[{"data":1,"prerenderedAt":25003},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":374,"course-wordcounts":24872,"ref-card-index":24928},[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":33,"blurb":376,"body":377,"description":24830,"extension":24831,"meta":24832,"module":29,"navigation":24834,"path":34,"practice":24835,"rawbody":24854,"readingTime":24855,"seo":24860,"sources":24861,"status":24868,"stem":24869,"summary":37,"topics":24870,"__hash__":24871},"course\u002F01.algorithms\u002F02.divide-and-conquer\u002F01.mergesort.md","",{"type":378,"value":379,"toc":24815},"minimark",[380,393,423,449,592,728,813,818,821,1415,1485,1488,1633,2417,2462,2574,2687,2731,2889,3328,3500,3505,3683,3862,4404,4504,4833,4837,4965,5107,5216,5384,5661,6084,6261,6443,6759,6763,6880,6884,7428,7471,7475,7648,7873,8220,8278,8436,8646,8862,9133,9254,9817,9878,10221,10233,10283,10429,10564,10620,10624,10735,10922,11251,11416,11603,11763,12514,12879,12928,13241,13352,13412,13474,13567,13743,14047,14730,15198,15202,15481,16975,17249,17289,19126,19132,20100,20268,20491,20900,20986,20990,21197,22085,22248,23327,23595,23599,24544,24811],[381,382,383,384,388,389,392],"p",{},"Some problems are easiest to solve by reducing them to ",[385,386,387],"em",{},"smaller versions of\nthemselves",". This is the ",[390,391,30],"strong",{}," paradigm, and it is one of the\nmost productive ideas in all of algorithm design. Every divide-and-conquer\nalgorithm has the same three-part skeleton:",[394,395,396,407,417],"ul",{},[397,398,399,402,403,406],"li",{},[390,400,401],{},"Divide"," the problem into one or more subproblems that are smaller\ninstances of the ",[385,404,405],{},"same"," problem.",[397,408,409,412,413,416],{},[390,410,411],{},"Conquer"," the subproblems by solving them recursively. When a subproblem is\nsmall enough (the ",[390,414,415],{},"base case","), solve it directly without recursing.",[397,418,419,422],{},[390,420,421],{},"Combine"," the subproblem solutions into a solution for the original\nproblem.",[381,424,425,426,430,441,442,445,446],{},"Erickson's advice captures the mindset: assume the recursion already\nworks, so that the recursive calls correctly solve the smaller instances, and\nfocus your energy on the divide and combine steps. This ",[427,428,429],"q",{},"recursion fairy",[431,432,433],"sup",{},[434,435,440],"a",{"href":436,"ariaDescribedBy":437,"dataFootnoteRef":376,"id":439},"#user-content-fn-erickson-rec",[438],"footnote-label","user-content-fnref-erickson-rec","1","\nstance turns a single hard problem into two manageable questions: ",[385,443,444],{},"how do I split?"," and\n",[385,447,448],{},"how do I merge?",[381,450,451,452,455,456,481,482,497,498,522,523,591],{},"The payoff is always a ",[390,453,454],{},"recurrence",". If an instance of size ",[457,458,461],"span",{"className":459},[460],"katex",[457,462,466],{"className":463,"ariaHidden":465},[464],"katex-html","true",[457,467,470,475],{"className":468},[469],"base",[457,471],{"className":472,"style":474},[473],"strut","height:0.4306em;",[457,476,480],{"className":477},[478,479],"mord","mathnormal","n"," spawns ",[457,483,485],{"className":484},[460],[457,486,488],{"className":487,"ariaHidden":465},[464],[457,489,491,494],{"className":490},[469],[457,492],{"className":493,"style":474},[473],[457,495,434],{"className":496},[478,479],"\nsubproblems each of size ",[457,499,501],{"className":500},[460],[457,502,504],{"className":503,"ariaHidden":465},[464],[457,505,507,511,514,518],{"className":506},[469],[457,508],{"className":509,"style":510},[473],"height:1em;vertical-align:-0.25em;",[457,512,480],{"className":513},[478,479],[457,515,517],{"className":516},[478],"\u002F",[457,519,521],{"className":520},[478,479],"b",", and the divide-plus-combine work costs\n",[457,524,526],{"className":525},[460],[457,527,529],{"className":528,"ariaHidden":465},[464],[457,530,532,535,539,544,586],{"className":531},[469],[457,533],{"className":534,"style":510},[473],[457,536,538],{"className":537},[478],"Θ",[457,540,543],{"className":541},[542],"mopen","(",[457,545,547,550],{"className":546},[478],[457,548,480],{"className":549},[478,479],[457,551,554],{"className":552},[553],"msupsub",[457,555,558],{"className":556},[557],"vlist-t",[457,559,562],{"className":560},[561],"vlist-r",[457,563,567],{"className":564,"style":566},[565],"vlist","height:0.6644em;",[457,568,570,575],{"style":569},"top:-3.063em;margin-right:0.05em;",[457,571],{"className":572,"style":574},[573],"pstrut","height:2.7em;",[457,576,582],{"className":577},[578,579,580,581],"sizing","reset-size6","size3","mtight",[457,583,585],{"className":584},[478,479,581],"c",[457,587,590],{"className":588},[589],"mclose",")",", then the total cost obeys",[457,593,596],{"className":594},[595],"katex-display",[457,597,599],{"className":598},[460],[457,600,602,635,678],{"className":601,"ariaHidden":465},[464],[457,603,605,608,613,616,619,622,627,632],{"className":604},[469],[457,606],{"className":607,"style":510},[473],[457,609,612],{"className":610,"style":611},[478,479],"margin-right:0.1389em;","T",[457,614,543],{"className":615},[542],[457,617,480],{"className":618},[478,479],[457,620,590],{"className":621},[589],[457,623],{"className":624,"style":626},[625],"mspace","margin-right:0.2778em;",[457,628,631],{"className":629},[630],"mrel","=",[457,633],{"className":634,"style":626},[625],[457,636,638,641,644,648,651,654,657,660,663,666,670,675],{"className":637},[469],[457,639],{"className":640,"style":510},[473],[457,642,434],{"className":643},[478,479],[457,645],{"className":646,"style":647},[625],"margin-right:0.1667em;",[457,649,612],{"className":650,"style":611},[478,479],[457,652,543],{"className":653},[542],[457,655,480],{"className":656},[478,479],[457,658,517],{"className":659},[478],[457,661,521],{"className":662},[478,479],[457,664,590],{"className":665},[589],[457,667],{"className":668,"style":669},[625],"margin-right:0.2222em;",[457,671,674],{"className":672},[673],"mbin","+",[457,676],{"className":677,"style":669},[625],[457,679,681,684,687,690,721,724],{"className":680},[469],[457,682],{"className":683,"style":510},[473],[457,685,538],{"className":686},[478],[457,688,543],{"className":689},[542],[457,691,693,696],{"className":692},[478],[457,694,480],{"className":695},[478,479],[457,697,699],{"className":698},[553],[457,700,702],{"className":701},[557],[457,703,705],{"className":704},[561],[457,706,709],{"className":707,"style":708},[565],"height:0.7144em;",[457,710,712,715],{"style":711},"top:-3.113em;margin-right:0.05em;",[457,713],{"className":714,"style":574},[573],[457,716,718],{"className":717},[578,579,580,581],[457,719,585],{"className":720},[478,479,581],[457,722,590],{"className":723},[589],[457,725,727],{"className":726},[478],".",[381,729,730,731,746,747,763,764,779,780,804,805,808,809,812],{},"Almost every algorithm in this module is an exercise in choosing ",[457,732,734],{"className":733},[460],[457,735,737],{"className":736,"ariaHidden":465},[464],[457,738,740,743],{"className":739},[469],[457,741],{"className":742,"style":474},[473],[457,744,434],{"className":745},[478,479],", ",[457,748,750],{"className":749},[460],[457,751,753],{"className":752,"ariaHidden":465},[464],[457,754,756,760],{"className":755},[469],[457,757],{"className":758,"style":759},[473],"height:0.6944em;",[457,761,521],{"className":762},[478,479],", and\n",[457,765,767],{"className":766},[460],[457,768,770],{"className":769,"ariaHidden":465},[464],[457,771,773,776],{"className":772},[469],[457,774],{"className":775,"style":474},[473],[457,777,585],{"className":778},[478,479]," wisely and then reading off ",[457,781,783],{"className":782},[460],[457,784,786],{"className":785,"ariaHidden":465},[464],[457,787,789,792,795,798,801],{"className":788},[469],[457,790],{"className":791,"style":510},[473],[457,793,612],{"className":794,"style":611},[478,479],[457,796,543],{"className":797},[542],[457,799,480],{"className":800},[478,479],[457,802,590],{"className":803},[589],". The ",[390,806,807],{},"master theorem"," (stated at the\nend of this lesson) turns that reading-off into a mechanical three-case rule;\nthe ",[434,810,811],{"href":23},"recursion tree"," is the picture behind it. Mergesort is the cleanest first\nexample, so we start there.",[814,815,817],"h2",{"id":816},"the-sorting-problem-revisited","The sorting problem, revisited",[381,819,820],{},"Recall the specification from the previous module:",[822,823,825],"callout",{"type":824},"problem",[381,826,827,830,831,1012,1013,1028,1029,1032,1033,1187,1188,727],{},[390,828,829],{},"Input:"," a sequence ",[457,832,834],{"className":833},[460],[457,835,837],{"className":836,"ariaHidden":465},[464],[457,838,840,843],{"className":839},[469],[457,841],{"className":842,"style":510},[473],[457,844,847,853,899,904,907,948,951,954,958,961,964,967,1008],{"className":845},[846],"minner",[457,848,852],{"className":849,"style":851},[542,850],"delimcenter","top:0em;","⟨",[457,854,856,859],{"className":855},[478],[457,857,434],{"className":858},[478,479],[457,860,862],{"className":861},[553],[457,863,866,890],{"className":864},[557,865],"vlist-t2",[457,867,869,885],{"className":868},[561],[457,870,873],{"className":871,"style":872},[565],"height:0.3011em;",[457,874,876,879],{"style":875},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[457,877],{"className":878,"style":574},[573],[457,880,882],{"className":881},[578,579,580,581],[457,883,440],{"className":884},[478,581],[457,886,889],{"className":887},[888],"vlist-s","​",[457,891,893],{"className":892},[561],[457,894,897],{"className":895,"style":896},[565],"height:0.15em;",[457,898],{},[457,900,903],{"className":901},[902],"mpunct",",",[457,905],{"className":906,"style":647},[625],[457,908,910,913],{"className":909},[478],[457,911,434],{"className":912},[478,479],[457,914,916],{"className":915},[553],[457,917,919,940],{"className":918},[557,865],[457,920,922,937],{"className":921},[561],[457,923,925],{"className":924,"style":872},[565],[457,926,927,930],{"style":875},[457,928],{"className":929,"style":574},[573],[457,931,933],{"className":932},[578,579,580,581],[457,934,936],{"className":935},[478,581],"2",[457,938,889],{"className":939},[888],[457,941,943],{"className":942},[561],[457,944,946],{"className":945,"style":896},[565],[457,947],{},[457,949,903],{"className":950},[902],[457,952],{"className":953,"style":647},[625],[457,955,957],{"className":956},[846],"…",[457,959],{"className":960,"style":647},[625],[457,962,903],{"className":963},[902],[457,965],{"className":966,"style":647},[625],[457,968,970,973],{"className":969},[478],[457,971,434],{"className":972},[478,479],[457,974,976],{"className":975},[553],[457,977,979,1000],{"className":978},[557,865],[457,980,982,997],{"className":981},[561],[457,983,986],{"className":984,"style":985},[565],"height:0.1514em;",[457,987,988,991],{"style":875},[457,989],{"className":990,"style":574},[573],[457,992,994],{"className":993},[578,579,580,581],[457,995,480],{"className":996},[478,479,581],[457,998,889],{"className":999},[888],[457,1001,1003],{"className":1002},[561],[457,1004,1006],{"className":1005,"style":896},[565],[457,1007],{},[457,1009,1011],{"className":1010,"style":851},[589,850],"⟩"," of ",[457,1014,1016],{"className":1015},[460],[457,1017,1019],{"className":1018,"ariaHidden":465},[464],[457,1020,1022,1025],{"className":1021},[469],[457,1023],{"className":1024,"style":474},[473],[457,1026,480],{"className":1027},[478,479]," numbers.