[{"data":1,"prerenderedAt":11518},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":374,"course-wordcounts":11386,"ref-card-index":11442},[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":261,"blurb":376,"body":377,"description":11347,"extension":11348,"meta":11349,"module":231,"navigation":11351,"path":262,"practice":11352,"rawbody":11367,"readingTime":11368,"seo":11373,"sources":11374,"status":11382,"stem":11383,"summary":264,"topics":11384,"__hash__":11385},"course\u002F01.algorithms\u002F08.dynamic-programming\u002F06.interval-dp.md","",{"type":378,"value":379,"toc":11333},"minimark",[380,530,533,538,1170,1522,1662,2156,2881,3184,3189,3547,3629,4045,4686,5050,5230,5234,5237,5696,5703,5707,6358,6613,6699,7122,7183,7191,7202,7546,8086,8418,8840,8936,9253,9257,9643,9964,10018,10073,10283,10287,11180,11329],[381,382,383,384,388,389,419,420,424,425,466,467,470,471,489,490,520,521,524,525,529],"p",{},"The sequence DPs we have met so far walked a string or array from one end and let\nthe subproblem be a ",[385,386,387],"em",{},"prefix",": the state was ",[390,391,392,393,418],"q",{},"the best answer using the first ",[394,395,398],"span",{"className":396},[397],"katex",[394,399,403],{"className":400,"ariaHidden":402},[401],"katex-html","true",[394,404,407,412],{"className":405},[406],"base",[394,408],{"className":409,"style":411},[410],"strut","height:0.6595em;",[394,413,417],{"className":414},[415,416],"mord","mathnormal","i"," items."," A second, equally common family does not decompose by prefix at all. It\ndecomposes by ",[421,422,423],"strong",{},"contiguous range",". The natural subproblem is \"the best answer\nfor the slice ",[394,426,428],{"className":427},[397],[394,429,431],{"className":430,"ariaHidden":402},[401],[394,432,434,438,443,446,451,456,461],{"className":433},[406],[394,435],{"className":436,"style":437},[410],"height:1em;vertical-align:-0.25em;",[394,439,442],{"className":440},[441],"mopen","[",[394,444,417],{"className":445},[415,416],[394,447,450],{"className":448},[449],"mpunct",",",[394,452],{"className":453,"style":455},[454],"mspace","margin-right:0.1667em;",[394,457,460],{"className":458,"style":459},[415,416],"margin-right:0.0572em;","j",[394,462,465],{"className":463},[464],"mclose","]",",\" and the recurrence builds a long range out of two shorter\nranges by guessing where they meet: a ",[421,468,469],{},"split point"," ",[394,472,474],{"className":473},[397],[394,475,477],{"className":476,"ariaHidden":402},[401],[394,478,480,484],{"className":479},[406],[394,481],{"className":482,"style":483},[410],"height:0.6944em;",[394,485,488],{"className":486,"style":487},[415,416],"margin-right:0.0315em;","k"," inside ",[394,491,493],{"className":492},[397],[394,494,496],{"className":495,"ariaHidden":402},[401],[394,497,499,502,505,508,511,514,517],{"className":498},[406],[394,500],{"className":501,"style":437},[410],[394,503,442],{"className":504},[441],[394,506,417],{"className":507},[415,416],[394,509,450],{"className":510},[449],[394,512],{"className":513,"style":455},[454],[394,515,460],{"className":516,"style":459},[415,416],[394,518,465],{"className":519},[464],". This\nis ",[421,522,523],{},"interval dynamic programming",", one of the ",[526,527,528],"a",{"href":236},"core DP\npatterns",", and once you see its\nshape you will find it everywhere: parenthesising a product, building an optimal\nsearch tree, bursting balloons, cutting a stick, partitioning a string.",[381,531,532],{},"The archetype, and the cleanest place to learn the pattern, is the problem of\nparenthesising a matrix product.",[534,535,537],"h2",{"id":536},"matrix-chain-multiplication","Matrix-chain multiplication",[381,539,540,541,580,581,617,618,637,638,803,804,807,808,811,812,827,828,881,882,1001,1002,1160,1161],{},"Multiplying a ",[394,542,544],{"className":543},[397],[394,545,547,569],{"className":546,"ariaHidden":402},[401],[394,548,550,554,557,561,566],{"className":549},[406],[394,551],{"className":552,"style":553},[410],"height:0.7778em;vertical-align:-0.1944em;",[394,555,381],{"className":556},[415,416],[394,558],{"className":559,"style":560},[454],"margin-right:0.2222em;",[394,562,565],{"className":563},[564],"mbin","×",[394,567],{"className":568,"style":560},[454],[394,570,572,576],{"className":571},[406],[394,573],{"className":574,"style":575},[410],"height:0.625em;vertical-align:-0.1944em;",[394,577,390],{"className":578,"style":579},[415,416],"margin-right:0.0359em;"," matrix by a ",[394,582,584],{"className":583},[397],[394,585,587,605],{"className":586,"ariaHidden":402},[401],[394,588,590,593,596,599,602],{"className":589},[406],[394,591],{"className":592,"style":553},[410],[394,594,390],{"className":595,"style":579},[415,416],[394,597],{"className":598,"style":560},[454],[394,600,565],{"className":601},[564],[394,603],{"className":604,"style":560},[454],[394,606,608,612],{"className":607},[406],[394,609],{"className":610,"style":611},[410],"height:0.4306em;",[394,613,616],{"className":614,"style":615},[415,416],"margin-right:0.0278em;","r"," matrix takes ",[394,619,621],{"className":620},[397],[394,622,624],{"className":623,"ariaHidden":402},[401],[394,625,627,630,634],{"className":626},[406],[394,628],{"className":629,"style":575},[410],[394,631,633],{"className":632,"style":579},[415,416],"pq",[394,635,616],{"className":636,"style":615},[415,416]," scalar\nmultiplications. Matrix multiplication is associative, so for a chain\n",[394,639,641],{"className":640},[397],[394,642,644],{"className":643,"ariaHidden":402},[401],[394,645,647,651,709,750,753,758,761],{"className":646},[406],[394,648],{"className":649,"style":650},[410],"height:0.8333em;vertical-align:-0.15em;",[394,652,654,658],{"className":653},[415],[394,655,657],{"className":656},[415,416],"A",[394,659,662],{"className":660},[661],"msupsub",[394,663,667,700],{"className":664},[665,666],"vlist-t","vlist-t2",[394,668,671,695],{"className":669},[670],"vlist-r",[394,672,676],{"className":673,"style":675},[674],"vlist","height:0.3011em;",[394,677,679,684],{"style":678},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[394,680],{"className":681,"style":683},[682],"pstrut","height:2.7em;",[394,685,691],{"className":686},[687,688,689,690],"sizing","reset-size6","size3","mtight",[394,692,694],{"className":693},[415,690],"1",[394,696,699],{"className":697},[698],"vlist-s","​",[394,701,703],{"className":702},[670],[394,704,707],{"className":705,"style":706},[674],"height:0.15em;",[394,708],{},[394,710,712,715],{"className":711},[415],[394,713,657],{"className":714},[415,416],[394,716,718],{"className":717},[661],[394,719,721,742],{"className":720},[665,666],[394,722,724,739],{"className":723},[670],[394,725,727],{"className":726,"style":675},[674],[394,728,729,732],{"style":678},[394,730],{"className":731,"style":683},[682],[394,733,735],{"className":734},[687,688,689,690],[394,736,738],{"className":737},[415,690],"2",[394,740,699],{"className":741},[698],[394,743,745],{"className":744},[670],[394,746,748],{"className":747,"style":706},[674],[394,749],{},[394,751],{"className":752,"style":455},[454],[394,754,757],{"className":755},[756],"minner","⋯",[394,759],{"className":760,"style":455},[454],[394,762,764,767],{"className":763},[415],[394,765,657],{"className":766},[415,416],[394,768,770],{"className":769},[661],[394,771,773,795],{"className":772},[665,666],[394,774,776,792],{"className":775},[670],[394,777,780],{"className":778,"style":779},[674],"height:0.1514em;",[394,781,782,785],{"style":678},[394,783],{"className":784,"style":683},[682],[394,786,788],{"className":787},[687,688,689,690],[394,789,791],{"className":790},[415,416,690],"n",[394,793,699],{"className":794},[698],[394,796,798],{"className":797},[670],[394,799,801],{"className":800,"style":706},[674],[394,802],{}," the ",[385,805,806],{},"answer"," is the same however we parenthesise, but the\n",[385,809,810],{},"cost"," is not. Given a chain of ",[394,813,815],{"className":814},[397],[394,816,818],{"className":817,"ariaHidden":402},[401],[394,819,821,824],{"className":820},[406],[394,822],{"className":823,"style":611},[410],[394,825,791],{"className":826},[415,416]," matrices where ",[394,829,831],{"className":830},[397],[394,832,834],{"className":833,"ariaHidden":402},[401],[394,835,837,840],{"className":836},[406],[394,838],{"className":839,"style":650},[410],[394,841,843,846],{"className":842},[415],[394,844,657],{"className":845},[415,416],[394,847,849],{"className":848},[661],[394,850,852,873],{"className":851},[665,666],[394,853,855,870],{"className":854},[670],[394,856,859],{"className":857,"style":858},[674],"height:0.3117em;",[394,860,861,864],{"style":678},[394,862],{"className":863,"style":683},[682],[394,865,867],{"className":866},[687,688,689,690],[394,868,417],{"className":869},[415,416,690],[394,871,699],{"className":872},[698],[394,874,876],{"className":875},[670],[394,877,879],{"className":878,"style":706},[674],[394,880],{}," has dimensions\n",[394,883,885],{"className":884},[397],[394,886,888,955],{"className":887,"ariaHidden":402},[401],[394,889,891,895,946,949,952],{"className":890},[406],[394,892],{"className":893,"style":894},[410],"height:0.7917em;vertical-align:-0.2083em;",[394,896,898,901],{"className":897},[415],[394,899,381],{"className":900},[415,416],[394,902,904],{"className":903},[661],[394,905,907,937],{"className":906},[665,666],[394,908,910,934],{"className":909},[670],[394,911,913],{"className":912,"style":858},[674],[394,914,915,918],{"style":678},[394,916],{"className":917,"style":683},[682],[394,919,921],{"className":920},[687,688,689,690],[394,922,924,927,931],{"className":923},[415,690],[394,925,417],{"className":926},[415,416,690],[394,928,930],{"className":929},[564,690],"−",[394,932,694],{"className":933},[415,690],[394,935,699],{"className":936},[698],[394,938,940],{"className":939},[670],[394,941,944],{"className":942,"style":943},[674],"height:0.2083em;",[394,945],{},[394,947],{"className":948,"style":560},[454],[394,950,565],{"className":951},[564],[394,953],{"className":954,"style":560},[454],[394,956,958,961],{"className":957},[406],[394,959],{"className":960,"style":575},[410],[394,962,964,967],{"className":963},[415],[394,965,381],{"className":966},[415,416],[394,968,970],{"className":969},[661],[394,971,973,993],{"className":972},[665,666],[394,974,976,990],{"className":975},[670],[394,977,979],{"className":978,"style":858},[674],[394,980,981,984],{"style":678},[394,982],{"className":983,"style":683},[682],[394,985,987],{"className":986},[687,688,689,690],[394,988,417],{"className":989},[415,416,690],[394,991,699],{"className":992},[698],[394,994,996],{"className":995},[670],[394,997,999],{"className":998,"style":706},[674],[394,1000],{}," (so the dimension sequence is ",[394,1003,1005],{"className":1004},[397],[394,1006,1008],{"className":1007,"ariaHidden":402},[401],[394,1009,1011,1014,1055,1058,1061,1101,1104,1107,1111,1114,1117,1120],{"className":1010},[406],[394,1012],{"className":1013,"style":575},[410],[394,1015,1017,1020],{"className":1016},[415],[394,1018,381],{"className":1019},[415,416],[394,1021,1023],{"className":1022},[661],[394,1024,1026,1047],{"className":1025},[665,666],[394,1027,1029,1044],{"className":1028},[670],[394,1030,1032],{"className":1031,"style":675},[674],[394,1033,1034,1037],{"style":678},[394,1035],{"className":1036,"style":683},[682],[394,1038,1040],{"className":1039},[687,688,689,690],[394,1041,1043],{"className":1042},[415,690],"0",[394,1045,699],{"className":1046},[698],[394,1048,1050],{"className":1049},[670],[394,1051,1053],{"className":1052,"style":706},[674],[394,1054],{},[394,1056,450],{"className":1057},[449],[394,1059],{"className":1060,"style":455},[454],[394,1062,1064,1067],{"className":1063},[415],[394,1065,381],{"className":1066},[415,416],[394,1068,1070],{"className":1069},[661],[394,1071,1073,1093],{"className":1072},[665,666],[394,1074,1076,1090],{"className":1075},[670],[394,1077,1079],{"className":1078,"style":675},[674],[394,1080,1081,1084],{"style":678},[394,1082],{"className":1083,"style":683},[682],[394,1085,1087],{"className":1086},[687,688,689,690],[394,1088,694],{"className":1089},[415,690],[394,1091,699],{"className":1092},[698],[394,1094,1096],{"className":1095},[670],[394,1097,1099],{"className":1098,"style":706},[674],[394,1100],{},[394,1102,450],{"className":1103},[449],[394,1105],{"className":1106,"style":455},[454],[394,1108,1110],{"className":1109},[756],"…",[394,1112],{"className":1113,"style":455},[454],[394,1115,450],{"className":1116},[449],[394,1118],{"className":1119,"style":455},[454],[394,1121,1123,1126],{"className":1122},[415],[394,1124,381],{"className":1125},[415,416],[394,1127,1129],{"className":1128},[661],[394,1130,1132,1152],{"className":1131},[665,666],[394,1133,1135,1149],{"className":1134},[670],[394,1136,1138],{"className":1137,"style":779},[674],[394,1139,1140,1143],{"style":678},[394,1141],{"className":1142,"style":683},[682],[394,1144,1146],{"className":1145},[687,688,689,690],[394,1147,791],{"className":1148},[415,416,690],[394,1150,699],{"className":1151},[698],[394,1153,1155],{"className":1154},[670],[394,1156,1158],{"className":1157,"style":706},[674],[394,1159],{},"), we\nwant the parenthesisation that minimizes the total number of scalar\nmultiplications.",[1162,1163,1164],"sup",{},[526,1165,694],{"href":1166,"ariaDescribedBy":1167,"dataFootnoteRef":376,"id":1169},"#user-content-fn-clrs-matrix",[1168],"footnote-label","user-content-fnref-clrs-matrix",[381,1171,1172,1173,1176,1177,1280,1281,1296,1297,1513,1514,1517,1518,1521],{},"The number of parenthesisations grows like the Catalan numbers, exponentially,\nso brute force is hopeless. But the problem has the two features every DP needs.\n",[421,1174,1175],{},"Optimal substructure",": in the optimal parenthesisation of ",[394,1178,1180],{"className":1179},[397],[394,1181,1183],{"className":1182,"ariaHidden":402},[401],[394,1184,1186,1190,1230,1233,1236,1239],{"className":1185},[406],[394,1187],{"className":1188,"style":1189},[410],"height:0.9694em;vertical-align:-0.2861em;",[394,1191,1193,1196],{"className":1192},[415],[394,1194,657],{"className":1195},[415,416],[394,1197,1199],{"className":1198},[661],[394,1200,1202,1222],{"className":1201},[665,666],[394,1203,1205,1219],{"className":1204},[670],[394,1206,1208],{"className":1207,"style":858},[674],[394,1209,1210,1213],{"style":678},[394,1211],{"className":1212,"style":683},[682],[394,1214,1216],{"className":1215},[687,688,689,690],[394,1217,417],{"className":1218},[415,416,690],[394,1220,699],{"className":1221},[698],[394,1223,1225],{"className":1224},[670],[394,1226,1228],{"className":1227,"style":706},[674],[394,1229],{},[394,1231],{"className":1232,"style":455},[454],[394,1234,757],{"className":1235},[756],[394,1237],{"className":1238,"style":455},[454],[394,1240,1242,1245],{"className":1241},[415],[394,1243,657],{"className":1244},[415,416],[394,1246,1248],{"className":1247},[661],[394,1249,1251,1271],{"className":1250},[665,666],[394,1252,1254,1268],{"className":1253},[670],[394,1255,1257],{"className":1256,"style":858},[674],[394,1258,1259,1262],{"style":678},[394,1260],{"className":1261,"style":683},[682],[394,1263,1265],{"className":1264},[687,688,689,690],[394,1266,460],{"className":1267,"style":459},[415,416,690],[394,1269,699],{"className":1270},[698],[394,1272,1274],{"className":1273},[670],[394,1275,1278],{"className":1276,"style":1277},[674],"height:0.2861em;",[394,1279],{},",\nthe outermost multiplication splits the chain at some ",[394,1282,1284],{"className":1283},[397],[394,1285,1287],{"className":1286,"ariaHidden":402},[401],[394,1288,1290,1293],{"className":1289},[406],[394,1291],{"className":1292,"style":483},[410],[394,1294,488],{"className":1295,"style":487},[415,416]," into\n",[394,1298,1300],{"className":1299},[397],[394,1301,1303],{"className":1302,"ariaHidden":402},[401],[394,1304,1306,1310,1314,1354,1357,1360,1363,1404,1408,1411,1461,1464,1467,1470,1510],{"className":1305},[406],[394,1307],{"className":1308,"style":1309},[410],"height:1.0361em;vertical-align:-0.2861em;",[394,1311,1313],{"className":1312},[441],"(",[394,1315,1317,1320],{"className":1316},[415],[394,1318,657],{"className":1319},[415,416],[394,1321,1323],{"className":1322},[661],[394,1324,1326,1346],{"className":1325},[665,666],[394,1327,1329,1343],{"className":1328},[670],[394,1330,1332],{"className":1331,"style":858},[674],[394,1333,1334,1337],{"style":678},[394,1335],{"className":1336,"style":683},[682],[394,1338,1340],{"className":1339},[687,688,689,690],[394,1341,417],{"className":1342},[415,416,690],[394,1344,699],{"className":1345},[698],[394,1347,1349],{"className":1348},[670],[394,1350,1352],{"className":1351,"style":706},[674],[394,1353],{},[394,1355],{"className":1356,"style":455},[454],[394,1358,757],{"className":1359},[756],[394,1361],{"className":1362,"style":455},[454],[394,1364,1366,1369],{"className":1365},[415],[394,1367,657],{"className":1368},[415,416],[394,1370,1372],{"className":1371},[661],[394,1373,1375,1396],{"className":1374},[665,666],[394,1376,1378,1393],{"className":1377},[670],[394,1379,1382],{"className":1380,"style":1381},[674],"height:0.3361em;",[394,1383,1384,1387],{"style":678},[394,1385],{"className":1386,"style":683},[682],[394,1388,1390],{"className":1389},[687,688,689,690],[394,1391,488],{"className":1392,"style":487},[415,416,690],[394,1394,699],{"className":1395},[698],[394,1397,1399],{"className":1398},[670],[394,1400,1402],{"className":1401,"style":706},[674],[394,1403],{},[394,1405,1407],{"className":1406},[464],")",[394,1409,1313],{"className":1410},[441],[394,1412,1414,1417],{"className":1413},[415],[394,1415,657],{"className":1416},[415,416],[394,1418,1420],{"className":1419},[661],[394,1421,1423,1453],{"className":1422},[665,666],[394,1424,1426,1450],{"className":1425},[670],[394,1427,1429],{"className":1428,"style":1381},[674],[394,1430,1431,1434],{"style":678},[394,1432],{"className":1433,"style":683},[682],[394,1435,1437],{"className":1436},[687,688,689,690],[394,1438,1440,1443,1447],{"className":1439},[415,690],[394,1441,488],{"className":1442,"style":487},[415,416,690],[394,1444,1446],{"className":1445},[564,690],"+",[394,1448,694],{"className":1449},[415,690],[394,1451,699],{"className":1452},[698],[394,1454,1456],{"className":1455},[670],[394,1457,1459],{"className":1458,"style":943},[674],[394,1460],{},[394,1462],{"className":1463,"style":455},[454],[394,1465,757],{"className":1466},[756],[394,1468],{"className":1469,"style":455},[454],[394,1471,1473,1476],{"className":1472},[415],[394,1474,657],{"className":1475},[415,416],[394,1477,1479],{"className":1478},[661],[394,1480,1482,1502],{"className":1481},[665,666],[394,1483,1485,1499],{"className":1484},[670],[394,1486,1488],{"className":1487,"style":858},[674],[394,1489,1490,1493],{"style":678},[394,1491],{"className":1492,"style":683},[682],[394,1494,1496],{"className":1495},[687,688,689,690],[394,1497,460],{"className":1498,"style":459},[415,416,690],[394,1500,699],{"className":1501},[698],[394,1503,1505],{"className":1504},[670],[394,1506,1508],{"className":1507,"style":1277},[674],[394,1509],{},[394,1511,1407],{"className":1512},[464],", and ",[385,1515,1516],{},"each side must itself be optimally\nparenthesised",": if a cheaper parenthesisation of a side existed, substituting it\nwould lower the total. ",[421,1519,1520],{},"Overlapping subproblems",": both sides are again\ncontiguous subchains, and the same subchain is reached by many different outer\nchoices.",[381,1523,1524,1525,1559,1560,1661],{},"Let ",[394,1526,1528],{"className":1527},[397],[394,1529,1531],{"className":1530,"ariaHidden":402},[401],[394,1532,1534,1537,1541,1544,1547,1550,1553,1556],{"className":1533},[406],[394,1535],{"className":1536,"style":437},[410],[394,1538,1540],{"className":1539},[415,416],"m",[394,1542,442],{"className":1543},[441],[394,1545,417],{"className":1546},[415,416],[394,1548,450],{"className":1549},[449],[394,1551],{"className":1552,"style":455},[454],[394,1554,460],{"className":1555,"style":459},[415,416],[394,1557,465],{"className":1558},[464]," be the minimum number of scalar multiplications needed to