],{"className":9559,"style":489},[454],[427,9561,9563],{"className":9562},[814],[427,9564,4460],{"className":9565,"style":4459},[448,4458],[427,9567],{"className":9568,"style":489},[454],[427,9570,979],{"className":9571},[448,449],[427,9573,499],{"className":9574},[498],[9452,9576,9577],{},[427,9578,9580],{"className":9579},[430],[427,9581,9583],{"className":9582,"ariaHidden":435},[434],[427,9584,9586,9589,9592,9595,9598,9601,9607,9610,9613],{"className":9585},[439],[427,9587],{"className":9588,"style":470},[443],[427,9590,4448],{"className":9591,"style":4447},[448,449],[427,9593,475],{"className":9594},[474],[427,9596,7073],{"className":9597},[448,449],[427,9599],{"className":9600,"style":489},[454],[427,9602,9604],{"className":9603},[814],[427,9605,4460],{"className":9606,"style":4459},[448,4458],[427,9608],{"className":9609,"style":489},[454],[427,9611,979],{"className":9612},[448,449],[427,9614,499],{"className":9615},[498],[9434,9617,9618,9621,9624,9663],{},[9452,9619,9620],{},"Best variant",[9452,9622,9623],{},"parallel rounds",[9452,9625,9626,9662],{},[427,9627,9629],{"className":9628},[430],[427,9630,9632],{"className":9631,"ariaHidden":435},[434],[427,9633,9635,9638,9641,9644,9647,9650,9653,9656,9659],{"className":9634},[439],[427,9636],{"className":9637,"style":470},[443],[427,9639,4448],{"className":9640,"style":4447},[448,449],[427,9642,475],{"className":9643},[474],[427,9645,7073],{"className":9646},[448,449],[427,9648],{"className":9649,"style":489},[454],[427,9651,7264],{"className":9652,"style":7263},[448,449],[427,9654,475],{"className":9655},[474],[427,9657,979],{"className":9658},[448,449],[427,9660,7274],{"className":9661},[498]," work after sort",[9452,9664,9665,9722],{},[427,9666,9668],{"className":9667},[430],[427,9669,9671,9695],{"className":9670,"ariaHidden":435},[434],[427,9672,9674,9677,9680,9683,9686,9689,9692],{"className":9673},[439],[427,9675],{"className":9676,"style":470},[443],[427,9678,4448],{"className":9679,"style":4447},[448,449],[427,9681,475],{"className":9682},[474],[427,9684,7073],{"className":9685},[448,449],[427,9687],{"className":9688,"style":479},[454],[427,9690,1737],{"className":9691},[1028],[427,9693],{"className":9694,"style":479},[454],[427,9696,9698,9701,9704,9707,9713,9716,9719],{"className":9697},[439],[427,9699],{"className":9700,"style":470},[443],[427,9702,979],{"className":9703},[448,449],[427,9705],{"className":9706,"style":489},[454],[427,9708,9710],{"className":9709},[814],[427,9711,4460],{"className":9712,"style":4459},[448,4458],[427,9714],{"className":9715,"style":489},[454],[427,9717,979],{"className":9718},[448,449],[427,9720,499],{"className":9721},[498]," (Fib. heap)",[9434,9724,9725,9728,9731,9734],{},[9452,9726,9727],{},"Shines when",[9452,9729,9730],{},"parallel \u002F distributed",[9452,9732,9733],{},"edges already sorted; sparse",[9452,9735,9736],{},"dense; adjacency-rich data",[381,9738,9739,9740,9743,9744,9747,9748,9750],{},"All three are correct for the ",[401,9741,9742],{},"same"," reason, the cut property, and all three are\ngreedy algorithms whose greediness is ",[401,9745,9746],{},"provably"," optimal, a rarity worth\nsavoring. If edge weights are not distinct the MST may not be unique, but every\nchoice these algorithms make is still safe, so they always return ",[401,9749,398],{}," minimum\nspanning tree. (The fastest known practical approaches combine Borůvka's rounds\nwith Prim\u002FKruskal to shave the bound further.)",[405,9752,9754],{"id":9753},"takeaways","Takeaways",[6494,9756,9757,9820,9868,10000,10136,10283],{},[1116,9758,9759,9760,9762,9763,9778,9779,9812,9813,9815,9816,9819],{},"A ",[385,9761,395],{}," connects all ",[427,9764,9766],{"className":9765},[430],[427,9767,9769],{"className":9768,"ariaHidden":435},[434],[427,9770,9772,9775],{"className":9771},[439],[427,9773],{"className":9774,"style":513},[443],[427,9776,979],{"className":9777},[448,449]," vertices of a connected weighted\ngraph with exactly ",[427,9780,9782],{"className":9781},[430],[427,9783,9785,9803],{"className":9784,"ariaHidden":435},[434],[427,9786,9788,9791,9794,9797,9800],{"className":9787},[439],[427,9789],{"className":9790,"style":1018},[443],[427,9792,979],{"className":9793},[448,449],[427,9795],{"className":9796,"style":479},[454],[427,9798,1029],{"className":9799},[1028],[427,9801],{"className":9802,"style":479},[454],[427,9804,9806,9809],{"className":9805},[439],[427,9807],{"className":9808,"style":1039},[443],[427,9810,1043],{"className":9811},[448]," edges of least total weight; ",[1052,9814,387],{}," means\n",[401,9817,9818],{},"connected and acyclic",", no root.",[1116,9821,9822,9823,9826,9827,9829,9830,9832,9833,565,9848,9851,9852,9867],{},"The ",[385,9824,9825],{},"cut property"," (cut rule) is the one lemma behind every MST algorithm: a\n",[401,9828,2060],{}," edge crossing a cut that respects the current edge set is always\n",[385,9831,1912],{},". The exchange argument must swap out the edge ",[427,9834,9836],{"className":9835},[430],[427,9837,9839],{"className":9838,"ariaHidden":435},[434],[427,9840,9842,9845],{"className":9841},[439],[427,9843],{"className":9844,"style":3108},[443],[427,9846,1462],{"className":9847,"style":2362},[448,449],[401,9849,9850],{},"on the cycle","\ncreated by adding ",[427,9853,9855],{"className":9854},[430],[427,9856,9858],{"className":9857,"ariaHidden":435},[434],[427,9859,9861,9864],{"className":9860},[439],[427,9862],{"className":9863,"style":513},[443],[427,9865,851],{"className":9866},[448,449],"; swapping an arbitrary crossing edge is a tempting\nmistake.",[1116,9869,9870,9925,9926,9959,9960,9999],{},[427,9871,9873],{"className":9872},[430],[427,9874,9876],{"className":9875,"ariaHidden":435},[434],[427,9877,9879,9882],{"className":9878},[439],[427,9880],{"className":9881,"style":4616},[443],[427,9883,9885],{"className":9884},[4620,4621],[427,9886,9888,9891,9922],{"className":9887},[448,781],[427,9889,4628],{"className":9890},[448],[427,9892,9894],{"className":9893},[448,4632],[427,9895,9897],{"className":9896},[633],[427,9898,9900],{"className":9899},[637],[427,9901,9903,9911],{"className":9902,"style":4616},[641],[427,9904,9905,9908],{"style":4644},[427,9906],{"className":9907,"style":4648},[649],[427,9909,2352],{"className":9910},[448],[427,9912,9913,9916],{"style":4644},[427,9914],{"className":9915,"style":4648},[649],[427,9917,9919],{"className":9918,"style":4661},[4660],[427,9920,4665],{"className":9921},[448],[427,9923,4669],{"className":9924},[448]," adds every component's cheapest exit edge each round; halving\nthe component count gives ",[427,9927,9929],{"className":9928},[430],[427,9930,9932],{"className":9931,"ariaHidden":435},[434],[427,9933,9935,9938,9941,9944,9950,9953,9956],{"className":9934},[439],[427,9936],{"className":9937,"style":470},[443],[427,9939,4448],{"className":9940,"style":4447},[448,449],[427,9942,475],{"className":9943},[474],[427,9945,9947],{"className":9946},[814],[427,9948,4460],{"className":9949,"style":4459},[448,4458],[427,9951],{"className":9952,"style":489},[454],[427,9954,979],{"className":9955},[448,449],[427,9957,499],{"className":9958},[498]," rounds and ",[427,9961,9963],{"className":9962},[430],[427,9964,9966],{"className":9965,"ariaHidden":435},[434],[427,9967,9969,9972,9975,9978,9981,9984,9990,9993,9996],{"className":9968},[439],[427,9970],{"className":9971,"style":470},[443],[427,9973,4448],{"className":9974,"style":4447},[448,449],[427,9976,475],{"className":9977},[474],[427,9979,7073],{"className":9980},[448,449],[427,9982],{"className":9983,"style":489},[454],[427,9985,9987],{"className":9986},[814],[427,9988,4460],{"className":9989,"style":4459},[448,4458],[427,9991],{"className":9992,"style":489},[454],[427,9994,979],{"className":9995},[448,449],[427,9997,499],{"className":9998},[498]," total, with\nnaturally parallel rounds.",[1116,10001,10002,10023,10024,10027,10028,4257,10067,10070,10071,10110,10111,10135],{},[427,10003,10005],{"className":10004},[430],[427,10006,10008],{"className":10007,"ariaHidden":435},[434],[427,10009,10011,10014],{"className":10010},[439],[427,10012],{"className":10013,"style":4616},[443],[427,10015,10017],{"className":10016},[4620,4621],[427,10018,10020],{"className":10019},[448,781],[427,10021,5050],{"className":10022},[448]," adds edges cheapest-first, skipping cycle-forming ones, using\n",[385,10025,10026],{},"union-find","; sorting dominates at ",[427,10029,10031],{"className":10030},[430],[427,10032,10034],{"className":10033,"ariaHidden":435},[434],[427,10035,10037,10040,10043,10046,10049,10052,10058,10061,10064],{"className":10036},[439],[427,10038],{"className":10039,"style":470},[443],[427,10041,4448],{"className":10042,"style":4447},[448,449],[427,10044,475],{"className":10045},[474],[427,10047,7073],{"className":10048},[448,449],[427,10050],{"className":10051,"style":489},[454],[427,10053,10055],{"className":10054},[814],[427,10056,4460],{"className":10057,"style":4459},[448,4458],[427,10059],{"className":10060,"style":489},[454],[427,10062,979],{"className":10063},[448,449],[427,10065,499],{"className":10066},[498],[385,10068,10069],{},"union by size"," keeps\nthe relabelling work to ",[427,10072,10074],{"className":10073},[430],[427,10075,10077],{"className":10076,"ariaHidden":435},[434],[427,10078,10080,10083,10086,10089,10092,10095,10101,10104,10107],{"className":10079},[439],[427,10081],{"className":10082,"style":470},[443],[427,10084,4448],{"className":10085,"style":4447},[448,449],[427,10087,475],{"className":10088},[474],[427,10090,979],{"className":10091},[448,449],[427,10093],{"className":10094,"style":489},[454],[427,10096,10098],{"className":10097},[814],[427,10099,4460],{"className":10100,"style":4459},[448,4458],[427,10102],{"className":10103,"style":489},[454],[427,10105,979],{"className":10106},[448,449],[427,10108,499],{"className":10109},[498]," because each vertex's component can double\nonly ",[427,10112,10114],{"className":10113},[430],[427,10115,10117],{"className":10116,"ariaHidden":435},[434],[427,10118,10120,10123,10129,10132],{"className":10119},[439],[427,10121],{"className":10122,"style":2772},[443],[427,10124,10126],{"className":10125},[814],[427,10127,4460],{"className":10128,"style":4459},[448,4458],[427,10130],{"className":10131,"style":489},[454],[427,10133,979],{"className":10134},[448,449]," times.",[1116,10137,10138,4023,10159,10180,10181,10184,10185,10224,10225,10282],{},[427,10139,10141],{"className":10140},[430],[427,10142,10144],{"className":10143,"ariaHidden":435},[434],[427,10145,10147,10150],{"className":10146},[439],[427,10148],{"className":10149,"style":444},[443],[427,10151,10153],{"className":10152},[4620,4621],[427,10154,10156],{"className":10155},[448,781],[427,10157,7406],{"className":10158},[448],[427,10160,10162],{"className":10161},[430],[427,10163,10165],{"className":10164,"ariaHidden":435},[434],[427,10166,10168,10171],{"className":10167},[439],[427,10169],{"className":10170,"style":2772},[443],[427,10172,10174],{"className":10173},[4620,4621],[427,10175,10177],{"className":10176},[448,781],[427,10178,7552],{"className":10179},[448]," rekeyed