\n",[390,1030,1031],{},"Output:"," a permutation ",[457,1034,1036],{"className":1035},[460],[457,1037,1039],{"className":1038,"ariaHidden":465},[464],[457,1040,1042,1046],{"className":1041},[469],[457,1043],{"className":1044,"style":1045},[473],"height:1.0019em;vertical-align:-0.25em;",[457,1047,1049,1052,1110,1113,1116,1119,1122,1125,1128,1184],{"className":1048},[846],[457,1050,852],{"className":1051,"style":851},[542,850],[457,1053,1055,1058],{"className":1054},[478],[457,1056,434],{"className":1057},[478,479],[457,1059,1061],{"className":1060},[553],[457,1062,1064,1101],{"className":1063},[557,865],[457,1065,1067,1098],{"className":1066},[561],[457,1068,1071,1083],{"className":1069,"style":1070},[565],"height:0.7519em;",[457,1072,1074,1077],{"style":1073},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[457,1075],{"className":1076,"style":574},[573],[457,1078,1080],{"className":1079},[578,579,580,581],[457,1081,440],{"className":1082},[478,581],[457,1084,1085,1088],{"style":569},[457,1086],{"className":1087,"style":574},[573],[457,1089,1091],{"className":1090},[578,579,580,581],[457,1092,1094],{"className":1093},[478,581],[457,1095,1097],{"className":1096},[478,581],"′",[457,1099,889],{"className":1100},[888],[457,1102,1104],{"className":1103},[561],[457,1105,1108],{"className":1106,"style":1107},[565],"height:0.2481em;",[457,1109],{},[457,1111,903],{"className":1112},[902],[457,1114],{"className":1115,"style":647},[625],[457,1117,957],{"className":1118},[846],[457,1120],{"className":1121,"style":647},[625],[457,1123,903],{"className":1124},[902],[457,1126],{"className":1127,"style":647},[625],[457,1129,1131,1134],{"className":1130},[478],[457,1132,434],{"className":1133},[478,479],[457,1135,1137],{"className":1136},[553],[457,1138,1140,1175],{"className":1139},[557,865],[457,1141,1143,1172],{"className":1142},[561],[457,1144,1146,1158],{"className":1145,"style":1070},[565],[457,1147,1149,1152],{"style":1148},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[457,1150],{"className":1151,"style":574},[573],[457,1153,1155],{"className":1154},[578,579,580,581],[457,1156,480],{"className":1157},[478,479,581],[457,1159,1160,1163],{"style":569},[457,1161],{"className":1162,"style":574},[573],[457,1164,1166],{"className":1165},[578,579,580,581],[457,1167,1169],{"className":1168},[478,581],[457,1170,1097],{"className":1171},[478,581],[457,1173,889],{"className":1174},[888],[457,1176,1178],{"className":1177},[561],[457,1179,1182],{"className":1180,"style":1181},[565],"height:0.247em;",[457,1183],{},[457,1185,1011],{"className":1186,"style":851},[589,850]," with\n",[457,1189,1191],{"className":1190},[460],[457,1192,1194,1265,1334,1354],{"className":1193,"ariaHidden":465},[464],[457,1195,1197,1201,1255,1258,1262],{"className":1196},[469],[457,1198],{"className":1199,"style":1200},[473],"height:1em;vertical-align:-0.2481em;",[457,1202,1204,1207],{"className":1203},[478],[457,1205,434],{"className":1206},[478,479],[457,1208,1210],{"className":1209},[553],[457,1211,1213,1247],{"className":1212},[557,865],[457,1214,1216,1244],{"className":1215},[561],[457,1217,1219,1230],{"className":1218,"style":1070},[565],[457,1220,1221,1224],{"style":1073},[457,1222],{"className":1223,"style":574},[573],[457,1225,1227],{"className":1226},[578,579,580,581],[457,1228,440],{"className":1229},[478,581],[457,1231,1232,1235],{"style":569},[457,1233],{"className":1234,"style":574},[573],[457,1236,1238],{"className":1237},[578,579,580,581],[457,1239,1241],{"className":1240},[478,581],[457,1242,1097],{"className":1243},[478,581],[457,1245,889],{"className":1246},[888],[457,1248,1250],{"className":1249},[561],[457,1251,1253],{"className":1252,"style":1107},[565],[457,1254],{},[457,1256],{"className":1257,"style":626},[625],[457,1259,1261],{"className":1260},[630],"≤",[457,1263],{"className":1264,"style":626},[625],[457,1266,1268,1271,1325,1328,1331],{"className":1267},[469],[457,1269],{"className":1270,"style":1200},[473],[457,1272,1274,1277],{"className":1273},[478],[457,1275,434],{"className":1276},[478,479],[457,1278,1280],{"className":1279},[553],[457,1281,1283,1317],{"className":1282},[557,865],[457,1284,1286,1314],{"className":1285},[561],[457,1287,1289,1300],{"className":1288,"style":1070},[565],[457,1290,1291,1294],{"style":1073},[457,1292],{"className":1293,"style":574},[573],[457,1295,1297],{"className":1296},[578,579,580,581],[457,1298,936],{"className":1299},[478,581],[457,1301,1302,1305],{"style":569},[457,1303],{"className":1304,"style":574},[573],[457,1306,1308],{"className":1307},[578,579,580,581],[457,1309,1311],{"className":1310},[478,581],[457,1312,1097],{"className":1313},[478,581],[457,1315,889],{"className":1316},[888],[457,1318,1320],{"className":1319},[561],[457,1321,1323],{"className":1322,"style":1107},[565],[457,1324],{},[457,1326],{"className":1327,"style":626},[625],[457,1329,1261],{"className":1330},[630],[457,1332],{"className":1333,"style":626},[625],[457,1335,1337,1341,1345,1348,1351],{"className":1336},[469],[457,1338],{"className":1339,"style":1340},[473],"height:0.7719em;vertical-align:-0.136em;",[457,1342,1344],{"className":1343},[846],"⋯",[457,1346],{"className":1347,"style":626},[625],[457,1349,1261],{"className":1350},[630],[457,1352],{"className":1353,"style":626},[625],[457,1355,1357,1361],{"className":1356},[469],[457,1358],{"className":1359,"style":1360},[473],"height:0.9989em;vertical-align:-0.247em;",[457,1362,1364,1367],{"className":1363},[478],[457,1365,434],{"className":1366},[478,479],[457,1368,1370],{"className":1369},[553],[457,1371,1373,1407],{"className":1372},[557,865],[457,1374,1376,1404],{"className":1375},[561],[457,1377,1379,1390],{"className":1378,"style":1070},[565],[457,1380,1381,1384],{"style":1148},[457,1382],{"className":1383,"style":574},[573],[457,1385,1387],{"className":1386},[578,579,580,581],[457,1388,480],{"className":1389},[478,479,581],[457,1391,1392,1395],{"style":569},[457,1393],{"className":1394,"style":574},[573],[457,1396,1398],{"className":1397},[578,579,580,581],[457,1399,1401],{"className":1400},[478,581],[457,1402,1097],{"className":1403},[478,581],[457,1405,889],{"className":1406},[888],[457,1408,1410],{"className":1409},[561],[457,1411,1413],{"className":1412,"style":1181},[565],[457,1414],{},[381,1416,1417,1418,1470,1471,1474,1475,727,1478],{},"Insertion sort grew a sorted prefix one element at a time, costing\n",[457,1419,1421],{"className":1420},[460],[457,1422,1424],{"className":1423,"ariaHidden":465},[464],[457,1425,1427,1431,1434,1437,1467],{"className":1426},[469],[457,1428],{"className":1429,"style":1430},[473],"height:1.0641em;vertical-align:-0.25em;",[457,1432,538],{"className":1433},[478],[457,1435,543],{"className":1436},[542],[457,1438,1440,1443],{"className":1439},[478],[457,1441,480],{"className":1442},[478,479],[457,1444,1446],{"className":1445},[553],[457,1447,1449],{"className":1448},[557],[457,1450,1452],{"className":1451},[561],[457,1453,1456],{"className":1454,"style":1455},[565],"height:0.8141em;",[457,1457,1458,1461],{"style":569},[457,1459],{"className":1460,"style":574},[573],[457,1462,1464],{"className":1463},[578,579,580,581],[457,1465,936],{"className":1466},[478,581],[457,1468,590],{"className":1469},[589]," in the worst case. Divide and conquer does much better.\nThe trick is to ask: ",[385,1472,1473],{},"if I already had two sorted halves, could I finish the\njob cheaply?"," The answer, yes, by merging, gives us ",[390,1476,1477],{},"mergesort",[431,1479,1480],{},[434,1481,936],{"href":1482,"ariaDescribedBy":1483,"dataFootnoteRef":376,"id":1484},"#user-content-fn-clrs-merge",[438],"user-content-fnref-clrs-merge",[814,1486,1487],{"id":1477},"Mergesort",[381,1489,1490,1491,1527,1528,1596,1597,1632],{},"To sort the subarray ",[457,1492,1494],{"className":1493},[460],[457,1495,1497],{"className":1496,"ariaHidden":465},[464],[457,1498,1500,1503,1507,1511,1514,1518,1523],{"className":1499},[469],[457,1501],{"className":1502,"style":510},[473],[457,1504,1506],{"className":1505},[478,479],"A",[457,1508,1510],{"className":1509},[542],"[",[457,1512,381],{"className":1513},[478,479],[457,1515,1517],{"className":1516},[478],"..",[457,1519,1522],{"className":1520,"style":1521},[478,479],"margin-right:0.0278em;","r",[457,1524,1526],{"className":1525},[589],"]",", split it at the midpoint\n",[457,1529,1531],{"className":1530},[460],[457,1532,1534,1554],{"className":1533,"ariaHidden":465},[464],[457,1535,1537,1541,1545,1548,1551],{"className":1536},[469],[457,1538],{"className":1539,"style":1540},[473],"height:0.625em;vertical-align:-0.1944em;",[457,1542,427],{"className":1543,"style":1544},[478,479],"margin-right:0.0359em;",[457,1546],{"className":1547,"style":626},[625],[457,1549,631],{"className":1550},[630],[457,1552],{"className":1553,"style":626},[625],[457,1555,1557,1560],{"className":1556},[469],[457,1558],{"className":1559,"style":510},[473],[457,1561,1563,1567,1570,1573,1576,1579,1582,1585,1588,1592],{"className":1562},[846],[457,1564,1566],{"className":1565,"style":851},[542,850],"⌊",[457,1568,543],{"className":1569},[542],[457,1571,381],{"className":1572},[478,479],[457,1574],{"className":1575,"style":669},[625],[457,1577,674],{"className":1578},[673],[457,1580],{"className":1581,"style":669},[625],[457,1583,1522],{"className":1584,"style":1521},[478,479],[457,1586,590],{"className":1587},[589],[457,1589,1591],{"className":1590},[478],"\u002F2",[457,1593,1595],{"className":1594,"style":851},[589,850],"⌋",", recursively sort the two halves, and merge them back\ntogether. A single element (",[457,1598,1600],{"className":1599},[460],[457,1601,1603,1623],{"className":1602,"ariaHidden":465},[464],[457,1604,1606,1610,1613,1616,1620],{"className":1605},[469],[457,1607],{"className":1608,"style":1609},[473],"height:0.8304em;vertical-align:-0.1944em;",[457,1611,381],{"className":1612},[478,479],[457,1614],{"className":1615,"style":626},[625],[457,1617,1619],{"className":1618},[630],"≥",[457,1621],{"className":1622,"style":626},[625],[457,1624,1626,1629],{"className":1625},[469],[457,1627],{"className":1628,"style":474},[473],[457,1630,1522],{"className":1631,"style":1521},[478,479],") is already sorted, so it is the base\ncase.",[1634,1635,1639,2322],"figure",{"className":1636},[1637,1638],"tikz-figure","tikz-diagram-rendered",[1640,1641,1646],"svg",{"xmlns":1642,"width":1643,"height":1644,"viewBox":1645},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","426.182","286.007","-75 -75 319.637 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21.748Q124.674 21.837 124.360 21.837Q124.042 21.837 123.638 21.748Q123.235 21.659 122.939 21.457Q122.644 21.255 122.644 20.934M123.098 20.934Q123.098 21.163 123.317 21.312Q123.536 21.461 123.828 21.529Q124.120 21.597 124.360 21.597Q124.524 21.597 124.732 21.561Q124.941 21.526 125.147 21.445Q125.354 21.365 125.486 21.237Q125.617 21.109 125.617 20.934Q125.617 20.582 125.236 20.488Q124.855 20.394 124.353 20.394L123.734 20.394Q123.495 20.394 123.297 20.545Q123.098 20.695 123.098 20.934M124.148 19.147Q124.814 19.147 124.814 18.350Q124.814 17.550 124.148 17.550Q123.478 17.550 123.478 18.350Q123.478 19.147 124.148 19.147M126.626 18.866Q126.626 18.545 126.751 18.256Q126.875 17.967 127.101 17.744Q127.326 17.520 127.622 17.400Q127.918 17.280 128.236 17.280Q128.564 17.280 128.825 17.380Q129.087 17.479 129.263 17.661Q129.439 17.844 129.533 18.102Q129.627 18.360 129.627 18.692Q129.627 18.784 129.545 18.805L127.289 18.805L127.289 18.866Q127.289 19.454 127.573 19.837Q127.856 20.220 128.424 20.220Q128.745 20.220 129.013 20.027Q129.282 19.834 129.370 19.519Q129.377 19.478 129.452 19.464L129.545 19.464Q129.627 19.488 129.627 19.560Q129.627 19.567 129.620 19.594Q129.507 19.991 129.136 20.230Q128.765 20.469 128.342 20.469Q127.904 20.469 127.504 20.261Q127.104 20.052 126.865 19.685Q126.626 19.318 126.626 18.866M127.296 18.596L129.111 18.596Q129.111 18.319 129.013 18.067Q128.916 17.814 128.718 17.658Q128.519 17.503 128.236 17.503Q127.959 17.503 127.745 17.661Q127.532 17.820 127.414 18.075Q127.296 18.330 127.296 18.596",[1667],[2323,2324,2327,2328,2416],"figcaption",{"className":2325},[2326],"tikz-cap","Mergesort on ",[457,2329,2331],{"className":2330},[460],[457,2332,2334],{"className":2333,"ariaHidden":465},[464],[457,2335,2337,2340,2343,2347,2350,2353,2356,2359,2362,2366,2369,2372,2375,2378,2381,2384,2387,2390,2394,2397,2400,2403,2406,2409,2413],{"className":2336},[469],[457,2338],{"className":2339,"style":510},[473],[457,2341,852],{"className":2342},[542],[457,2344,2346],{"className":2345},[478],"5",[457,2348,903],{"className":2349},[902],[457,2351],{"className":2352,"style":647},[625],[457,2354,936],{"className":2355},[478],[457,2357,903],{"className":2358},[902],[457,2360],{"className":2361,"style":647},[625],[457,2363,2365],{"className":2364},[478],"4",[457,2367,903],{"className":2368},[902],[457,2370],{"className":2371,"style":647},[625],[457,2373,1659],{"className":2374},[478],[457,2376,903],{"className":2377},[902],[457,2379],{"className":2380,"style":647},[625],[457,2382,440],{"className":2383},[478],[457,2385,903],{"className":2386},[902],[457,2388],{"className":2389,"style":647},[625],[457,2391,2393],{"className":2392},[478],"3",[457,2395,903],{"className":2396},[902],[457,2398],{"className":2399,"style":647},[625],[457,2401,936],{"className":2402},[478],[457,2404,903],{"className":2405},[902],[457,2407],{"className":2408,"style":647},[625],[457,2410,2412],{"className":2411},[478],"6",[457,2414,1011],{"className":2415},[589],". The top half divides (black arrows) down to singletons — the base cases; the bottom half merges (blue arrows) those sorted runs back up, pair by pair, to the final sorted list.",[2418,2419,2423],"pre",{"className":2420,"code":2421,"language":2422,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Merge-Sort}(A, p, r)$ — sort $A[p..r]$ in increasing order\nnumber: 1\nif $p \u003C r$ then\n  