compute\n",[394,1561,1563],{"className":1562},[397],[394,1564,1566],{"className":1565,"ariaHidden":402},[401],[394,1567,1569,1572,1612,1615,1618,1621],{"className":1568},[406],[394,1570],{"className":1571,"style":1189},[410],[394,1573,1575,1578],{"className":1574},[415],[394,1576,657],{"className":1577},[415,416],[394,1579,1581],{"className":1580},[661],[394,1582,1584,1604],{"className":1583},[665,666],[394,1585,1587,1601],{"className":1586},[670],[394,1588,1590],{"className":1589,"style":858},[674],[394,1591,1592,1595],{"style":678},[394,1593],{"className":1594,"style":683},[682],[394,1596,1598],{"className":1597},[687,688,689,690],[394,1599,417],{"className":1600},[415,416,690],[394,1602,699],{"className":1603},[698],[394,1605,1607],{"className":1606},[670],[394,1608,1610],{"className":1609,"style":706},[674],[394,1611],{},[394,1613],{"className":1614,"style":455},[454],[394,1616,757],{"className":1617},[756],[394,1619],{"className":1620,"style":455},[454],[394,1622,1624,1627],{"className":1623},[415],[394,1625,657],{"className":1626},[415,416],[394,1628,1630],{"className":1629},[661],[394,1631,1633,1653],{"className":1632},[665,666],[394,1634,1636,1650],{"className":1635},[670],[394,1637,1639],{"className":1638,"style":858},[674],[394,1640,1641,1644],{"style":678},[394,1642],{"className":1643,"style":683},[682],[394,1645,1647],{"className":1646},[687,688,689,690],[394,1648,460],{"className":1649,"style":459},[415,416,690],[394,1651,699],{"className":1652},[698],[394,1654,1656],{"className":1655},[670],[394,1657,1659],{"className":1658,"style":1277},[674],[394,1660],{},". A single matrix needs no multiplications, and otherwise we try\nevery split:",[394,1663,1666],{"className":1664},[1665],"katex-display",[394,1667,1669],{"className":1668},[397],[394,1670,1672,1711],{"className":1671,"ariaHidden":402},[401],[394,1673,1675,1678,1681,1684,1687,1690,1693,1696,1699,1703,1708],{"className":1674},[406],[394,1676],{"className":1677,"style":437},[410],[394,1679,1540],{"className":1680},[415,416],[394,1682,442],{"className":1683},[441],[394,1685,417],{"className":1686},[415,416],[394,1688,450],{"className":1689},[449],[394,1691],{"className":1692,"style":455},[454],[394,1694,460],{"className":1695,"style":459},[415,416],[394,1697,465],{"className":1698},[464],[394,1700],{"className":1701,"style":1702},[454],"margin-right:0.2778em;",[394,1704,1707],{"className":1705},[1706],"mrel","=",[394,1709],{"className":1710,"style":1702},[454],[394,1712,1714,1718],{"className":1713},[406],[394,1715],{"className":1716,"style":1717},[410],"height:3.28em;vertical-align:-1.39em;",[394,1719,1721,1732,2152],{"className":1720},[756],[394,1722,1726],{"className":1723,"style":1725},[441,1724],"delimcenter","top:0em;",[394,1727,1731],{"className":1728},[1729,1730],"delimsizing","size4","{",[394,1733,1735],{"className":1734},[415],[394,1736,1739,2071,2076],{"className":1737},[1738],"mtable",[394,1740,1743],{"className":1741},[1742],"col-align-l",[394,1744,1746,2062],{"className":1745},[665,666],[394,1747,1749,2059],{"className":1748},[670],[394,1750,1753,1766],{"className":1751,"style":1752},[674],"height:1.89em;",[394,1754,1756,1760],{"style":1755},"top:-3.89em;",[394,1757],{"className":1758,"style":1759},[682],"height:3.008em;",[394,1761,1763],{"className":1762},[415],[394,1764,1043],{"className":1765},[415],[394,1767,1769,1772],{"style":1768},"top:-2.05em;",[394,1770],{"className":1771,"style":1759},[682],[394,1773,1775,1839,1846,1849,1852,1855,1858,1861,1864,1867,1870,1873,1876,1879,1882,1885,1888,1891,1894,1897,1900,1903,1906,1909,1912,1915,1918,1967,1970,2010,2013,2053],{"className":1774},[415],[394,1776,1779,1787],{"className":1777},[1778],"mop",[394,1780,1782],{"className":1781},[1778],[394,1783,1786],{"className":1784},[415,1785],"mathrm","min",[394,1788,1790],{"className":1789},[661],[394,1791,1793,1831],{"className":1792},[665,666],[394,1794,1796,1828],{"className":1795},[670],[394,1797,1799],{"className":1798,"style":1381},[674],[394,1800,1802,1805],{"style":1801},"top:-2.55em;margin-right:0.05em;",[394,1803],{"className":1804,"style":683},[682],[394,1806,1808],{"className":1807},[687,688,689,690],[394,1809,1811,1814,1818,1821,1825],{"className":1810},[415,690],[394,1812,417],{"className":1813},[415,416,690],[394,1815,1817],{"className":1816},[1706,690],"≤",[394,1819,488],{"className":1820,"style":487},[415,416,690],[394,1822,1824],{"className":1823},[1706,690],"\u003C",[394,1826,460],{"className":1827,"style":459},[415,416,690],[394,1829,699],{"className":1830},[698],[394,1832,1834],{"className":1833},[670],[394,1835,1837],{"className":1836,"style":1277},[674],[394,1838],{},[394,1840,1842],{"className":1841},[441],[394,1843,1313],{"className":1844},[1729,1845],"size1",[394,1847,1540],{"className":1848},[415,416],[394,1850,442],{"className":1851},[441],[394,1853,417],{"className":1854},[415,416],[394,1856,450],{"className":1857},[449],[394,1859],{"className":1860,"style":455},[454],[394,1862,488],{"className":1863,"style":487},[415,416],[394,1865,465],{"className":1866},[464],[394,1868],{"className":1869,"style":560},[454],[394,1871,1446],{"className":1872},[564],[394,1874],{"className":1875,"style":560},[454],[394,1877,1540],{"className":1878},[415,416],[394,1880,442],{"className":1881},[441],[394,1883,488],{"className":1884,"style":487},[415,416],[394,1886],{"className":1887,"style":560},[454],[394,1889,1446],{"className":1890},[564],[394,1892],{"className":1893,"style":560},[454],[394,1895,694],{"className":1896},[415],[394,1898,450],{"className":1899},[449],[394,1901],{"className":1902,"style":455},[454],[394,1904,460],{"className":1905,"style":459},[415,416],[394,1907,465],{"className":1908},[464],[394,1910],{"className":1911,"style":560},[454],[394,1913,1446],{"className":1914},[564],[394,1916],{"className":1917,"style":560},[454],[394,1919,1921,1924],{"className":1920},[415],[394,1922,381],{"className":1923},[415,416],[394,1925,1927],{"className":1926},[661],[394,1928,1930,1959],{"className":1929},[665,666],[394,1931,1933,1956],{"className":1932},[670],[394,1934,1936],{"className":1935,"style":858},[674],[394,1937,1938,1941],{"style":678},[394,1939],{"className":1940,"style":683},[682],[394,1942,1944],{"className":1943},[687,688,689,690],[394,1945,1947,1950,1953],{"className":1946},[415,690],[394,1948,417],{"className":1949},[415,416,690],[394,1951,930],{"className":1952},[564,690],[394,1954,694],{"className":1955},[415,690],[394,1957,699],{"className":1958},[698],[394,1960,1962],{"className":1961},[670],[394,1963,1965],{"className":1964,"style":943},[674],[394,1966],{},[394,1968],{"className":1969,"style":455},[454],[394,1971,1973,1976],{"className":1972},[415],[394,1974,381],{"className":1975},[415,416],[394,1977,1979],{"className":1978},[661],[394,1980,1982,2002],{"className":1981},[665,666],[394,1983,1985,1999],{"className":1984},[670],[394,1986,1988],{"className":1987,"style":1381},[674],[394,1989,1990,1993],{"style":678},[394,1991],{"className":1992,"style":683},[682],[394,1994,1996],{"className":1995},[687,688,689,690],[394,1997,488],{"className":1998,"style":487},[415,416,690],[394,2000,699],{"className":2001},[698],[394,2003,2005],{"className":2004},[670],[394,2006,2008],{"className":2007,"style":706},[674],[394,2009],{},[394,2011],{"className":2012,"style":455},[454],[394,2014,2016,2019],{"className":2015},[415],[394,2017,381],{"className":2018},[415,416],[394,2020,2022],{"className":2021},[661],[394,2023,2025,2045],{"className":2024},[665,666],[394,2026,2028,2042],{"className":2027},[670],[394,2029,2031],{"className":2030,"style":858},[674],[394,2032,2033,2036],{"style":678},[394,2034],{"className":2035,"style":683},[682],[394,2037,2039],{"className":2038},[687,688,689,690],[394,2040,460],{"className":2041,"style":459},[415,416,690],[394,2043,699],{"className":2044},[698],[394,2046,2048],{"className":2047},[670],[394,2049,2051],{"className":2050,"style":1277},[674],[394,2052],{},[394,2054,2056],{"className":2055},[464],[394,2057,1407],{"className":2058},[1729,1845],[394,2060,699],{"className":2061},[698],[394,2063,2065],{"className":2064},[670],[394,2066,2069],{"className":2067,"style":2068},[674],"height:1.39em;",[394,2070],{},[394,2072],{"className":2073,"style":2075},[2074],"arraycolsep","width:1em;",[394,2077,2079],{"className":2078},[1742],[394,2080,2082,2144],{"className":2081},[665,666],[394,2083,2085,2141],{"className":2084},[670],[394,2086,2088,2114],{"className":2087,"style":1752},[674],[394,2089,2090,2093],{"style":1755},[394,2091],{"className":2092,"style":1759},[682],[394,2094,2096,2099,2102,2105,2108,2111],{"className":2095},[415],[394,2097,417],{"className":2098},[415,416],[394,2100],{"className":2101,"style":1702},[454],[394,2103,1707],{"className":2104},[1706],[394,2106],{"className":2107,"style":1702},[454],[394,2109,460],{"className":2110,"style":459},[415,416],[394,2112,450],{"className":2113},[449],[394,2115,2116,2119],{"style":1768},[394,2117],{"className":2118,"style":1759},[682],[394,2120,2122,2125,2128,2131,2134,2137],{"className":2121},[415],[394,2123,417],{"className":2124},[415,416],[394,2126],{"className":2127,"style":1702},[454],[394,2129,1824],{"className":2130},[1706],[394,2132],{"className":2133,"style":1702},[454],[394,2135,460],{"className":2136,"style":459},[415,416],[394,2138,2140],{"className":2139},[415],".",[394,2142,699],{"className":2143},[698],[394,2145,2147],{"className":2146},[670],[394,2148,2150],{"className":2149,"style":2068},[674],[394,2151],{},[394,2153],{"className":2154},[464,2155],"nulldelimiter",[381,2157,2158,2159,2301,2302,2403,2404,2520,2521,2631,2632,2739,2740,2140],{},"The term 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is the cost of the final multiplication: the left\nblock 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at a split ",[394,3169,3171],{"className":3170},[397],[394,3172,3174],{"className":3173,"ariaHidden":402},[401],[394,3175,3177,3180],{"className":3176},[406],[394,3178],{"className":3179,"style":483},[410],[394,3181,488],{"className":3182,"style":487},[415,416],"; the chosen split is highlighted",[3185,3186,3188],"h3",{"id":3187},"filling-the-table","Filling the table",[381,3190,3191,3192,3225,3226,3229,3230,3260,3261,3352,3353,3423,3424,3495,3496,3530,3531,3546],{},"The recurrence for ",[394,3193,3195],{"className":3194},[397],[394,3196,3198],{"className":3197,"ariaHidden":402},[401],[394,3199,3201,3204,3207,3210,3213,3216,3219,3222],{"className":3200},[406],[394,3202],{"className":3203,"style":437},[410],[394,3205,1540],{"className":3206},[415,416],[394,3208,442],{"className":3209},[441],[394,3211,417],{"className":3212},[415,416],[394,3214,450],{"className":3215},[449],[394,3217],{"className":3218,"style":455},[454],[394,3220,460],{"className":3221,"style":459},[415,416],[394,3223,465],{"className":3224},[464]," depends only on intervals ",[385,3227,3228],{},"strictly shorter"," than\n",[394,3231,3233],{"className":3232},[397],[394,3234,3236],{"className":3235,"ariaHidden":402},[401],[394,3237,3239,3242,3245,3248,3251,3254,3257],{"className":3238},[406],[394,3240],{"className":3241,"style":437},[410],[394,3243,442],{"className":3244},[441],[394,3246,417],{"className":3247},[415,416],[394,3249,450],{"className":3250},[449],[394,3252],{"className":3253,"style":455},[454],[394,3255,460],{"className":3256,"style":459},[415,416],[394,3258,465],{"className":3259},[464]," (the left side has length ",[394,3262,3264],{"className":3263},[397],[394,3265,3267,3286,3305,3324,3343],{"className":3266,"ariaHidden":402},[401],[394,3268,3270,3274,3277,3280,3283],{"className":3269},[406],[394,3271],{"className":3272,"style":3273},[410],"height:0.7778em;vertical-align:-0.0833em;",[394,3275,488],{"className":3276,"style":487},[415,416],[394,3278],{"className":3279,"style":560},[454],[394,3281,930],{"className":3282},[564],[394,3284],{"className":3285,"style":560},[454],[394,3287,3289,3293,3296,3299,3302],{"className":3288},[406],[394,3290],{"className":3291,"style":3292},[410],"height:0.7429em;vertical-align:-0.0833em;",[394,3294,417],{"className":3295},[415,416],[394,3297],{"className":3298,"style":560},[454],[394,3300,1446],{"className":3301},[564],[394,3303],{"className":3304,"style":560},[454],[394,3306,3308,3312,3315,3318,3321],{"className":3307},[406],[394,3309],{"className":3310,"style":3311},[410],"height:0.7804em;vertical-align:-0.136em;",[394,3313,694],{"className":3314},[415],[394,3316],{"className":3317,"style":1702},[454],[394,3319,1817],{"className":3320},[1706],[394,3322],{"className":3323,"style":1702},[454],[394,3325,3327,3331,3334,3337,3340],{"className":3326},[406],[394,3328],{"className":3329,"style":3330},[410],"height:0.854em;vertical-align:-0.1944em;",[394,3332,460],{"className":3333,"style":459},[415,416],[394,3335],{"className":3336,"style":560},[454],[394,3338,930],{"className":3339},[564],[394,3341],{"className":3342,"style":560},[454],[394,3344,3346,3349],{"className":3345},[406],[394,3347],{"className":3348,"style":411},[410],[394,3350,417],{"className":3351},[415,416],", the right side\n",[394,3354,3356],{"className":3355},[397],[394,3357,3359,3377,3396,3414],{"className":3358,"ariaHidden":402},[401],[394,3360,3362,3365,3368,3371,3374],{"className":3361},[406],[394,3363],{"className":3364,"style":3330},[410],[394,3366,460],{"className":3367,"style":459},[415,416],[394,3369],{"className":3370,"style":560},[454],[394,3372,930],{"className":3373},[564],[394,3375],{"className":3376,"style":560},[454],[394,3378,3380,3384,3387,3390,3393],{"className":3379},[406],[394,3381],{"className":3382,"style":3383},[410],"height:0.8304em;vertical-align:-0.136em;",[394,3385,488],{"className":3386,"style":487},[415,416],[394,3388],{"className":3389,"style":1702},[454],[394,3391,1817],{"className":3392},[1706],[394,3394],{"className":3395,"style":1702},[454],[394,3397,3399,3402,3405,3408,3411],{"className":3398},[406],[394,3400],{"className":3401,"style":3330},[410],[394,3403,460],{"className":3404,"style":459},[415,416],[394,3406],{"className":3407,"style":560},[454],[394,3409,930],{"className":3410},[564],[394,3412],{"className":3413,"style":560},[454],[394,3415,3417,3420],{"className":3416},[406],[394,3418],{"className":3419,"style":411},[410],[394,3421,417],{"className":3422},[415,416],"). So if we fill the table in order of increasing interval length\n",[394,3425,3427],{"className":3426},[397],[394,3428,3430,3449,3467,3485],{"className":3429,"ariaHidden":402},[401],[394,3431,3433,3436,3440,3443,3446],{"className":3432},[406],[394,3434],{"className":3435,"style":483},[410],[394,3437,3439],{"className":3438},[415],"ℓ",[394,3441],{"className":3442,"style":1702},[454],[394,3444,1707],{"className":3445},[1706],[394,3447],{"className":3448,"style":1702},[454],[394,3450,3452,3455,3458,3461,3464],{"className":3451},[406],[394,3453],{"className":3454,"style":3330},[410],[394,3456,460],{"className":3457,"style":459},[415,416],[394,3459],{"className":3460,"style":560},[454],[394,3462,930],{"className":3463},[564],[394,3465],{"className":3466,"style":560},[454],[394,3468,3470,3473,3476,3479,3482],{"className":3469},[406],[394,3471],{"className":3472,"style":3292},[410],[394,3474,417],{"className":3475},[415,416],[394,3477],{"className":3478,"style":560},[454],[394,3480,1446],{"className":3481},[564],[394,3483],{"className":3484,"style":560},[454],[394,3486,3488,3492],{"className":3487},[406],[394,3489],{"className":3490,"style":3491},[410],"height:0.6444em;",[394,3493,694],{"className":3494},[415],", every value we read is already computed. We also record, in a\nsplit table ",[394,3497,3499],{"className":3498},[397],[394,3500,3502],{"className":3501,"ariaHidden":402},[401],[394,3503,3505,3508,3512,3515,3518,3521,3524,3527],{"className":3504},[406],[394,3506],{"className":3507,"style":437},[410],[394,3509,3511],{"className":3510},[415,416],"s",[394,3513,442],{"className":3514},[441],[394,3516,417],{"className":3517},[415,416],[394,3519,450],{"className":3520},[449],[394,3522],{"className":3523,"style":455},[454],[394,3525,460],{"className":3526,"style":459},[415,416],[394,3528,465],{"className":3529},[464],", the ",[394,3532,3534],{"className":3533},[397],[394,3535,3537],{"className":3536,"ariaHidden":402},[401],[394,3538,3540,3543],{"className":3539},[406],[394,3541],{"className":3542,"style":483},[410],[394,3544,488],{"className":3545,"style":487},[415,416]," that achieved the minimum, so we can reconstruct the\nparenthesisation afterwards.",[3548,3549,3553],"pre",{"className":3550,"code":3551,"language":3552,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Matrix-Chain-Order}(p)$ — fill $m$, $s$ by increasing chain length\n$n \\gets \\text{length}(p) - 1$\nfor $i \\gets 1$ to $n$ do\n  $m[i,i] \\gets 0$\nfor $\\ell \\gets 2$ to $n$ do            \u002F\u002F $\\ell$ = chain length\n  for $i \\gets 1$ to $n - \\ell + 1$ do\n    $j \\gets i + \\ell - 1$\n    $m[i,j] \\gets \\infty$\n    for $k \\gets i$ to $j - 1$ do        \u002F\u002F try every split\n      $q \\gets m[i,k] + m[k+1,j] + p_{i-1}\\cdot p_k \\cdot p_j$\n      if $q \u003C m[i,j]$ then\n        $m[i,j] \\gets q$\n        $s[i,j] \\gets k$\nreturn $m,\\ s$\n","algorithm",[3554,3555,3556,3562,3567,3572,3577,3582,3587,3592,3597,3602,3607,3612,3617,3623],"code",{"__ignoreMap":376},[394,3557,3559],{"class":3558,"line":6},"line",[394,3560,3561],{},"caption: $\\textsc{Matrix-Chain-Order}(p)$ — fill $m$, $s$ by increasing chain length\n",[394,3563,3564],{"class":3558,"line":18},[394,3565,3566],{},"$n \\gets \\text{length}(p) - 1$\n",[394,3568,3569],{"class":3558,"line":24},[394,3570,3571],{},"for $i \\gets 1$ to $n$ do\n",[394,3573,3574],{"class":3558,"line":73},[394,3575,3576],{},"  $m[i,i] \\gets 0$\n",[394,3578,3579],{"class":3558,"line":102},[394,3580,3581],{},"for $\\ell \\gets 2$ to $n$ do            \u002F\u002F $\\ell$ = chain length\n",[394,3583,3584],{"class":3558,"line":108},[394,3585,3586],{},"  for $i \\gets 1$ to $n - \\ell + 1$ do\n",[394,3588,3589],{"class":3558,"line":116},[394,3590,3591],{},"    $j \\gets i + \\ell - 1$\n",[394,3593,3594],{"class":3558,"line":196},[394,3595,3596],{},"    $m[i,j] \\gets \\infty$\n",[394,3598,3599],{"class":3558,"line":202},[394,3600,3601],{},"    for $k \\gets i$ to $j - 1$ do        \u002F\u002F try every split\n",[394,3603,3604],{"class":3558,"line":283},[394,3605,3606],{},"      $q \\gets m[i,k] + m[k+1,j] + p_{i-1}\\cdot p_k \\cdot p_j$\n",[394,3608,3609],{"class":3558,"line":333},[394,3610,3611],{},"      if $q \u003C m[i,j]$ then\n",[394,3613,3614],{"class":3558,"line":354},[394,3615,3616],{},"        $m[i,j] \\gets q$\n",[394,3618,3620],{"class":3558,"line":3619},13,[394,3621,3622],{},"        $s[i,j] \\gets k$\n",[394,3624,3626],{"class":3558,"line":3625},14,[394,3627,3628],{},"return $m,\\ s$\n",[381,3630,3631,3632,3686,3687,3712,3713,3728,3729,3780,3781,3831,3832,3865,3866,3881,3882,3915,3916,3968,3969,4044],{},"There are ",[394,3633,3635],{"className":3634},[397],[394,3636,3638],{"className":3637,"ariaHidden":402},[401],[394,3639,3641,3645,3649,3652,3683],{"className":3640},[406],[394,3642],{"className":3643,"style":3644},[410],"height:1.0641em;vertical-align:-0.25em;",[394,3646,3648],{"className":3647},[415],"Θ",[394,3650,1313],{"className":3651},[441],[394,3653,3655,3658],{"className":3654},[415],[394,3656,791],{"className":3657},[415,416],[394,3659,3661],{"className":3660},[661],[394,3662,3664],{"className":3663},[665],[394,3665,3667],{"className":3666},[670],[394,3668,3671],{"className":3669,"style":3670},[674],"height:0.8141em;",[394,3672,3674,3677],{"style":3673},"top:-3.063em;margin-right:0.05em;",[394,3675],{"className":3676,"style":683},[682],[394,3678,3680],{"className":3679},[687,688,689,690],[394,3681,738],{"className":3682},[415,690],[394,3684,1407],{"className":3685},[464]," entries and each costs ",[394,3688,3690],{"className":3689},[397],[394,3691,3693],{"className":3692,"ariaHidden":402},[401],[394,3694,3696,3699,3703,3706,3709],{"className":3695},[406],[394,3697],{"className":3698,"style":437},[410],[394,3700,3702],{"className":3701,"style":615},[415,416],"O",[394,3704,1313],{"className":3705},[441],[394,3707,791],{"className":3708},[415,416],[394,3710,1407],{"className":3711},[464]," to