by attachment cost: grow one tree\nfrom a start vertex, attaching the lightest crossing edge via a ",[385,10182,10183],{},"priority\nqueue","; ",[427,10186,10188],{"className":10187},[430],[427,10189,10191],{"className":10190,"ariaHidden":435},[434],[427,10192,10194,10197,10200,10203,10206,10209,10215,10218,10221],{"className":10193},[439],[427,10195],{"className":10196,"style":470},[443],[427,10198,4448],{"className":10199,"style":4447},[448,449],[427,10201,475],{"className":10202},[474],[427,10204,7073],{"className":10205},[448,449],[427,10207],{"className":10208,"style":489},[454],[427,10210,10212],{"className":10211},[814],[427,10213,4460],{"className":10214,"style":4459},[448,4458],[427,10216],{"className":10217,"style":489},[454],[427,10219,979],{"className":10220},[448,449],[427,10222,499],{"className":10223},[498]," with a binary heap, ",[427,10226,10228],{"className":10227},[430],[427,10229,10231,10255],{"className":10230,"ariaHidden":435},[434],[427,10232,10234,10237,10240,10243,10246,10249,10252],{"className":10233},[439],[427,10235],{"className":10236,"style":470},[443],[427,10238,4448],{"className":10239,"style":4447},[448,449],[427,10241,475],{"className":10242},[474],[427,10244,7073],{"className":10245},[448,449],[427,10247],{"className":10248,"style":479},[454],[427,10250,1737],{"className":10251},[1028],[427,10253],{"className":10254,"style":479},[454],[427,10256,10258,10261,10264,10267,10273,10276,10279],{"className":10257},[439],[427,10259],{"className":10260,"style":470},[443],[427,10262,979],{"className":10263},[448,449],[427,10265],{"className":10266,"style":489},[454],[427,10268,10270],{"className":10269},[814],[427,10271,4460],{"className":10272,"style":4459},[448,4458],[427,10274],{"className":10275,"style":489},[454],[427,10277,979],{"className":10278},[448,449],[427,10280,499],{"className":10281},[498]," with a Fibonacci\nheap, linear on dense graphs.",[1116,10284,10285],{},"Greedy is optimal here, a clean, provable success of the greedy paradigm.",[10287,10288,10291,10296],"section",{"className":10289,"dataFootnotes":376},[10290],"footnotes",[405,10292,10295],{"className":10293,"id":1960},[10294],"sr-only","Footnotes",[1113,10297,10298,10312,10358],{},[1116,10299,10301,10304,10305],{"id":10300},"user-content-fn-clrs-cut",[385,10302,10303],{},"CLRS",", Ch. 23 — Minimum Spanning Trees — the cut property identifying a safe edge for the greedy MST template. ",[398,10306,10311],{"href":10307,"ariaLabel":10308,"className":10309,"dataFootnoteBackref":376},"#user-content-fnref-clrs-cut","Back to reference 1",[10310],"data-footnote-backref","↩",[1116,10313,10315,10318,10319,10352,10353],{"id":10314},"user-content-fn-erickson-mst",[385,10316,10317],{},"Erickson",", Ch. 8 — Minimum Spanning Trees — Borůvka's component-merging rounds in ",[427,10320,10322],{"className":10321},[430],[427,10323,10325],{"className":10324,"ariaHidden":435},[434],[427,10326,10328,10331,10334,10337,10343,10346,10349],{"className":10327},[439],[427,10329],{"className":10330,"style":470},[443],[427,10332,4448],{"className":10333,"style":4447},[448,449],[427,10335,475],{"className":10336},[474],[427,10338,10340],{"className":10339},[814],[427,10341,4460],{"className":10342,"style":4459},[448,4458],[427,10344],{"className":10345,"style":489},[454],[427,10347,480],{"className":10348,"style":479},[448,449],[427,10350,499],{"className":10351},[498]," phases. ",[398,10354,10311],{"href":10355,"ariaLabel":10356,"className":10357,"dataFootnoteBackref":376},"#user-content-fnref-erickson-mst","Back to reference 2",[10310],[1116,10359,10361,10364,10365],{"id":10360},"user-content-fn-skiena-mst",[385,10362,10363],{},"Skiena",", §6 — Weighted Graph Algorithms — Prim's algorithm growing one tree via a priority queue. ",[398,10366,10311],{"href":10367,"ariaLabel":10368,"className":10369,"dataFootnoteBackref":376},"#user-content-fnref-skiena-mst","Back to reference 3",[10310],[10371,10372,10373],"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":10375},[10376,10377,10378,10379,10382,10383,10384,10385],{"id":407,"depth":18,"text":408},{"id":1884,"depth":18,"text":1885},{"id":4263,"depth":18,"text":4264},{"id":5025,"depth":18,"text":5026,"children":10380},[10381],{"id":6303,"depth":24,"text":6304},{"id":7381,"depth":18,"text":7382},{"id":9344,"depth":18,"text":9345},{"id":9753,"depth":18,"text":9754},{"id":1960,"depth":18,"text":10295},"Suppose you must lay cable to connect a set of towns, and every possible\nconnection has a known cost. You want all the towns linked — any town reachable\nfrom any other — while spending as little as possible. Laying a redundant link\nwould only waste money, so the cheapest solution can contain no cycle: it is a\ntree that spans every town. Finding the cheapest such tree is the\nminimum spanning tree problem, and it is the first place in this course\nwhere a greedy