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F split point\n  call $\\textsc{Merge-Sort}(A, p, q)$ \u002F\u002F sort left half\n  call $\\textsc{Merge-Sort}(A, q + 1, r)$ \u002F\u002F sort right half\n  call $\\textsc{Merge}(A, p, q, r)$ \u002F\u002F combine halves\n","algorithm",[2424,2425,2426,2432,2437,2442,2447,2452,2457],"code",{"__ignoreMap":376},[457,2427,2429],{"class":2428,"line":6},"line",[457,2430,2431],{},"caption: $\\textsc{Merge-Sort}(A, p, r)$ — sort $A[p..r]$ in increasing order\n",[457,2433,2434],{"class":2428,"line":18},[457,2435,2436],{},"number: 1\n",[457,2438,2439],{"class":2428,"line":24},[457,2440,2441],{},"if $p \u003C r$ then\n",[457,2443,2444],{"class":2428,"line":73},[457,2445,2446],{},"  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F split point\n",[457,2448,2449],{"class":2428,"line":102},[457,2450,2451],{},"  call $\\textsc{Merge-Sort}(A, p, q)$ \u002F\u002F sort left half\n",[457,2453,2454],{"class":2428,"line":108},[457,2455,2456],{},"  call $\\textsc{Merge-Sort}(A, q + 1, r)$ \u002F\u002F sort right half\n",[457,2458,2459],{"class":2428,"line":116},[457,2460,2461],{},"  call $\\textsc{Merge}(A, p, q, r)$ \u002F\u002F combine halves\n",[381,2463,2464,2465,2468,2469,2495,2496,2526,2527,2573],{},"All the real work lives in the ",[390,2466,2467],{},"combine"," step. ",[457,2470,2472],{"className":2471},[460],[457,2473,2475],{"className":2474,"ariaHidden":465},[464],[457,2476,2478,2482],{"className":2477},[469],[457,2479],{"className":2480,"style":2481},[473],"height:0.8778em;vertical-align:-0.1944em;",[457,2483,2487],{"className":2484},[2485,2486],"enclosing","textsc",[457,2488,2491],{"className":2489},[478,2490],"text",[457,2492,2494],{"className":2493},[478],"Merge"," takes two adjacent\nsorted runs, ",[457,2497,2499],{"className":2498},[460],[457,2500,2502],{"className":2501,"ariaHidden":465},[464],[457,2503,2505,2508,2511,2514,2517,2520,2523],{"className":2504},[469],[457,2506],{"className":2507,"style":510},[473],[457,2509,1506],{"className":2510},[478,479],[457,2512,1510],{"className":2513},[542],[457,2515,381],{"className":2516},[478,479],[457,2518,1517],{"className":2519},[478],[457,2521,427],{"className":2522,"style":1544},[478,479],[457,2524,1526],{"className":2525},[589]," and ",[457,2528,2530],{"className":2529},[460],[457,2531,2533,2557],{"className":2532,"ariaHidden":465},[464],[457,2534,2536,2539,2542,2545,2548,2551,2554],{"className":2535},[469],[457,2537],{"className":2538,"style":510},[473],[457,2540,1506],{"className":2541},[478,479],[457,2543,1510],{"className":2544},[542],[457,2546,427],{"className":2547,"style":1544},[478,479],[457,2549],{"className":2550,"style":669},[625],[457,2552,674],{"className":2553},[673],[457,2555],{"className":2556,"style":669},[625],[457,2558,2560,2563,2567,2570],{"className":2559},[469],[457,2561],{"className":2562,"style":510},[473],[457,2564,2566],{"className":2565},[478],"1..",[457,2568,1522],{"className":2569,"style":1521},[478,479],[457,2571,1526],{"className":2572},[589],", and interleaves them into a single\nsorted run in place. It copies each half into a scratch array, then repeatedly\ntakes the smaller of the two front elements and writes it back.",[2418,2575,2577],{"className":2420,"code":2576,"language":2422,"meta":376,"style":376},"caption: $\\textsc{Merge}(A, p, q, r)$ — merge sorted $A[p..q]$ and $A[q+1..r]$\nnumber: 2\n$n_1 \\gets q - p + 1$\n$n_2 \\gets r - q$\nlet $L[1..n_1 + 1]$ and $R[1..n_2 + 1]$ be new arrays\nfor $i \\gets 1$ to $n_1$ do\n  $L[i] \\gets A[p + i - 1]$ \u002F\u002F copy left half\nfor $j \\gets 1$ to $n_2$ do\n  $R[j] \\gets A[q + j]$ \u002F\u002F copy right half\n$L[n_1 + 1] \\gets \\infty$ \u002F\u002F sentinel guards the run end\n$R[n_2 + 1] \\gets \\infty$\n$i \\gets 1$\n$j \\gets 1$\nfor $k \\gets p$ to $r$ do\n  if $L[i] \\le R[j]$ then\n    $A[k] \\gets L[i]$\n    $i \\gets i + 1$\n  else\n    $A[k] \\gets R[j]$\n    $j \\gets j + 1$\n",[2424,2578,2579,2584,2589,2594,2599,2604,2609,2614,2619,2624,2629,2634,2639,2645,2651,2657,2663,2669,2675,2681],{"__ignoreMap":376},[457,2580,2581],{"class":2428,"line":6},[457,2582,2583],{},"caption: $\\textsc{Merge}(A, p, q, r)$ — merge sorted $A[p..q]$ and $A[q+1..r]$\n",[457,2585,2586],{"class":2428,"line":18},[457,2587,2588],{},"number: 2\n",[457,2590,2591],{"class":2428,"line":24},[457,2592,2593],{},"$n_1 \\gets q - p + 1$\n",[457,2595,2596],{"class":2428,"line":73},[457,2597,2598],{},"$n_2 \\gets r - q$\n",[457,2600,2601],{"class":2428,"line":102},[457,2602,2603],{},"let $L[1..n_1 + 1]$ and $R[1..n_2 + 1]$ be new arrays\n",[457,2605,2606],{"class":2428,"line":108},[457,2607,2608],{},"for $i \\gets 1$ to $n_1$ do\n",[457,2610,2611],{"class":2428,"line":116},[457,2612,2613],{},"  $L[i] \\gets A[p + i - 1]$ \u002F\u002F copy left half\n",[457,2615,2616],{"class":2428,"line":196},[457,2617,2618],{},"for $j \\gets 1$ to $n_2$ do\n",[457,2620,2621],{"class":2428,"line":202},[457,2622,2623],{},"  $R[j] \\gets A[q + j]$ \u002F\u002F copy right half\n",[457,2625,2626],{"class":2428,"line":283},[457,2627,2628],{},"$L[n_1 + 1] \\gets \\infty$ \u002F\u002F sentinel guards the run end\n",[457,2630,2631],{"class":2428,"line":333},[457,2632,2633],{},"$R[n_2 + 1] \\gets \\infty$\n",[457,2635,2636],{"class":2428,"line":354},[457,2637,2638],{},"$i \\gets 1$\n",[457,2640,2642],{"class":2428,"line":2641},13,[457,2643,2644],{},"$j \\gets 1$\n",[457,2646,2648],{"class":2428,"line":2647},14,[457,2649,2650],{},"for $k \\gets p$ to $r$ do\n",[457,2652,2654],{"class":2428,"line":2653},15,[457,2655,2656],{},"  if $L[i] \\le R[j]$ then\n",[457,2658,2660],{"class":2428,"line":2659},16,[457,2661,2662],{},"    $A[k] \\gets L[i]$\n",[457,2664,2666],{"class":2428,"line":2665},17,[457,2667,2668],{},"    $i \\gets i + 1$\n",[457,2670,2672],{"class":2428,"line":2671},18,[457,2673,2674],{},"  else\n",[457,2676,2678],{"class":2428,"line":2677},19,[457,2679,2680],{},"    $A[k] \\gets R[j]$\n",[457,2682,2684],{"class":2428,"line":2683},20,[457,2685,2686],{},"    $j \\gets j + 1$\n",[381,2688,2689,2690,2693,2694,2710,2711,2726,2727,2730],{},"The two ",[390,2691,2692],{},"sentinel"," values ",[457,2695,2697],{"className":2696},[460],[457,2698,2700],{"className":2699,"ariaHidden":465},[464],[457,2701,2703,2706],{"className":2702},[469],[457,2704],{"className":2705,"style":474},[473],[457,2707,2709],{"className":2708},[478],"∞"," are a small but useful device: once one\nhalf is used up, its front element is forever ",[457,2712,2714],{"className":2713},[460],[457,2715,2717],{"className":2716,"ariaHidden":465},[464],[457,2718,2720,2723],{"className":2719},[469],[457,2721],{"className":2722,"style":474},[473],[457,2724,2709],{"className":2725},[478],", so the comparison\nalways picks from the other half. This removes the need to test ",[427,2728,2729],{},"have we run out?"," on every iteration.",[381,2732,2733,2734,2526,2751,2768,2769,2526,2786,2804,2805,2822,2823,2838,2839,2863,2864,2888],{},"Picture the merge in flight. Two sorted runs ",[457,2735,2737],{"className":2736},[460],[457,2738,2740],{"className":2739,"ariaHidden":465},[464],[457,2741,2743,2747],{"className":2742},[469],[457,2744],{"className":2745,"style":2746},[473],"height:0.6833em;",[457,2748,2750],{"className":2749},[478,479],"L",[457,2752,2754],{"className":2753},[460],[457,2755,2757],{"className":2756,"ariaHidden":465},[464],[457,2758,2760,2763],{"className":2759},[469],[457,2761],{"className":2762,"style":2746},[473],[457,2764,2767],{"className":2765,"style":2766},[478,479],"margin-right:0.0077em;","R"," sit above the output;\nthe cursors ",[457,2770,2772],{"className":2771},[460],[457,2773,2775],{"className":2774,"ariaHidden":465},[464],[457,2776,2778,2782],{"className":2777},[469],[457,2779],{"className":2780,"style":2781},[473],"height:0.6595em;",[457,2783,2785],{"className":2784},[478,479],"i",[457,2787,2789],{"className":2788},[460],[457,2790,2792],{"className":2791,"ariaHidden":465},[464],[457,2793,2795,2799],{"className":2794},[469],[457,2796],{"className":2797,"style":2798},[473],"height:0.854em;vertical-align:-0.1944em;",[457,2800,2803],{"className":2801,"style":2802},[478,479],"margin-right:0.0572em;","j"," point at their smallest uncopied elements, and ",[457,2806,2808],{"className":2807},[460],[457,2809,2811],{"className":2810,"ariaHidden":465},[464],[457,2812,2814,2817],{"className":2813},[469],[457,2815],{"className":2816,"style":759},[473],[457,2818,2821],{"className":2819,"style":2820},[478,479],"margin-right:0.0315em;","k","\nmarks where the next winner lands in ",[457,2824,2826],{"className":2825},[460],[457,2827,2829],{"className":2828,"ariaHidden":465},[464],[457,2830,2832,2835],{"className":2831},[469],[457,2833],{"className":2834,"style":2746},[473],[457,2836,1506],{"className":2837},[478,479],". Each step compares ",[457,2840,2842],{"className":2841},[460],[457,2843,2845],{"className":2844,"ariaHidden":465},[464],[457,2846,2848,2851,2854,2857,2860],{"className":2847},[469],[457,2849],{"className":2850,"style":510},[473],[457,2852,2750],{"className":2853},[478,479],[457,2855,1510],{"className":2856},[542],[457,2858,2785],{"className":2859},[478,479],[457,2861,1526],{"className":2862},[589]," to ",[457,2865,2867],{"className":2866},[460],[457,2868,2870],{"className":2869,"ariaHidden":465},[464],[457,2871,2873,2876,2879,2882,2885],{"className":2872},[469],[457,2874],{"className":2875,"style":510},[473],[457,2877,2767],{"className":2878,"style":2766},[478,479],[457,2880,1510],{"className":2881},[542],[457,2883,2803],{"className":2884,"style":2802},[478,479],[457,2886,1526],{"className":2887},[589],",\nwrites the smaller, and advances that one cursor.",[1634,2890,2892,3231],{"className":2891},[1637,1638],[1640,2893,2897],{"xmlns":1642,"width":2894,"height":2895,"viewBox":2896},"340.352","212.902","-75 -75 255.264 159.676",[1647,2898,2899,2906,2909,2917,2920,2927,2939,2942,2949,2952,2959,2968,2977,2984,2987,2994,3006,3009,3016,3019,3025,3034,3042,3049,3061,3072,3083,3088,3091,3100,3108,3118],{"stroke":1649,"style":1650},[1647,2900,2902],{"transform":2901},"translate(-34.041 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0.228L24.247-6.088L23.513-6.088L23.513-6.430L24.589-6.430",[1667],[2323,3232,3234,3235,2526,3250,3265,3266,2526,3281,3296,3297,3312,3313,727],{"className":3233},[2326],"Merge step with cursors ",[457,3236,3238],{"className":3237},[460],[457,3239,3241],{"className":3240,"ariaHidden":465},[464],[457,3242,3244,3247],{"className":3243},[469],[457,3245],{"className":3246,"style":2781},[473],[457,3248,2785],{"className":3249},[478,479],[457,3251,3253],{"className":3252},[460],[457,3254,3256],{"className":3255,"ariaHidden":465},[464],[457,3257,3259,3262],{"className":3258},[469],[457,3260],{"className":3261,"style":2798},[473],[457,3263,2803],{"className":3264,"style":2802},[478,479]," on sorted runs ",[457,3267,3269],{"className":3268},[460],[457,3270,3272],{"className":3271,"ariaHidden":465},[464],[457,3273,3275,3278],{"className":3274},[469],[457,3276],{"className":3277,"style":2746},[473],[457,3279,2750],{"className":3280},[478,479],[457,3282,3284],{"className":3283},[460],[457,3285,3287],{"className":3286,"ariaHidden":465},[464],[457,3288,3290,3293],{"className":3289},[469],[457,3291],{"className":3292,"style":2746},[473],[457,3294,2767],{"className":3295,"style":2766},[478,479]," writing the smaller value into ",[457,3298,3300],{"className":3299},[460],[457,3301,3303],{"className":3302,"ariaHidden":465},[464],[457,3304,3306,3309],{"className":3305},[469],[457,3307],{"className":3308,"style":2746},[473],[457,3310,1506],{"className":3311},[478,479]," at ",[457,3314,3316],{"className":3315},[460],[457,3317,3319],{"className":3318,"ariaHidden":465},[464],[457,3320,3322,3325],{"className":3321},[469],[457,3323],{"className":3324,"style":759},[473],[457,3326,2821],{"className":3327,"style":2820},[478,479],[381,3329,3330,3331,2526,3375,3418,3419,3443,3444,2526,3468,3483,3484,3499],{},"Here ",[457,3332,3334],{"className":3333},[460],[457,3335,3337,3364],{"className":3336,"ariaHidden":465},[464],[457,3338,3340,3343,3346,3349,3352,3355,3358,3361],{"className":3339},[469],[457,3341],{"className":3342,"style":510},[473],[457,3344,2750],{"className":3345},[478,479],[457,3347,1510],{"className":3348},[542],[457,3350,2785],{"className":3351},[478,479],[457,3353,1526],{"className":3354},[589],[457,3356],{"className":3357,"style":626},[625],[457,3359,631],{"className":3360},[630],[457,3362],{"className":3363,"style":626},[625],[457,3365,3367,3371],{"className":3366},[469],[457,3368],{"className":3369,"style":3370},[473],"height:0.6444em;",[457,3372,3374],{"className":3373},[478],"11",[457,3376,3378],{"className":3377},[460],[457,3379,3381,3408],{"className":3380,"ariaHidden":465},[464],[457,3382,3384,3387,3390,3393,3396,3399,3402,3405],{"className":3383},[469],[457,3385],{"className":3386,"style":510},[473],[457,3388,2767],{"className":3389,"style":2766},[478,479],[457,3391,1510],{"className":3392},[542],[457,3394,2803],{"className":3395,"style":2802},[478,479],[457,3397,1526],{"className":3398},[589],[457,3400],{"className":3401,"style":626},[625],[457,3403,631],{"className":3404},[630],[457,3406],{"className":3407,"style":626},[625],[457,3409,3411,3414],{"className":3410},[469],[457,3412],{"className":3413,"style":3370},[473],[457,3415,3417],{"className":3416},[478],"8",", so ",[457,3420,3422],{"className":3421},[460],[457,3423,3425],{"className":3424,"ariaHidden":465},[464],[457,3426,3428,3431,3434,3437,3440],{"className":3427},[469],[457,3429],{"className":3430,"style":510},[473],[457,3432,2767],{"className":3433,"style":2766},[478,479],[457,3435,1510],{"className":3436},[542],[457,3438,2803],{"className":3439,"style":2802},[478,479],[457,3441,1526],{"className":3442},[589]," wins: it is written to ",[457,3445,3447],{"className":3446},[460],[457,3448,3450],{"className":3449,"ariaHidden":465},[464],[457,3451,3453,3456,3459,3462,3465],{"className":3452},[469],[457,3454],{"className":3455,"style":510},[473],[457,3457,1506],{"className":3458},[478,479],[457,3460,1510],{"className":3461},[542],[457,3463,2821],{"className":3464,"style":2820},[478,479],[457,3466,1526],{"className":3467},[589],[457,3469,3471],{"className":3470},[460],[457,3472,3474],{"className":3473,"ariaHidden":465},[464],[457,3475,3477,3480],{"className":3476},[469],[457,3478],{"className":3479,"style":2798},[473],[457,3481,2803],{"className":3482,"style":2802},[478,479],"\nadvances. The two ",[457,3485,3487],{"className":3486},[460],[457,3488,3490],{"className":3489,"ariaHidden":465},[464],[457,3491,3493,3496],{"className":3492},[469],[457,3494],{"className":3495,"style":474},[473],[457,3497,2709],{"className":3498},[478]," sentinels guard the right ends so the comparison is\nalways well-defined.",[3501,3502,3504],"h3",{"id":3503},"why-merge-is-correct","Why merge is correct",[381,3506,3507,3508,3532,3533,3605,3606,3621,3622,3625,3626,3651,3652,746,3667,3682],{},"Merge runs in ",[457,3509,3511],{"className":3510},[460],[457,3512,3514],{"className":3513,"ariaHidden":465},[464],[457,3515,3517,3520,3523,3526,3529],{"className":3516},[469],[457,3518],{"className":3519,"style":510},[473],[457,3521,538],{"className":3522},[478],[457,3524,543],{"className":3525},[542],[457,3527,480],{"className":3528},[478,479],[457,3530,590],{"className":3531},[589]," time on ",[457,3534,3536],{"className":3535},[460],[457,3537,3539,3557,3577,3596],{"className":3538,"ariaHidden":465},[464],[457,3540,3542,3545,3548,3551,3554],{"className":3541},[469],[457,3543],{"className":3544,"style":474},[473],[457,3546,480],{"className":3547},[478,479],[457,3549],{"className":3550,"style":626},[625],[457,3552,631],{"className":3553},[630],[457,3555],{"className":3556,"style":626},[625],[457,3558,3560,3564,3567,3570,3574],{"className":3559},[469],[457,3561],{"className":3562,"style":3563},[473],"height:0.6667em;vertical-align:-0.0833em;",[457,3565,1522],{"className":3566,"style":1521},[478,479],[457,3568],{"className":3569,"style":669},[625],[457,3571,3573],{"className":3572},[673],"−",[457,3575],{"className":3576,"style":669},[625],[457,3578,3580,3584,3587,3590,3593],{"className":3579},[469],[457,3581],{"className":3582,"style":3583},[473],"height:0.7778em;vertical-align:-0.1944em;",[457,3585,381],{"className":3586},[478,479],[457,3588],{"className":3589,"style":669},[625],[457,3591,674],{"className":3592},[673],[457,3594],{"className":3595,"style":669},[625],[457,3597,3599,3602],{"className":3598},[469],[457,3600],{"className":3601,"style":3370},[473],[457,3603,440],{"className":3604},[478]," elements: each of the ",[457,3607,3609],{"className":3608},[460],[457,3610,3612],{"className":3611,"ariaHidden":465},[464],[457,3613,3615,3618],{"className":3614},[469],[457,3616],{"className":3617,"style":474},[473],[457,3619,480],{"className":3620},[478,479],"\niterations of the final ",[390,3623,3624],{},"for"," loop does ",[457,3627,3629],{"className":3628},[460],[457,3630,3632],{"className":3631,"ariaHidden":465},[464],[457,3633,3635,3638,3642,3645,3648],{"className":3634},[469],[457,3636],{"className":3637,"style":510},[473],[457,3639,3641],{"className":3640,"style":1521},[478,479],"O",[457,3643,543],{"className":3644},[542],[457,3646,440],{"className":3647},[478],[457,3649,590],{"className":3650},[589]," work and advances exactly one\nof ",[457,3653,3655],{"className":3654},[460],[457,3656,3658],{"className":3657,"ariaHidden":465},[464],[457,3659,3661,3664],{"className":3660},[469],[457,3662],{"className":3663,"style":2781},[473],[457,3665,2785],{"className":3666},[478,479],[457,3668,3670],{"className":3669},[460],[457,3671,3673],{"className":3672,"ariaHidden":465},[464],[457,3674,3676,3679],{"className":3675},[469],[457,3677],{"className":3678,"style":2798},[473],[457,3680,2803],{"className":3681,"style":2802},[478,479],". Correctness rests on a loop invariant:",[822,3684,3686],{"type":3685},"lemma",[381,3687,3688,3691,3692],{},[390,3689,3690],{},"Invariant (Merge loop invariant)."," ",[385,3693,3694,3695,3697,3698,3746,3747,3781,3782,2526,3797,3812,3813,2526,3837,3861],{},"At the start of each iteration of the ",[390,3696,3624],{}," loop, the subarray\n",[457,3699,3701],{"className":3700},[460],[457,3702,3704,3734],{"className":3703,"ariaHidden":465},[464],[457,3705,3707,3710,3713,3716,3719,3722,3725,3728,3731],{"className":3706},[469],[457,3708],{"className":3709,"style":510},[473],[457,3711,1506],{"className":3712},[478,479],[457,3714,1510],{"className":3715},[542],[457,3717,381],{"className":3718},[478,479],[457,3720,1517],{"className":3721},[478],[457,3723,2821],{"className":3724,"style":2820},[478,479],[457,3726],{"className":3727,"style":669},[625],[457,3729,3573],{"className":3730},[673],[457,3732],{"className":3733,"style":669},[625],[457,3735,3737,3740,3743],{"className":3736},[469],[457,3738],{"className":3739,"style":510},[473],[457,3741,440],{"className":3742},[478],[457,3744,1526],{"className":3745},[589]," contains the ",[457,3748,3750],{"className":3749},[460],[457,3751,3753,3772],{"className":3752,"ariaHidden":465},[464],[457,3754,3756,3760,3763,3766,3769],{"className":3755},[469],[457,3757],{"className":3758,"style":3759},[473],"height:0.7778em;vertical-align:-0.0833em;",[457,3761,2821],{"className":3762,"style":2820},[478,479],[457,3764],{"className":3765,"style":669},[625],[457,3767,3573],{"className":3768},[673],[457,3770],{"className":3771,"style":669},[625],[457,3773,3775,3778],{"className":3774},[469],[457,3776],{"className":3777,"style":1540},[473],[457,3779,381],{"className":3780},[478,479]," smallest elements of ",[457,3783,3785],{"className":3784},[460],[457,3786,3788],{"className":3787,"ariaHidden":465},[464],[457,3789,3791,3794],{"className":3790},[469],[457,3792],{"className":3793,"style":2746},[473],[457,3795,2750],{"className":3796},[478,479],[457,3798,3800],{"className":3799},[460],[457,3801,3803],{"className":3802,"ariaHidden":465},[464],[457,3804,3806,3809],{"className":3805},[469],[457,3807],{"className":3808,"style":2746},[473],[457,3810,2767],{"className":3811,"style":2766},[478,479],", in sorted\norder. Also ",[457,3814,3816],{"className":3815},[460],[457,3817,3819],{"className":3818,"ariaHidden":465},[464],[457,3820,3822,3825,3828,3831,3834],{"className":3821},[469],[457,3823],{"className":3824,"style":510},[473],[457,3826,2750],{"className":3827},[478,479],[457,3829,1510],{"className":3830},[542],[457,3832,2785],{"className":3833},[478,479],[457,3835,1526],{"className":3836},[589],[457,3838,3840],{"className":3839},[460],[457,3841,3843],{"className":3842,"ariaHidden":465},[464],[457,3844,3846,3849,3852,3855,3858],{"className":3845},[469],[457,3847],{"className":3848,"style":510},[473],[457,3850,2767],{"className":3851,"style":2766},[478,479],[457,3853,1510],{"className":3854},[542],[457,3856,2803],{"className":3857,"style":2802},[478,479],[457,3859,1526],{"className":3860},[589]," are the smallest elements of their arrays\nnot yet copied back.",[394,3863,3864,4018,4264],{},[397,3865,3866,3869,3870,3418,3903,3951,3952,3968,3969,2526,3993,4017],{},[390,3867,3868],{},"Initialization."," Before the first iteration ",[457,3871,3873],{"className":3872},[460],[457,3874,3876,3894],{"className":3875,"ariaHidden":465},[464],[457,3877,3879,3882,3885,3888,3891],{"className":3878},[469],[457,3880],{"className":3881,"style":759},[473],[457,3883,2821],{"className":3884,"style":2820},[478,479],[457,3886],{"className":3887,"style":626},[625],[457,3889,631],{"className":3890},[630],[457,3892],{"className":3893,"style":626},[625],[457,3895,3897,3900],{"className":3896},[469],[457,3898],{"className":3899,"style":1540},[473],[457,3901,381],{"className":3902},[478,479],[457,3904,3906],{"className":3905},[460],[457,3907,3909,3939],{"className":3908,"ariaHidden":465},[464],[457,3910,3912,3915,3918,3921,3924,3927,3930,3933,3936],{"className":3911},[469],[457,3913],{"className":3914,"style":510},[473],[457,3916,1506],{"className":3917},[478,479],[457,3919,1510],{"className":3920},[542],[457,3922,381],{"className":3923},[478,479],[457,3925,1517],{"className":3926},[478],[457,3928,2821],{"className":3929,"style":2820},[478,479],[457,3931],{"className":3932,"style":669},[625],[457,3934,3573],{"className":3935},[673],[457,3937],{"className":3938,"style":669},[625],[457,3940,3942,3945,3948],{"className":3941},[469],[457,3943],{"className":3944,"style":510},[473],[457,3946,440],{"className":3947},[478],[457,3949,1526],{"className":3950},[589]," is\nempty: it holds the ",[457,3953,3955],{"className":3954},[460],[457,3956,3958],{"className":3957,"ariaHidden":465},[464],[457,3959,3961,3964],{"className":3960},[469],[457,3962],{"className":3963,"style":3370},[473],[457,3965,3967],{"className":3966},[478],"0"," smallest elements, vacuously sorted. Since nothing\nhas been copied, ",[457,3970,3972],{"className":3971},[460],[457,3973,3975],{"className":3974,"ariaHidden":465},[464],[457,3976,3978,3981,3984,3987,3990],{"className":3977},[469],[457,3979],{"className":3980,"style":510},[473],[457,3982,2750],{"className":3983},[478,479],[457,3985,1510],{"className":3986},[542],[457,3988,440],{"className":3989},[478],[457,3991,1526],{"className":3992},[589],[457,3994,3996],{"className":3995},[460],[457,3997,3999],{"className":3998,"ariaHidden":465},[464],[457,4000,4002,4005,4008,4011,4014],{"className":4001},[469],[457,4003],{"className":4004,"style":510},[473],[457,4006,2767],{"className":4007,"style":2766},[478,479],[457,4009,1510],{"className":4010},[542],[457,4012,440],{"className":4013},[478],[457,4015,1526],{"className":4016},[589]," are indeed the smallest uncopied elements.",[397,4019,4020,4023,4024,4075,4076,4100,4101,4149,4150,4180,4181,4232,4233,2526,4248,4263],{},[390,4021,4022],{},"Maintenance."," Suppose ",[457,4025,4027],{"className":4026},[460],[457,4028,4030,4057],{"className":4029,"ariaHidden":465},[464],[457,4031,4033,4036,4039,4042,4045,4048,4051,4054],{"className":4032},[469],[457,4034],{"className":4035,"style":510},[473],[457,4037,2750],{"className":4038},[478,479],[457,4040,1510],{"className":4041},[542],[457,4043,2785],{"className":4044},[478,479],[457,4046,1526],{"className":4047},[589],[457,4049],{"className":4050,"style":626},[625],[457,4052,1261],{"className":4053},[630],[457,4055],{"className":4056,"style":626},[625],[457,4058,4060,4063,4066,4069,4072],{"className":4059},[469],[457,4061],{"className":4062,"style":510},[473],[457,4064,2767],{"className":4065,"style":2766},[478,479],[457,4067,1510],{"className":4068},[542],[457,4070,2803],{"className":4071,"style":2802},[478,479],[457,4073,1526],{"className":4074},[589]," (the other case is symmetric). Then\n",[457,4077,4079],{"className":4078},[460],[457,4080,4082],{"className":4081,"ariaHidden":465},[464],[457,4083,4085,4088,4091,4094,4097],{"className":4084},[469],[457,4086],{"className":4087,"style":510},[473],[457,4089,2750],{"className":4090},[478,479],[457,4092,1510],{"className":4093},[542],[457,4095,2785],{"className":4096},[478,479],[457,4098,1526],{"className":4099},[589]," is the smallest uncopied element. Appending it to the