fill (the inner loop over\n",[394,3714,3716],{"className":3715},[397],[394,3717,3719],{"className":3718,"ariaHidden":402},[401],[394,3720,3722,3725],{"className":3721},[406],[394,3723],{"className":3724,"style":483},[410],[394,3726,488],{"className":3727,"style":487},[415,416],"), so the algorithm runs in ",[394,3730,3732],{"className":3731},[397],[394,3733,3735],{"className":3734,"ariaHidden":402},[401],[394,3736,3738,3741,3744,3747,3777],{"className":3737},[406],[394,3739],{"className":3740,"style":3644},[410],[394,3742,3648],{"className":3743},[415],[394,3745,1313],{"className":3746},[441],[394,3748,3750,3753],{"className":3749},[415],[394,3751,791],{"className":3752},[415,416],[394,3754,3756],{"className":3755},[661],[394,3757,3759],{"className":3758},[665],[394,3760,3762],{"className":3761},[670],[394,3763,3765],{"className":3764,"style":3670},[674],[394,3766,3767,3770],{"style":3673},[394,3768],{"className":3769,"style":683},[682],[394,3771,3773],{"className":3772},[687,688,689,690],[394,3774,3776],{"className":3775},[415,690],"3",[394,3778,1407],{"className":3779},[464]," time and ",[394,3782,3784],{"className":3783},[397],[394,3785,3787],{"className":3786,"ariaHidden":402},[401],[394,3788,3790,3793,3796,3799,3828],{"className":3789},[406],[394,3791],{"className":3792,"style":3644},[410],[394,3794,3648],{"className":3795},[415],[394,3797,1313],{"className":3798},[441],[394,3800,3802,3805],{"className":3801},[415],[394,3803,791],{"className":3804},[415,416],[394,3806,3808],{"className":3807},[661],[394,3809,3811],{"className":3810},[665],[394,3812,3814],{"className":3813},[670],[394,3815,3817],{"className":3816,"style":3670},[674],[394,3818,3819,3822],{"style":3673},[394,3820],{"className":3821,"style":683},[682],[394,3823,3825],{"className":3824},[687,688,689,690],[394,3826,738],{"className":3827},[415,690],[394,3829,1407],{"className":3830},[464]," space. The\nanswer is ",[394,3833,3835],{"className":3834},[397],[394,3836,3838],{"className":3837,"ariaHidden":402},[401],[394,3839,3841,3844,3847,3850,3853,3856,3859,3862],{"className":3840},[406],[394,3842],{"className":3843,"style":437},[410],[394,3845,1540],{"className":3846},[415,416],[394,3848,442],{"className":3849},[441],[394,3851,694],{"className":3852},[415],[394,3854,450],{"className":3855},[449],[394,3857],{"className":3858,"style":455},[454],[394,3860,791],{"className":3861},[415,416],[394,3863,465],{"className":3864},[464],"; the optimal parenthesisation is read back from ",[394,3867,3869],{"className":3868},[397],[394,3870,3872],{"className":3871,"ariaHidden":402},[401],[394,3873,3875,3878],{"className":3874},[406],[394,3876],{"className":3877,"style":611},[410],[394,3879,3511],{"className":3880},[415,416]," by\nrecursing: ",[394,3883,3885],{"className":3884},[397],[394,3886,3888],{"className":3887,"ariaHidden":402},[401],[394,3889,3891,3894,3897,3900,3903,3906,3909,3912],{"className":3890},[406],[394,3892],{"className":3893,"style":437},[410],[394,3895,3511],{"className":3896},[415,416],[394,3898,442],{"className":3899},[441],[394,3901,694],{"className":3902},[415],[394,3904,450],{"className":3905},[449],[394,3907],{"className":3908,"style":455},[454],[394,3910,791],{"className":3911},[415,416],[394,3913,465],{"className":3914},[464]," gives the outermost split, then ",[394,3917,3919],{"className":3918},[397],[394,3920,3922],{"className":3921,"ariaHidden":402},[401],[394,3923,3925,3928,3931,3934,3937,3940,3943,3946,3949,3952,3955,3958,3961,3964],{"className":3924},[406],[394,3926],{"className":3927,"style":437},[410],[394,3929,3511],{"className":3930},[415,416],[394,3932,442],{"className":3933},[441],[394,3935,694],{"className":3936},[415],[394,3938,450],{"className":3939},[449],[394,3941],{"className":3942,"style":455},[454],[394,3944],{"className":3945,"style":455},[454],[394,3947,3511],{"className":3948},[415,416],[394,3950,442],{"className":3951},[441],[394,3953,694],{"className":3954},[415],[394,3956,450],{"className":3957},[449],[394,3959],{"className":3960,"style":455},[454],[394,3962,791],{"className":3963},[415,416],[394,3965,3967],{"className":3966},[464],"]]"," and\n",[394,3970,3972],{"className":3971},[397],[394,3973,3975,4020],{"className":3974,"ariaHidden":402},[401],[394,3976,3978,3981,3984,3987,3990,3993,3996,3999,4002,4005,4008,4011,4014,4017],{"className":3977},[406],[394,3979],{"className":3980,"style":437},[410],[394,3982,3511],{"className":3983},[415,416],[394,3985,442],{"className":3986},[441],[394,3988],{"className":3989,"style":455},[454],[394,3991,3511],{"className":3992},[415,416],[394,3994,442],{"className":3995},[441],[394,3997,694],{"className":3998},[415],[394,4000,450],{"className":4001},[449],[394,4003],{"className":4004,"style":455},[454],[394,4006,791],{"className":4007},[415,416],[394,4009,465],{"className":4010},[464],[394,4012],{"className":4013,"style":560},[454],[394,4015,1446],{"className":4016},[564],[394,4018],{"className":4019,"style":560},[454],[394,4021,4023,4026,4029,4032,4035,4038,4041],{"className":4022},[406],[394,4024],{"className":4025,"style":437},[410],[394,4027,694],{"className":4028},[415],[394,4030,450],{"className":4031},[449],[394,4033],{"className":4034,"style":455},[454],[394,4036],{"className":4037,"style":455},[454],[394,4039,791],{"className":4040},[415,416],[394,4042,465],{"className":4043},[464]," give the next ones, and so on down to single matrices.",[2882,4046,4048,4424],{"className":4047},[2885,2886],[2888,4049,4053],{"xmlns":2890,"width":4050,"height":4051,"viewBox":4052},"328.505","197.652","-75 -75 246.379 148.239",[2895,4054,4055,4100,4103,4138,4141,4176,4179,4213,4216,4251,4280,4283,4286,4313,4316,4319],{"stroke":2897,"style":2898},[2895,4056,4058,4061],{"fill":4057},"var(--tk-soft-accent)",[2900,4059],{"d":4060},"M44.253-26.97h59.524v-22.763H44.253Z",[2895,4062,4063,4070,4076,4082,4088,4094],{"fill":2897,"stroke":2902,"fontSize":2963},[2895,4064,4066],{"transform":4065},"translate(87.382 -83.108)",[2900,4067],{"d":4068,"fill":2897,"stroke":2897,"className":4069,"style":2971},"M-39.058 46.836Q-39.058 46.783-39.049 46.748L-38.381 44.080Q-38.328 43.882-38.328 43.685Q-38.328 43.289-38.592 43.289Q-38.878 43.289-39.012 43.612Q-39.146 43.935-39.264 44.441Q-39.282 44.524-39.357 44.524L-39.462 44.524Q-39.510 44.524-39.532 44.485Q-39.554 44.445-39.554 44.405Q-39.392 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41.013Q-26.391 41.013-26.358 41.046Q-26.325 41.079-26.325 41.114L-26.325 46.396Q-26.325 46.691-25.174 46.691",[2911],[2895,4077,4078],{"transform":4065},[2900,4079],{"d":4080,"fill":2897,"stroke":2897,"className":4081,"style":2971},"M-23.551 48.611Q-23.551 48.571-23.516 48.536Q-23.191 48.224-23.011 47.818Q-22.830 47.411-22.830 46.963L-22.830 46.880Q-22.971 47.007-23.173 47.007Q-23.318 47.007-23.432 46.941Q-23.547 46.875-23.613 46.763Q-23.679 46.651-23.679 46.502Q-23.679 46.282-23.538 46.141Q-23.397 46.001-23.173 46.001Q-22.852 46.001-22.712 46.299Q-22.571 46.598-22.571 46.963Q-22.571 47.473-22.775 47.928Q-22.980 48.382-23.345 48.734Q-23.380 48.752-23.406 48.752Q-23.463 48.752-23.507 48.708Q-23.551 48.664-23.551 48.611",[2911],[2895,4083,4084],{"transform":4065},[2900,4085],{"d":4086,"fill":2897,"stroke":2897,"className":4087,"style":2971},"M-17.643 45.530L-20.082 45.530L-20.082 45.214L-17.256 41.066Q-17.212 41.013-17.146 41.013L-16.992 41.013Q-16.953 41.013-16.920 41.046Q-16.887 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upper-triangular ",[394,5001,5003],{"className":5002},[397],[394,5004,5006],{"className":5005,"ariaHidden":402},[401],[394,5007,5009,5012],{"className":5008},[406],[394,5010],{"className":5011,"style":611},[410],[394,5013,1540],{"className":5014},[415,416]," table, filled along diagonals of increasing length; cell ",[394,5017,5019],{"className":5018},[397],[394,5020,5022],{"className":5021,"ariaHidden":402},[401],[394,5023,5025,5028,5031,5034,5037,5040,5043,5046],{"className":5024},[406],[394,5026],{"className":5027,"style":437},[410],[394,5029,1540],{"className":5030},[415,416],[394,5032,442],{"className":5033},[441],[394,5035,417],{"className":5036},[415,416],[394,5038,450],{"className":5039},[449],[394,5041],{"className":5042,"style":455},[454],[394,5044,460],{"className":5045,"style":459},[415,416],[394,5047,465],{"className":5048},[464]," depends on its row to the left and its column below",[381,5051,5052,5053,5086,5087,5102,5103,5136,5137,5140,5141,5174,5175,5140,5178,5229],{},"The diagonal ",[394,5054,5056],{"className":5055},[397],[394,5057,5059,5077],{"className":5058,"ariaHidden":402},[401],[394,5060,5062,5065,5068,5071,5074],{"className":5061},[406],[394,5063],{"className":5064,"style":411},[410],[394,5066,417],{"className":5067},[415,416],[394,5069],{"className":5070,"style":1702},[454],[394,5072,1707],{"className":5073},[1706],[394,5075],{"className":5076,"style":1702},[454],[394,5078,5080,5083],{"className":5079},[406],[394,5081],{"className":5082,"style":3330},[410],[394,5084,460],{"className":5085,"style":459},[415,416]," (length ",[394,5088,5090],{"className":5089},[397],[394,5091,5093],{"className":5092,"ariaHidden":402},[401],[394,5094,5096,5099],{"className":5095},[406],[394,5097],{"className":5098,"style":3491},[410],[394,5100,694],{"className":5101},[415],") is the base case; each successive diagonal\nmoving toward the top-right corner holds longer intervals, and cell ",[394,5104,5106],{"className":5105},[397],[394,5107,5109],{"className":5108,"ariaHidden":402},[401],[394,5110,5112,5115,5118,5121,5124,5127,5130,5133],{"className":5111},[406],[394,5113],{"className":5114,"style":437},[410],[394,5116,1540],{"className":5117},[415,416],[394,5119,442],{"className":5120},[441],[394,5122,417],{"className":5123},[415,416],[394,5125,450],{"className":5126},[449],[394,5128],{"className":5129,"style":455},[454],[394,5131,460],{"className":5132,"style":459},[415,416],[394,5134,465],{"className":5135},[464],"\ndraws on cells in its own ",[421,5138,5139],{},"row to the left"," (the ",[394,5142,5144],{"className":5143},[397],[394,5145,5147],{"className":5146,"ariaHidden":402},[401],[394,5148,5150,5153,5156,5159,5162,5165,5168,5171],{"className":5149},[406],[394,5151],{"className":5152,"style":437},[410],[394,5154,1540],{"className":5155},[415,416],[394,5157,442],{"className":5158},[441],[394,5160,417],{"className":5161},[415,416],[394,5163,450],{"className":5164},[449],[394,5166],{"className":5167,"style":455},[454],[394,5169,488],{"className":5170,"style":487},[415,416],[394,5172,465],{"className":5173},[464]," terms) and its own\n",[421,5176,5177],{},"column below",[394,5179,5181],{"className":5180},[397],[394,5182,5184,5208],{"className":5183,"ariaHidden":402},[401],[394,5185,5187,5190,5193,5196,5199,5202,5205],{"className":5186},[406],[394,5188],{"className":5189,"style":437},[410],[394,5191,1540],{"className":5192},[415,416],[394,5194,442],{"className":5195},[441],[394,5197,488],{"className":5198,"style":487},[415,416],[394,5200],{"className":5201,"style":560},[454],[394,5203,1446],{"className":5204},[564],[394,5206],{"className":5207,"style":560},[454],[394,5209,5211,5214,5217,5220,5223,5226],{"className":5210},[406],[394,5212],{"className":5213,"style":437},[410],[394,5215,694],{"className":5216},[415],[394,5218,450],{"className":5219},[449],[394,5221],{"className":5222,"style":455},[454],[394,5224,460],{"className":5225,"style":459},[415,416],[394,5227,465],{"className":5228},[464]," terms). That dependency shape, left along the\nrow and down the column, is the signature of an interval DP.",[534,5231,5233],{"id":5232},"the-interval-dp-recipe","The interval-DP recipe",[381,5235,5236],{},"Strip matrix-chain of its specifics and a reusable pattern remains.",[5238,5239,5241,5278,5551],"callout",{"type":5240},"remark",[381,5242,5243,5246,5247,5277],{},[421,5244,5245],{},"Remark (The interval-DP recipe)."," Let the state be a contiguous range ",[394,5248,5250],{"className":5249},[397],[394,5251,5253],{"className":5252,"ariaHidden":402},[401],[394,5254,5256,5259,5262,5265,5268,5271,5274],{"className":5255},[406],[394,5257],{"className":5258,"style":437},[410],[394,5260,442],{"className":5261},[441],[394,5263,417],{"className":5264},[415,416],[394,5266,450],{"className":5267},[449],[394,5269],{"className":5270,"style":455},[454],[394,5272,460],{"className":5273,"style":459},[415,416],[394,5275,465],{"className":5276},[464]," of the input.",[5279,5280,5281,5356,5394,5511],"ol",{},[5282,5283,5284,470,5287,5325,5326,2140],"li",{},[421,5285,5286],{},"Subproblem.",[394,5288,5290],{"className":5289},[397],[394,5291,5293],{"className":5292,"ariaHidden":402},[401],[394,5294,5296,5299,5307,5310,5313,5316,5319,5322],{"className":5295},[406],[394,5297],{"className":5298,"style":437},[410],[394,5300,5303],{"className":5301},[415,5302],"text",[394,5304,5306],{"className":5305},[415],"dp",[394,5308,442],{"className":5309},[441],[394,5311,417],{"className":5312},[415,416],[394,5314,450],{"className":5315},[449],[394,5317],{"className":5318,"style":455},[454],[394,5320,460],{"className":5321,"style":459},[415,416],[394,5323,465],{"className":5324},[464]," = the optimum over the slice ",[394,5327,5329],{"className":5328},[397],[394,5330,5332],{"className":5331,"ariaHidden":402},[401],[394,5333,5335,5338,5341,5344,5347,5350,5353],{"className":5334},[406],[394,5336],{"className":5337,"style":437},[410],[394,5339,442],{"className":5340},[441],[394,5342,417],{"className":5343},[415,416],[394,5345,450],{"className":5346},[449],[394,5348],{"className":5349,"style":455},[454],[394,5351,460],{"className":5352,"style":459},[415,416],[394,5354,465],{"className":5355},[464],[5282,5357,5358,5361,5362,5377,5378,5393],{},[421,5359,5360],{},"Base case."," Length-",[394,5363,5365],{"className":5364},[397],[394,5366,5368],{"className":5367,"ariaHidden":402},[401],[394,5369,5371,5374],{"className":5370},[406],[394,5372],{"className":5373,"style":3491},[410],[394,5375,694],{"className":5376},[415]," (or length-",[394,5379,5381],{"className":5380},[397],[394,5382,5384],{"className":5383,"ariaHidden":402},[401],[394,5385,5387,5390],{"className":5386},[406],[394,5388],{"className":5389,"style":3491},[410],[394,5391,1043],{"className":5392},[415],") intervals are trivial.",[5282,5395,5396,5399,5400,470,5403,489,5418,5448,5449,5464,5465,5464,5480,5495,5496,2140],{},[421,5397,5398],{},"Recurrence."," Choose a ",[421,5401,5402],{},"split or pivot",[394,5404,5406],{"className":5405},[397],[394,5407,5409],{"className":5408,"ariaHidden":402},[401],[394,5410,5412,5415],{"className":5411},[406],[394,5413],{"className":5414,"style":483},[410],[394,5416,488],{"className":5417,"style":487},[415,416],[394,5419,5421],{"className":5420},[397],[394,5422,5424],{"className":5423,"ariaHidden":402},[401],[394,5425,5427,5430,5433,5436,5439,5442,5445],{"className":5426},[406],[394,5428],{"className":5429,"style":437},[410],[394,5431,442],{"className":5432},[441],[394,5434,417],{"className":5435},[415,416],[394,5437,450],{"className":5438},[449],[394,5440],{"className":5441,"style":455},[454],[394,5443,460],{"className":5444,"style":459},[415,416],[394,5446,465],{"className":5447},[464]," that breaks\nthe range into two independent subranges, combine their optima plus a cost\nthat depends only on ",[394,5450,5452],{"className":5451},[397],[394,5453,5455],{"className":5454,"ariaHidden":402},[401],[394,5456,5458,5461],{"className":5457},[406],[394,5459],{"className":5460,"style":411},[410],[394,5462,417],{"className":5463},[415,416],", ",[394,5466,5468],{"className":5467},[397],[394,5469,5471],{"className":5470,"ariaHidden":402},[401],[394,5472,5474,5477],{"className":5473},[406],[394,5475],{"className":5476,"style":3330},[410],[394,5478,460],{"className":5479,"style":459},[415,416],[394,5481,5483],{"className":5482},[397],[394,5484,5486],{"className":5485,"ariaHidden":402},[401],[394,5487,5489,5492],{"className":5488},[406],[394,5490],{"className":5491,"style":483},[410],[394,5493,488],{"className":5494,"style":487},[415,416],", and minimize (or maximize) over ",[394,5497,5499],{"className":5498},[397],[394,5500,5502],{"className":5501,"ariaHidden":402},[401],[394,5503,5505,5508],{"className":5504},[406],[394,5506],{"className":5507,"style":483},[410],[394,5509,488],{"className":5510,"style":487},[415,416],[5282,5512,5513,5516,5517,5520,5521,2140],{},[421,5514,5515],{},"Order."," Fill by ",[421,5518,5519],{},"increasing interval length",", so both subranges are\nalready solved when you reach ",[394,5522,5524],{"className":5523},[397],[394,5525,5527],{"className":5526,"ariaHidden":402},[401],[394,5528,5530,5533,5536,5539,5542,5545,5548],{"className":5529},[406],[394,5531],{"className":5532,"style":437},[410],[394,5534,442],{"className":5535},[441],[394,5537,417],{"className":5538},[415,416],[394,5540,450],{"className":5541},[449],[394,5543],{"className":5544,"style":455},[454],[394,5546,460],{"className":5547,"style":459},[415,416],[394,5549,465],{"className":5550},[464],[381,5552,5553,5554,5604,5605,5629,5630,5645,5646,2140],{},"With ",[394,5555,5557],{"className":5556},[397],[394,5558,5560],{"className":5559,"ariaHidden":402},[401],[394,5561,5563,5566,5569,5572,5601],{"className":5562},[406],[394,5564],{"className":5565,"style":3644},[410],[394,5567,3648],{"className":5568},[415],[394,5570,1313],{"className":5571},[441],[394,5573,5575,5578],{"className":5574},[415],[394,5576,791],{"className":5577},[415,416],[394,5579,5581],{"className":5580},[661],[394,5582,5584],{"className":5583},[665],[394,5585,5587],{"className":5586},[670],[394,5588,5590],{"className":5589,"style":3670},[674],[394,5591,5592,5595],{"style":3673},[394,5593],{"className":5594,"style":683},[682],[394,5596,5598],{"className":5597},[687,688,689,690],[394,5599,738],{"className":5600},[415,690],[394,5602,1407],{"className":5603},[464]," states and an ",[394,5606,5608],{"className":5607},[397],[394,5609,5611],{"className":5610,"ariaHidden":402},[401],[394,5612,5614,5617,5620,5623,5626],{"className":5613},[406],[394,5615],{"className":5616,"style":437},[410],[394,5618,3702],{"className":5619,"style":615},[415,416],[394,5621,1313],{"className":5622},[441],[394,5624,791],{"className":5625},[415,416],[394,5627,1407],{"className":5628},[464]," choice of ",[394,5631,5633],{"className":5632},[397],[394,5634,5636],{"className":5635,"ariaHidden":402},[401],[394,5637,5639,5642],{"className":5638},[406],[394,5640],{"className":5641,"style":483},[410],[394,5643,488],{"className":5644,"style":487},[415,416]," per state, the running\ntime is the characteristic ",[394,5647,5649],{"className":5648},[397],[394,5650,5652],{"className":5651,"ariaHidden":402},[401],[394,5653,5655,5658,5661,5664,5693],{"className":5654},[406],[394,5656],{"className":5657,"style":3644},[410],[394,5659,3702],{"className":5660,"style":615},[415,416],[394,5662,1313],{"className":5663},[441],[394,5665,5667,5670],{"className":5666},[415],[394,5668,791],{"className":5669},[415,416],[394,5671,5673],{"className":5672},[661],[394,5674,5676],{"className":5675},[665],[394,5677,5679],{"className":5678},[670],[394,5680,5682],{"className":5681,"style":3670},[674],[394,5683,5684,5687],{"style":3673},[394,5685],{"className":5686,"style":683},[682],[394,5688,5690],{"className":5689},[687,688,689,690],[394,5691,3776],{"className":5692},[415,690],[394,5694,1407],{"className":5695},[464],[381,5697,5698,5699,5702],{},"The art in any specific problem is steps 1 and 3: defining the slice so that the\ntwo sides really are ",[421,5700,5701],{},"independent",", and finding the cost term that depends only\non the endpoints and the split. The rest is bookkeeping.",[534,5704,5706],{"id":5705},"optimal-binary-search-tree","Optimal binary search tree",[381,5708,5709,5710,5725,5726,5836,5837,5843,5844,5847,5848,6001,6002,6017,6018,6048,6049,6097,6098,6146,6147,6162,6163,6166,6167,6197,6198,6357],{},"A first variation keeps the split idea but changes what the cost term measures.