strategy is provably\noptimal.","md",{"moduleNumber":102,"lessonNumber":24,"order":10389},503,true,[10392,10396,10399,10403],{"title":10393,"slug":10394,"difficulty":10395},"Min Cost to Connect All Points","min-cost-to-connect-all-points","Medium",{"title":10397,"slug":10398,"difficulty":10395},"Path With Minimum Effort","path-with-minimum-effort",{"title":10400,"slug":10401,"difficulty":10402},"Swim in Rising Water","swim-in-rising-water","Hard",{"title":10404,"slug":10405,"difficulty":10402},"Find Critical and Pseudo-Critical Edges in MST","find-critical-and-pseudo-critical-edges-in-minimum-spanning-tree","---\ntitle: Minimum Spanning Trees\nmodule: Graphs\nmoduleNumber: 5\nlessonNumber: 3\norder: 503\nsummary: >-\n  Given a weighted network, how do we connect everything as cheaply as possible?\n  The answer is a minimum spanning tree. One lemma, the cut property, justifies\n  _every_ correct MST algorithm, and from it two famous greedy methods fall out:\n  Kruskal's, which grows a forest edge by edge with a union-find structure, and\n  Prim's, which grows a single tree using a priority queue.\ntopics: [Minimum Spanning Trees]\nsources:\n  - book: CLRS\n    ref: \"Ch. 23 — Minimum Spanning Trees\"\n  - book: Skiena\n    ref: \"§6 — Weighted Graph Algorithms\"\n  - book: Erickson\n    ref: \"Ch. 8 — Minimum Spanning Trees\"\npractice:\n  - title: 'Min Cost to Connect All Points'\n    slug: min-cost-to-connect-all-points\n    difficulty: Medium\n  - title: 'Path With Minimum Effort'\n    slug: path-with-minimum-effort\n    difficulty: Medium\n  - title: 'Swim in Rising Water'\n    slug: swim-in-rising-water\n    difficulty: Hard\n  - title: 'Find Critical and Pseudo-Critical Edges in MST'\n    slug: find-critical-and-pseudo-critical-edges-in-minimum-spanning-tree\n    difficulty: Hard\n---\n\nSuppose you must lay cable to connect a set of towns, and every possible\nconnection has a known cost. You want all the towns linked — any town reachable\nfrom any other — while spending as little as possible. Laying a redundant link\nwould only waste money, so the cheapest solution can contain no cycle: it is a\n**tree** that **spans** every town. Finding the cheapest such tree is the\n**minimum spanning tree** problem, and it is the first place in this course\nwhere a [_greedy_](\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method) strategy is provably\noptimal.\n\n## The problem\n\nStated precisely, in the shape the rest of this lesson will use:\n\n> **Input.** A weighted, _undirected_ graph $G = (V, E)$ with a weight function\n> $c : E \\to \\mathbb{R}$. Negative weights are allowed.\n>\n> **Precondition.** $G$ is connected.\n>\n> **Output.** A **minimum spanning tree** (MST) of $G$: a subset\n> $E' \\subseteq E$ such that $T = (V, E')$ is a tree and\n> $\\text{cost}(T) = \\sum_{e \\in E'} c_e$ is minimized.\n\nHere a **tree** means a _connected, acyclic_ graph, not to be confused with a\nrooted or rooted-and-ordered tree; an MST has no distinguished root. A spanning\ntree on $n = |V|$ vertices always has exactly $n - 1$ edges: enough to connect\neverything, one fewer than would create a cycle.\n\nIt is worth recalling _why_ \"tree\" is exactly the right object. For a graph\n$G = (V, E)$ the following are equivalent, and any one of them could serve as the\ndefinition.\n\n> **Theorem (Tree characterizations).**\n>\n> 1. $G$ is a tree (connected and acyclic).\n> 2. $G$ is connected and every edge is a **bridge** (an edge whose removal\n>    increases the number of connected components).\n> 3. $G$ is acyclic, and adding any edge in $\\binom{V}{2} \\setminus E$ creates a\n>    cycle.\n> 4. $G$ is connected and $|E| = |V| - 1$.\n> 5. $G$ is acyclic and $|E| = |V| - 1$.\n\nBelow, a weighted graph (left) and one of its minimum spanning trees (the\nthick colored edges):\n\n$$\n% caption: A weighted graph with one minimum spanning tree shown in thick edges.\n\\begin{tikzpicture}[\n  every node\u002F.style={circle, draw, minimum size=7mm, inner sep=0pt, font=\\small},\n  mst\u002F.style={line width=1.6pt, draw=blue!70!black},\n  wt\u002F.style={font=\\scriptsize, inner sep=1pt, draw=none}]\n  \\node (b) at (1.4,1.8) {$b$};\n  \\node (c) at (3.4,1.8) {$c$};\n  \\node (d) at (5.4,1.8) {$d$};\n  \\node (a) at (0.4,0.6) {$a$};\n  \\node (i) at (3.0,0.6) {$i$};\n  \\node (e) at (6.2,0.4) {$e$};\n  \\node (h) at (1.6,-0.7) {$h$};\n  \\node (g) at (3.4,-0.9) {$g$};\n  \\node (f) at (5.2,-0.7) {$f$};\n  % non-tree edges\n  \\draw (a) -- node[wt,left]{$8$} (h);\n  \\draw (b) -- node[wt,above]{$11$} (i);\n  \\draw (d) -- node[wt,left]{$14$} (f);\n  \\draw (e) -- node[wt,right]{$10$} (f);\n  \\draw (i) -- node[wt,left]{$7$} (h);\n  % tree edges (MST)\n  \\draw[mst] (a) -- node[wt,above left]{$4$} (b);\n  \\draw[mst] (b) -- node[wt,above]{$7$} (c);\n  \\draw[mst] (c) -- node[wt,above]{$7$} (d);\n  \\draw[mst] (c) -- node[wt,right]{$2$} (i);\n  \\draw[mst] (d) -- node[wt,above]{$9$} (e);\n  \\draw[mst] (i) -- node[wt,right]{$6$} (g);\n  \\draw[mst] (h) -- node[wt,below]{$1$} (g);\n  \\draw[mst] (g) -- node[wt,below]{$2$} (f);\n\\end{tikzpicture}\n$$\n\nThe thick tree spans all nine towns with total weight\n$1 + 2 + 2 + 4 + 6 + 7 + 7 + 9 = 38$; no spanning tree is cheaper.