sorted\n",[457,4102,4104],{"className":4103},[460],[457,4105,4107,4137],{"className":4106,"ariaHidden":465},[464],[457,4108,4110,4113,4116,4119,4122,4125,4128,4131,4134],{"className":4109},[469],[457,4111],{"className":4112,"style":510},[473],[457,4114,1506],{"className":4115},[478,479],[457,4117,1510],{"className":4118},[542],[457,4120,381],{"className":4121},[478,479],[457,4123,1517],{"className":4124},[478],[457,4126,2821],{"className":4127,"style":2820},[478,479],[457,4129],{"className":4130,"style":669},[625],[457,4132,3573],{"className":4133},[673],[457,4135],{"className":4136,"style":669},[625],[457,4138,4140,4143,4146],{"className":4139},[469],[457,4141],{"className":4142,"style":510},[473],[457,4144,440],{"className":4145},[478],[457,4147,1526],{"className":4148},[589]," keeps ",[457,4151,4153],{"className":4152},[460],[457,4154,4156],{"className":4155,"ariaHidden":465},[464],[457,4157,4159,4162,4165,4168,4171,4174,4177],{"className":4158},[469],[457,4160],{"className":4161,"style":510},[473],[457,4163,1506],{"className":4164},[478,479],[457,4166,1510],{"className":4167},[542],[457,4169,381],{"className":4170},[478,479],[457,4172,1517],{"className":4173},[478],[457,4175,2821],{"className":4176,"style":2820},[478,479],[457,4178,1526],{"className":4179},[589]," sorted and now containing the ",[457,4182,4184],{"className":4183},[460],[457,4185,4187,4205,4223],{"className":4186,"ariaHidden":465},[464],[457,4188,4190,4193,4196,4199,4202],{"className":4189},[469],[457,4191],{"className":4192,"style":3759},[473],[457,4194,2821],{"className":4195,"style":2820},[478,479],[457,4197],{"className":4198,"style":669},[625],[457,4200,3573],{"className":4201},[673],[457,4203],{"className":4204,"style":669},[625],[457,4206,4208,4211,4214,4217,4220],{"className":4207},[469],[457,4209],{"className":4210,"style":3583},[473],[457,4212,381],{"className":4213},[478,479],[457,4215],{"className":4216,"style":669},[625],[457,4218,674],{"className":4219},[673],[457,4221],{"className":4222,"style":669},[625],[457,4224,4226,4229],{"className":4225},[469],[457,4227],{"className":4228,"style":3370},[473],[457,4230,440],{"className":4231},[478],"\nsmallest elements. Incrementing ",[457,4234,4236],{"className":4235},[460],[457,4237,4239],{"className":4238,"ariaHidden":465},[464],[457,4240,4242,4245],{"className":4241},[469],[457,4243],{"className":4244,"style":2781},[473],[457,4246,2785],{"className":4247},[478,479],[457,4249,4251],{"className":4250},[460],[457,4252,4254],{"className":4253,"ariaHidden":465},[464],[457,4255,4257,4260],{"className":4256},[469],[457,4258],{"className":4259,"style":759},[473],[457,4261,2821],{"className":4262,"style":2820},[478,479]," restores the invariant.",[397,4265,4266,4269,4270,3418,4321,4351,4352,4403],{},[390,4267,4268],{},"Termination."," The loop ends with ",[457,4271,4273],{"className":4272},[460],[457,4274,4276,4294,4312],{"className":4275,"ariaHidden":465},[464],[457,4277,4279,4282,4285,4288,4291],{"className":4278},[469],[457,4280],{"className":4281,"style":759},[473],[457,4283,2821],{"className":4284,"style":2820},[478,479],[457,4286],{"className":4287,"style":626},[625],[457,4289,631],{"className":4290},[630],[457,4292],{"className":4293,"style":626},[625],[457,4295,4297,4300,4303,4306,4309],{"className":4296},[469],[457,4298],{"className":4299,"style":3563},[473],[457,4301,1522],{"className":4302,"style":1521},[478,479],[457,4304],{"className":4305,"style":669},[625],[457,4307,674],{"className":4308},[673],[457,4310],{"className":4311,"style":669},[625],[457,4313,4315,4318],{"className":4314},[469],[457,4316],{"className":4317,"style":3370},[473],[457,4319,440],{"className":4320},[478],[457,4322,4324],{"className":4323},[460],[457,4325,4327],{"className":4326,"ariaHidden":465},[464],[457,4328,4330,4333,4336,4339,4342,4345,4348],{"className":4329},[469],[457,4331],{"className":4332,"style":510},[473],[457,4334,1506],{"className":4335},[478,479],[457,4337,1510],{"className":4338},[542],[457,4340,381],{"className":4341},[478,479],[457,4343,1517],{"className":4344},[478],[457,4346,1522],{"className":4347,"style":1521},[478,479],[457,4349,1526],{"className":4350},[589]," holds all\n",[457,4353,4355],{"className":4354},[460],[457,4356,4358,4376,4394],{"className":4357,"ariaHidden":465},[464],[457,4359,4361,4364,4367,4370,4373],{"className":4360},[469],[457,4362],{"className":4363,"style":3563},[473],[457,4365,1522],{"className":4366,"style":1521},[478,479],[457,4368],{"className":4369,"style":669},[625],[457,4371,3573],{"className":4372},[673],[457,4374],{"className":4375,"style":669},[625],[457,4377,4379,4382,4385,4388,4391],{"className":4378},[469],[457,4380],{"className":4381,"style":3583},[473],[457,4383,381],{"className":4384},[478,479],[457,4386],{"className":4387,"style":669},[625],[457,4389,674],{"className":4390},[673],[457,4392],{"className":4393,"style":669},[625],[457,4395,4397,4400],{"className":4396},[469],[457,4398],{"className":4399,"style":3370},[473],[457,4401,440],{"className":4402},[478]," elements in sorted order. The sentinels guarantee we never read\npast the real data.",[381,4405,4406,4407,445,4455,4503],{},"Run the loop to completion on the two halves ",[457,4408,4410],{"className":4409},[460],[457,4411,4413],{"className":4412,"ariaHidden":465},[464],[457,4414,4416,4419,4422,4425,4428,4431,4434,4437,4440,4443,4446,4449,4452],{"className":4415},[469],[457,4417],{"className":4418,"style":510},[473],[457,4420,852],{"className":4421},[542],[457,4423,936],{"className":4424},[478],[457,4426,903],{"className":4427},[902],[457,4429],{"className":4430,"style":647},[625],[457,4432,2365],{"className":4433},[478],[457,4435,903],{"className":4436},[902],[457,4438],{"className":4439,"style":647},[625],[457,4441,2346],{"className":4442},[478],[457,4444,903],{"className":4445},[902],[457,4447],{"className":4448,"style":647},[625],[457,4450,1659],{"className":4451},[478],[457,4453,1011],{"className":4454},[589],[457,4456,4458],{"className":4457},[460],[457,4459,4461],{"className":4460,"ariaHidden":465},[464],[457,4462,4464,4467,4470,4473,4476,4479,4482,4485,4488,4491,4494,4497,4500],{"className":4463},[469],[457,4465],{"className":4466,"style":510},[473],[457,4468,852],{"className":4469},[542],[457,4471,440],{"className":4472},[478],[457,4474,903],{"className":4475},[902],[457,4477],{"className":4478,"style":647},[625],[457,4480,936],{"className":4481},[478],[457,4483,903],{"className":4484},[902],[457,4486],{"className":4487,"style":647},[625],[457,4489,2393],{"className":4490},[478],[457,4492,903],{"className":4493},[902],[457,4495],{"className":4496,"style":647},[625],[457,4498,2412],{"className":4499},[478],[457,4501,1011],{"className":4502},[589]," and every element lands in its sorted slot below:",[1634,4505,4507,4732],{"className":4506},[1637,1638],[1640,4508,4512],{"xmlns":1642,"width":4509,"height":4510,"viewBox":4511},"399.491","88.201","-75 -75 299.618 66.151",[1647,4513,4514,4536,4539,4546,4549,4556,4559,4566,4569,4576,4588,4599,4611,4623,4644,4655,4666,4677,4688,4699,4710,4721],{"stroke":1649,"style":1650},[1647,4515,4517,4524,4530],{"stroke":1654,"fontFamily":4516,"fontSize":1659},"cmti7",[1647,4518,4520],{"transform":4519},"translate(-62.91 -26.022)",[1652,4521],{"d":4522,"fill":1649,"stroke":1649,"className":4523,"style":1668},"M1.628-34.120Q1.788-33.813 2.356-33.813Q2.588-33.813 2.814-33.886Q3.039-33.960 3.192-34.117Q3.344-34.274 3.344-34.507Q3.344-34.671 3.203-34.782Q3.063-34.893 2.879-34.934L2.489-35.009Q2.212-35.064 2.028-35.248Q1.843-35.433 1.843-35.703Q1.843-35.946 1.947-36.139Q2.052-36.332 2.231-36.470Q2.411-36.609 2.634-36.677Q2.858-36.745 3.084-36.745Q3.419-36.745 3.689-36.598Q3.959-36.451 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14.28)",[1652,4719],{"d":4621,"fill":1649,"stroke":1649,"className":4720,"style":2916},[1667],[1647,4722,4723,4726],{"fill":3051},[1652,4724],{"d":4725},"M189.85-12.32h19.918v-19.916H189.85Z",[1647,4727,4729],{"transform":4728},"translate(196.857 14.28)",[1652,4730],{"d":4574,"fill":1649,"stroke":1649,"className":4731,"style":2916},[1667],[2323,4733,4735,4736,2526,4784,4832],{"className":4734},[2326],"The completed merge of sorted halves ",[457,4737,4739],{"className":4738},[460],[457,4740,4742],{"className":4741,"ariaHidden":465},[464],[457,4743,4745,4748,4751,4754,4757,4760,4763,4766,4769,4772,4775,4778,4781],{"className":4744},[469],[457,4746],{"className":4747,"style":510},[473],[457,4749,852],{"className":4750},[542],[457,4752,936],{"className":4753},[478],[457,4755,903],{"className":4756},[902],[457,4758],{"className":4759,"style":647},[625],[457,4761,2365],{"className":4762},[478],[457,4764,903],{"className":4765},[902],[457,4767],{"className":4768,"style":647},[625],[457,4770,2346],{"className":4771},[478],[457,4773,903],{"className":4774},[902],[457,4776],{"className":4777,"style":647},[625],[457,4779,1659],{"className":4780},[478],[457,4782,1011],{"className":4783},[589],[457,4785,4787],{"className":4786},[460],[457,4788,4790],{"className":4789,"ariaHidden":465},[464],[457,4791,4793,4796,4799,4802,4805,4808,4811,4814,4817,4820,4823,4826,4829],{"className":4792},[469],[457,4794],{"className":4795,"style":510},[473],[457,4797,852],{"className":4798},[542],[457,4800,440],{"className":4801},[478],[457,4803,903],{"className":4804},[902],[457,4806],{"className":4807,"style":647},[625],[457,4809,936],{"className":4810},[478],[457,4812,903],{"className":4813},[902],[457,4815],{"className":4816,"style":647},[625],[457,4818,2393],{"className":4819},[478],[457,4821,903],{"className":4822},[902],[457,4824],{"className":4825,"style":647},[625],[457,4827,2412],{"className":4828},[478],[457,4830,1011],{"className":4831},[589]," interleaves into one sorted run.",[814,4834,4836],{"id":4835},"analyzing-the-cost","Analyzing the cost",[381,4838,4839,4840,4864,4865,4880,4881,4905,4906,4939,4940,4964],{},"Let ",[457,4841,4843],{"className":4842},[460],[457,4844,4846],{"className":4845,"ariaHidden":465},[464],[457,4847,4849,4852,4855,4858,4861],{"className":4848},[469],[457,4850],{"className":4851,"style":510},[473],[457,4853,612],{"className":4854,"style":611},[478,479],[457,4856,543],{"className":4857},[542],[457,4859,480],{"className":4860},[478,479],[457,4862,590],{"className":4863},[589]," be the worst-case running time of mergesort on ",[457,4866,4868],{"className":4867},[460],[457,4869,4871],{"className":4870,"ariaHidden":465},[464],[457,4872,4874,4877],{"className":4873},[469],[457,4875],{"className":4876,"style":474},[473],[457,4878,480],{"className":4879},[478,479]," elements.\nSplitting costs ",[457,4882,4884],{"className":4883},[460],[457,4885,4887],{"className":4886,"ariaHidden":465},[464],[457,4888,4890,4893,4896,4899,4902],{"className":4889},[469],[457,4891],{"className":4892,"style":510},[473],[457,4894,538],{"className":4895},[478],[457,4897,543],{"className":4898},[542],[457,4900,440],{"className":4901},[478],[457,4903,590],{"className":4904},[589],", the two recursive calls cost ",[457,4907,4909],{"className":4908},[460],[457,4910,4912],{"className":4911,"ariaHidden":465},[464],[457,4913,4915,4918,4921,4924,4927,4930,4933,4936],{"className":4914},[469],[457,4916],{"className":4917,"style":510},[473],[457,4919,936],{"className":4920},[478],[457,4922],{"className":4923,"style":647},[625],[457,4925,612],{"className":4926,"style":611},[478,479],[457,4928,543],{"className":4929},[542],[457,4931,480],{"className":4932},[478,479],[457,4934,1591],{"className":4935},[478],[457,4937,590],{"className":4938},[589],", and the\nmerge costs ",[457,4941,4943],{"className":4942},[460],[457,4944,4946],{"className":4945,"ariaHidden":465},[464],[457,4947,4949,4952,4955,4958,4961],{"className":4948},[469],[457,4950],{"className":4951,"style":510},[473],[457,4953,538],{"className":4954},[478],[457,4956,543],{"className":4957},[542],[457,4959,480],{"className":4960},[478,479],[457,4962,590],{"className":4963},[589],". So",[457,4966,4968],{"className":4967},[595],[457,4969,4971],{"className":4970},[460],[457,4972,4974,5001,5037,5086],{"className":4973,"ariaHidden":465},[464],[457,4975,4977,4980,4983,4986,4989,4992,4995,4998],{"className":4976},[469],[457,4978],{"className":4979,"style":510},[473],[457,4981,612],{"className":4982,"style":611},[478,479],[457,4984,543],{"className":4985},[542],[457,4987,480],{"className":4988},[478,479],[457,4990,590],{"className":4991},[589],[457,4993],{"className":4994,"style":626},[625],[457,4996,631],{"className":4997},[630],[457,4999],{"className":5000,"style":626},[625],[457,5002,5004,5007,5010,5013,5016,5019,5022,5025,5028,5031,5034],{"className":5003},[469],[457,5005],{"className":5006,"style":510},[473],[457,5008,936],{"className":5009},[478],[457,5011],{"className":5012,"style":647},[625],[457,5014,612],{"className":5015,"style":611},[478,479],[457,5017,543],{"className":5018},[542],[457,5020,480],{"className":5021},[478,479],[457,5023,1591],{"className":5024},[478],[457,5026,590],{"className":5027},[589],[457,5029],{"className":5030,"style":669},[625],[457,5032,674],{"className":5033},[673],[457,5035],{"className":5036,"style":669},[625],[457,5038,5040,5043,5046,5049,5052,5055,5058,5062,5065,5068,5071,5074,5077,5080,5083],{"className":5039},[469],[457,5041],{"className":5042,"style":510},[473],[457,5044,538],{"className":5045},[478],[457,5047,543],{"className":5048},[542],[457,5050,480],{"className":5051},[478,479],[457,5053,590],{"className":5054},[589],[457,5056,903],{"className":5057},[902],[457,5059],{"className":5060,"style":5061},[625],"margin-right:2em;",[457,5063],{"className":5064,"style":647},[625],[457,5066,612],{"className":5067,"style":611},[478,479],[457,5069,543],{"className":5070},[542],[457,5072,440],{"className":5073},[478],[457,5075,590],{"className":5076},[589],[457,5078],{"className":5079,"style":626},[625],[457,5081,631],{"className":5082},[630],[457,5084],{"className":5085,"style":626},[625],[457,5087,5089,5092,5095,5098,5101,5104],{"className":5088},[469],[457,5090],{"className":5091,"style":510},[473],[457,5093,538],{"className":5094},[478],[457,5096,543],{"className":5097},[542],[457,5099,440],{"className":5100},[478],[457,5102,590],{"className":5103},[589],[457,5105,727],{"className":5106},[478],[381,5108,5109,5110,5153,5154,5156,5157,5160,5161,5176,5177,5195,5196,5215],{},"To