\nGiven ",[394,5711,5713],{"className":5712},[397],[394,5714,5716],{"className":5715,"ariaHidden":402},[401],[394,5717,5719,5722],{"className":5718},[406],[394,5720],{"className":5721,"style":611},[410],[394,5723,791],{"className":5724},[415,416]," sorted keys with search probabilities ",[394,5727,5729],{"className":5728},[397],[394,5730,5732],{"className":5731,"ariaHidden":402},[401],[394,5733,5735,5738,5778,5781,5784,5787,5790,5793,5796],{"className":5734},[406],[394,5736],{"className":5737,"style":575},[410],[394,5739,5741,5744],{"className":5740},[415],[394,5742,381],{"className":5743},[415,416],[394,5745,5747],{"className":5746},[661],[394,5748,5750,5770],{"className":5749},[665,666],[394,5751,5753,5767],{"className":5752},[670],[394,5754,5756],{"className":5755,"style":675},[674],[394,5757,5758,5761],{"style":678},[394,5759],{"className":5760,"style":683},[682],[394,5762,5764],{"className":5763},[687,688,689,690],[394,5765,694],{"className":5766},[415,690],[394,5768,699],{"className":5769},[698],[394,5771,5773],{"className":5772},[670],[394,5774,5776],{"className":5775,"style":706},[674],[394,5777],{},[394,5779,450],{"className":5780},[449],[394,5782],{"className":5783,"style":455},[454],[394,5785,1110],{"className":5786},[756],[394,5788],{"className":5789,"style":455},[454],[394,5791,450],{"className":5792},[449],[394,5794],{"className":5795,"style":455},[454],[394,5797,5799,5802],{"className":5798},[415],[394,5800,381],{"className":5801},[415,416],[394,5803,5805],{"className":5804},[661],[394,5806,5808,5828],{"className":5807},[665,666],[394,5809,5811,5825],{"className":5810},[670],[394,5812,5814],{"className":5813,"style":779},[674],[394,5815,5816,5819],{"style":678},[394,5817],{"className":5818,"style":683},[682],[394,5820,5822],{"className":5821},[687,688,689,690],[394,5823,791],{"className":5824},[415,416,690],[394,5826,699],{"className":5827},[698],[394,5829,5831],{"className":5830},[670],[394,5832,5834],{"className":5833,"style":706},[674],[394,5835],{},", an ",[421,5838,5839,5840],{},"optimal\n",[526,5841,5842],{"href":90},"binary search tree"," is the\nBST minimizing the ",[385,5845,5846],{},"expected"," search cost\n",[394,5849,5851],{"className":5850},[397],[394,5852,5854,5961,5989],{"className":5853,"ariaHidden":402},[401],[394,5855,5857,5861,5908,5911,5951,5954,5958],{"className":5856},[406],[394,5858],{"className":5859,"style":5860},[410],"height:1.0497em;vertical-align:-0.2997em;",[394,5862,5864,5871],{"className":5863},[1778],[394,5865,5870],{"className":5866,"style":5869},[1778,5867,5868],"op-symbol","small-op","position:relative;top:0em;","∑",[394,5872,5874],{"className":5873},[661],[394,5875,5877,5899],{"className":5876},[665,666],[394,5878,5880,5896],{"className":5879},[670],[394,5881,5884],{"className":5882,"style":5883},[674],"height:0.162em;",[394,5885,5887,5890],{"style":5886},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[394,5888],{"className":5889,"style":683},[682],[394,5891,5893],{"className":5892},[687,688,689,690],[394,5894,417],{"className":5895},[415,416,690],[394,5897,699],{"className":5898},[698],[394,5900,5902],{"className":5901},[670],[394,5903,5906],{"className":5904,"style":5905},[674],"height:0.2997em;",[394,5907],{},[394,5909],{"className":5910,"style":455},[454],[394,5912,5914,5917],{"className":5913},[415],[394,5915,381],{"className":5916},[415,416],[394,5918,5920],{"className":5919},[661],[394,5921,5923,5943],{"className":5922},[665,666],[394,5924,5926,5940],{"className":5925},[670],[394,5927,5929],{"className":5928,"style":858},[674],[394,5930,5931,5934],{"style":678},[394,5932],{"className":5933,"style":683},[682],[394,5935,5937],{"className":5936},[687,688,689,690],[394,5938,417],{"className":5939},[415,416,690],[394,5941,699],{"className":5942},[698],[394,5944,5946],{"className":5945},[670],[394,5947,5949],{"className":5948,"style":706},[674],[394,5950],{},[394,5952],{"className":5953,"style":560},[454],[394,5955,5957],{"className":5956},[564],"⋅",[394,5959],{"className":5960,"style":560},[454],[394,5962,5964,5967,5970,5977,5980,5983,5986],{"className":5963},[406],[394,5965],{"className":5966,"style":437},[410],[394,5968,1313],{"className":5969},[441],[394,5971,5973],{"className":5972},[415,5302],[394,5974,5976],{"className":5975},[415],"depth of key ",[394,5978,417],{"className":5979},[415,416],[394,5981],{"className":5982,"style":560},[454],[394,5984,1446],{"className":5985},[564],[394,5987],{"className":5988,"style":560},[454],[394,5990,5992,5995,5998],{"className":5991},[406],[394,5993],{"className":5994,"style":437},[410],[394,5996,694],{"className":5997},[415],[394,5999,1407],{"className":6000},[464],". Choosing key ",[394,6003,6005],{"className":6004},[397],[394,6006,6008],{"className":6007,"ariaHidden":402},[401],[394,6009,6011,6014],{"className":6010},[406],[394,6012],{"className":6013,"style":483},[410],[394,6015,488],{"className":6016,"style":487},[415,416]," as the root of\nthe subtree on keys ",[394,6019,6021],{"className":6020},[397],[394,6022,6024],{"className":6023,"ariaHidden":402},[401],[394,6025,6027,6030,6033,6036,6039,6042,6045],{"className":6026},[406],[394,6028],{"className":6029,"style":437},[410],[394,6031,442],{"className":6032},[441],[394,6034,417],{"className":6035},[415,416],[394,6037,450],{"className":6038},[449],[394,6040],{"className":6041,"style":455},[454],[394,6043,460],{"className":6044,"style":459},[415,416],[394,6046,465],{"className":6047},[464]," splits the rest into a left subtree on ",[394,6050,6052],{"className":6051},[397],[394,6053,6055,6085],{"className":6054,"ariaHidden":402},[401],[394,6056,6058,6061,6064,6067,6070,6073,6076,6079,6082],{"className":6057},[406],[394,6059],{"className":6060,"style":437},[410],[394,6062,442],{"className":6063},[441],[394,6065,417],{"className":6066},[415,416],[394,6068,450],{"className":6069},[449],[394,6071],{"className":6072,"style":455},[454],[394,6074,488],{"className":6075,"style":487},[415,416],[394,6077],{"className":6078,"style":560},[454],[394,6080,930],{"className":6081},[564],[394,6083],{"className":6084,"style":560},[454],[394,6086,6088,6091,6094],{"className":6087},[406],[394,6089],{"className":6090,"style":437},[410],[394,6092,694],{"className":6093},[415],[394,6095,465],{"className":6096},[464]," and a\nright subtree on ",[394,6099,6101],{"className":6100},[397],[394,6102,6104,6125],{"className":6103,"ariaHidden":402},[401],[394,6105,6107,6110,6113,6116,6119,6122],{"className":6106},[406],[394,6108],{"className":6109,"style":437},[410],[394,6111,442],{"className":6112},[441],[394,6114,488],{"className":6115,"style":487},[415,416],[394,6117],{"className":6118,"style":560},[454],[394,6120,1446],{"className":6121},[564],[394,6123],{"className":6124,"style":560},[454],[394,6126,6128,6131,6134,6137,6140,6143],{"className":6127},[406],[394,6129],{"className":6130,"style":437},[410],[394,6132,694],{"className":6133},[415],[394,6135,450],{"className":6136},[449],[394,6138],{"className":6139,"style":455},[454],[394,6141,460],{"className":6142,"style":459},[415,416],[394,6144,465],{"className":6145},[464],", again two independent ranges combined at a pivot.\nMaking ",[394,6148,6150],{"className":6149},[397],[394,6151,6153],{"className":6152,"ariaHidden":402},[401],[394,6154,6156,6159],{"className":6155},[406],[394,6157],{"className":6158,"style":483},[410],[394,6160,488],{"className":6161,"style":487},[415,416]," the root pushes ",[385,6164,6165],{},"every"," key in ",[394,6168,6170],{"className":6169},[397],[394,6171,6173],{"className":6172,"ariaHidden":402},[401],[394,6174,6176,6179,6182,6185,6188,6191,6194],{"className":6175},[406],[394,6177],{"className":6178,"style":437},[410],[394,6180,442],{"className":6181},[441],[394,6183,417],{"className":6184},[415,416],[394,6186,450],{"className":6187},[449],[394,6189],{"className":6190,"style":455},[454],[394,6192,460],{"className":6193,"style":459},[415,416],[394,6195,465],{"className":6196},[464]," down one level, which adds the\ntotal weight ",[394,6199,6201],{"className":6200},[397],[394,6202,6204,6242],{"className":6203,"ariaHidden":402},[401],[394,6205,6207,6210,6215,6218,6221,6224,6227,6230,6233,6236,6239],{"className":6206},[406],[394,6208],{"className":6209,"style":437},[410],[394,6211,6214],{"className":6212,"style":6213},[415,416],"margin-right:0.0269em;","w",[394,6216,442],{"className":6217},[441],[394,6219,417],{"className":6220},[415,416],[394,6222,450],{"className":6223},[449],[394,6225],{"className":6226,"style":455},[454],[394,6228,460],{"className":6229,"style":459},[415,416],[394,6231,465],{"className":6232},[464],[394,6234],{"className":6235,"style":1702},[454],[394,6237,1707],{"className":6238},[1706],[394,6240],{"className":6241,"style":1702},[454],[394,6243,6245,6249,6314,6317],{"className":6244},[406],[394,6246],{"className":6247,"style":6248},[410],"height:1.2643em;vertical-align:-0.2997em;",[394,6250,6252,6255],{"className":6251},[1778],[394,6253,5870],{"className":6254,"style":5869},[1778,5867,5868],[394,6256,6258],{"className":6257},[661],[394,6259,6261,6306],{"className":6260},[665,666],[394,6262,6264,6303],{"className":6263},[670],[394,6265,6268,6288],{"className":6266,"style":6267},[674],"height:0.9646em;",[394,6269,6270,6273],{"style":5886},[394,6271],{"className":6272,"style":683},[682],[394,6274,6276],{"className":6275},[687,688,689,690],[394,6277,6279,6282,6285],{"className":6278},[415,690],[394,6280,616],{"className":6281,"style":615},[415,416,690],[394,6283,1707],{"className":6284},[1706,690],[394,6286,417],{"className":6287},[415,416,690],[394,6289,6291,6294],{"style":6290},"top:-3.2029em;margin-right:0.05em;",[394,6292],{"className":6293,"style":683},[682],[394,6295,6297],{"className":6296},[687,688,689,690],[394,6298,6300],{"className":6299},[415,690],[394,6301,460],{"className":6302,"style":459},[415,416,690],[394,6304,699],{"className":6305},[698],[394,6307,6309],{"className":6308},[670],[394,6310,6312],{"className":6311,"style":5905},[674],[394,6313],{},[394,6315],{"className":6316,"style":455},[454],[394,6318,6320,6323],{"className":6319},[415],[394,6321,381],{"className":6322},[415,416],[394,6324,6326],{"className":6325},[661],[394,6327,6329,6349],{"className":6328},[665,666],[394,6330,6332,6346],{"className":6331},[670],[394,6333,6335],{"className":6334,"style":779},[674],[394,6336,6337,6340],{"style":678},[394,6338],{"className":6339,"style":683},[682],[394,6341,6343],{"className":6342},[687,688,689,690],[394,6344,616],{"className":6345,"style":615},[415,416,690],[394,6347,699],{"className":6348},[698],[394,6350,6352],{"className":6351},[670],[394,6353,6355],{"className":6354,"style":706},[674],[394,6356],{}," to the 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root ",[394,6617,6619],{"className":6618},[397],[394,6620,6622],{"className":6621,"ariaHidden":402},[401],[394,6623,6625,6628],{"className":6624},[406],[394,6626],{"className":6627,"style":483},[410],[394,6629,488],{"className":6630,"style":487},[415,416]," hangs the two solved subtrees beneath it, and the act of adding a\nroot pushes ",[385,6633,6165],{},[394,6635,6637],{"className":6636},[397],[394,6638,6640],{"className":6639,"ariaHidden":402},[401],[394,6641,6643,6646,6649,6652,6655,6658,6661],{"className":6642},[406],[394,6644],{"className":6645,"style":437},[410],[394,6647,442],{"className":6648},[441],[394,6650,417],{"className":6651},[415,416],[394,6653,450],{"className":6654},[449],[394,6656],{"className":6657,"style":455},[454],[394,6659,460],{"className":6660,"style":459},[415,416],[394,6662,465],{"className":6663},[464]," one level deeper, so its whole 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",[394,6955,6957],{"className":6956},[397],[394,6958,6960],{"className":6959,"ariaHidden":402},[401],[394,6961,6963,6966,6969,6972,6975,6978,6981],{"className":6962},[406],[394,6964],{"className":6965,"style":437},[410],[394,6967,442],{"className":6968},[441],[394,6970,417],{"className":6971},[415,416],[394,6973,450],{"className":6974},[449],[394,6976],{"className":6977,"style":455},[454],[394,6979,460],{"className":6980,"style":459},[415,416],[394,6982,465],{"className":6983},[464],": the optimal subtrees on ",[394,6986,6988],{"className":6987},[397],[394,6989,6991,7021],{"className":6990,"ariaHidden":402},[401],[394,6992,6994,6997,7000,7003,7006,7009,7012,7015,7018],{"className":6993},[406],[394,6995],{"className":6996,"style":437},[410],[394,6998,442],{"className":6999},[441],[394,7001,417],{"className":7002},[415,416],[394,7004,450],{"className":7005},[449],[394,7007],{"className":7008,"style":455},[454],[394,7010,488],{"className":7011,"style":487},[415,416],[394,7013],{"className":7014,"style":560},[454],[394,7016,930],{"className":7017},[564],[394,7019],{"className":7020,"style":560},[454],[394,7022,7024,7027,7030],{"className":7023},[406],[394,7025],{"className":7026,"style":437},[410],[394,7028,694],{"className":7029},[415],[394,7031,465],{"className":7032},[464]," and ",[394,7035,7037],{"className":7036},[397],[394,7038,7040,7061],{"className":7039,"ariaHidden":402},[401],[394,7041,7043,7046,7049,7052,7055,7058],{"className":7042},[406],[394,7044],{"className":7045,"style":437},[410],[394,7047,442],{"className":7048},[441],[394,7050,488],{"className":7051,"style":487},[415,416],[394,7053],{"className":7054,"style":560},[454],[394,7056,1446],{"className":7057},[564],[394,7059],{"className":7060,"style":560},[454],[394,7062,7064,7067,7070,7073,7076,7079],{"className":7063},[406],[394,7065],{"className":7066,"style":437},[410],[394,7068,694],{"className":7069},[415],[394,7071,450],{"className":7072},[449],[394,7074],{"className":7075,"style":455},[454],[394,7077,460],{"className":7078,"style":459},[415,416],[394,7080,465],{"className":7081},[464]," hang below, and rooting adds one level to all keys, charging ",[394,7084,7086],{"className":7085},[397],[394,7087,7089],{"className":7088,"ariaHidden":402},[401],[394,7090,7092,7095,7098,7101,7104,7107,7110,7113,7116,7119],{"className":7091},[406],[394,7093],{"className":7094,"style":437},[410],[394,7096,1446],{"className":7097},[415],[394,7099],{"className":7100,"style":455},[454],[394,7102,6214],{"className":7103,"style":6213},[415,416],[394,7105,442],{"className":7106},[441],[394,7108,417],{"className":7109},[415,416],[394,7111,450],{"className":7112},[449],[394,7114],{"className":7115,"style":455},[454],[394,7117,460],{"className":7118,"style":459},[415,416],[394,7120,465],{"className":7121},[464],[381,7123,7124,7125,7175,7176],{},"The shape is identical to matrix-chain, with the same diagonals and the same\n",[394,7126,7128],{"className":7127},[397],[394,7129,7131],{"className":7130,"ariaHidden":402},[401],[394,7132,7134,7137,7140,7143,7172],{"className":7133},[406],[394,7135],{"className":7136,"style":3644},[410],[394,7138,3702],{"className":7139,"style":615},[415,416],[394,7141,1313],{"className":7142},[441],[394,7144,7146,7149],{"className":7145},[415],[394,7147,791],{"className":7148},[415,416],[394,7150,7152],{"className":7151},[661],[394,7153,7155],{"className":7154},[665],[394,7156,7158],{"className":7157},[670],[394,7159,7161],{"className":7160,"style":3670},[674],[394,7162,7163,7166],{"style":3673},[394,7164],{"className":7165,"style":683},[682],[394,7167,7169],{"className":7168},[687,688,689,690],[394,7170,3776],{"className":7171},[415,690],[394,7173,1407],{"className":7174},[464]," fill, the combine term now being the swept-down weight of the whole\ninterval rather than a product of dimensions.",[1162,7177,7178],{},[526,7179,738],{"href":7180,"ariaDescribedBy":7181,"dataFootnoteRef":376,"id":7182},"#user-content-fn-clrs-obst",[1168],"user-content-fnref-clrs-obst",[534,7184,7186,7187,7190],{"id":7185},"the-last-operation-trick-burst-balloons-and-cutting-a-stick","The ",[390,7188,7189],{},"last operation"," trick: Burst Balloons and cutting a stick",[381,7192,7193,7194,7197,7198,7201],{},"Some problems resist the naive split because the cost of a piece depends on what\nis ",[385,7195,7196],{},"adjacent"," to it, and the first split severs exactly the adjacency that sets\nthe cost. The fix is to guess the ",[421,7199,7200],{},"last"," operation instead of the first.",[381,7203,7204,7205,7208,7209,7236,7237,7292,7293,7308,7309,7486,7487,7490,7491,7506,7507,7510,7511,7541,7542,7545],{},"Take ",[421,7206,7207],{},"Burst Balloons",": balloons ",[394,7210,7212],{"className":7211},[397],[394,7213,7215],{"className":7214,"ariaHidden":402},[401],[394,7216,7218,7221,7224,7227,7230,7233],{"className":7217},[406],[394,7219],{"className":7220,"style":3491},[410],[394,7222,694],{"className":7223},[415],[394,7225],{"className":7226,"style":455},[454],[394,7228,1110],{"className":7229},[756],[394,7231],{"className":7232,"style":455},[454],[394,7234,791],{"className":7235},[415,416]," have values ",[394,7238,7240],{"className":7239},[397],[394,7241,7243],{"className":7242,"ariaHidden":402},[401],[394,7244,7246,7250],{"className":7245},[406],[394,7247],{"className":7248,"style":7249},[410],"height:0.5806em;vertical-align:-0.15em;",[394,7251,7253,7257],{"className":7252},[415],[394,7254,7256],{"className":7255,"style":579},[415,416],"v",[394,7258,7260],{"className":7259},[661],[394,7261,7263,7284],{"className":7262},[665,666],[394,7264,7266,7281],{"className":7265},[670],[394,7267,7269],{"className":7268,"style":858},[674],[394,7270,7272,7275],{"style":7271},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[394,7273],{"className":7274,"style":683},[682],[394,7276,7278],{"className":7277},[687,688,689,690],[394,7279,417],{"className":7280},[415,416,690],[394,7282,699],{"className":7283},[698],[394,7285,7287],{"className":7286},[670],[394,7288,7290],{"className":7289,"style":706},[674],[394,7291],{},", and bursting\nballoon ",[394,7294,7296],{"className":7295},[397],[394,7297,7299],{"className":7298,"ariaHidden":402},[401],[394,7300,7302,7305],{"className":7301},[406],[394,7303],{"className":7304,"style":411},[410],[394,7306,417],{"className":7307},[415,416]," earns ",[394,7310,7312],{"className":7311},[397],[394,7313,7315,7378,7433],{"className":7314,"ariaHidden":402},[401],[394,7316,7318,7322,7369,7372,7375],{"className":7317},[406],[394,7319],{"className":7320,"style":7321},[410],"height:0.5945em;vertical-align:-0.15em;",[394,7323,7325,7328],{"className":7324},[415],[394,7326,7256],{"className":7327,"style":579},[415,416],[394,7329,7331],{"className":7330},[661],[394,7332,7334,7361],{"className":7333},[665,666],[394,7335,7337,7358],{"className":7336},[670],[394,7338,7340],{"className":7339,"style":1381},[674],[394,7341,7342,7345],{"style":7271},[394,7343],{"className":7344,"style":683},[682],[394,7346,7348],{"className":7347},[687,688,689,690],[394,7349,7351],{"className":7350},[415,690],[394,7352,7354],{"className":7353},[415,5302,690],[394,7355,7357],{"className":7356},[415,690],"left",[394,7359,699],{"className":7360},[698],[394,7362,7364],{"className":7363},[670],[394,7365,7367],{"className":7366,"style":706},[674],[394,7368],{},[394,7370],{"className":7371,"style":560},[454],[394,7373,5957],{"className":7374},[564],[394,7376],{"className":7377,"style":560},[454],[394,7379,7381,7384,7424,7427,7430],{"className":7380},[406],[394,7382],{"className":7383,"style":7321},[410],[394,7385,7387,7390],{"className":7386},[415],[394,7388,7256],{"className":7389,"style":579},[415,416],[394,7391,7393],{"className":7392},[661],[394,7394,7396,7416],{"className":7395},[665,666],[394,7397,7399,7413],{"className":7398},[670],[394,7400,7402],{"className":7401,"style":858},[674],[394,7403,7404,7407],{"style":7271},[394,7405],{"className":7406,"style":683},[682],[394,7408,7410],{"className":7409},[687,688,689,690],[394,7411,417],{"className":7412},[415,416,690],[394,7414,699],{"className":7415},[698],[394,7417,7419],{"className":7418},[670],[394,7420,7422],{"className":7421,"style":706},[674],[394,7423],{},[394,7425],{"className":7426,"style":560},[454],[394,7428,5957],{"className":7429},[564],[394,7431],{"className":7432,"style":560},[454],[394,7434,7436,7439],{"className":7435},[406],[394,7437],{"className":7438,"style":2171},[410],[394,7440,7442,7445],{"className":7441},[415],[394,7443,7256],{"className":7444,"style":579},[415,416],[394,7446,7448],{"className":7447},[661],[394,7449,7451,7478],{"className":7450},[665,666],[394,7452,7454,7475],{"className":7453},[670],[394,7455,7457],{"className":7456,"style":1381},[674],[394,7458,7459,7462],{"style":7271},[394,7460],{"className":7461,"style":683},[682],[394,7463,7465],{"className":7464},[687,688,689,690],[394,7466,7468],{"className":7467},[415,690],[394,7469,7471],{"className":7470},[415,5302,690],[394,7472,7474],{"className":7473},[415,690],"right",[394,7476,699],{"className":7477},[698],[394,7479,7481],{"className":7480},[670],[394,7482,7484],{"className":7483,"style":1277},[674],[394,7485],{}," where