\n\n## The cut property\n\nAll three algorithms below are instances of a single greedy template: maintain a\nset $A$ of edges that is always a subset of _some_ MST, and at each step add one\nmore **safe** edge, meaning an edge that can be added to $A$ while keeping\n$A \\subseteq$ (some MST). The entire theory of MSTs reduces to recognizing safe\nedges, and one lemma does that for us.[^clrs-cut]\n\nFirst, the vocabulary. A **cut** $(S, V \\setminus S)$ is a partition of the\nvertices into two groups. An edge **crosses** the cut if its endpoints lie on\nopposite sides. A cut **respects** an edge set $A$ if no edge of $A$ crosses it.\nAn edge crossing the cut is **light** (or _cheapest_) if it has the minimum\nweight of all crossing edges.\n\n$$\n% caption: A cut splitting vertices into $S$ and its complement, with the light crossing\n%          edge highlighted.\n\\begin{tikzpicture}[\n  v\u002F.style={circle, fill=black, inner sep=1.6pt},\n  cross\u002F.style={line width=1.4pt, draw=red!75!black}]\n  % left blob S\n  \\draw[rounded corners=10pt, draw=gray] (-0.2,-1.1) rectangle (3.0,1.3);\n  \\node at (1.4,-1.45) {$S$};\n  \\node[v] (u) at (2.6,0.2) {};\n  \\node[below=1pt of u, font=\\scriptsize] {$u$};\n  \\node[v] (s1) at (0.4,0.7) {};\n  \\node[v] (s2) at (1.0,-0.4) {};\n  \\node[v] (s3) at (1.7,0.6) {};\n  \\draw (s1)--(s3); \\draw (s2)--(s3); \\draw (s3)--(u);\n  % right blob V\\S\n  \\draw[rounded corners=10pt, draw=gray] (4.0,-1.1) rectangle (7.2,1.3);\n  \\node at (5.6,-1.45) {$V \\setminus S$};\n  \\node[v] (vv) at (4.4,0.2) {};\n  \\node[above=1pt of vv, font=\\scriptsize] {$v$};\n  \\node[v] (t1) at (6.0,0.7) {};\n  \\node[v] (t2) at (6.8,-0.3) {};\n  \\node[v] (t3) at (5.5,-0.5) {};\n  \\draw (vv)--(t3); \\draw (t3)--(t2); \\draw (t1)--(t2);\n  % crossing light edge\n  \\draw[cross] (u) -- node[midway, above, font=\\scriptsize, draw=none]{light}\n    node[midway, below, font=\\scriptsize, draw=none]{$c_{uv}$ min} (vv);\n\\end{tikzpicture}\n$$\n\n> **Property (Cut rule).** Let $A$ be a subset of some MST of $G$. Let\n> $(S, V \\setminus S)$ be any cut that respects $A$, and let $e = (u, v)$ with\n> $u \\in S$, $v \\in V \\setminus S$ be a light edge crossing that cut. Then $e$ is\n> safe for $A$: there is an MST containing $A \\cup \\set{e}$.\n\n> **Proof (an exchange argument — done carefully).** Let $T^*$ be an MST\n> containing $A$, and suppose, for contradiction, that _no_ MST contains $e$. Since\n> $T^*$ is connected, there is at least one edge of $T^*$ crossing the cut\n> $(S, V \\setminus S)$.\n>\n> It is tempting to grab _any_ such crossing edge $f \\in T^*$, swap it for $e$, and\n> argue $T^* - f + e$ is a cheaper tree. **This is a mistake**: removing an\n> arbitrary crossing edge $f$ need not reconnect into a tree once we add $e$; the\n> result can be disconnected or contain a cycle. We must remove the _right_ edge.\n>\n> So instead: adding $e$ to $T^*$ creates a unique cycle, and because $e$ crosses\n> the cut, that cycle must cross back at some edge $g \\in T^*$, with $g$ also\n> crossing $(S, V \\setminus S)$. Because the cut **respects** $A$, this $g \\notin A$.\n> Form\n> $$\n> T' = T^* - \\set{g} + \\set{e}.\n> $$\n> Deleting $g$ breaks the unique cycle, so $T'$ is again a spanning tree, and it\n> still contains $A \\cup \\set{e}$. Since $e$ is the _light_ crossing edge,\n> $c_e \\le c_g$, hence\n> $$\n> \\text{cost}(T') = \\text{cost}(T^*) - c_g + c_e \\le \\text{cost}(T^*).\n> $$\n> But $T^*$ was minimum, so equality holds: $T'$ is _also_ an MST, and it contains\n> $e$. Contradiction. Therefore some MST contains $A \\cup \\set{e}$, i.e. $e$ is\n> safe. $\\quad\\blacksquare$\n\n$$\n% caption: The exchange argument: adding edge $e$ creates a cycle whose crossing edge $g$\n%          is swapped out.\n\\begin{tikzpicture}[\n  v\u002F.style={circle, fill=black, inner sep=1.6pt},\n  emph\u002F.style={line width=1.5pt, draw=acc},\n  swap\u002F.style={line width=1.5pt, draw=red!75!black, dashed}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[dashed, gray] (3.1,-1.4) -- (3.1,1.4);\n  \\node[v] (u) at (1.8,0.7) {}; \\node[above=1pt of u, font=\\scriptsize]{$u$};\n  \\node[v] (v) at (4.4,0.7) {}; \\node[above=1pt of v, font=\\scriptsize]{$v$};\n  \\node[v] (x) at (1.2,-0.6) {};\n  \\node[v] (y) at (4.9,-0.6) {};\n  \\node[v] (p) at (5.9,0.3) {};\n  % cycle: e (u-v) plus tree path back through g\n  \\draw[emph] (u) -- node[above, font=\\scriptsize, draw=none, fill=white, inner sep=1.5pt]{$e$} (v);\n  \\draw (u) -- (x);\n  \\draw[swap] (x) -- node[below, font=\\scriptsize, draw=none]{$g$} (y);\n  \\draw (y) -- (v);\n  \\draw (v) -- (p);\n  \\node[font=\\scriptsize] at (6.6,1.0) {$T^*$};\n\\end{tikzpicture}\n$$\n\nThe cut property is the engine. Every correct MST algorithm, whether Borůvka's,\nPrim's, or Kruskal's, is just a strategy for _choosing which cut to apply it\nto_, and **all three only ever add edges that obey the cut rule.**\n\n## Borůvka's algorithm\n\nThe oldest MST algorithm (Otakar Borůvka, 1926) is also the most directly cut-rule\ndriven, and it parallelises well.[^erickson-mst] The idea: every component, in every\nround, _simultaneously_ reaches out along its own cheapest outgoing edge.