see why this resolves to ",[457,5111,5113],{"className":5112},[460],[457,5114,5116],{"className":5115,"ariaHidden":465},[464],[457,5117,5119,5122,5125,5128,5131,5134,5144,5147,5150],{"className":5118},[469],[457,5120],{"className":5121,"style":510},[473],[457,5123,538],{"className":5124},[478],[457,5126,543],{"className":5127},[542],[457,5129,480],{"className":5130},[478,479],[457,5132],{"className":5133,"style":647},[625],[457,5135,5138],{"className":5136},[5137],"mop",[457,5139,5143],{"className":5140,"style":5142},[478,5141],"mathrm","margin-right:0.0139em;","log",[457,5145],{"className":5146,"style":647},[625],[457,5148,480],{"className":5149},[478,479],[457,5151,590],{"className":5152},[589],", draw the ",[390,5155,811],{},".\nEach node is labeled with the ",[385,5158,5159],{},"non-recursive"," work it does, the cost of its\nown merge. The root merges ",[457,5162,5164],{"className":5163},[460],[457,5165,5167],{"className":5166,"ariaHidden":465},[464],[457,5168,5170,5173],{"className":5169},[469],[457,5171],{"className":5172,"style":474},[473],[457,5174,480],{"className":5175},[478,479]," elements; its two children each merge ",[457,5178,5180],{"className":5179},[460],[457,5181,5183],{"className":5182,"ariaHidden":465},[464],[457,5184,5186,5189,5192],{"className":5185},[469],[457,5187],{"className":5188,"style":510},[473],[457,5190,480],{"className":5191},[478,479],[457,5193,1591],{"className":5194},[478],"; the\nnext level has four nodes each merging ",[457,5197,5199],{"className":5198},[460],[457,5200,5202],{"className":5201,"ariaHidden":465},[464],[457,5203,5205,5208,5211],{"className":5204},[469],[457,5206],{"className":5207,"style":510},[473],[457,5209,480],{"className":5210},[478,479],[457,5212,5214],{"className":5213},[478],"\u002F4","; and so on.",[1634,5217,5219,5357],{"className":5218},[1637,1638],[1640,5220,5224],{"xmlns":1642,"width":5221,"height":5222,"viewBox":5223},"301.299","141.313","-75 -75 225.974 105.985",[1647,5225,5226,5229,5236,5239,5258,5261,5278,5281,5297,5300,5316,5319,5335,5338,5354],{"stroke":1649,"style":1650},[1652,5227],{"fill":1654,"d":5228},"M45.342-72.07H33.425a4 4 0 0 0-4 4v11.917a4 4 0 0 0 4 4h11.917a4 4 0 0 0 4-4V-68.07a4 4 0 0 0-4-4ZM29.425-52.153",[1647,5230,5232],{"transform":5231},"translate(-4.792 1.937)",[1652,5233],{"d":5234,"fill":1649,"stroke":1649,"className":5235,"style":2916},"M40.474-63.219Q40.474-62.824 40.687-62.549Q40.900-62.275 41.282-62.275Q41.827-62.275 42.333-62.510Q42.838-62.745 43.155-63.167Q43.176-63.202 43.238-63.202Q43.295-63.202 43.341-63.151Q43.387-63.101 43.387-63.048Q43.387-63.013 43.361-62.987Q43.014-62.512 42.451-62.261Q41.889-62.011 41.265-62.011Q40.834-62.011 40.485-62.213Q40.135-62.415 39.944-62.771Q39.753-63.127 39.753-63.553Q39.753-64.015 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16.724h7.973v.38h-7.973z",[1647,5273,5274],{"transform":5265},[1652,5275],{"d":5276,"fill":1649,"stroke":1649,"className":5277,"style":5248},"M44.867-60.037L43.012-60.037L43.012-60.294L45.107-63.001Q45.145-63.048 45.204-63.048L45.336-63.048Q45.377-63.048 45.404-63.020Q45.432-62.993 45.432-62.952L45.432-60.294L46.121-60.294L46.121-60.037L45.432-60.037L45.432-59.489Q45.432-59.316 46.109-59.316L46.109-59.058L44.190-59.058L44.190-59.316Q44.867-59.316 44.867-59.489L44.867-60.037M44.908-62.377L43.302-60.294L44.908-60.294",[1667],[1652,5279],{"fill":1654,"d":5280},"M-34.4-12.12-50.436 7.399M12.622 7.598H.705a4 4 0 0 0-4 4v11.917a4 4 0 0 0 4 4h11.917a4 4 0 0 0 4-4V11.598a4 4 0 0 0-4-4ZM-3.295 27.515",[1647,5282,5283,5289,5292],{"stroke":1654},[1647,5284,5286],{"transform":5285},"translate(-37.907 81.276)",[1652,5287],{"d":5246,"fill":1649,"stroke":1649,"className":5288,"style":5248},[1667],[1652,5290],{"d":5291},"M2.677 16.724h7.973v.38H2.677z",[1647,5293,5294],{"transform":5285},[1652,5295],{"d":5276,"fill":1649,"stroke":1649,"className":5296,"style":5248},[1667],[1652,5298],{"fill":1654,"d":5299},"M-17.714-12.12-1.68 7.399M110.784-32.236H98.867a4 4 0 0 0-4 4v11.917a4 4 0 0 0 4 4h11.917a4 4 0 0 0 4-4v-11.917a4 4 0 0 0-4-4ZM94.867-12.32",[1647,5301,5302,5308,5311],{"stroke":1654},[1647,5303,5305],{"transform":5304},"translate(60.255 41.442)",[1652,5306],{"d":5246,"fill":1649,"stroke":1649,"className":5307,"style":5248},[1667],[1652,5309],{"d":5310},"M100.839-23.11h7.973v.38h-7.973z",[1647,5312,5313],{"transform":5304},[1652,5314],{"d":5256,"fill":1649,"stroke":1649,"className":5315,"style":5248},[1667],[1652,5317],{"fill":1654,"d":5318},"m49.542-55.93 45.125 27.472M78.063 7.598H66.146a4 4 0 0 0-4 4v11.917a4 4 0 0 0 4 4h11.917a4 4 0 0 0 4-4V11.598a4 4 0 0 0-4-4ZM62.146 27.515",[1647,5320,5321,5327,5330],{"stroke":1654},[1647,5322,5324],{"transform":5323},"translate(27.534 81.276)",[1652,5325],{"d":5246,"fill":1649,"stroke":1649,"className":5326,"style":5248},[1667],[1652,5328],{"d":5329},"M68.118 16.724h7.973v.38h-7.973z",[1647,5331,5332],{"transform":5323},[1652,5333],{"d":5276,"fill":1649,"stroke":1649,"className":5334,"style":5248},[1667],[1652,5336],{"fill":1654,"d":5337},"M96.482-12.12 80.448 7.399M143.504 7.598h-11.917a4 4 0 0 0-4 4v11.917a4 4 0 0 0 4 4h11.917a4 4 0 0 0 4-4V11.598a4 4 0 0 0-4-4Zm-15.917 19.917",[1647,5339,5340,5346,5349],{"stroke":1654},[1647,5341,5343],{"transform":5342},"translate(92.976 81.276)",[1652,5344],{"d":5246,"fill":1649,"stroke":1649,"className":5345,"style":5248},[1667],[1652,5347],{"d":5348},"M133.56 16.724h7.973v.38h-7.973z",[1647,5350,5351],{"transform":5342},[1652,5352],{"d":5276,"fill":1649,"stroke":1649,"className":5353,"style":5248},[1667],[1652,5355],{"fill":1654,"d":5356},"m113.168-12.12 16.035 19.518",[2323,5358,5360,5361,5364,5365,5383],{"className":5359},[2326],"Recursion tree for mergesort with ",[385,5362,5363],{},"each"," node showing its merge cost, summing to ",[457,5366,5368],{"className":5367},[460],[457,5369,5371],{"className":5370,"ariaHidden":465},[464],[457,5372,5374,5377,5380],{"className":5373},[469],[457,5375],{"className":5376,"style":474},[473],[457,5378,585],{"className":5379},[478,479],[457,5381,480],{"className":5382},[478,479]," per level.",[381,5385,5386,5387,5390,5391,5409,5410,5409,5471,5531,5532,5547,5548,5590,5591,5642,5643,727],{},"The key observation: ",[390,5388,5389],{},"each level sums to the same amount."," The root level is\n",[457,5392,5394],{"className":5393},[460],[457,5395,5397],{"className":5396,"ariaHidden":465},[464],[457,5398,5400,5403,5406],{"className":5399},[469],[457,5401],{"className":5402,"style":474},[473],[457,5404,585],{"className":5405},[478,479],[457,5407,480],{"className":5408},[478,479],"; the next is ",[457,5411,5413],{"className":5412},[460],[457,5414,5416,5435,5459],{"className":5415,"ariaHidden":465},[464],[457,5417,5419,5422,5425,5428,5432],{"className":5418},[469],[457,5420],{"className":5421,"style":3370},[473],[457,5423,936],{"className":5424},[478],[457,5426],{"className":5427,"style":669},[625],[457,5429,5431],{"className":5430},[673],"⋅",[457,5433],{"className":5434,"style":669},[625],[457,5436,5438,5441,5444,5447,5450,5453,5456],{"className":5437},[469],[457,5439],{"className":5440,"style":510},[473],[457,5442,585],{"className":5443},[478,479],[457,5445,480],{"className":5446},[478,479],[457,5448,1591],{"className":5449},[478],[457,5451],{"className":5452,"style":626},[625],[457,5454,631],{"className":5455},[630],[457,5457],{"className":5458,"style":626},[625],[457,5460,5462,5465,5468],{"className":5461},[469],[457,5463],{"className":5464,"style":474},[473],[457,5466,585],{"className":5467},[478,479],[457,5469,480],{"className":5470},[478,479],[457,5472,5474],{"className":5473},[460],[457,5475,5477,5495,5519],{"className":5476,"ariaHidden":465},[464],[457,5478,5480,5483,5486,5489,5492],{"className":5479},[469],[457,5481],{"className":5482,"style":3370},[473],[457,5484,2365],{"className":5485},[478],[457,5487],{"className":5488,"style":669},[625],[457,5490,5431],{"className":5491},[673],[457,5493],{"className":5494,"style":669},[625],[457,5496,5498,5501,5504,5507,5510,5513,5516],{"className":5497},[469],[457,5499],{"className":5500,"style":510},[473],[457,5502,585],{"className":5503},[478,479],[457,5505,480],{"className":5506},[478,479],[457,5508,5214],{"className":5509},[478],[457,5511],{"className":5512,"style":626},[625],[457,5514,631],{"className":5515},[630],[457,5517],{"className":5518,"style":626},[625],[457,5520,5522,5525,5528],{"className":5521},[469],[457,5523],{"className":5524,"style":474},[473],[457,5526,585],{"className":5527},[478,479],[457,5529,480],{"className":5530},[478,479],"; in\ngeneral level ",[457,5533,5535],{"className":5534},[460],[457,5536,5538],{"className":5537,"ariaHidden":465},[464],[457,5539,5541,5544],{"className":5540},[469],[457,5542],{"className":5543,"style":2781},[473],[457,5545,2785],{"className":5546},[478,479]," has ",[457,5549,5551],{"className":5550},[460],[457,5552,5554],{"className":5553,"ariaHidden":465},[464],[457,5555,5557,5561],{"className":5556},[469],[457,5558],{"className":5559,"style":5560},[473],"height:0.8247em;",[457,5562,5564,5567],{"className":5563},[478],[457,5565,936],{"className":5566},[478],[457,5568,5570],{"className":5569},[553],[457,5571,5573],{"className":5572},[557],[457,5574,5576],{"className":5575},[561],[457,5577,5579],{"className":5578,"style":5560},[565],[457,5580,5581,5584],{"style":569},[457,5582],{"className":5583,"style":574},[573],[457,5585,5587],{"className":5586},[578,579,580,581],[457,5588,2785],{"className":5589},[478,479,581]," nodes each doing 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of\n",[457,5644,5646],{"className":5645},[460],[457,5647,5649],{"className":5648,"ariaHidden":465},[464],[457,5650,5652,5655,5658],{"className":5651},[469],[457,5653],{"className":5654,"style":474},[473],[457,5656,585],{"className":5657},[478,479],[457,5659,480],{"className":5660},[478,479],[1634,5662,5664,5938],{"className":5663},[1637,1638],[1640,5665,5669],{"xmlns":1642,"width":5666,"height":5667,"viewBox":5668},"342.134","110.493","-75 -75 256.600 82.870",[1647,5670,5671,5683,5686,5705,5708,5724,5727,5744,5747,5763,5766,5782,5785,5801,5804,5807,5810,5813,5816,5820,5823,5827,5830,5833,5836,5839,5843,5878,5908],{"stroke":1649,"style":1650},[1647,5672,5673,5676],{"fill":1837},[1652,5674],{"d":5675},"M2.883-54.623h28.453v-9.68H2.883Z",[1647,5677,5679],{"transform":5678},"translate(-4.259 -55.399)",[1652,5680],{"d":5681,"fill":1649,"stroke":1649,"className":5682,"style":1668},"M18.139-3.460Q18.139-3.136 18.327-2.924Q18.515-2.712 18.833-2.712Q19.284-2.712 19.694-2.878Q20.104-3.043 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rows give ",[457,6046,6048],{"className":6047},[460],[457,6049,6051],{"className":6050,"ariaHidden":465},[464],[457,6052,6054,6057,6060,6063,6066,6069,6075,6078,6081],{"className":6053},[469],[457,6055],{"className":6056,"style":510},[473],[457,6058,538],{"className":6059},[478],[457,6061,543],{"className":6062},[542],[457,6064,480],{"className":6065},[478,479],[457,6067],{"className":6068,"style":647},[625],[457,6070,6072],{"className":6071},[5137],[457,6073,5143],{"className":6074,"style":5142},[478,5141],[457,6076],{"className":6077,"style":647},[625],[457,6079,480],{"className":6080},[478,479],[457,6082,590],{"className":6083},[589],[381,6085,6086,6087,6102,6103,6118,6119,6180,6181,6260],{},"Halving from ",[457,6088,6090],{"className":6089},[460],[457,6091,6093],{"className":6092,"ariaHidden":465},[464],[457,6094,6096,6099],{"className":6095},[469],[457,6097],{"className":6098,"style":474},[473],[457,6100,480],{"className":6101},[478,479]," down