left and\nright are its ",[385,7488,7489],{},"current"," surviving neighbors; bursting removes ",[394,7492,7494],{"className":7493},[397],[394,7495,7497],{"className":7496,"ariaHidden":402},[401],[394,7498,7500,7503],{"className":7499},[406],[394,7501],{"className":7502,"style":411},[410],[394,7504,417],{"className":7505},[415,416]," and rejoins the\nneighbors. We want the maximum total earnings. If we try to fix the ",[385,7508,7509],{},"first","\nballoon burst in ",[394,7512,7514],{"className":7513},[397],[394,7515,7517],{"className":7516,"ariaHidden":402},[401],[394,7518,7520,7523,7526,7529,7532,7535,7538],{"className":7519},[406],[394,7521],{"className":7522,"style":437},[410],[394,7524,442],{"className":7525},[441],[394,7527,417],{"className":7528},[415,416],[394,7530,450],{"className":7531},[449],[394,7533],{"className":7534,"style":455},[454],[394,7536,460],{"className":7537,"style":459},[415,416],[394,7539,465],{"className":7540},[464],", the two halves are ",[421,7543,7544],{},"not"," independent: a balloon in the\nleft half can later have a right neighbor in the right half, so the halves keep\ninteracting through the shared boundary.",[381,7547,7548,7549,7564,7565,7567,7568,7598,7599,7697,7698,7713,7714,7748,7749,7033,7764,7779,7780,7942,7943,7973,7974,470,7977,7992,7993,7033,8008,8023,8024,8054,8055,7033,8070,8085],{},"Now fix the balloon ",[394,7550,7552],{"className":7551},[397],[394,7553,7555],{"className":7554,"ariaHidden":402},[401],[394,7556,7558,7561],{"className":7557},[406],[394,7559],{"className":7560,"style":483},[410],[394,7562,488],{"className":7563,"style":487},[415,416]," that is burst ",[421,7566,7200],{}," in the open interval ",[394,7569,7571],{"className":7570},[397],[394,7572,7574],{"className":7573,"ariaHidden":402},[401],[394,7575,7577,7580,7583,7586,7589,7592,7595],{"className":7576},[406],[394,7578],{"className":7579,"style":437},[410],[394,7581,1313],{"className":7582},[441],[394,7584,417],{"className":7585},[415,416],[394,7587,450],{"className":7588},[449],[394,7590],{"className":7591,"style":455},[454],[394,7593,460],{"className":7594,"style":459},[415,416],[394,7596,1407],{"className":7597},[464]," (using\nsentinels ",[394,7600,7602],{"className":7601},[397],[394,7603,7605],{"className":7604,"ariaHidden":402},[401],[394,7606,7608,7611,7651,7654,7657],{"className":7607},[406],[394,7609],{"className":7610,"style":2171},[410],[394,7612,7614,7617],{"className":7613},[415],[394,7615,7256],{"className":7616,"style":579},[415,416],[394,7618,7620],{"className":7619},[661],[394,7621,7623,7643],{"className":7622},[665,666],[394,7624,7626,7640],{"className":7625},[670],[394,7627,7629],{"className":7628,"style":858},[674],[394,7630,7631,7634],{"style":7271},[394,7632],{"className":7633,"style":683},[682],[394,7635,7637],{"className":7636},[687,688,689,690],[394,7638,417],{"className":7639},[415,416,690],[394,7641,699],{"className":7642},[698],[394,7644,7646],{"className":7645},[670],[394,7647,7649],{"className":7648,"style":706},[674],[394,7650],{},[394,7652,450],{"className":7653},[449],[394,7655],{"className":7656,"style":455},[454],[394,7658,7660,7663],{"className":7659},[415],[394,7661,7256],{"className":7662,"style":579},[415,416],[394,7664,7666],{"className":7665},[661],[394,7667,7669,7689],{"className":7668},[665,666],[394,7670,7672,7686],{"className":7671},[670],[394,7673,7675],{"className":7674,"style":858},[674],[394,7676,7677,7680],{"style":7271},[394,7678],{"className":7679,"style":683},[682],[394,7681,7683],{"className":7682},[687,688,689,690],[394,7684,460],{"className":7685,"style":459},[415,416,690],[394,7687,699],{"className":7688},[698],[394,7690,7692],{"className":7691},[670],[394,7693,7695],{"className":7694,"style":1277},[674],[394,7696],{}," just outside the range that are never burst). When ",[394,7699,7701],{"className":7700},[397],[394,7702,7704],{"className":7703,"ariaHidden":402},[401],[394,7705,7707,7710],{"className":7706},[406],[394,7708],{"className":7709,"style":483},[410],[394,7711,488],{"className":7712,"style":487},[415,416]," is\nburst last, ",[385,7715,7716,7717,7747],{},"every other balloon in ",[394,7718,7720],{"className":7719},[397],[394,7721,7723],{"className":7722,"ariaHidden":402},[401],[394,7724,7726,7729,7732,7735,7738,7741,7744],{"className":7725},[406],[394,7727],{"className":7728,"style":437},[410],[394,7730,1313],{"className":7731},[441],[394,7733,417],{"className":7734},[415,416],[394,7736,450],{"className":7737},[449],[394,7739],{"className":7740,"style":455},[454],[394,7742,460],{"className":7743,"style":459},[415,416],[394,7745,1407],{"className":7746},[464]," is already gone",", so its two\nneighbors at that moment are exactly the boundaries ",[394,7750,7752],{"className":7751},[397],[394,7753,7755],{"className":7754,"ariaHidden":402},[401],[394,7756,7758,7761],{"className":7757},[406],[394,7759],{"className":7760,"style":411},[410],[394,7762,417],{"className":7763},[415,416],[394,7765,7767],{"className":7766},[397],[394,7768,7770],{"className":7769,"ariaHidden":402},[401],[394,7771,7773,7776],{"className":7772},[406],[394,7774],{"className":7775,"style":3330},[410],[394,7777,460],{"className":7778,"style":459},[415,416],". The earnings for\nthat final burst are ",[394,7781,7783],{"className":7782},[397],[394,7784,7786,7841,7896],{"className":7785,"ariaHidden":402},[401],[394,7787,7789,7792,7832,7835,7838],{"className":7788},[406],[394,7790],{"className":7791,"style":7321},[410],[394,7793,7795,7798],{"className":7794},[415],[394,7796,7256],{"className":7797,"style":579},[415,416],[394,7799,7801],{"className":7800},[661],[394,7802,7804,7824],{"className":7803},[665,666],[394,7805,7807,7821],{"className":7806},[670],[394,7808,7810],{"className":7809,"style":858},[674],[394,7811,7812,7815],{"style":7271},[394,7813],{"className":7814,"style":683},[682],[394,7816,7818],{"className":7817},[687,688,689,690],[394,7819,417],{"className":7820},[415,416,690],[394,7822,699],{"className":7823},[698],[394,7825,7827],{"className":7826},[670],[394,7828,7830],{"className":7829,"style":706},[674],[394,7831],{},[394,7833],{"className":7834,"style":560},[454],[394,7836,5957],{"className":7837},[564],[394,7839],{"className":7840,"style":560},[454],[394,7842,7844,7847,7887,7890,7893],{"className":7843},[406],[394,7845],{"className":7846,"style":7321},[410],[394,7848,7850,7853],{"className":7849},[415],[394,7851,7256],{"className":7852,"style":579},[415,416],[394,7854,7856],{"className":7855},[661],[394,7857,7859,7879],{"className":7858},[665,666],[394,7860,7862,7876],{"className":7861},[670],[394,7863,7865],{"className":7864,"style":1381},[674],[394,7866,7867,7870],{"style":7271},[394,7868],{"className":7869,"style":683},[682],[394,7871,7873],{"className":7872},[687,688,689,690],[394,7874,488],{"className":7875,"style":487},[415,416,690],[394,7877,699],{"className":7878},[698],[394,7880,7882],{"className":7881},[670],[394,7883,7885],{"className":7884,"style":706},[674],[394,7886],{},[394,7888],{"className":7889,"style":560},[454],[394,7891,5957],{"className":7892},[564],[394,7894],{"className":7895,"style":560},[454],[394,7897,7899,7902],{"className":7898},[406],[394,7900],{"className":7901,"style":2171},[410],[394,7903,7905,7908],{"className":7904},[415],[394,7906,7256],{"className":7907,"style":579},[415,416],[394,7909,7911],{"className":7910},[661],[394,7912,7914,7934],{"className":7913},[665,666],[394,7915,7917,7931],{"className":7916},[670],[394,7918,7920],{"className":7919,"style":858},[674],[394,7921,7922,7925],{"style":7271},[394,7923],{"className":7924,"style":683},[682],[394,7926,7928],{"className":7927},[687,688,689,690],[394,7929,460],{"className":7930,"style":459},[415,416,690],[394,7932,699],{"className":7933},[698],[394,7935,7937],{"className":7936},[670],[394,7938,7940],{"className":7939,"style":1277},[674],[394,7941],{},", fixed and independent of order;\nthe balloons in ",[394,7944,7946],{"className":7945},[397],[394,7947,7949],{"className":7948,"ariaHidden":402},[401],[394,7950,7952,7955,7958,7961,7964,7967,7970],{"className":7951},[406],[394,7953],{"className":7954,"style":437},[410],[394,7956,1313],{"className":7957},[441],[394,7959,417],{"className":7960},[415,416],[394,7962,450],{"className":7963},[449],[394,7965],{"className":7966,"style":455},[454],[394,7968,488],{"className":7969,"style":487},[415,416],[394,7971,1407],{"className":7972},[464]," were all burst ",[385,7975,7976],{},"before",[394,7978,7980],{"className":7979},[397],[394,7981,7983],{"className":7982,"ariaHidden":402},[401],[394,7984,7986,7989],{"className":7985},[406],[394,7987],{"className":7988,"style":483},[410],[394,7990,488],{"className":7991,"style":487},[415,416]," while ",[394,7994,7996],{"className":7995},[397],[394,7997,7999],{"className":7998,"ariaHidden":402},[401],[394,8000,8002,8005],{"className":8001},[406],[394,8003],{"className":8004,"style":411},[410],[394,8006,417],{"className":8007},[415,416],[394,8009,8011],{"className":8010},[397],[394,8012,8014],{"className":8013,"ariaHidden":402},[401],[394,8015,8017,8020],{"className":8016},[406],[394,8018],{"className":8019,"style":483},[410],[394,8021,488],{"className":8022,"style":487},[415,416],"\nstood as fixed walls, and likewise ",[394,8025,8027],{"className":8026},[397],[394,8028,8030],{"className":8029,"ariaHidden":402},[401],[394,8031,8033,8036,8039,8042,8045,8048,8051],{"className":8032},[406],[394,8034],{"className":8035,"style":437},[410],[394,8037,1313],{"className":8038},[441],[394,8040,488],{"className":8041,"style":487},[415,416],[394,8043,450],{"className":8044},[449],[394,8046],{"className":8047,"style":455},[454],[394,8049,460],{"className":8050,"style":459},[415,416],[394,8052,1407],{"className":8053},[464]," played out against walls 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They are 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14.594-39.462Q14.400-39.553 14.400-39.767M16.510-44.232Q16.510-44.372 16.621-44.476Q16.732-44.580 16.870-44.580Q16.973-44.580 17.047-44.509Q17.122-44.437 17.122-44.334Q17.122-44.194 17.012-44.090Q16.902-43.986 16.762-43.986Q16.659-43.986 16.584-44.060Q16.510-44.135 16.510-44.232",[2911],[3163,8617,8619,8620,8635,8636,8666,8667,8682,8683,8707,8708],{"className":8618},[3166],"Bursting ",[394,8621,8623],{"className":8622},[397],[394,8624,8626],{"className":8625,"ariaHidden":402},[401],[394,8627,8629,8632],{"className":8628},[406],[394,8630],{"className":8631,"style":483},[410],[394,8633,488],{"className":8634,"style":487},[415,416]," last in ",[394,8637,8639],{"className":8638},[397],[394,8640,8642],{"className":8641,"ariaHidden":402},[401],[394,8643,8645,8648,8651,8654,8657,8660,8663],{"className":8644},[406],[394,8646],{"className":8647,"style":437},[410],[394,8649,1313],{"className":8650},[441],[394,8652,417],{"className":8653},[415,416],[394,8655,450],{"className":8656},[449],[394,8658],{"className":8659,"style":455},[454],[394,8661,460],{"className":8662,"style":459},[415,416],[394,8664,1407],{"className":8665},[464],": every other balloon is already gone, so ",[394,8668,8670],{"className":8669},[397],[394,8671,8673],{"className":8672,"ariaHidden":402},[401],[394,8674,8676,8679],{"className":8675},[406],[394,8677],{"className":8678,"style":483},[410],[394,8680,488],{"className":8681,"style":487},[415,416],"'s neighbors are pinned to the walls ",[394,8684,8686],{"className":8685},[397],[394,8687,8689],{"className":8688,"ariaHidden":402},[401],[394,8690,8692,8695,8698,8701,8704],{"className":8691},[406],[394,8693],{"className":8694,"style":3330},[410],[394,8696,417],{"className":8697},[415,416],[394,8699,450],{"className":8700},[449],[394,8702],{"className":8703,"style":455},[454],[394,8705,460],{"className":8706,"style":459},[415,416]," and earn ",[394,8709,8711],{"className":8710},[397],[394,8712,8714],{"className":8713,"ariaHidden":402},[401],[394,8715,8717,8720,8760,8800],{"className":8716},[406],[394,8718],{"className":8719,"style":2171},[410],[394,8721,8723,8726],{"className":8722},[415],[394,8724,7256],{"className":8725,"style":579},[415,416],[394,8727,8729],{"className":8728},[661],[394,8730,8732,8752],{"className":8731},[665,666],[394,8733,8735,8749],{"className":8734},[670],[394,8736,8738],{"className":8737,"style":858},[674],[394,8739,8740,8743],{"style":7271},[394,8741],{"className":8742,"style":683},[682],[394,8744,8746],{"className":8745},[687,688,689,690],[394,8747,417],{"className":8748},[415,416,690],[394,8750,699],{"className":8751},[698],[394,8753,8755],{"className":8754},[670],[394,8756,8758],{"className":8757,"style":706},[674],[394,8759],{},[394,8761,8763,8766],{"className":8762},[415],[394,8764,7256],{"className":8765,"style":579},[415,416],[394,8767,8769],{"className":8768},[661],[394,8770,8772,8792],{"className":8771},[665,666],[394,8773,8775,8789],{"className":8774},[670],[394,8776,8778],{"className":8777,"style":1381},[674],[394,8779,8780,8783],{"style":7271},[394,8781],{"className":8782,"style":683},[682],[394,8784,8786],{"className":8785},[687,688,689,690],[394,8787,488],{"className":8788,"style":487},[415,416,690],[394,8790,699],{"className":8791},[698],[394,8793,8795],{"className":8794},[670],[394,8796,8798],{"className":8797,"style":706},[674],[394,8799],{},[394,8801,8803,8806],{"className":8802},[415],[394,8804,7256],{"className":8805,"style":579},[415,416],[394,8807,8809],{"className":8808},[661],[394,8810,8812,8832],{"className":8811},[665,666],[394,8813,8815,8829],{"className":8814},[670],[394,8816,8818],{"className":8817,"style":858},[674],[394,8819,8820,8823],{"style":7271},[394,8821],{"className":8822,"style":683},[682],[394,8824,8826],{"className":8825},[687,688,689,690],[394,8827,460],{"className":8828,"style":459},[415,416,690],[394,8830,699],{"className":8831},[698],[394,8833,8835],{"className":8834},[670],[394,8836,8838],{"className":8837,"style":1277},[674],[394,8839],{},[5238,8841,8843],{"type":8842},"note",[381,8844,8845,8848,8849,8851,8852,8854,8855,8870,8871,7033,8901,8931,8932,8935],{},[421,8846,8847],{},"Intuition."," Splitting on the ",[385,8850,7509],{}," balloon leaves the two halves coupled,\nbecause a later burst can reach across the cut. Splitting on the ",[385,8853,7200],{}," balloon\ndecouples them: once ",[394,8856,8858],{"className":8857},[397],[394,8859,8861],{"className":8860,"ariaHidden":402},[401],[394,8862,8864,8867],{"className":8863},[406],[394,8865],{"className":8866,"style":483},[410],[394,8868,488],{"className":8869,"style":487},[415,416]," is the only survivor, its neighbors are pinned to the\ninterval's walls, so whatever happened inside ",[394,8872,8874],{"className":8873},[397],[394,8875,8877],{"className":8876,"ariaHidden":402},[401],[394,8878,8880,8883,8886,8889,8892,8895,8898],{"className":8879},[406],[394,8881],{"className":8882,"style":437},[410],[394,8884,1313],{"className":8885},[441],[394,8887,417],{"className":8888},[415,416],[394,8890,450],{"className":8891},[449],[394,8893],{"className":8894,"style":455},[454],[394,8896,488],{"className":8897,"style":487},[415,416],[394,8899,1407],{"className":8900},[464],[394,8902,8904],{"className":8903},[397],[394,8905,8907],{"className":8906,"ariaHidden":402},[401],[394,8908,8910,8913,8916,8919,8922,8925,8928],{"className":8909},[406],[394,8911],{"className":8912,"style":437},[410],[394,8914,1313],{"className":8915},[441],[394,8917,488],{"className":8918,"style":487},[415,416],[394,8920,450],{"className":8921},[449],[394,8923],{"className":8924,"style":455},[454],[394,8926,460],{"className":8927,"style":459},[415,416],[394,8929,1407],{"className":8930},[464]," could not have\ndepended on the other side. ",[390,8933,8934],{},"Last"," turns a moving boundary into a fixed one.",[381,8937,8938,8941,8942,8945,8946,8961,8962,8992,8993,8995,8996,9011,9012,9027,9028,9137,9138,7033,9168,9198,9199,9249,9250,9252],{},[421,8939,8940],{},"Minimum Cost to Cut a Stick"," is the same idea in dual form. A stick has cut\npositions inside it; making a cut costs the ",[385,8943,8944],{},"current length"," of the piece being\ncut. Fix which cut ",[394,8947,8949],{"className":8948},[397],[394,8950,8952],{"className":8951,"ariaHidden":402},[401],[394,8953,8955,8958],{"className":8954},[406],[394,8956],{"className":8957,"style":483},[410],[394,8959,488],{"className":8960,"style":487},[415,416]," in ",[394,8963,8965],{"className":8964},[397],[394,8966,8968],{"className":8967,"ariaHidden":402},[401],[394,8969,8971,8974,8977,8980,8983,8986,8989],{"className":8970},[406],[394,8972],{"className":8973,"style":437},[410],[394,8975,442],{"className":8976},[441],[394,8978,417],{"className":8979},[415,416],[394,8981,450],{"className":8982},[449],[394,8984],{"className":8985,"style":455},[454],[394,8987,460],{"className":8988,"style":459},[415,416],[394,8990,465],{"className":8991},[464]," is performed ",[421,8994,7200],{},": at that moment the piece\nruns uncut from wall ",[394,8997,8999],{"className":8998},[397],[394,9000,9002],{"className":9001,"ariaHidden":402},[401],[394,9003,9005,9008],{"className":9004},[406],[394,9006],{"className":9007,"style":411},[410],[394,9009,417],{"className":9010},[415,416]," to wall ",[394,9013,9015],{"className":9014},[397],[394,9016,9018],{"className":9017,"ariaHidden":402},[401],[394,9019,9021,9024],{"className":9020},[406],[394,9022],{"className":9023,"style":3330},[410],[394,9025,460],{"className":9026,"style":459},[415,416],", so the cost is exactly the length\n",[394,9029,9031],{"className":9030},[397],[394,9032,9034,9091],{"className":9033,"ariaHidden":402},[401],[394,9035,9037,9041,9082,9085,9088],{"className":9036},[406],[394,9038],{"className":9039,"style":9040},[410],"height:0.8694em;vertical-align:-0.2861em;",[394,9042,9044,9048],{"className":9043},[415],[394,9045,9047],{"className":9046},[415,416],"x",[394,9049,9051],{"className":9050},[661],[394,9052,9054,9074],{"className":9053},[665,666],[394,9055,9057,9071],{"className":9056},[670],[394,9058,9060],{"className":9059,"style":858},[674],[394,9061,9062,9065],{"style":678},[394,9063],{"className":9064,"style":683},[682],[394,9066,9068],{"className":9067},[687,688,689,690],[394,9069,460],{"className":9070,"style":459},[415,416,690],[394,9072,699],{"className":9073},[698],[394,9075,9077],{"className":9076},[670],[394,9078,9080],{"className":9079,"style":1277},[674],[394,9081],{},[394,9083],{"className":9084,"style":560},[454],[394,9086,930],{"className":9087},[564],[394,9089],{"className":9090,"style":560},[454],[394,9092,9094,9097],{"className":9093},[406],[394,9095],{"className":9096,"style":7249},[410],[394,9098,9100,9103],{"className":9099},[415],[394,9101,9047],{"className":9102},[415,416],[394,9104,9106],{"className":9105},[661],[394,9107,9109,9129],{"className":9108},[665,666],[394,9110,9112,9126],{"className":9111},[670],[394,9113,9115],{"className":9114,"style":858},[674],[394,9116,9117,9120],{"style":678},[394,9118],{"className":9119,"style":683},[682],[394,9121,9123],{"className":9122},[687,688,689,690],[394,9124,417],{"className":9125},[415,416,690],[394,9127,699],{"className":9128},[698],[394,9130,9132],{"className":9131},[670],[394,9133,9135],{"className":9134,"style":706},[674],[394,9136],{},", fixed; and the cuts in ",[394,9139,9141],{"className":9140},[397],[394,9142,9144],{"className":9143,"ariaHidden":402},[401],[394,9145,9147,9150,9153,9156,9159,9162,9165],{"className":9146},[406],[394,9148],{"className":9149,"style":437},[410],[394,9151,1313],{"className":9152},[441],[394,9154,417],{"className":9155},[415,416],[394,9157,450],{"className":9158},[449],[394,9160],{"className":9161,"style":455},[454],[394,9163,488],{"className":9164,"style":487},[415,416],[394,9166,1407],{"className":9167},[464],[394,9169,9171],{"className":9170},[397],[394,9172,9174],{"className":9173,"ariaHidden":402},[401],[394,9175,9177,9180,9183,9186,9189,9192,9195],{"className":9176},[406],[394,9178],{"className":9179,"style":437},[410],[394,9181,1313],{"className":9182},[441],[394,9184,488],{"className":9185,"style":487},[415,416],[394,9187,450],{"className":9188},[449],[394,9190],{"className":9191,"style":455},[454],[394,9193,460],{"className":9194,"style":459},[415,416],[394,9196,1407],{"className":9197},[464]," were made earlier, each\nwithin its own sub-piece, independently. Same recurrence, ",[394,9200,9202],{"className":9201},[397],[394,9203,9205],{"className":9204,"ariaHidden":402},[401],[394,9206,9208,9211,9214,9217,9246],{"className":9207},[406],[394,9209],{"className":9210,"style":3644},[410],[394,9212,3702],{"className":9213,"style":615},[415,416],[394,9215,1313],{"className":9216},[441],[394,9218,9220,9223],{"className":9219},[415],[394,9221,791],{"className":9222},[415,416],[394,9224,9226],{"className":9225},[661],[394,9227,9229],{"className":9228},[665],[394,9230,9232],{"className":9231},[670],[394,9233,9235],{"className":9234,"style":3670},[674],[394,9236,9237,9240],{"style":3673},[394,9238],{"className":9239,"style":683},[682],[394,9241,9243],{"className":9242},[687,688,689,690],[394,9244,3776],{"className":9245},[415,690],[394,9247,1407],{"className":9248},[464]," over the cut\npositions. The lesson generalizes: when the per-step cost depends on neighbors, ask\nwhich step is ",[385,9251,7200],{},", since the last step is the one whose context is fully\ndetermined by the interval endpoints.",[534,9254,9256],{"id":9255},"palindrome-partitioning-over-a-string","Palindrome partitioning over a string",[381,9258,9259,9260,9263,9264,9279,9280,9317,9318,9349,9350,9380,9381,7033,9488,9552,9553,9568,9569,9597,9598,9626,9627,9642],{},"Interval DP also runs over strings. In ",[421,9261,9262],{},"Palindrome Partitioning II"," we cut a\nstring ",[394,9265,9267],{"className":9266},[397],[394,9268,9270],{"className":9269,"ariaHidden":402},[401],[394,9271,9273,9276],{"className":9272},[406],[394,9274],{"className":9275,"style":611},[410],[394,9277,3511],{"className":9278},[415,416]," into pieces that are each palindromes, minimizing the number of cuts.