\n\nMaintain a forest $A$, initially the $n$ isolated vertices. In each **round**,\neach current component $C$ looks at the cut $(C, V \\setminus C)$, which respects\n$A$, and selects its lightest crossing edge. By the cut property every such edge\nis safe, so we add them all at once and merge the components they join. Because\nevery component at least halves in count each round (each one merges with at\nleast one neighbor), the number of components drops by a factor $\\ge 2$ per\nround, so only $O(\\log V)$ rounds are needed.\n\n```algorithm\ncaption: $\\textsc{Borůvka}(G, c)$ — every component grabs its cheapest exit edge\nnumber: 1\n$A \\gets \\emptyset$ \u002F\u002F $n$ singleton components\nwhile $A$ has more than one component do\n  foreach component $C$ of $(V, A)$ do\n    $e_C \\gets$ the lightest edge crossing $(C,\\, V \\setminus C)$ \u002F\u002F safe by cut rule\n  foreach distinct edge $e_C$ chosen do\n    $A \\gets A \\cup \\set{e_C}$ \u002F\u002F add all exit edges at once\nreturn $A$\n```\n\n> **Remark (A subtlety).** To keep the result acyclic when weights tie, break ties by a\n> fixed total order on edges (e.g. by index). Otherwise two components could pick\n> _each other's_ edge as different edges of equal weight and create a cycle.\n\n**Running time.** Each round scans all edges to find component-minimum exits in\n$O(E)$ time and there are $O(\\log V)$ rounds, so $\\textsc{Borůvka}$ runs in\n$O(E \\log V)$, the same headline bound as Prim and Kruskal, but with the\nuseful property that the per-round work is fully parallel.\n\nOne round on six isolated vertices shows the parallel grab. Each singleton (a\ncomponent of one) points an arrow along its own cheapest incident edge; the chosen\nedges, taken together, merge the six components down to two in a single sweep:\n\n$$\n% caption: One Borůvka round: every component (here singletons) selects its cheapest exit\n%          edge (red arrows). The five chosen edges merge six components into two.\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n  V\u002F.style={circle, draw, minimum size=8mm, font=\\small},\n  pick\u002F.style={->, line width=1.3pt, draw=red!75!black},\n  wt\u002F.style={font=\\scriptsize, inner sep=1.5pt, fill=white}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[V] (a) at (0,1.4) {$a$};\n  \\node[V] (b) at (2.2,1.4) {$b$};\n  \\node[V] (c) at (4.4,1.4) {$c$};\n  \\node[V] (d) at (0,0) {$d$};\n  \\node[V] (e) at (2.2,0) {$e$};\n  \\node[V] (f) at (4.4,0) {$f$};\n  % undirected edges with weights (thin grey)\n  \\draw[black!45] (a) -- node[wt]{$3$} (b);\n  \\draw[black!45] (b) -- node[wt]{$5$} (c);\n  \\draw[black!45] (a) -- node[wt]{$8$} (d);\n  \\draw[black!45] (b) -- node[wt]{$6$} (e);\n  \\draw[black!45] (c) -- node[wt]{$4$} (f);\n  \\draw[black!45] (d) -- node[wt]{$2$} (e);\n  \\draw[black!45] (e) -- node[wt]{$7$} (f);\n  % each component's cheapest exit (red), drawn slightly offset\n  \\draw[pick] (a) to[bend left=14] (b);   % a: cheapest 3\n  \\draw[pick] (b) to[bend left=14] (a);   % b: cheapest 3\n  \\draw[pick] (c) to[bend left=14] (f);   % c: cheapest 4\n  \\draw[pick] (d) to[bend left=14] (e);   % d: cheapest 2\n  \\draw[pick] (e) to[bend left=14] (d);   % e: cheapest 2\n  \\draw[pick] (f) to[bend left=14] (c);   % f: cheapest 4\n  \\node[font=\\scriptsize, red!75!black, align=center] at (6.5,0.7)\n    {chosen edges\\\\merge $6 \\to 2$\\\\components};\n\\end{tikzpicture}\n$$\n\n## Kruskal's algorithm\n\n$\\textsc{Kruskal}$'s strategy is global and edge-centric: consider the edges in order\nof increasing weight, and add each one _unless_ it would form a cycle. The set\n$A$ is a **forest** (an acyclic graph) that gradually coalesces into a single\ntree once $G$ is fully connected.\n\nWhy is each added edge safe? When Kruskal accepts the cheapest remaining edge\n$(u, v)$, it connects two different trees of the current forest. Take $S$ to be\nthe vertices of $u$'s tree. This cut respects $A$ (no forest edge crosses out of\na tree), and $(u, v)$ is the _lightest_ edge crossing it; any lighter crossing\nedge would have been considered earlier and would have joined the trees already.\nBy the cut property, $(u, v)$ is safe.\n\nThe cycle test, \"are $u$ and $v$ already in the same tree?\", is exactly the\nquestion $\\text{comp}(u) \\stackrel{?}{=} \\text{comp}(v)$: does adding $(u, v)$\nkeep the graph acyclic? This is what the\n[**disjoint-set** (union-find)](\u002Falgorithms\u002Fdata-structures\u002Funion-find) data\nstructure answers efficiently. It supports $\\textsc{Make-Set}$, $\\textsc{Find}$\n($\\text{comp}(x)$, which component is $x$ in?), and $\\textsc{Union}$ (merge two\ncomponents).\n\n```algorithm\ncaption: $\\textsc{Kruskal}(G, c)$ — grow a forest, cheapest safe edge first\nnumber: 2\n$A \\gets \\emptyset$\nforeach vertex $v \\in V$ do\n  call $\\textsc{Make-Set}(v)$\nsort the edges of $E$ into nondecreasing order by weight $c$\nforeach edge $(u, v) \\in E$ in sorted order do\n  if $\\textsc{Find}(u) \\neq \\textsc{Find}(v)$ then \u002F\u002F stays acyclic\n    $A \\gets A \\cup \\set{(u, v)}$ \u002F\u002F safe edge\n    call $\\textsc{Union}(u, v)$\nreturn $A$\n```\n\nRunning $\\textsc{Kruskal}$ on the nine-town graph above makes the accept\u002Freject\nrhythm visible. We scan the thirteen edges in nondecreasing weight; the first\neight that join _distinct_ trees are accepted, and the five that would close a\ncycle are skipped. The eight accepted edges are exactly the MST of weight $38$:\n\n$$\n% caption: Kruskal's edge-by-edge trace on the nine-town graph: accept an edge unless both\n%          ends already share a tree. Eight accepts total $38$.