to the base case of ",[457,6104,6106],{"className":6105},[460],[457,6107,6109],{"className":6108,"ariaHidden":465},[464],[457,6110,6112,6115],{"className":6111},[469],[457,6113],{"className":6114,"style":3370},[473],[457,6116,440],{"className":6117},[478]," takes ",[457,6120,6122],{"className":6121},[460],[457,6123,6125],{"className":6124,"ariaHidden":465},[464],[457,6126,6128,6131,6174,6177],{"className":6127},[469],[457,6129],{"className":6130,"style":5973},[473],[457,6132,6134,6140],{"className":6133},[5137],[457,6135,6137],{"className":6136},[5137],[457,6138,5143],{"className":6139,"style":5142},[478,5141],[457,6141,6143],{"className":6142},[553],[457,6144,6146,6166],{"className":6145},[557,865],[457,6147,6149,6163],{"className":6148},[561],[457,6150,6152],{"className":6151,"style":5995},[565],[457,6153,6154,6157],{"style":5998},[457,6155],{"className":6156,"style":574},[573],[457,6158,6160],{"className":6159},[578,579,580,581],[457,6161,936],{"className":6162},[478,581],[457,6164,889],{"className":6165},[888],[457,6167,6169],{"className":6168},[561],[457,6170,6172],{"className":6171,"style":6017},[565],[457,6173],{},[457,6175],{"className":6176,"style":647},[625],[457,6178,480],{"className":6179},[478,479]," steps, so\nthere are ",[457,6182,6184],{"className":6183},[460],[457,6185,6187,6251],{"className":6186,"ariaHidden":465},[464],[457,6188,6190,6193,6236,6239,6242,6245,6248],{"className":6189},[469],[457,6191],{"className":6192,"style":5973},[473],[457,6194,6196,6202],{"className":6195},[5137],[457,6197,6199],{"className":6198},[5137],[457,6200,5143],{"className":6201,"style":5142},[478,5141],[457,6203,6205],{"className":6204},[553],[457,6206,6208,6228],{"className":6207},[557,865],[457,6209,6211,6225],{"className":6210},[561],[457,6212,6214],{"className":6213,"style":5995},[565],[457,6215,6216,6219],{"style":5998},[457,6217],{"className":6218,"style":574},[573],[457,6220,6222],{"className":6221},[578,579,580,581],[457,6223,936],{"className":6224},[478,581],[457,6226,889],{"className":6227},[888],[457,6229,6231],{"className":6230},[561],[457,6232,6234],{"className":6233,"style":6017},[565],[457,6235],{},[457,6237],{"className":6238,"style":647},[625],[457,6240,480],{"className":6241},[478,479],[457,6243],{"className":6244,"style":669},[625],[457,6246,674],{"className":6247},[673],[457,6249],{"className":6250,"style":669},[625],[457,6252,6254,6257],{"className":6253},[469],[457,6255],{"className":6256,"style":3370},[473],[457,6258,440],{"className":6259},[478]," levels. Multiplying the per-level cost by the number\nof levels:",[457,6262,6264],{"className":6263},[595],[457,6265,6267],{"className":6266},[460],[457,6268,6270,6297,6319,6386,6407],{"className":6269,"ariaHidden":465},[464],[457,6271,6273,6276,6279,6282,6285,6288,6291,6294],{"className":6272},[469],[457,6274],{"className":6275,"style":510},[473],[457,6277,612],{"className":6278,"style":611},[478,479],[457,6280,543],{"className":6281},[542],[457,6283,480],{"className":6284},[478,479],[457,6286,590],{"className":6287},[589],[457,6289],{"className":6290,"style":626},[625],[457,6292,631],{"className":6293},[630],[457,6295],{"className":6296,"style":626},[625],[457,6298,6300,6304,6307,6310,6313,6316],{"className":6299},[469],[457,6301],{"className":6302,"style":6303},[473],"height:0.4445em;",[457,6305,585],{"className":6306},[478,479],[457,6308,480],{"className":6309},[478,479],[457,6311],{"className":6312,"style":669},[625],[457,6314,5431],{"className":6315},[673],[457,6317],{"className":6318,"style":669},[625],[457,6320,6322,6325,6328,6371,6374,6377,6380,6383],{"className":6321},[469],[457,6323],{"className":6324,"style":510},[473],[457,6326,543],{"className":6327},[542],[457,6329,6331,6337],{"className":6330},[5137],[457,6332,6334],{"className":6333},[5137],[457,6335,5143],{"className":6336,"style":5142},[478,5141],[457,6338,6340],{"className":6339},[553],[457,6341,6343,6363],{"className":6342},[557,865],[457,6344,6346,6360],{"className":6345},[561],[457,6347,6349],{"className":6348,"style":5995},[565],[457,6350,6351,6354],{"style":5998},[457,6352],{"className":6353,"style":574},[573],[457,6355,6357],{"className":6356},[578,579,580,581],[457,6358,936],{"className":6359},[478,581],[457,6361,889],{"className":6362},[888],[457,6364,6366],{"className":6365},[561],[457,6367,6369],{"className":6368,"style":6017},[565],[457,6370],{},[457,6372],{"className":6373,"style":647},[625],[457,6375,480],{"className":6376},[478,479],[457,6378],{"className":6379,"style":669},[625],[457,6381,674],{"className":6382},[673],[457,6384],{"className":6385,"style":669},[625],[457,6387,6389,6392,6395,6398,6401,6404],{"className":6388},[469],[457,6390],{"className":6391,"style":510},[473],[457,6393,440],{"className":6394},[478],[457,6396,590],{"className":6397},[589],[457,6399],{"className":6400,"style":626},[625],[457,6402,631],{"className":6403},[630],[457,6405],{"className":6406,"style":626},[625],[457,6408,6410,6413,6416,6419,6422,6425,6431,6434,6437,6440],{"className":6409},[469],[457,6411],{"className":6412,"style":510},[473],[457,6414,538],{"className":6415},[478],[457,6417,543],{"className":6418},[542],[457,6420,480],{"className":6421},[478,479],[457,6423],{"className":6424,"style":647},[625],[457,6426,6428],{"className":6427},[5137],[457,6429,5143],{"className":6430,"style":5142},[478,5141],[457,6432],{"className":6433,"style":647},[625],[457,6435,480],{"className":6436},[478,479],[457,6438,590],{"className":6439},[589],[457,6441,727],{"className":6442},[478],[381,6444,6445,6446,6448,6449,746,6482,6515,6516,3418,6569,6685,6686,6717,6718,6742,6743,6758],{},"This is the canonical application of the ",[390,6447,807],{}," (",[457,6450,6452],{"className":6451},[460],[457,6453,6455,6473],{"className":6454,"ariaHidden":465},[464],[457,6456,6458,6461,6464,6467,6470],{"className":6457},[469],[457,6459],{"className":6460,"style":474},[473],[457,6462,434],{"className":6463},[478,479],[457,6465],{"className":6466,"style":626},[625],[457,6468,631],{"className":6469},[630],[457,6471],{"className":6472,"style":626},[625],[457,6474,6476,6479],{"className":6475},[469],[457,6477],{"className":6478,"style":3370},[473],[457,6480,936],{"className":6481},[478],[457,6483,6485],{"className":6484},[460],[457,6486,6488,6506],{"className":6487,"ariaHidden":465},[464],[457,6489,6491,6494,6497,6500,6503],{"className":6490},[469],[457,6492],{"className":6493,"style":759},[473],[457,6495,521],{"className":6496},[478,479],[457,6498],{"className":6499,"style":626},[625],[457,6501,631],{"className":6502},[630],[457,6504],{"className":6505,"style":626},[625],[457,6507,6509,6512],{"className":6508},[469],[457,6510],{"className":6511,"style":3370},[473],[457,6513,936],{"className":6514},[478],",\n",[457,6517,6519],{"className":6518},[460],[457,6520,6522,6551],{"className":6521,"ariaHidden":465},[464],[457,6523,6525,6528,6533,6536,6539,6542,6545,6548],{"className":6524},[469],[457,6526],{"className":6527,"style":510},[473],[457,6529,6532],{"className":6530,"style":6531},[478,479],"margin-right:0.1076em;","f",[457,6534,543],{"className":6535},[542],[457,6537,480],{"className":6538},[478,479],[457,6540,590],{"className":6541},[589],[457,6543],{"className":6544,"style":626},[625],[457,6546,631],{"className":6547},[630],[457,6549],{"className":6550,"style":626},[625],[457,6552,6554,6557,6560,6563,6566],{"className":6553},[469],[457,6555],{"className":6556,"style":510},[473],[457,6558,538],{"className":6559},[478],[457,6561,543],{"className":6562},[542],[457,6564,480],{"className":6565},[478,479],[457,6567,590],{"className":6568},[589],[457,6570,6572],{"className":6571},[460],[457,6573,6575,6676],{"className":6574,"ariaHidden":465},[464],[457,6576,6578,6582,6667,6670,6673],{"className":6577},[469],[457,6579],{"className":6580,"style":6581},[473],"height:0.8491em;",[457,6583,6585,6588],{"className":6584},[478],[457,6586,480],{"className":6587},[478,479],[457,6589,6591],{"className":6590},[553],[457,6592,6594],{"className":6593},[557],[457,6595,6597],{"className":6596},[561],[457,6598,6600],{"className":6599,"style":6581},[565],[457,6601,6602,6605],{"style":569},[457,6603],{"className":6604,"style":574},[573],[457,6606,6608],{"className":6607},[578,579,580,581],[457,6609,6611,6660,6664],{"className":6610},[478,581],[457,6612,6614,6620],{"className":6613},[5137,581],[457,6615,6617],{"className":6616},[5137,581],[457,6618,5143],{"className":6619,"style":5142},[478,5141,581],[457,6621,6623],{"className":6622},[553],[457,6624,6626,6651],{"className":6625},[557,865],[457,6627,6629,6648],{"className":6628},[561],[457,6630,6633],{"className":6631,"style":6632},[565],"height:0.2302em;",[457,6634,6636,6640],{"style":6635},"top:-2.2341em;margin-right:0.0714em;",[457,6637],{"className":6638,"style":6639},[573],"height:2.5em;",[457,6641,6645],{"className":6642},[578,6643,6644,581],"reset-size3","size1",[457,6646,521],{"className":6647},[478,479,581],[457,6649,889],{"className":6650},[888],[457,6652,6654],{"className":6653},[561],[457,6655,6658],{"className":6656,"style":6657},[565],"height:0.2659em;",[457,6659],{},[457,6661],{"className":6662,"style":6663},[625,581],"margin-right:0.1952em;",[457,6665,434],{"className":6666},[478,479,581],[457,6668],{"className":6669,"style":626},[625],[457,6671,631],{"className":6672},[630],[457,6674],{"className":6675,"style":626},[625],[457,6677,6679,6682],{"className":6678},[469],[457,6680],{"className":6681,"style":474},[473],[457,6683,480],{"className":6684},[478,479]," and we land in the balanced case),\nbut the recursion tree makes the ",[457,6687,6689],{"className":6688},[460],[457,6690,6692],{"className":6691,"ariaHidden":465},[464],[457,6693,6695,6699,6702,6705,6711,6714],{"className":6694},[469],[457,6696],{"className":6697,"style":6698},[473],"height:0.8889em;vertical-align:-0.1944em;",[457,6700,480],{"className":6701},[478,479],[457,6703],{"className":6704,"style":647},[625],[457,6706,6708],{"className":6707},[5137],[457,6709,5143],{"className":6710,"style":5142},[478,5141],[457,6712],{"className":6713,"style":647},[625],[457,6715,480],{"className":6716},[478,479]," concrete: ",[457,6719,6721],{"className":6720},[460],[457,6722,6724],{"className":6723,"ariaHidden":465},[464],[457,6725,6727,6730,6736,6739],{"className":6726},[469],[457,6728],{"className":6729,"style":6698},[473],[457,6731,6733],{"className":6732},[5137],[457,6734,5143],{"className":6735,"style":5142},[478,5141],[457,6737],{"className":6738,"style":647},[625],[457,6740,480],{"className":6741},[478,479]," levels, ",[457,6744,6746],{"className":6745},[460],[457,6747,6749],{"className":6748,"ariaHidden":465},[464],[457,6750,6752,6755],{"className":6751},[469],[457,6753],{"className":6754,"style":474},[473],[457,6756,480],{"className":6757},[478,479]," work\napiece.",[814,6760,6762],{"id":6761},"stability","Stability",[381,6764,6765,6766,6769,6770,6785,6786,6807,6808,6859,6860,6875,6876,6879],{},"Mergesort is ",[390,6767,6768],{},"stable",": equal elements keep their original relative order.