\nPrecompute ",[394,9281,9283],{"className":9282},[397],[394,9284,9286],{"className":9285,"ariaHidden":402},[401],[394,9287,9289,9292,9299,9302,9305,9308,9311,9314],{"className":9288},[406],[394,9290],{"className":9291,"style":437},[410],[394,9293,9295],{"className":9294},[415,5302],[394,9296,9298],{"className":9297},[415],"pal",[394,9300,442],{"className":9301},[441],[394,9303,417],{"className":9304},[415,416],[394,9306,450],{"className":9307},[449],[394,9309],{"className":9310,"style":455},[454],[394,9312,460],{"className":9313,"style":459},[415,416],[394,9315,465],{"className":9316},[464],", whether ",[394,9319,9321],{"className":9320},[397],[394,9322,9324],{"className":9323,"ariaHidden":402},[401],[394,9325,9327,9330,9333,9336,9339,9343,9346],{"className":9326},[406],[394,9328],{"className":9329,"style":437},[410],[394,9331,3511],{"className":9332},[415,416],[394,9334,442],{"className":9335},[441],[394,9337,417],{"className":9338},[415,416],[394,9340,9342],{"className":9341},[415],"..",[394,9344,460],{"className":9345,"style":459},[415,416],[394,9347,465],{"className":9348},[464]," is a palindrome, itself an\ninterval DP, since ",[394,9351,9353],{"className":9352},[397],[394,9354,9356],{"className":9355,"ariaHidden":402},[401],[394,9357,9359,9362,9365,9368,9371,9374,9377],{"className":9358},[406],[394,9360],{"className":9361,"style":437},[410],[394,9363,3511],{"className":9364},[415,416],[394,9366,442],{"className":9367},[441],[394,9369,417],{"className":9370},[415,416],[394,9372,9342],{"className":9373},[415],[394,9375,460],{"className":9376,"style":459},[415,416],[394,9378,465],{"className":9379},[464]," is a palindrome iff ",[394,9382,9384],{"className":9383},[397],[394,9385,9387,9442],{"className":9386,"ariaHidden":402},[401],[394,9388,9390,9393,9433,9436,9439],{"className":9389},[406],[394,9391],{"className":9392,"style":7249},[410],[394,9394,9396,9399],{"className":9395},[415],[394,9397,3511],{"className":9398},[415,416],[394,9400,9402],{"className":9401},[661],[394,9403,9405,9425],{"className":9404},[665,666],[394,9406,9408,9422],{"className":9407},[670],[394,9409,9411],{"className":9410,"style":858},[674],[394,9412,9413,9416],{"style":678},[394,9414],{"className":9415,"style":683},[682],[394,9417,9419],{"className":9418},[687,688,689,690],[394,9420,417],{"className":9421},[415,416,690],[394,9423,699],{"className":9424},[698],[394,9426,9428],{"className":9427},[670],[394,9429,9431],{"className":9430,"style":706},[674],[394,9432],{},[394,9434],{"className":9435,"style":1702},[454],[394,9437,1707],{"className":9438},[1706],[394,9440],{"className":9441,"style":1702},[454],[394,9443,9445,9448],{"className":9444},[406],[394,9446],{"className":9447,"style":2171},[410],[394,9449,9451,9454],{"className":9450},[415],[394,9452,3511],{"className":9453},[415,416],[394,9455,9457],{"className":9456},[661],[394,9458,9460,9480],{"className":9459},[665,666],[394,9461,9463,9477],{"className":9462},[670],[394,9464,9466],{"className":9465,"style":858},[674],[394,9467,9468,9471],{"style":678},[394,9469],{"className":9470,"style":683},[682],[394,9472,9474],{"className":9473},[687,688,689,690],[394,9475,460],{"className":9476,"style":459},[415,416,690],[394,9478,699],{"className":9479},[698],[394,9481,9483],{"className":9482},[670],[394,9484,9486],{"className":9485,"style":1277},[674],[394,9487],{},[394,9489,9491],{"className":9490},[397],[394,9492,9494,9518,9540],{"className":9493,"ariaHidden":402},[401],[394,9495,9497,9500,9503,9506,9509,9512,9515],{"className":9496},[406],[394,9498],{"className":9499,"style":437},[410],[394,9501,3511],{"className":9502},[415,416],[394,9504,442],{"className":9505},[441],[394,9507,417],{"className":9508},[415,416],[394,9510],{"className":9511,"style":560},[454],[394,9513,1446],{"className":9514},[564],[394,9516],{"className":9517,"style":560},[454],[394,9519,9521,9524,9528,9531,9534,9537],{"className":9520},[406],[394,9522],{"className":9523,"style":3330},[410],[394,9525,9527],{"className":9526},[415],"1..",[394,9529,460],{"className":9530,"style":459},[415,416],[394,9532],{"className":9533,"style":560},[454],[394,9535,930],{"className":9536},[564],[394,9538],{"className":9539,"style":560},[454],[394,9541,9543,9546,9549],{"className":9542},[406],[394,9544],{"className":9545,"style":437},[410],[394,9547,694],{"className":9548},[415],[394,9550,465],{"className":9551},[464]," is\n(a length-",[394,9554,9556],{"className":9555},[397],[394,9557,9559],{"className":9558,"ariaHidden":402},[401],[394,9560,9562,9565],{"className":9561},[406],[394,9563],{"className":9564,"style":3491},[410],[394,9566,738],{"className":9567},[415]," shorter interval). Then let ",[394,9570,9572],{"className":9571},[397],[394,9573,9575],{"className":9574,"ariaHidden":402},[401],[394,9576,9578,9581,9588,9591,9594],{"className":9577},[406],[394,9579],{"className":9580,"style":437},[410],[394,9582,9584],{"className":9583},[415,5302],[394,9585,9587],{"className":9586},[415],"cut",[394,9589,442],{"className":9590},[441],[394,9592,460],{"className":9593,"style":459},[415,416],[394,9595,465],{"className":9596},[464]," be the fewest cuts for\nthe prefix ",[394,9599,9601],{"className":9600},[397],[394,9602,9604],{"className":9603,"ariaHidden":402},[401],[394,9605,9607,9610,9613,9616,9620,9623],{"className":9606},[406],[394,9608],{"className":9609,"style":437},[410],[394,9611,3511],{"className":9612},[415,416],[394,9614,442],{"className":9615},[441],[394,9617,9619],{"className":9618},[415],"0..",[394,9621,460],{"className":9622,"style":459},[415,416],[394,9624,465],{"className":9625},[464],"; for each ",[394,9628,9630],{"className":9629},[397],[394,9631,9633],{"className":9632,"ariaHidden":402},[401],[394,9634,9636,9639],{"className":9635},[406],[394,9637],{"className":9638,"style":3330},[410],[394,9640,460],{"className":9641,"style":459},[415,416]," we look back to the last palindromic piece:",[394,9644,9646],{"className":9645},[1665],[394,9647,9649],{"className":9648},[397],[394,9650,9652,9682],{"className":9651,"ariaHidden":402},[401],[394,9653,9655,9658,9664,9667,9670,9673,9676,9679],{"className":9654},[406],[394,9656],{"className":9657,"style":437},[410],[394,9659,9661],{"className":9660},[415,5302],[394,9662,9587],{"className":9663},[415],[394,9665,442],{"className":9666},[441],[394,9668,460],{"className":9669,"style":459},[415,416],[394,9671,465],{"className":9672},[464],[394,9674],{"className":9675,"style":1702},[454],[394,9677,1707],{"className":9678},[1706],[394,9680],{"className":9681,"style":1702},[454],[394,9683,9685,9689,9781,9784],{"className":9684},[406],[394,9686],{"className":9687,"style":9688},[410],"height:3em;vertical-align:-1.25em;",[394,9690,9692,9698],{"className":9691},[1778],[394,9693,9695],{"className":9694},[1778],[394,9696,1786],{"className":9697},[415,1785],[394,9699,9701],{"className":9700},[661],[394,9702,9704,9772],{"className":9703},[665,666],[394,9705,9707,9769],{"className":9706},[670],[394,9708,9711],{"className":9709,"style":9710},[674],"height:0.3448em;",[394,9712,9714,9717],{"style":9713},"top:-2.5198em;margin-right:0.05em;",[394,9715],{"className":9716,"style":683},[682],[394,9718,9720],{"className":9719},[687,688,689,690],[394,9721,9723,9726,9729,9732,9735,9738,9741,9748,9754,9757,9760,9763,9766],{"className":9722},[415,690],[394,9724,1043],{"className":9725},[415,690],[394,9727,1817],{"className":9728},[1706,690],[394,9730,417],{"className":9731},[415,416,690],[394,9733,1817],{"className":9734},[1706,690],[394,9736,460],{"className":9737,"style":459},[415,416,690],[394,9739,450],{"className":9740},[449,690],[394,9742,9744],{"className":9743},[454,690],[394,9745,9747],{"className":9746},[690]," ",[394,9749,9751],{"className":9750},[415,5302,690],[394,9752,9298],{"className":9753},[415,690],[394,9755,442],{"className":9756},[441,690],[394,9758,417],{"className":9759},[415,416,690],[394,9761,450],{"className":9762},[449,690],[394,9764,460],{"className":9765,"style":459},[415,416,690],[394,9767,465],{"className":9768},[464,690],[394,9770,699],{"className":9771},[698],[394,9773,9775],{"className":9774},[670],[394,9776,9779],{"className":9777,"style":9778},[674],"height:0.3552em;",[394,9780],{},[394,9782],{"className":9783,"style":455},[454],[394,9785,9787,9793,9961],{"className":9786},[756],[394,9788,9790],{"className":9789,"style":1725},[441,1724],[394,9791,1731],{"className":9792},[1729,1730],[394,9794,9796],{"className":9795},[415],[394,9797,9799,9884,9887],{"className":9798},[1738],[394,9800,9802],{"className":9801},[1742],[394,9803,9805,9875],{"className":9804},[665,666],[394,9806,9808,9872],{"className":9807},[670],[394,9809,9812,9824],{"className":9810,"style":9811},[674],"height:1.69em;",[394,9813,9815,9818],{"style":9814},"top:-3.69em;",[394,9816],{"className":9817,"style":1759},[682],[394,9819,9821],{"className":9820},[415],[394,9822,1043],{"className":9823},[415],[394,9825,9827,9830],{"style":9826},"top:-2.25em;",[394,9828],{"className":9829,"style":1759},[682],[394,9831,9833,9839,9842,9845,9848,9851,9854,9857,9860,9863,9866,9869],{"className":9832},[415],[394,9834,9836],{"className":9835},[415,5302],[394,9837,9587],{"className":9838},[415],[394,9840,442],{"className":9841},[441],[394,9843,417],{"className":9844},[415,416],[394,9846],{"className":9847,"style":560},[454],[394,9849,930],{"className":9850},[564],[394,9852],{"className":9853,"style":560},[454],[394,9855,694],{"className":9856},[415],[394,9858,465],{"className":9859},[464],[394,9861],{"className":9862,"style":560},[454],[394,9864,1446],{"className":9865},[564],[394,9867],{"className":9868,"style":560},[454],[394,9870,694],{"className":9871},[415],[394,9873,699],{"className":9874},[698],[394,9876,9878],{"className":9877},[670],[394,9879,9882],{"className":9880,"style":9881},[674],"height:1.19em;",[394,9883],{},[394,9885],{"className":9886,"style":2075},[2074],[394,9888,9890],{"className":9889},[1742],[394,9891,9893,9953],{"className":9892},[665,666],[394,9894,9896,9950],{"className":9895},[670],[394,9897,9899,9925],{"className":9898,"style":9811},[674],[394,9900,9901,9904],{"style":9814},[394,9902],{"className":9903,"style":1759},[682],[394,9905,9907,9910,9913,9916,9919,9922],{"className":9906},[415],[394,9908,417],{"className":9909},[415,416],[394,9911],{"className":9912,"style":1702},[454],[394,9914,1707],{"className":9915},[1706],[394,9917],{"className":9918,"style":1702},[454],[394,9920,1043],{"className":9921},[415],[394,9923,450],{"className":9924},[449],[394,9926,9927,9930],{"style":9826},[394,9928],{"className":9929,"style":1759},[682],[394,9931,9933,9936,9939,9943,9946],{"className":9932},[415],[394,9934,417],{"className":9935},[415,416],[394,9937],{"className":9938,"style":1702},[454],[394,9940,9942],{"className":9941},[1706],">",[394,9944],{"className":9945,"style":1702},[454],[394,9947,9949],{"className":9948},[415],"0.",[394,9951,699],{"className":9952},[698],[394,9954,9956],{"className":9955},[670],[394,9957,9959],{"className":9958,"style":9881},[674],[394,9960],{},[394,9962],{"className":9963},[464,2155],[381,9965,9966,9967,10017],{},"The palindrome table is the interval DP (filled by increasing length); the cut\ncount is then a ",[394,9968,9970],{"className":9969},[397],[394,9971,9973],{"className":9972,"ariaHidden":402},[401],[394,9974,9976,9979,9982,9985,10014],{"className":9975},[406],[394,9977],{"className":9978,"style":3644},[410],[394,9980,3702],{"className":9981,"style":615},[415,416],[394,9983,1313],{"className":9984},[441],[394,9986,9988,9991],{"className":9987},[415],[394,9989,791],{"className":9990},[415,416],[394,9992,9994],{"className":9993},[661],[394,9995,9997],{"className":9996},[665],[394,9998,10000],{"className":9999},[670],[394,10001,10003],{"className":10002,"style":3670},[674],[394,10004,10005,10008],{"style":3673},[394,10006],{"className":10007,"style":683},[682],[394,10009,10011],{"className":10010},[687,688,689,690],[394,10012,738],{"className":10013},[415,690],[394,10015,1407],{"className":10016},[464]," sweep over it, a clean example of one interval DP feeding\na second, simpler one.",[534,10019,10021,10022,10072],{"id":10020},"when-on3-is-too-slow","When ",[394,10023,10025],{"className":10024},[397],[394,10026,10028],{"className":10027,"ariaHidden":402},[401],[394,10029,10031,10034,10037,10040,10069],{"className":10030},[406],[394,10032],{"className":10033,"style":3644},[410],[394,10035,3702],{"className":10036,"style":615},[415,416],[394,10038,1313],{"className":10039},[441],[394,10041,10043,10046],{"className":10042},[415],[394,10044,791],{"className":10045},[415,416],[394,10047,10049],{"className":10048},[661],[394,10050,10052],{"className":10051},[665],[394,10053,10055],{"className":10054},[670],[394,10056,10058],{"className":10057,"style":3670},[674],[394,10059,10060,10063],{"style":3673},[394,10061],{"className":10062,"style":683},[682],[394,10064,10066],{"className":10065},[687,688,689,690],[394,10067,3776],{"className":10068},[415,690],[394,10070,1407],{"className":10071},[464]," is too slow",[381,10074,7186,10075,10125,10126,10141,10142,10145,10146,10149,10150,10165,10166,10169,10170,10220,10221,10224,10225,10275,10276],{},[394,10076,10078],{"className":10077},[397],[394,10079,10081],{"className":10080,"ariaHidden":402},[401],[394,10082,10084,10087,10090,10093,10122],{"className":10083},[406],[394,10085],{"className":10086,"style":3644},[410],[394,10088,3702],{"className":10089,"style":615},[415,416],[394,10091,1313],{"className":10092},[441],[394,10094,10096,10099],{"className":10095},[415],[394,10097,791],{"className":10098},[415,416],[394,10100,10102],{"className":10101},[661],[394,10103,10105],{"className":10104},[665],[394,10106,10108],{"className":10107},[670],[394,10109,10111],{"className":10110,"style":3670},[674],[394,10112,10113,10116],{"style":3673},[394,10114],{"className":10115,"style":683},[682],[394,10117,10119],{"className":10118},[687,688,689,690],[394,10120,3776],{"className":10121},[415,690],[394,10123,1407],{"className":10124},[464]," cost comes from the inner scan over all splits ",[394,10127,10129],{"className":10128},[397],[394,10130,10132],{"className":10131,"ariaHidden":402},[401],[394,10133,10135,10138],{"className":10134},[406],[394,10136],{"className":10137,"style":483},[410],[394,10139,488],{"className":10140,"style":487},[415,416],". For a class of\ninterval DPs (matrix-chain, optimal BST, and others whose cost obeys the\n",[421,10143,10144],{},"quadrangle inequality",") the optimal split point is ",[385,10147,10148],{},"monotone"," in the\nendpoints, so the search for ",[394,10151,10153],{"className":10152},[397],[394,10154,10156],{"className":10155,"ariaHidden":402},[401],[394,10157,10159,10162],{"className":10158},[406],[394,10160],{"className":10161,"style":483},[410],[394,10163,488],{"className":10164,"style":487},[415,416]," can be confined to a shrinking window. This is\n",[421,10167,10168],{},"Knuth's optimization",", and it drops the running time to ",[394,10171,10173],{"className":10172},[397],[394,10174,10176],{"className":10175,"ariaHidden":402},[401],[394,10177,10179,10182,10185,10188,10217],{"className":10178},[406],[394,10180],{"className":10181,"style":3644},[410],[394,10183,3702],{"className":10184,"style":615},[415,416],[394,10186,1313],{"className":10187},[441],[394,10189,10191,10194],{"className":10190},[415],[394,10192,791],{"className":10193},[415,416],[394,10195,10197],{"className":10196},[661],[394,10198,10200],{"className":10199},[665],[394,10201,10203],{"className":10202},[670],[394,10204,10206],{"className":10205,"style":3670},[674],[394,10207,10208,10211],{"style":3673},[394,10209],{"className":10210,"style":683},[682],[394,10212,10214],{"className":10213},[687,688,689,690],[394,10215,738],{"className":10216},[415,690],[394,10218,1407],{"className":10219},[464],". We treat the\nconditions and the proof in the lesson on ",[526,10222,10223],{"href":277},"DP\noptimizations","; for now, note\nonly that the ",[394,10226,10228],{"className":10227},[397],[394,10229,10231],{"className":10230,"ariaHidden":402},[401],[394,10232,10234,10237,10240,10243,10272],{"className":10233},[406],[394,10235],{"className":10236,"style":3644},[410],[394,10238,3702],{"className":10239,"style":615},[415,416],[394,10241,1313],{"className":10242},[441],[394,10244,10246,10249],{"className":10245},[415],[394,10247,791],{"className":10248},[415,416],[394,10250,10252],{"className":10251},[661],[394,10253,10255],{"className":10254},[665],[394,10256,10258],{"className":10257},[670],[394,10259,10261],{"className":10260,"style":3670},[674],[394,10262,10263,10266],{"style":3673},[394,10264],{"className":10265,"style":683},[682],[394,10267,10269],{"className":10268},[687,688,689,690],[394,10270,3776],{"className":10271},[415,690],[394,10273,1407],{"className":10274},[464]," here is not always the last word.",[1162,10277,10278],{},[526,10279,3776],{"href":10280,"ariaDescribedBy":10281,"dataFootnoteRef":376,"id":10282},"#user-content-fn-skiena-dp",[1168],"user-content-fnref-skiena-dp",[534,10284,10286],{"id":10285},"takeaways","Takeaways",[10288,10289,10290,10503,11033,11073,11116,11121],"ul",{},[5282,10291,10292,10294,10295,10325,10326,470,10328,10343,10344,10347,10348,10398,10399,10415,10439,10440,2140],{},[421,10293,261],{}," solves problems over a contiguous range: the state is a slice\n",[394,10296,10298],{"className":10297},[397],[394,10299,10301],{"className":10300,"ariaHidden":402},[401],[394,10302,10304,10307,10310,10313,10316,10319,10322],{"className":10303},[406],[394,10305],{"className":10306,"style":437},[410],[394,10308,442],{"className":10309},[441],[394,10311,417],{"className":10312},[415,416],[394,10314,450],{"className":10315},[449],[394,10317],{"className":10318,"style":455},[454],[394,10320,460],{"className":10321,"style":459},[415,416],[394,10323,465],{"className":10324},[464],", the recurrence picks a ",[421,10327,5402],{},[394,10329,10331],{"className":10330},[397],[394,10332,10334],{"className":10333,"ariaHidden":402},[401],[394,10335,10337,10340],{"className":10336},[406],[394,10338],{"className":10339,"style":483},[410],[394,10341,488],{"className":10342,"style":487},[415,416]," that