\n\\begin{tikzpicture}[font=\\small, y=6mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[font=\\footnotesize\\bfseries] at (0.0,1) {weight};\n  \\node[font=\\footnotesize\\bfseries] at (1.3,1) {edge};\n  \\node[font=\\footnotesize\\bfseries] at (3.1,1) {action};\n  \\draw[acc] (-0.5,0.5) -- (4.2,0.5);\n  \\foreach \\w\u002F\\e\u002F\\y in {1\u002F{h\\text{--}g}\u002F0, 2\u002F{c\\text{--}i}\u002F-1, 2\u002F{g\\text{--}f}\u002F-2,\n    4\u002F{a\\text{--}b}\u002F-3, 6\u002F{i\\text{--}g}\u002F-4, 7\u002F{b\\text{--}c}\u002F-5, 7\u002F{c\\text{--}d}\u002F-6, 9\u002F{d\\text{--}e}\u002F-9}\n  {\n    \\node[font=\\scriptsize] at (0.0,\\y) {$\\w$};\n    \\node[font=\\scriptsize] at (1.3,\\y) {$\\e$};\n    \\node[font=\\scriptsize, acc] at (3.1,\\y) {accept (safe)};\n  }\n  \\foreach \\w\u002F\\e\u002F\\y in {7\u002F{i\\text{--}h}\u002F-7, 8\u002F{a\\text{--}h}\u002F-8, 10\u002F{e\\text{--}f}\u002F-10, 11\u002F{b\\text{--}i}\u002F-11, 14\u002F{d\\text{--}f}\u002F-12}\n  {\n    \\node[font=\\scriptsize] at (0.0,\\y) {$\\w$};\n    \\node[font=\\scriptsize] at (1.3,\\y) {$\\e$};\n    \\node[font=\\scriptsize, gray] at (3.1,\\y) {reject (cycle)};\n  }\n  \\node[font=\\footnotesize, acc] at (1.8,-13.4) {total MST weight $= 38$};\n\\end{tikzpicture}\n$$\n\n### Making it fast: union by size\n\nThe simplest implementation keeps an array $\\text{comp}[1 \\ldots n]$\nwhere $\\text{comp}[v]$ names $v$'s current component. Then $\\textsc{Find}$ is\n$O(1)$, but a naive $\\textsc{Union}$ that relabels one whole side costs $O(n)$ in\nthe worst case. The fix is the classic **union-by-size** argument:\n\n- Alongside $\\text{comp}[\\cdot]$, keep $\\text{members}[c]$, a linked list of the\n  vertices currently in component $c$, so we can enumerate a component cheaply.\n- On $\\textsc{Union}(u, v)$, relabel the **smaller** component: choose the side\n  with $|\\text{comp}_u| \\le |\\text{comp}_v|$ and rewrite $\\text{comp}[x]$ for the\n  vertices $x$ in that smaller side.\n\nWhy this wins: whenever a vertex $x$ has its label $\\text{comp}[x]$ rewritten, it\nsat in the _smaller_ of the two merged components, so the component containing\n$x$ **at least doubles** in size. A vertex can be doubled at most $\\log_2 n$\ntimes before it is in a component of all $n$ vertices. Hence each vertex is\nrelabelled $O(\\log n)$ times across the whole run, and the total work spent\nupdating $\\text{comp}[\\cdot]$ over _all_ unions is $O(n \\log n)$.\n\n**Running time.** Sorting the edges dominates: $O(m \\log m) = O(m \\log n)$ since\n$m \\le n^2$. The union-find work adds only $O(n \\log n)$ with union by size (and\nis effectively linear, $O(m\\,\\alpha(n))$, with the inverse-Ackermann tricks of\nunion-by-rank plus path compression). So overall $\\textsc{Kruskal}$ runs in\n$$\nT(n, m) = O(m \\log n).\n$$\n\n## Prim's algorithm\n\n$\\textsc{Prim}$'s strategy is local and vertex-centric: grow a _single_ tree outward\nfrom an arbitrary root, repeatedly attaching the cheapest edge that links a tree\nvertex to a non-tree vertex. Here the set $A$ is always one connected tree.[^skiena-mst]\n\nSafety is again the cut property. Let $S$ be the vertices currently in the tree.\nThe cut $(S, V \\setminus S)$ respects $A$, and Prim deliberately picks the\n_lightest_ edge crossing it, a light edge, so the chosen edge is safe.\n\nThis is **exactly [$\\textsc{Dijkstra}$](\u002Falgorithms\u002Fgraphs\u002Fshortest-paths) with\nthe relaxation rule changed**. Where\nDijkstra keys a frontier vertex by $d[x] + c_{xv}$ (distance from the source),\nPrim keys it by $c_{xv}$ alone (cost to attach to the _finished_ set). Prim keeps\nevery non-tree vertex $v$ in a\n[**min-priority queue**](\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort) keyed by\n$\\text{key}[v]$,\nmaintaining the invariant\n$$\n\\forall v \\in Q : \\quad \\text{key}[v] = \\min_{x \\notin Q}\\; c_{xv}\n\\qquad\\text{(the cheapest way to connect $v$ to a finished vertex).}\n$$\nExtracting the minimum yields the next vertex to absorb; absorbing it may make\nits neighbors cheaper to reach, so we relax their keys with $\\textsc{Decrease-Key}$.\n\n```algorithm\ncaption: $\\textsc{Prim}(G, c, s)$ — grow one tree from an arbitrary start $s$\nnumber: 3\nforeach vertex $u \\in V$ do\n  $\\text{key}[u] \\gets \\infty$\n  $\\pi[u] \\gets \\text{nil}$\n$\\text{key}[s] \\gets 0$ \u002F\u002F start tree at $s$\n$Q \\gets V$ \u002F\u002F min-PQ keyed by $\\text{key}$\n$E' \\gets \\emptyset$\nwhile $Q \\neq \\emptyset$ do\n  $u \\gets \\textsc{Extract-Min}(Q)$ \u002F\u002F cheapest to attach\n  if $\\pi[u] \\neq \\text{nil}$ then\n    $E' \\gets E' \\cup \\set{(\\pi[u], u)}$ \u002F\u002F commit safe edge\n  foreach $v$ adjacent to $u$ with $v \\in Q$ do\n    if $c_{uv} \u003C \\text{key}[v]$ then\n      $\\pi[v] \\gets u$ \u002F\u002F $v$'s best link to tree\n      $\\text{key}[v] \\gets c_{uv}$ \u002F\u002F Decrease-Key\nreturn $E'$\n```\n\nA single Prim step looks like this. The tree $S = \\set{a, b, c}$ has been grown\nfrom root $a$; every frontier vertex carries a key equal to its cheapest edge\ninto $S$. The cut $(S, V \\setminus S)$ respects the tree, and $\\textsc{Extract-Min}$\nreturns the endpoint of the lightest crossing edge — here $c\\text{--}i$ at weight\n$2$ — which the cut property certifies as safe:\n\n$$\n% caption: A Prim snapshot: the grown tree $S$ (shaded, dashed box) and the frontier;\n%          $\\textsc{Extract-Min}$ pulls the lightest crossing edge ($c\\text{--}i$, weight\n%          $2$).\n\\begin{tikzpicture}[>=Stealth, font=\\small,\n  T\u002F.style={circle, draw, minimum size=8mm, fill=acc!25, font=\\small},\n  F\u002F.style={circle, draw, minimum size=8mm, font=\\small},\n  wt\u002F.style={font=\\scriptsize, inner sep=1pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[dashed, gray, rounded corners, fill=black!4] (-0.7,-0.7) rectangle (4.3,1.7);\n  \\node[font=\\scriptsize, gray] at (-0.25,1.4) {$S$};\n  \\node[T] (a) at (0,0.5) {$a$};\n  \\node[T] (b) at (1.7,1.0) {$b$};\n  \\node[T] (c) at (3.4,0.5) {$c$};\n  \\node[F] (i) at (3.2,-1.8) {$i$};\n  \\node[F] (d) at (6.0,1.0) {$d$};\n  \\node[F] (g) at (5.6,-1.8) {$g$};\n  \\draw[line width=1.4pt, acc] (a) -- node[wt, above left]{$4$} (b);\n  \\draw[line width=1.4pt, acc] (b) -- node[wt, above right]{$7$} (c);\n  \\draw[red!80!black, line width=1.3pt] (c) -- node[wt, left, fill=white, inner sep=1.5pt]{$2$} (i);\n  \\draw (c) -- node[wt, above]{$7$} (d);\n  \\draw (i) -- node[wt, below]{$6$} (g);\n  \\node[font=\\scriptsize, red!80!black] at (1.4,-1.8) {light edge: key$[i]=2$};\n\\end{tikzpicture}\n$$\n\n**Running time.** Prim performs $n$ $\\textsc{Extract-Min}$ operations and up to\n$m$ $\\textsc{Decrease-Key}$ operations, so in general\n$T(n, m) = O\\big(n\\,(T_{\\text{ins}} + T_{\\text{ext}}) + m\\,T_{\\text{dec}}\\big)$.\nWith a **binary heap** all three operations cost $O(\\log n)$, giving\n$O(m \\log n)$, the same as Kruskal. With a **Fibonacci heap**,\n$\\textsc{Decrease-Key}$ drops to $O(1)$ amortized while $\\textsc{Extract-Min}$\nstays $O(\\log n)$, improving the total to\n$$\nT(n, m) = O(m + n \\log n).\n$$\nNotice that when the graph is dense, $m = \\Omega(n \\log n)$, this is\n$O(m)$, that is, **linear**, and\n[asymptotically](\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis) the best known.\n\n## Which to use?\n\nThroughout, $n = |V|$ and $m = |E|$.\n\n| | Borůvka | Kruskal | Prim |\n| --- | --- | --- | --- |\n| Grows | every **component** at once | a **forest**, globally | a single **tree**, from a root |\n| Core structure | component scan \u002F union-find | union-find (union by size) | priority queue |\n| Headline time | $O(m \\log n)$ | $O(m \\log n)$ | $O(m \\log n)$ |\n| Best variant | parallel rounds | $O(m\\,\\alpha(n))$ work after sort | $O(m + n \\log n)$ (Fib. heap) |\n| Shines when | parallel \u002F distributed | edges already sorted; sparse | dense; adjacency-rich data |\n\nAll three are correct for the _same_ reason, the cut property, and all three are\ngreedy algorithms whose greediness is _provably_ optimal, a rarity worth\nsavoring. If edge weights are not distinct the MST may not be unique, but every\nchoice these algorithms make is still safe, so they always return _a_ minimum\nspanning tree. (The fastest known practical approaches combine Borůvka's rounds\nwith Prim\u002FKruskal to shave the bound further.)\n\n## Takeaways\n\n- A **minimum spanning tree** connects all $n$ vertices of a connected weighted\n  graph with exactly $n - 1$ edges of least total weight; \"tree\" means\n  _connected and acyclic_, no root.\n- The **cut property** (cut rule) is the one lemma behind every MST algorithm: a\n  _light_ edge crossing a cut that respects the current edge set is always\n  **safe**. The exchange argument must swap out the edge $g$ _on the cycle_\n  created by adding $e$; swapping an arbitrary crossing edge is a tempting\n  mistake.\n- $\\textsc{Borůvka}$ adds every component's cheapest exit edge each round; halving\n  the component count gives $O(\\log n)$ rounds and $O(m \\log n)$ total, with\n  naturally parallel rounds.\n- $\\textsc{Kruskal}$ adds edges cheapest-first, skipping cycle-forming ones, using\n  **union-find**; sorting dominates at $O(m \\log n)$, and **union by size** keeps\n  the relabelling work to $O(n \\log n)$ because each vertex's component can double\n  only $\\log n$ times.\n- $\\textsc{Prim}$ is $\\textsc{Dijkstra}$ rekeyed by attachment cost: grow one tree\n  from a start vertex, attaching the lightest crossing edge via a **priority\n  queue**; $O(m \\log n)$ with a binary heap, $O(m + n \\log n)$ with a Fibonacci\n  heap, linear on dense graphs.\n- Greedy is optimal here, a clean, provable success of the greedy paradigm.\n\n[^clrs-cut]: **CLRS**, Ch. 23 — Minimum Spanning Trees — the cut property identifying a safe edge for the greedy MST template.\n[^erickson-mst]: **Erickson**, Ch. 8 — Minimum Spanning Trees — Borůvka's component-merging rounds in $O(\\log V)$ phases.\n[^skiena-mst]: **Skiena**, §6 — Weighted Graph Algorithms — Prim's algorithm growing one tree via a priority queue.\n",{"text":10408,"minutes":10409,"time":10410,"words":10411},"11 min read",10.35,621000,2070,{"title":169,"description":10386},[10414,10416,10418],{"book":10303,"ref":10415},"Ch. 23 — Minimum Spanning Trees",{"book":10363,"ref":10417},"§6 — Weighted Graph Algorithms",{"book":10317,"ref":10419},"Ch. 8 — Minimum Spanning 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