breaks the range\ninto two independent parts, and the table is filled by ",[421,10345,10346],{},"increasing interval\nlength"," so both parts are ready, typically ",[394,10349,10351],{"className":10350},[397],[394,10352,10354],{"className":10353,"ariaHidden":402},[401],[394,10355,10357,10360,10363,10366,10395],{"className":10356},[406],[394,10358],{"className":10359,"style":3644},[410],[394,10361,3648],{"className":10362},[415],[394,10364,1313],{"className":10365},[441],[394,10367,10369,10372],{"className":10368},[415],[394,10370,791],{"className":10371},[415,416],[394,10373,10375],{"className":10374},[661],[394,10376,10378],{"className":10377},[665],[394,10379,10381],{"className":10380},[670],[394,10382,10384],{"className":10383,"style":3670},[674],[394,10385,10386,10389],{"style":3673},[394,10387],{"className":10388,"style":683},[682],[394,10390,10392],{"className":10391},[687,688,689,690],[394,10393,738],{"className":10394},[415,690],[394,10396,1407],{"className":10397},[464]," states ",[394,10400,10402],{"className":10401},[397],[394,10403,10405],{"className":10404,"ariaHidden":402},[401],[394,10406,10408,10412],{"className":10407},[406],[394,10409],{"className":10410,"style":10411},[410],"height:0.6667em;vertical-align:-0.0833em;",[394,10413,565],{"className":10414},[415],[394,10416,10418],{"className":10417},[397],[394,10419,10421],{"className":10420,"ariaHidden":402},[401],[394,10422,10424,10427,10430,10433,10436],{"className":10423},[406],[394,10425],{"className":10426,"style":437},[410],[394,10428,3702],{"className":10429,"style":615},[415,416],[394,10431,1313],{"className":10432},[441],[394,10434,791],{"className":10435},[415,416],[394,10437,1407],{"className":10438},[464]," splits ",[394,10441,10443],{"className":10442},[397],[394,10444,10446,10459],{"className":10445,"ariaHidden":402},[401],[394,10447,10449,10453,10456],{"className":10448},[406],[394,10450],{"className":10451,"style":10452},[410],"height:0.3669em;",[394,10454,1707],{"className":10455},[1706],[394,10457],{"className":10458,"style":1702},[454],[394,10460,10462,10465,10468,10471,10500],{"className":10461},[406],[394,10463],{"className":10464,"style":3644},[410],[394,10466,3702],{"className":10467,"style":615},[415,416],[394,10469,1313],{"className":10470},[441],[394,10472,10474,10477],{"className":10473},[415],[394,10475,791],{"className":10476},[415,416],[394,10478,10480],{"className":10479},[661],[394,10481,10483],{"className":10482},[665],[394,10484,10486],{"className":10485},[670],[394,10487,10489],{"className":10488,"style":3670},[674],[394,10490,10491,10494],{"style":3673},[394,10492],{"className":10493,"style":683},[682],[394,10495,10497],{"className":10496},[687,688,689,690],[394,10498,3776],{"className":10499},[415,690],[394,10501,1407],{"className":10502},[464],[5282,10504,10505,10507,10508,10845,10846,10897,10898,10931,10932,3780,10982,11032],{},[421,10506,537],{}," is the archetype:\n",[394,10509,10511],{"className":10510},[397],[394,10512,10514,10550,10650,10674,10704],{"className":10513,"ariaHidden":402},[401],[394,10515,10517,10520,10523,10526,10529,10532,10535,10538,10541,10544,10547],{"className":10516},[406],[394,10518],{"className":10519,"style":437},[410],[394,10521,1540],{"className":10522},[415,416],[394,10524,442],{"className":10525},[441],[394,10527,417],{"className":10528},[415,416],[394,10530,450],{"className":10531},[449],[394,10533],{"className":10534,"style":455},[454],[394,10536,460],{"className":10537,"style":459},[415,416],[394,10539,465],{"className":10540},[464],[394,10542],{"className":10543,"style":1702},[454],[394,10545,1707],{"className":10546},[1706],[394,10548],{"className":10549,"style":1702},[454],[394,10551,10553,10556,10614,10620,10623,10626,10629,10632,10635,10638,10641,10644,10647],{"className":10552},[406],[394,10554],{"className":10555,"style":6410},[410],[394,10557,10559,10565],{"className":10558},[1778],[394,10560,10562],{"className":10561},[1778],[394,10563,1786],{"className":10564},[415,1785],[394,10566,10568],{"className":10567},[661],[394,10569,10571,10606],{"className":10570},[665,666],[394,10572,10574,10603],{"className":10573},[670],[394,10575,10577],{"className":10576,"style":1381},[674],[394,10578,10579,10582],{"style":1801},[394,10580],{"className":10581,"style":683},[682],[394,10583,10585],{"className":10584},[687,688,689,690],[394,10586,10588,10591,10594,10597,10600],{"className":10587},[415,690],[394,10589,417],{"className":10590},[415,416,690],[394,10592,1817],{"className":10593},[1706,690],[394,10595,488],{"className":10596,"style":487},[415,416,690],[394,10598,1824],{"className":10599},[1706,690],[394,10601,460],{"className":10602,"style":459},[415,416,690],[394,10604,699],{"className":10605},[698],[394,10607,10609],{"className":10608},[670],[394,10610,10612],{"className":10611,"style":1277},[674],[394,10613],{},[394,10615,10617],{"className":10616},[441],[394,10618,1313],{"className":10619},[1729,1845],[394,10621,1540],{"className":10622},[415,416],[394,10624,442],{"className":10625},[441],[394,10627,417],{"className":10628},[415,416],[394,10630,450],{"className":10631},[449],[394,10633],{"className":10634,"style":455},[454],[394,10636,488],{"className":10637,"style":487},[415,416],[394,10639,465],{"className":10640},[464],[394,10642],{"className":10643,"style":560},[454],[394,10645,1446],{"className":10646},[564],[394,10648],{"className":10649,"style":560},[454],[394,10651,10653,10656,10659,10662,10665,10668,10671],{"className":10652},[406],[394,10654],{"className":10655,"style":437},[410],[394,10657,1540],{"className":10658},[415,416],[394,10660,442],{"className":10661},[441],[394,10663,488],{"className":10664,"style":487},[415,416],[394,10666],{"className":10667,"style":560},[454],[394,10669,1446],{"className":10670},[564],[394,10672],{"className":10673,"style":560},[454],[394,10675,10677,10680,10683,10686,10689,10692,10695,10698,10701],{"className":10676},[406],[394,10678],{"className":10679,"style":437},[410],[394,10681,694],{"className":10682},[415],[394,10684,450],{"className":10685},[449],[394,10687],{"className":10688,"style":455},[454],[394,10690,460],{"className":10691,"style":459},[415,416],[394,10693,465],{"className":10694},[464],[394,10696],{"className":10697,"style":560},[454],[394,10699,1446],{"className":10700},[564],[394,10702],{"className":10703,"style":560},[454],[394,10705,10707,10710,10759,10799,10839],{"className":10706},[406],[394,10708],{"className":10709,"style":6410},[410],[394,10711,10713,10716],{"className":10712},[415],[394,10714,381],{"className":10715},[415,416],[394,10717,10719],{"className":10718},[661],[394,10720,10722,10751],{"className":10721},[665,666],[394,10723,10725,10748],{"className":10724},[670],[394,10726,10728],{"className":10727,"style":858},[674],[394,10729,10730,10733],{"style":678},[394,10731],{"className":10732,"style":683},[682],[394,10734,10736],{"className":10735},[687,688,689,690],[394,10737,10739,10742,10745],{"className":10738},[415,690],[394,10740,417],{"className":10741},[415,416,690],[394,10743,930],{"className":10744},[564,690],[394,10746,694],{"className":10747},[415,690],[394,10749,699],{"className":10750},[698],[394,10752,10754],{"className":10753},[670],[394,10755,10757],{"className":10756,"style":943},[674],[394,10758],{},[394,10760,10762,10765],{"className":10761},[415],[394,10763,381],{"className":10764},[415,416],[394,10766,10768],{"className":10767},[661],[394,10769,10771,10791],{"className":10770},[665,666],[394,10772,10774,10788],{"className":10773},[670],[394,10775,10777],{"className":10776,"style":1381},[674],[394,10778,10779,10782],{"style":678},[394,10780],{"className":10781,"style":683},[682],[394,10783,10785],{"className":10784},[687,688,689,690],[394,10786,488],{"className":10787,"style":487},[415,416,690],[394,10789,699],{"className":10790},[698],[394,10792,10794],{"className":10793},[670],[394,10795,10797],{"className":10796,"style":706},[674],[394,10798],{},[394,10800,10802,10805],{"className":10801},[415],[394,10803,381],{"className":10804},[415,416],[394,10806,10808],{"className":10807},[661],[394,10809,10811,10831],{"className":10810},[665,666],[394,10812,10814,10828],{"className":10813},[670],[394,10815,10817],{"className":10816,"style":858},[674],[394,10818,10819,10822],{"style":678},[394,10820],{"className":10821,"style":683},[682],[394,10823,10825],{"className":10824},[687,688,689,690],[394,10826,460],{"className":10827,"style":459},[415,416,690],[394,10829,699],{"className":10830},[698],[394,10832,10834],{"className":10833},[670],[394,10835,10837],{"className":10836,"style":1277},[674],[394,10838],{},[394,10840,10842],{"className":10841},[464],[394,10843,1407],{"className":10844},[1729,1845],",\n",[394,10847,10849],{"className":10848},[397],[394,10850,10852,10888],{"className":10851,"ariaHidden":402},[401],[394,10853,10855,10858,10861,10864,10867,10870,10873,10876,10879,10882,10885],{"className":10854},[406],[394,10856],{"className":10857,"style":437},[410],[394,10859,1540],{"className":10860},[415,416],[394,10862,442],{"className":10863},[441],[394,10865,417],{"className":10866},[415,416],[394,10868,450],{"className":10869},[449],[394,10871],{"className":10872,"style":455},[454],[394,10874,417],{"className":10875},[415,416],[394,10877,465],{"className":10878},[464],[394,10880],{"className":10881,"style":1702},[454],[394,10883,1707],{"className":10884},[1706],[394,10886],{"className":10887,"style":1702},[454],[394,10889,10891,10894],{"className":10890},[406],[394,10892],{"className":10893,"style":3491},[410],[394,10895,1043],{"className":10896},[415],", with a split table ",[394,10899,10901],{"className":10900},[397],[394,10902,10904],{"className":10903,"ariaHidden":402},[401],[394,10905,10907,10910,10913,10916,10919,10922,10925,10928],{"className":10906},[406],[394,10908],{"className":10909,"style":437},[410],[394,10911,3511],{"className":10912},[415,416],[394,10914,442],{"className":10915},[441],[394,10917,417],{"className":10918},[415,416],[394,10920,450],{"className":10921},[449],[394,10923],{"className":10924,"style":455},[454],[394,10926,460],{"className":10927,"style":459},[415,416],[394,10929,465],{"className":10930},[464]," to reconstruct the parenthesisation in\n",[394,10933,10935],{"className":10934},[397],[394,10936,10938],{"className":10937,"ariaHidden":402},[401],[394,10939,10941,10944,10947,10950,10979],{"className":10940},[406],[394,10942],{"className":10943,"style":3644},[410],[394,10945,3648],{"className":10946},[415],[394,10948,1313],{"className":10949},[441],[394,10951,10953,10956],{"className":10952},[415],[394,10954,791],{"className":10955},[415,416],[394,10957,10959],{"className":10958},[661],[394,10960,10962],{"className":10961},[665],[394,10963,10965],{"className":10964},[670],[394,10966,10968],{"className":10967,"style":3670},[674],[394,10969,10970,10973],{"style":3673},[394,10971],{"className":10972,"style":683},[682],[394,10974,10976],{"className":10975},[687,688,689,690],[394,10977,3776],{"className":10978},[415,690],[394,10980,1407],{"className":10981},[464],[394,10983,10985],{"className":10984},[397],[394,10986,10988],{"className":10987,"ariaHidden":402},[401],[394,10989,10991,10994,10997,11000,11029],{"className":10990},[406],[394,10992],{"className":10993,"style":3644},[410],[394,10995,3648],{"className":10996},[415],[394,10998,1313],{"className":10999},[441],[394,11001,11003,11006],{"className":11002},[415],[394,11004,791],{"className":11005},[415,416],[394,11007,11009],{"className":11008},[661],[394,11010,11012],{"className":11011},[665],[394,11013,11015],{"className":11014},[670],[394,11016,11018],{"className":11017,"style":3670},[674],[394,11019,11020,11023],{"style":3673},[394,11021],{"className":11022,"style":683},[682],[394,11024,11026],{"className":11025},[687,688,689,690],[394,11027,738],{"className":11028},[415,690],[394,11030,1407],{"className":11031},[464]," space.",[5282,11034,11035,11038,11039,11072],{},[421,11036,11037],{},"Optimal BSTs"," reuse the same shape, replacing the combine term with the swept\nweight ",[394,11040,11042],{"className":11041},[397],[394,11043,11045],{"className":11044,"ariaHidden":402},[401],[394,11046,11048,11051,11054,11057,11060,11063,11066,11069],{"className":11047},[406],[394,11049],{"className":11050,"style":437},[410],[394,11052,6214],{"className":11053,"style":6213},[415,416],[394,11055,442],{"className":11056},[441],[394,11058,417],{"className":11059},[415,416],[394,11061,450],{"className":11062},[449],[394,11064],{"className":11065,"style":455},[454],[394,11067,460],{"className":11068,"style":459},[415,416],[394,11070,465],{"className":11071},[464]," of the interval.",[5282,11074,7186,11075,11080,11081,8961,11083,11113,11114,2140],{},[421,11076,11077,11079],{},[390,11078,7189],{}," trick"," (Burst Balloons, Minimum Cost to Cut a Stick)\nfixes which move happens ",[385,11082,7200],{},[394,11084,11086],{"className":11085},[397],[394,11087,11089],{"className":11088,"ariaHidden":402},[401],[394,11090,11092,11095,11098,11101,11104,11107,11110],{"className":11091},[406],[394,11093],{"className":11094,"style":437},[410],[394,11096,442],{"className":11097},[441],[394,11099,417],{"className":11100},[415,416],[394,11102,450],{"className":11103},[449],[394,11105],{"className":11106,"style":455},[454],[394,11108,460],{"className":11109,"style":459},[415,416],[394,11111,465],{"className":11112},[464],", not first, pinning the moving\nneighbor cost to the fixed interval walls and making the two sides truly\n",[421,11115,5701],{},[5282,11117,11118,11120],{},[421,11119,9262],{}," runs interval DP over a string: a palindrome\ntable by increasing length, then a linear cut-count sweep.",[5282,11122,11123,11125,11126,11176,11177,11179],{},[421,11124,10168],{}," cuts matrix-chain-style DPs to ",[394,11127,11129],{"className":11128},[397],[394,11130,11132],{"className":11131,"ariaHidden":402},[401],[394,11133,11135,11138,11141,11144,11173],{"className":11134},[406],[394,11136],{"className":11137,"style":3644},[410],[394,11139,3702],{"className":11140,"style":615},[415,416],[394,11142,1313],{"className":11143},[441],[394,11145,11147,11150],{"className":11146},[415],[394,11148,791],{"className":11149},[415,416],[394,11151,11153],{"className":11152},[661],[394,11154,11156],{"className":11155},[665],[394,11157,11159],{"className":11158},[670],[394,11160,11162],{"className":11161,"style":3670},[674],[394,11163,11164,11167],{"style":3673},[394,11165],{"className":11166,"style":683},[682],[394,11168,11170],{"className":11169},[687,688,689,690],[394,11171,738],{"className":11172},[415,690],[394,11174,1407],{"className":11175},[464]," when the\n",[421,11178,10144],{}," makes the optimal split monotone — see the\nDP-optimizations lesson.",[11181,11182,11185,11190],"section",{"className":11183,"dataFootnotes":376},[11184],"footnotes",[534,11186,11189],{"className":11187,"id":1168},[11188],"sr-only","Footnotes",[5279,11191,11192,11306,11317],{},[5282,11193,11195,11198,11199,11232,11233,8961,11248,11298,11299],{"id":11194},"user-content-fn-clrs-matrix",[421,11196,11197],{},"CLRS",", Ch. 15 — Dynamic Programming (§15.2): matrix-chain multiplication, the ",[394,11200,11202],{"className":11201},[397],[394,11203,11205],{"className":11204,"ariaHidden":402},[401],[394,11206,11208,11211,11214,11217,11220,11223,11226,11229],{"className":11207},[406],[394,11209],{"className":11210,"style":437},[410],[394,11212,1540],{"className":11213},[415,416],[394,11215,442],{"className":11216},[441],[394,11218,417],{"className":11219},[415,416],[394,11221,450],{"className":11222},[449],[394,11224],{"className":11225,"style":455},[454],[394,11227,460],{"className":11228,"style":459},[415,416],[394,11230,465],{"className":11231},[464]," recurrence, and reconstruction from the split table ",[394,11234,11236],{"className":11235},[397],[394,11237,11239],{"className":11238,"ariaHidden":402},[401],[394,11240,11242,11245],{"className":11241},[406],[394,11243],{"className":11244,"style":611},[410],[394,11246,3511],{"className":11247},[415,416],[394,11249,11251],{"className":11250},[397],[394,11252,11254],{"className":11253,"ariaHidden":402},[401],[394,11255,11257,11260,11263,11266,11295],{"className":11256},[406],[394,11258],{"className":11259,"style":3644},[410],[394,11261,3648],{"className":11262},[415],[394,11264,1313],{"className":11265},[441],[394,11267,11269,11272],{"className":11268},[415],[394,11270,791],{"className":11271},[415,416],[394,11273,11275],{"className":11274},[661],[394,11276,11278],{"className":11277},[665],[394,11279,11281],{"className":11280},[670],[394,11282,11284],{"className":11283,"style":3670},[674],[394,11285,11286,11289],{"style":3673},[394,11287],{"className":11288,"style":683},[682],[394,11290,11292],{"className":11291},[687,688,689,690],[394,11293,3776],{"className":11294},[415,690],[394,11296,1407],{"className":11297},[464],". ",[526,11300,11305],{"href":11301,"ariaLabel":11302,"className":11303,"dataFootnoteBackref":376},"#user-content-fnref-clrs-matrix","Back to reference 1",[11304],"data-footnote-backref","↩",[5282,11307,11309,11311,11312],{"id":11308},"user-content-fn-clrs-obst",[421,11310,11197],{},", Ch. 15 — Dynamic Programming (§15.5): optimal binary search trees as the same interval DP with the interval-weight combine term. ",[526,11313,11305],{"href":11314,"ariaLabel":11315,"className":11316,"dataFootnoteBackref":376},"#user-content-fnref-clrs-obst","Back to reference 2",[11304],[5282,11318,11320,11323,11324],{"id":11319},"user-content-fn-skiena-dp",[421,11321,11322],{},"Skiena",", § — Dynamic Programming: interval DPs as range subproblems combined at a split, and when monotonicity (Knuth) lowers the cubic cost. ",[526,11325,11305],{"href":11326,"ariaLabel":11327,"className":11328,"dataFootnoteBackref":376},"#user-content-fnref-skiena-dp","Back to reference 3",[11304],[11330,11331,11332],"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":11334},[11335,11338,11339,11340,11342,11343,11345,11346],{"id":536,"depth":18,"text":537,"children":11336},[11337],{"id":3187,"depth":24,"text":3188},{"id":5232,"depth":18,"text":5233},{"id":5705,"depth":18,"text":5706},{"id":7185,"depth":18,"text":11341},"The last operation trick: Burst Balloons and cutting a stick",{"id":9255,"depth":18,"text":9256},{"id":10020,"depth":18,"text":11344},"When O(n3) is too slow",{"id":10285,"depth":18,"text":10286},{"id":1168,"depth":18,"text":11189},"The sequence DPs we have met so far walked a string or array from one end and let\nthe subproblem be a prefix: the state was the best answer using the first i items. A second, equally common family does not decompose by prefix at all. It\ndecomposes by contiguous range. The natural subproblem is \"the best answer\nfor the slice [i,j],\" and the recurrence builds a long range out of two shorter\nranges by guessing where they meet: a split point k inside [i,j]. This\nis interval dynamic programming, one of the core DP\npatterns, and once you see its\nshape you will find it everywhere: parenthesising a product, building an optimal\nsearch tree, bursting balloons, cutting a stick, partitioning a string.","md",{"moduleNumber":196,"lessonNumber":108,"order":11350},806,true,[11353,11356,11358,11360,11364],{"title":8940,"slug":11354,"difficulty":11355},"minimum-cost-to-cut-a-stick","Hard",{"title":7207,"slug":11357,"difficulty":11355},"burst-balloons",{"title":9262,"slug":11359,"difficulty":11355},"palindrome-partitioning-ii",{"title":11361,"slug":11362,"difficulty":11363},"Minimum Cost Tree From Leaf Values","minimum-cost-tree-from-leaf-values","Medium",{"title":11365,"slug":11366,"difficulty":11355},"Strange Printer","strange-printer","---\ntitle: Interval DP\nmodule: Dynamic Programming\nmoduleNumber: 8\nlessonNumber: 6\norder: 806\nsummary: >-\n  Many problems ask for the best way to combine a contiguous range of items, and\n  the answer is a dynamic program over subintervals $[i,j]$ that chooses a split\n  point $k$. We derive the pattern from matrix-chain multiplication —\n  parenthesising a product to minimize scalar multiplications in $O(n^3)$ — distil\n  it into a reusable template filled by increasing interval length, and then meet\n  its sharpest variant: the \"last operation\" trick behind Burst Balloons and\n  cutting a stick, where fixing the _last_ move (not the first) makes the two\n  sides independent.\ntopics: [Dynamic Programming]\nsources:\n  - book: CLRS\n    ref: \"Ch. 15 — Dynamic Programming (§15.2 Matrix-chain, §15.5 Optimal BST)\"\n  - book: Skiena\n    ref: \"§ — Dynamic Programming\"\n  - book: Erickson\n    ref: \"Ch. — Dynamic Programming\"\npractice:\n  - title: 'Minimum Cost to Cut a Stick'\n    slug: minimum-cost-to-cut-a-stick\n    difficulty: Hard\n  - title: 'Burst Balloons'\n    slug: burst-balloons\n    difficulty: Hard\n  - title: 'Palindrome Partitioning II'\n    slug: palindrome-partitioning-ii\n    difficulty: Hard\n  - title: 'Minimum Cost Tree From Leaf Values'\n    slug: minimum-cost-tree-from-leaf-values\n    difficulty: Medium\n  - title: 'Strange Printer'\n    slug: strange-printer\n    difficulty: Hard\n---\n\nThe sequence DPs we have met so far walked a string or array from one end and let\nthe subproblem be a _prefix_: the state was \"the best answer using the first $i$\nitems.\" A second, equally common family does not decompose by prefix at all. It\ndecomposes by **contiguous range**. The natural subproblem is \"the best answer\nfor the slice $[i,j]$,\" and the recurrence builds a long range out of two shorter\nranges by guessing where they meet: a **split point** $k$ inside $[i,j]$. This\nis **interval dynamic programming**, one of the [core DP\npatterns](\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples), and once you see its\nshape you will find it everywhere: parenthesising a product, building an optimal\nsearch tree, bursting balloons, cutting a stick, partitioning a string.\n\nThe archetype, and the cleanest place to learn the pattern, is the problem of\nparenthesising a matrix product.\n\n## Matrix-chain multiplication\n\nMultiplying a $p \\times q$ matrix by a $q \\times r$ matrix takes $pqr$ scalar\nmultiplications. Matrix multiplication is associative, so for a chain\n$A_1 A_2 \\cdots A_n$ the _answer_ is the same however we parenthesise, but the\n_cost_ is not. Given a chain of $n$ matrices where $A_i$ has dimensions\n$p_{i-1} \\times p_i$ (so the dimension sequence is $p_0, p_1, \\dots, p_n$), we\nwant the parenthesisation that minimizes the total number of scalar\nmultiplications.[^clrs-matrix]\n\nThe number of parenthesisations grows like the Catalan numbers, exponentially,\nso brute force is hopeless. But the problem has the two features every DP needs.\n**Optimal substructure**: in the optimal parenthesisation of $A_i \\cdots A_j$,\nthe outermost multiplication splits the chain at some $k$ into\n$(A_i \\cdots A_k)(A_{k+1} \\cdots A_j)$, and _each side must itself be optimally\nparenthesised_: if a cheaper parenthesisation of a side existed, substituting it\nwould lower the total. **Overlapping subproblems**: both sides are again\ncontiguous subchains, and the same subchain is reached by many different outer\nchoices.\n\nLet $m[i,j]$ be the minimum number of scalar multiplications needed to compute\n$A_i \\cdots A_j$. A single matrix needs no multiplications, and otherwise we try\nevery split:\n\n$$\nm[i,j] =\n\\begin{cases}\n0 & i = j,\\\\[4pt]\n\\displaystyle\\min_{i \\le k \u003C j}\\bigl(m[i,k] + m[k+1,j] + p_{i-1}\\,p_k\\,p_j\\bigr) & i \u003C j.\n\\end{cases}\n$$\n\nThe term $p_{i-1} p_k p_j$ is the cost of the final multiplication: the left\nblock $A_i \\cdots A_k$ is a $p_{i-1} \\times p_k$ matrix, the right block\n$A_{k+1} \\cdots A_j$ is $p_k \\times p_j$, and combining them costs\n$p_{i-1} p_k p_j$.\n\n$$\n% caption: Combine two solved subintervals at a split $k$; the chosen split is highlighted\n\\begin{tikzpicture}[\n  >=stealth,\n  every node\u002F.style={font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % the row of matrices i..j\n  \\foreach \\x\u002F\\lab in {0\u002Fi, 1\u002F{}, 2\u002Fk, 3\u002F{k{+}1}, 4\u002F{}, 5\u002Fj} {\n    \\node[draw, minimum size=7mm] (n\\x) at (\\x*1.2, 0) {};\n    \\node[font=\\footnotesize] at (\\x*1.2, -0.62) {$\\lab$};\n  }\n  % full bracket over [i,j]\n  \\draw[thick] (-0.45, 0.55) -- (6.45, 0.55);\n  \\draw[thick] (-0.45, 0.55) -- (-0.45, 0.4);\n  \\draw[thick] (6.45, 0.55) -- (6.45, 0.4);\n  \\node at (3.0, 0.9) {$m[i,j]$};\n  % left sub-bracket [i,k]\n  \\draw[acc, thick] (-0.45, -1.05) -- (2.85, -1.05);\n  \\draw[acc, thick] (-0.45, -1.05) -- (-0.45, -0.9);\n  \\draw[acc, thick] (2.85, -1.05) -- (2.85, -0.9);\n  \\node[acc] at (1.2, -1.42) {$m[i,k]$};\n  % right sub-bracket [k+1,j]\n  \\draw[acc, thick] (3.15, -1.05) -- (6.45, -1.05);\n  \\draw[acc, thick] (3.15, -1.05) -- (3.15, -0.9);\n  \\draw[acc, thick] (6.45, -1.05) -- (6.45, -0.9);\n  \\node[acc] at (4.8, -1.42) {$m[k{+}1,j]$};\n  % combine cost\n  \\node at (3.0, -2.15) {combine cost $= p_{i-1}\\,p_k\\,p_j$};\n\\end{tikzpicture}\n$$\n\n### Filling the table\n\nThe recurrence for $m[i,j]$ depends only on intervals _strictly shorter_ than\n$[i,j]$ (the left side has length $k-i+1 \\le j-i$, the right side\n$j-k \\le j-i$). So if we fill the table in order of increasing interval length\n$\\ell = j - i + 1$, every value we read is already computed. We also record, in a\nsplit table $s[i,j]$, the $k$ that achieved the minimum, so we can reconstruct the\nparenthesisation afterwards.\n\n```algorithm\ncaption: $\\textsc{Matrix-Chain-Order}(p)$ — fill $m$, $s$ by increasing chain length\n$n \\gets \\text{length}(p) - 1$\nfor $i \\gets 1$ to $n$ do\n  $m[i,i] \\gets 0$\nfor $\\ell \\gets 2$ to $n$ do            \u002F\u002F $\\ell$ = chain length\n  for $i \\gets 1$ to $n - \\ell + 1$ do\n    $j \\gets i + \\ell - 1$\n    $m[i,j] \\gets \\infty$\n    for $k \\gets i$ to $j - 1$ do        \u002F\u002F try every split\n      $q \\gets m[i,k] + m[k+1,j] + p_{i-1}\\cdot p_k \\cdot p_j$\n      if $q \u003C m[i,j]$ then\n        $m[i,j] \\gets q$\n        $s[i,j] \\gets k$\nreturn $m,\\ s$\n```\n\nThere are $\\Theta(n^2)$ entries and each costs $O(n)$ to fill (the inner loop over\n$k$), so the algorithm runs in $\\Theta(n^3)$ time and $\\Theta(n^2)$ space. The\nanswer is $m[1,n]$; the optimal parenthesisation is read back from $s$ by\nrecursing: $s[1,n]$ gives the outermost split, then $s[1,\\,s[1,n]]$ and\n$s[\\,s[1,n]+1,\\,n]$ give the next ones, and so on down to single matrices.\n\n$$\n% caption: Optimal split of $A_1\\!\\cdots\\!A_4$ at $k=3$; each node shows its interval cost $m[i,j]$, dims $p=\\langle 5,4,6,2,7\\rangle$\n\\begin{tikzpicture}[\n  >=stealth,\n  every node\u002F.style={font=\\small},\n  box\u002F.style={draw, minimum width=18mm, minimum height=8mm, font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[box, fill=acc!12] (root) at (4,3) {$m[1,4]=158$};\n  \\node[box] (l) at (1.6,1.3) {$m[1,3]=88$};\n  \\node[box] (r) at (6.4,1.3) {$m[4,4]=0$};\n  \\node[box] (ll) at (0,-0.4) {$m[1,1]=0$};\n  \\node[box] (lr) at (3.2,-0.4) {$m[2,3]=48$};\n  \\draw[->, acc, thick] (root) -- node[above left=-1mm, font=\\footnotesize]{$k{=}3$} (l);\n  \\draw[->] (root) -- (r);\n  \\draw[->, acc, thick] (l) -- node[above left=-1mm, font=\\footnotesize]{$k{=}1$} (ll);\n  \\draw[->] (l) -- (lr);\n  \\node[font=\\footnotesize] at (4,3.95) {combine $+\\,p_0 p_3 p_4 = 5\\cdot2\\cdot7=70$};\n\\end{tikzpicture}\n$$\n\n$$\n% caption: The upper-triangular $m$ table, filled along diagonals of increasing length; cell $m[i,j]$ depends on its row to the left and its column below\n\\begin{tikzpicture}[\n  >=stealth,\n  every node\u002F.style={font=\\footnotesize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % grid of cells for a 5-matrix chain, upper triangle i\u003C=j\n  \\foreach \\i in {1,...,5} {\n    \\foreach \\j in {1,...,5} {\n      \\ifnum\\j\u003C\\i\\else\n        \\node[draw, minimum size=8mm] (c\\i\\j) at (\\j*0.9, -\\i*0.9) {};\n      \\fi\n    }\n  }\n  % the target cell m[2,4]\n  \\node[draw, fill=acc!18, minimum size=8mm] at (4*0.9, -2*0.9) {$m_{24}$};\n  % its row to the left (m[2,2], m[2,3]) and column below (m[3,4], m[4,4])\n  \\node[draw, acc, very thick, minimum size=8mm] at (2*0.9, -2*0.9) {};\n  \\node[draw, acc, very thick, minimum size=8mm] at (3*0.9, -2*0.9) {};\n  \\node[draw, acc, very thick, minimum size=8mm] at (4*0.9, -3*0.9) {};\n  \\node[draw, acc, very thick, minimum size=8mm] at (4*0.9, -4*0.9) {};\n  % axis labels\n  \\foreach \\j in {1,...,5} { \\node at (\\j*0.9, -0.35) {$j{=}\\j$}; }\n  \\foreach \\i in {1,...,5} { \\node at (0.25, -\\i*0.9) {$i{=}\\i$}; }\n  % diagonal fill-order arrow (red process annotation), routed through the empty\n  % lower-left triangle so it clears the cell digits and bordered dependency cells;\n  % horizontal label placed in clear space below, with a thin red leader\n  \\draw[->, thick, red!75!black] (1.0*0.9, -4.6*0.9) -- (3.2*0.9, -1.15*0.9);\n  \\node[font=\\footnotesize, align=center, red!75!black, anchor=west] at (1.15*0.9, -4.95*0.9)\n    {increasing\\\\$\\text{len} = j-i+1$};\n\\end{tikzpicture}\n$$\n\nThe diagonal $i = j$ (length $1$) is the base case; each successive diagonal\nmoving toward the top-right corner holds longer intervals, and cell $m[i,j]$\ndraws on cells in its own **row to the left** (the $m[i,k]$ terms) and its own\n**column below** (the $m[k+1,j]$ terms). That dependency shape, left along the\nrow and down the column, is the signature of an interval DP.\n\n## The interval-DP recipe\n\nStrip matrix-chain of its specifics and a reusable pattern remains.\n\n> **Remark (The interval-DP recipe).** Let the state be a contiguous range $[i,j]$ of the input.\n> 1. **Subproblem.** $\\text{dp}[i,j]$ = the optimum over the slice $[i,j]$.\n> 2. **Base case.** Length-$1$ (or length-$0$) intervals are trivial.\n> 3. **Recurrence.** Choose a **split or pivot** $k$ inside $[i,j]$ that breaks\n>    the range into two independent subranges, combine their optima plus a cost\n>    that depends only on $i$, $j$, $k$, and minimize (or maximize) over $k$.\n> 4. **Order.** Fill by **increasing interval length**, so both subranges are\n>    already solved when you reach $[i,j]$.\n>\n> With $\\Theta(n^2)$ states and an $O(n)$ choice of $k$ per state, the running\n> time is the characteristic $O(n^3)$.\n\nThe art in any specific problem is steps 1 and 3: defining the slice so that the\ntwo sides really are **independent**, and finding the cost term that depends only\non the endpoints and the split. The rest is bookkeeping.\n\n## Optimal binary search tree\n\nA first variation keeps the split idea but changes what the cost term measures.\nGiven $n$ sorted keys with search probabilities $p_1, \\dots, p_n$, an **optimal\n[binary search tree](\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees)** is the\nBST minimizing the _expected_ search cost\n$\\sum_i p_i \\cdot (\\text{depth of key } i + 1)$. Choosing key $k$ as the root of\nthe subtree on keys $[i,j]$ splits the rest into a left subtree on $[i,k-1]$ and a\nright subtree on $[k+1,j]$, again two independent ranges combined at a pivot.\nMaking $k$ the root pushes _every_ key in $[i,j]$ down one level, which adds the\ntotal weight $w[i,j] = \\sum_{r=i}^{j} p_r$ to the cost:\n\n$$\ne[i,j] = \\min_{i \\le k \\le j}\\bigl(e[i,k-1] + e[k+1,j] + w[i,j]\\bigr).\n$$\n\nPicking root $k$ hangs the two solved subtrees beneath it, and the act of adding a\nroot pushes _every_ key in $[i,j]$ one level deeper, so its whole weight\n$w[i,j]$ is charged once more:\n\n$$\n% caption: Root $k$ on keys $[i,j]$: the optimal subtrees on $[i,k-1]$ and $[k+1,j]$ hang below, and rooting adds one level to all keys, charging $+\\,w[i,j]$.\n\\begin{tikzpicture}[\n  >=stealth, every node\u002F.style={font=\\small},\n  key\u002F.style={circle, draw=acc, very thick, minimum size=8mm, inner sep=0pt},\n  sub\u002F.style={draw, isosceles triangle, shape border rotate=90, minimum width=20mm, minimum height=11mm, inner sep=1pt, anchor=north}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[key] (k) at (0,0) {$k$};\n  \\node[sub] (L) at (-1.6,-0.85) {};\n  \\node[sub] (R) at (1.6,-0.85) {};\n  \\node[font=\\footnotesize] at (-1.6,-2.3) {$e[i,k{-}1]$};\n  \\node[font=\\footnotesize] at (1.6,-2.3) {$e[k{+}1,j]$};\n  \\draw[acc, thick] (k.south) -- (L.north);\n  \\draw[acc, thick] (k.south) -- (R.north);\n  \\node[font=\\footnotesize, acc, align=center, anchor=west] at (1.3,0.35)\n    {root adds 1 level\\\\to all of $[i,j]$:\\\\$+\\,w[i,j]$};\n\\end{tikzpicture}\n$$\n\nThe shape is identical to matrix-chain, with the same diagonals and the same\n$O(n^3)$ fill, the combine term now being the swept-down weight of the whole\ninterval rather than a product of dimensions.[^clrs-obst]\n\n## The \"last operation\" trick: Burst Balloons and cutting a stick\n\nSome problems resist the naive split because the cost of a piece depends on what\nis _adjacent_ to it, and the first split severs exactly the adjacency that sets\nthe cost. The fix is to guess the **last** operation instead of the first.\n\nTake **Burst Balloons**: balloons $1 \\dots n$ have values $v_i$, and bursting\nballoon $i$ earns $v_{\\text{left}} \\cdot v_i \\cdot v_{\\text{right}}$ where left and\nright are its _current_ surviving neighbors; bursting removes $i$ and rejoins the\nneighbors. We want the maximum total earnings. If we try to fix the _first_\nballoon burst in $[i,j]$, the two halves are **not** independent: a balloon in the\nleft half can later have a right neighbor in the right half, so the halves keep\ninteracting through the shared boundary.\n\nNow fix the balloon $k$ that is burst **last** in the open interval $(i,j)$ (using\nsentinels $v_i, v_j$ just outside the range that are never burst). When $k$ is\nburst last, _every other balloon in $(i,j)$ is already gone_, so its two\nneighbors at that moment are exactly the boundaries $i$ and $j$. The earnings for\nthat final burst are $v_i \\cdot v_k \\cdot v_j$, fixed and independent of order;\nthe balloons in $(i,k)$ were all burst _before_ $k$ while $i$ and $k$\nstood as fixed walls, and likewise $(k,j)$ played out against walls $k$ and $j$.\nThe two sides never touch. They are independent:\n\n$$\n\\text{dp}[i,j] = \\max_{i \u003C k \u003C j}\\bigl(\\text{dp}[i,k] + \\text{dp}[k,j] + v_i\\, v_k\\, v_j\\bigr).\n$$\n\n$$\n% caption: Bursting $k$ last in $(i,j)$: every other balloon is already gone, so $k$'s neighbors are pinned to the walls $i,j$ and earn $v_i v_k v_j$\n\\begin{tikzpicture}[\n  >=stealth,\n  every node\u002F.style={font=\\small},\n  bln\u002F.style={circle, draw, minimum size=7mm, inner sep=0},\n  wall\u002F.style={circle, draw, fill=black!8, minimum size=7mm, inner sep=0}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[wall] (i) at (0,0) {$i$};\n  \\node[bln, dashed, text=black!45] (a) at (1.4,0) {};\n  \\node[bln, dashed, text=black!45] (b) at (2.8,0) {};\n  \\node[bln, draw=acc, very thick, text=acc] (k) at (4.2,0) {$k$};\n  \\node[bln, dashed, text=black!45] (c) at (5.6,0) {};\n  \\node[wall] (j) at (7.0,0) {$j$};\n  \\draw[->, acc, thick] (i) to[bend left=28] (k);\n  \\draw[->, acc, thick] (j) to[bend right=28] (k);\n  \\node[font=\\footnotesize, text=black!45, fill=white, inner sep=1.5pt] at (2.1,0.92) {burst before $k$};\n  \\node[font=\\footnotesize, text=black!45, fill=white, inner sep=1.5pt] at (5.6,0.92) {before $k$};\n  \\node[font=\\footnotesize, acc] at (4.2,-1.3) {final earn $= v_i\\,v_k\\,v_j$};\n\\end{tikzpicture}\n$$\n\n> **Intuition.** Splitting on the _first_ balloon leaves the two halves coupled,\n> because a later burst can reach across the cut. Splitting on the _last_ balloon\n> decouples them: once $k$ is the only survivor, its neighbors are pinned to the\n> interval's walls, so whatever happened inside $(i,k)$ and $(k,j)$ could not have\n> depended on the other side. \"Last\" turns a moving boundary into a fixed one.\n\n**Minimum Cost to Cut a Stick** is the same idea in dual form. A stick has cut\npositions inside it; making a cut costs the _current length_ of the piece being\ncut. Fix which cut $k$ in $[i,j]$ is performed **last**: at that moment the piece\nruns uncut from wall $i$ to wall $j$, so the cost is exactly the length\n$x_j - x_i$, fixed; and the cuts in $(i,k)$ and $(k,j)$ were made earlier, each\nwithin its own sub-piece, independently. Same recurrence, $O(n^3)$ over the cut\npositions. The lesson generalizes: when the per-step cost depends on neighbors, ask\nwhich step is _last_, since the last step is the one whose context is fully\ndetermined by the interval endpoints.\n\n## Palindrome partitioning over a string\n\nInterval DP also runs over strings. In **Palindrome Partitioning II** we cut a\nstring $s$ into pieces that are each palindromes, minimizing the number of cuts.\nPrecompute $\\text{pal}[i,j]$, whether $s[i..j]$ is a palindrome, itself an\ninterval DP, since $s[i..j]$ is a palindrome iff $s_i = s_j$ and $s[i+1..j-1]$ is\n(a length-$2$ shorter interval). Then let $\\text{cut}[j]$ be the fewest cuts for\nthe prefix $s[0..j]$; for each $j$ we look back to the last palindromic piece:\n\n$$\n\\text{cut}[j] = \\min_{0 \\le i \\le j,\\ \\text{pal}[i,j]}\n\\begin{cases}\n0 & i = 0,\\\\\n\\text{cut}[i-1] + 1 & i > 0.\n\\end{cases}\n$$\n\nThe palindrome table is the interval DP (filled by increasing length); the cut\ncount is then a $O(n^2)$ sweep over it, a clean example of one interval DP feeding\na second, simpler one.\n\n## When $O(n^3)$ is too slow\n\nThe $O(n^3)$ cost comes from the inner scan over all splits $k$. For a class of\ninterval DPs (matrix-chain, optimal BST, and others whose cost obeys the\n**quadrangle inequality**) the optimal split point is _monotone_ in the\nendpoints, so the search for $k$ can be confined to a shrinking window. This is\n**Knuth's optimization**, and it drops the running time to $O(n^2)$. We treat the\nconditions and the proof in the lesson on [DP\noptimizations](\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations); for now, note\nonly that the $O(n^3)$ here is not always the last word.[^skiena-dp]\n\n## Takeaways\n\n- **Interval DP** solves problems over a contiguous range: the state is a slice\n  $[i,j]$, the recurrence picks a **split or pivot** $k$ that breaks the range\n  into two independent parts, and the table is filled by **increasing interval\n  length** so both parts are ready, typically $\\Theta(n^2)$ states $\\times$\n  $O(n)$ splits $= O(n^3)$.\n- **Matrix-chain multiplication** is the archetype:\n  $m[i,j] = \\min_{i \\le k \u003C j}\\bigl(m[i,k] + m[k+1,j] + p_{i-1}p_k p_j\\bigr)$,\n  $m[i,i]=0$, with a split table $s[i,j]$ to reconstruct the parenthesisation in\n  $\\Theta(n^3)$ time and $\\Theta(n^2)$ space.\n- **Optimal BSTs** reuse the same shape, replacing the combine term with the swept\n  weight $w[i,j]$ of the interval.\n- The **\"last operation\" trick** (Burst Balloons, Minimum Cost to Cut a Stick)\n  fixes which move happens _last_ in $[i,j]$, not first, pinning the moving\n  neighbor cost to the fixed interval walls and making the two sides truly\n  **independent**.\n- **Palindrome Partitioning II** runs interval DP over a string: a palindrome\n  table by increasing length, then a linear cut-count sweep.\n- **Knuth's optimization** cuts matrix-chain-style DPs to $O(n^2)$ when the\n  **quadrangle inequality** makes the optimal split monotone — see the\n  DP-optimizations lesson.\n\n[^clrs-matrix]: **CLRS**, Ch. 15 — Dynamic Programming (§15.2): matrix-chain multiplication, the $m[i,j]$ recurrence, and reconstruction from the split table $s$ in $\\Theta(n^3)$.\n[^clrs-obst]: **CLRS**, Ch. 15 — Dynamic Programming (§15.5): optimal binary search trees as the same interval DP with the interval-weight combine term.\n[^skiena-dp]: **Skiena**, § — Dynamic Programming: interval DPs as range subproblems combined at a split, and when monotonicity (Knuth) lowers the cubic cost.\n",{"text":11369,"minutes":11370,"time":11371,"words":11372},"9 min read",8.285,497100,1657,{"title":261,"description":11347},[11375,11377,11379],{"book":11197,"ref":11376},"Ch. 15 — Dynamic Programming (§15.2 Matrix-chain, §15.5 Optimal BST)",{"book":11322,"ref":11378},"§ — Dynamic Programming",{"book":11380,"ref":11381},"Erickson","Ch. — Dynamic 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