\nThis falls out of the ",[457,6771,6773],{"className":6772},[460],[457,6774,6776],{"className":6775,"ariaHidden":465},[464],[457,6777,6779,6782],{"className":6778},[469],[457,6780],{"className":6781,"style":1340},[473],[457,6783,1261],{"className":6784},[630]," in ",[457,6787,6789],{"className":6788},[460],[457,6790,6792],{"className":6791,"ariaHidden":465},[464],[457,6793,6795,6798],{"className":6794},[469],[457,6796],{"className":6797,"style":2481},[473],[457,6799,6801],{"className":6800},[2485,2486],[457,6802,6804],{"className":6803},[478,2490],[457,6805,2494],{"className":6806},[478],": when ",[457,6809,6811],{"className":6810},[460],[457,6812,6814,6841],{"className":6813,"ariaHidden":465},[464],[457,6815,6817,6820,6823,6826,6829,6832,6835,6838],{"className":6816},[469],[457,6818],{"className":6819,"style":510},[473],[457,6821,2750],{"className":6822},[478,479],[457,6824,1510],{"className":6825},[542],[457,6827,2785],{"className":6828},[478,479],[457,6830,1526],{"className":6831},[589],[457,6833],{"className":6834,"style":626},[625],[457,6836,631],{"className":6837},[630],[457,6839],{"className":6840,"style":626},[625],[457,6842,6844,6847,6850,6853,6856],{"className":6843},[469],[457,6845],{"className":6846,"style":510},[473],[457,6848,2767],{"className":6849,"style":2766},[478,479],[457,6851,1510],{"className":6852},[542],[457,6854,2803],{"className":6855,"style":2802},[478,479],[457,6857,1526],{"className":6858},[589]," we take from\n",[457,6861,6863],{"className":6862},[460],[457,6864,6866],{"className":6865,"ariaHidden":465},[464],[457,6867,6869,6872],{"className":6868},[469],[457,6870],{"className":6871,"style":2746},[473],[457,6873,2750],{"className":6874},[478,479],", the ",[385,6877,6878],{},"left"," (earlier) half, first. Stability matters when records are\nsorted on one key but carry others: a stable sort lets you sort by secondary\nkey, then primary key, and trust that ties on the primary preserve the\nsecondary ordering.",[814,6881,6883],{"id":6882},"mergesort-versus-other-sorts","Mergesort versus other sorts",[6885,6886,6887,6907],"table",{},[6888,6889,6890],"thead",{},[6891,6892,6893,6897,6899,6902,6905],"tr",{},[6894,6895,6896],"th",{},"Property",[6894,6898,1487],{},[6894,6900,6901],{},"Insertion sort",[6894,6903,6904],{},"Heapsort",[6894,6906,39],{},[6908,6909,6910,7102,7282,7400,7415],"tbody",{},[6891,6911,6912,6916,6957,7009,7050],{},[6913,6914,6915],"td",{},"Worst case",[6913,6917,6918],{},[457,6919,6921],{"className":6920},[460],[457,6922,6924],{"className":6923,"ariaHidden":465},[464],[457,6925,6927,6930,6933,6936,6939,6942,6948,6951,6954],{"className":6926},[469],[457,6928],{"className":6929,"style":510},[473],[457,6931,538],{"className":6932},[478],[457,6934,543],{"className":6935},[542],[457,6937,480],{"className":6938},[478,479],[457,6940],{"className":6941,"style":647},[625],[457,6943,6945],{"className":6944},[5137],[457,6946,5143],{"className":6947,"style":5142},[478,5141],[457,6949],{"className":6950,"style":647},[625],[457,6952,480],{"className":6953},[478,479],[457,6955,590],{"className":6956},[589],[6913,6958,6959],{},[457,6960,6962],{"className":6961},[460],[457,6963,6965],{"className":6964,"ariaHidden":465},[464],[457,6966,6968,6971,6974,6977,7006],{"className":6967},[469],[457,6969],{"className":6970,"style":1430},[473],[457,6972,538],{"className":6973},[478],[457,6975,543],{"className":6976},[542],[457,6978,6980,6983],{"className":6979},[478],[457,6981,480],{"className":6982},[478,479],[457,6984,6986],{"className":6985},[553],[457,6987,6989],{"className":6988},[557],[457,6990,6992],{"className":6991},[561],[457,6993,6995],{"className":6994,"style":1455},[565],[457,6996,6997,7000],{"style":569},[457,6998],{"className":6999,"style":574},[573],[457,7001,7003],{"className":7002},[578,579,580,581],[457,7004,936],{"className":7005},[478,581],[457,7007,590],{"className":7008},[589],[6913,7010,7011],{},[457,7012,7014],{"className":7013},[460],[457,7015,7017],{"className":7016,"ariaHidden":465},[464],[457,7018,7020,7023,7026,7029,7032,7035,7041,7044,7047],{"className":7019},[469],[457,7021],{"className":7022,"style":510},[473],[457,7024,538],{"className":7025},[478],[457,7027,543],{"className":7028},[542],[457,7030,480],{"className":7031},[478,479],[457,7033],{"className":7034,"style":647},[625],[457,7036,7038],{"className":7037},[5137],[457,7039,5143],{"className":7040,"style":5142},[478,5141],[457,7042],{"className":7043,"style":647},[625],[457,7045,480],{"className":7046},[478,479],[457,7048,590],{"className":7049},[589],[6913,7051,7052],{},[457,7053,7055],{"className":7054},[460],[457,7056,7058],{"className":7057,"ariaHidden":465},[464],[457,7059,7061,7064,7067,7070,7099],{"className":7060},[469],[457,7062],{"className":7063,"style":1430},[473],[457,7065,538],{"className":7066},[478],[457,7068,543],{"className":7069},[542],[457,7071,7073,7076],{"className":7072},[478],[457,7074,480],{"className":7075},[478,479],[457,7077,7079],{"className":7078},[553],[457,7080,7082],{"className":7081},[557],[457,7083,7085],{"className":7084},[561],[457,7086,7088],{"className":7087,"style":1455},[565],[457,7089,7090,7093],{"style":569},[457,7091],{"className":7092,"style":574},[573],[457,7094,7096],{"className":7095},[578,579,580,581],[457,7097,936],{"className":7098},[478,581],[457,7100,590],{"className":7101},[589],[6891,7103,7104,7107,7148,7200,7241],{},[6913,7105,7106],{},"Average case",[6913,7108,7109],{},[457,7110,7112],{"className":7111},[460],[457,7113,7115],{"className":7114,"ariaHidden":465},[464],[457,7116,7118,7121,7124,7127,7130,7133,7139,7142,7145],{"className":7117},[469],[457,7119],{"className":7120,"style":510},[473],[457,7122,538],{"className":7123},[478],[457,7125,543],{"className":7126},[542],[457,7128,480],{"className":7129},[478,479],[457,7131],{"className":7132,"style":647},[625],[457,7134,7136],{"className":7135},[5137],[457,7137,5143],{"className":7138,"style":5142},[478,5141],[457,7140],{"className":7141,"style":647},[625],[457,7143,480],{"className":7144},[478,479],[457,7146,590],{"className":7147},[589],[6913,7149,7150],{},[457,7151,7153],{"className":7152},[460],[457,7154,7156],{"className":7155,"ariaHidden":465},[464],[457,7157,7159,7162,7165,7168,7197],{"className":7158},[469],[457,7160],{"className":7161,"style":1430},[473],[457,7163,538],{"className":7164},[478],[457,7166,543],{"className":7167},[542],[457,7169,7171,7174],{"className":7170},[478],[457,7172,480],{"className":7173},[478,479],[457,7175,7177],{"className":7176},[553],[457,7178,7180],{"className":7179},[557],[457,7181,7183],{"className":7182},[561],[457,7184,7186],{"className":7185,"style":1455},[565],[457,7187,7188,7191],{"style":569},[457,7189],{"className":7190,"style":574},[573],[457,7192,7194],{"className":7193},[578,579,580,581],[457,7195,936],{"className":7196},[478,581],[457,7198,590],{"className":7199},[589],[6913,7201,7202],{},[457,7203,7205],{"className":7204},[460],[457,7206,7208],{"className":7207,"ariaHidden":465},[464],[457,7209,7211,7214,7217,7220,7223,7226,7232,7235,7238],{"className":7210},[469],[457,7212],{"className":7213,"style":510},[473],[457,7215,538],{"className":7216},[478],[457,7218,543],{"className":7219},[542],[457,7221,480],{"className":7222},[478,479],[457,7224],{"className":7225,"style":647},[625],[457,7227,7229],{"className":7228},[5137],[457,7230,5143],{"className":7231,"style":5142},[478,5141],[457,7233],{"className":7234,"style":647},[625],[457,7236,480],{"className":7237},[478,479],[457,7239,590],{"className":7240},[589],[6913,7242,7243],{},[457,7244,7246],{"className":7245},[460],[457,7247,7249],{"className":7248,"ariaHidden":465},[464],[457,7250,7252,7255,7258,7261,7264,7267,7273,7276,7279],{"className":7251},[469],[457,7253],{"className":7254,"style":510},[473],[457,7256,538],{"className":7257},[478],[457,7259,543],{"className":7260},[542],[457,7262,480],{"className":7263},[478,479],[457,7265],{"className":7266,"style":647},[625],[457,7268,7270],{"className":7269},[5137],[457,7271,5143],{"className":7272,"style":5142},[478,5141],[457,7274],{"className":7275,"style":647},[625],[457,7277,480],{"className":7278},[478,479],[457,7280,590],{"className":7281},[589],[6891,7283,7284,7287,7313,7339,7365],{},[6913,7285,7286],{},"Extra space",[6913,7288,7289],{},[457,7290,7292],{"className":7291},[460],[457,7293,7295],{"className":7294,"ariaHidden":465},[464],[457,7296,7298,7301,7304,7307,7310],{"className":7297},[469],[457,7299],{"className":7300,"style":510},[473],[457,7302,538],{"className":7303},[478],[457,7305,543],{"className":7306},[542],[457,7308,480],{"className":7309},[478,479],[457,7311,590],{"className":7312},[589],[6913,7314,7315],{},[457,7316,7318],{"className":7317},[460],[457,7319,7321],{"className":7320,"ariaHidden":465},[464],[457,7322,7324,7327,7330,7333,7336],{"className":7323},[469],[457,7325],{"className":7326,"style":510},[473],[457,7328,538],{"className":7329},[478],[457,7331,543],{"className":7332},[542],[457,7334,440],{"className":7335},[478],[457,7337,590],{"className":7338},[589],[6913,7340,7341],{},[457,7342,7344],{"className":7343},[460],[457,7345,7347],{"className":7346,"ariaHidden":465},[464],[457,7348,7350,7353,7356,7359,7362],{"className":7349},[469],[457,7351],{"className":7352,"style":510},[473],[457,7354,538],{"className":7355},[478],[457,7357,543],{"className":7358},[542],[457,7360,440],{"className":7361},[478],[457,7363,590],{"className":7364},[589],[6913,7366,7367],{},[457,7368,7370],{"className":7369},[460],[457,7371,7373],{"className":7372,"ariaHidden":465},[464],[457,7374,7376,7379,7382,7385,7391,7394,7397],{"className":7375},[469],[457,7377],{"className":7378,"style":510},[473],[457,7380,538],{"className":7381},[478],[457,7383,543],{"className":7384},[542],[457,7386,7388],{"className":7387},[5137],[457,7389,5143],{"className":7390,"style":5142},[478,5141],[457,7392],{"className":7393,"style":647},[625],[457,7395,480],{"className":7396},[478,479],[457,7398,590],{"className":7399},[589],[6891,7401,7402,7405,7408,7410,7413],{},[6913,7403,7404],{},"Stable",[6913,7406,7407],{},"yes",[6913,7409,7407],{},[6913,7411,7412],{},"no",[6913,7414,7412],{},[6891,7416,7417,7420,7422,7424,7426],{},[6913,7418,7419],{},"In place",[6913,7421,7412],{},[6913,7423,7407],{},[6913,7425,7407],{},[6913,7427,7407],{},[381,7429,7430,7431,7434,7435,7442,7443,7467,7468,7470],{},"Mergesort's worst-case guarantee and stability make it the sort of choice when\npredictability matters or when data does not fit in memory. Its sequential,\nmerge-based access pattern is ideal for sorting linked lists and for\n",[390,7432,7433],{},"external sorting"," of data streamed from disk.",[431,7436,7437],{},[434,7438,2393],{"href":7439,"ariaDescribedBy":7440,"dataFootnoteRef":376,"id":7441},"#user-content-fn-skiena-sort",[438],"user-content-fnref-skiena-sort"," Its cost is the ",[457,7444,7446],{"className":7445},[460],[457,7447,7449],{"className":7448,"ariaHidden":465},[464],[457,7450,7452,7455,7458,7461,7464],{"className":7451},[469],[457,7453],{"className":7454,"style":510},[473],[457,7456,538],{"className":7457},[478],[457,7459,543],{"className":7460},[542],[457,7462,480],{"className":7463},[478,479],[457,7465,590],{"className":7466},[589],"\nauxiliary array. ",[434,7469,39],{"href":40},", the subject of the next lesson, trades that\nguarantee for better constants and in-place operation.",[814,7472,7474],{"id":7473},"counting-inversions","Counting inversions",[381,7476,7477,7478,7643,7644,7647],{},"Here is a problem that has nothing to do with sorting on its surface, yet falls\nto the very machinery we just built. Given a list\n",[457,7479,7481],{"className":7480},[460],[457,7482,7484],{"className":7483,"ariaHidden":465},[464],[457,7485,7487,7490],{"className":7486},[469],[457,7488],{"className":7489,"style":510},[473],[457,7491,7493,7496,7536,7539,7542,7582,7585,7588,7591,7594,7597,7600,7640],{"className":7492},[846],[457,7494,852],{"className":7495,"style":851},[542,850],[457,7497,7499,7502],{"className":7498},[478],[457,7500,434],{"className":7501},[478,479],[457,7503,7505],{"className":7504},[553],[457,7506,7508,7528],{"className":7507},[557,865],[457,7509,7511,7525],{"className":7510},[561],[457,7512,7514],{"className":7513,"style":872},[565],[457,7515,7516,7519],{"style":875},[457,7517],{"className":7518,"style":574},[573],[457,7520,7522],{"className":7521},[578,579,580,581],[457,7523,440],{"className":7524},[478,581],[457,7526,889],{"className":7527},[888],[457,7529,7531],{"className":7530},[561],[457,7532,7534],{"className":7533,"style":896},[565],[457,7535],{},[457,7537,903],{"className":7538},[902],[457,7540],{"className":7541,"style":647},[625],[457,7543,7545,7548],{"className":7544},[478],[457,7546,434],{"className":7547},[478,479],[457,7549,7551],{"className":7550},[553],[457,7552,7554,7574],{"className":7553},[557,865],[457,7555,7557,7571],{"className":7556},[561],[457,7558,7560],{"className":7559,"style":872},[565],[457,7561,7562,7565],{"style":875},[457,7563],{"className":7564,"style":574},[573],[457,7566,7568],{"className":7567},[578,579,580,581],[457,7569,936],{"className":7570},[478,581],[457,7572,889],{"className":7573},[888],[457,7575,7577],{"className":7576},[561],[457,7578,7580],{"className":7579,"style":896},[565],[457,7581],{},[457,7583,903],{"className":7584},[902],[457,7586],{"className":7587,"style":647},[625],[457,7589,957],{"className":7590},[846],[457,7592],{"className":7593,"style":647},[625],[457,7595,903],{"className":7596},[902],[457,7598],{"className":7599,"style":647},[625],[457,7601,7603,7606],{"className":7602},[478],[457,7604,434],{"className":7605},[478,479],[457,7607,7609],{"className":7608},[553],[457,7610,7612,7632],{"className":7611},[557,865],[457,7613,7615,7629],{"className":7614},[561],[457,7616,7618],{"className":7617,"style":985},[565],[457,7619,7620,7623],{"style":875},[457,7621],{"className":7622,"style":574},[573],[457,7624,7626],{"className":7625},[578,579,580,581],[457,7627,480],{"className":7628},[478,479,581],[457,7630,889],{"className":7631},[888],[457,7633,7635],{"className":7634},[561],[457,7636,7638],{"className":7637,"style":896},[565],[457,7639],{},[457,7641,1011],{"className":7642,"style":851},[589,850],", how ",[385,7645,7646],{},"close to sorted"," is it? A natural measure\ncounts the pairs that are out of order.",[822,7649,7650],{"type":824},[381,7651,7652,7654,7655,7682,7683,3691,7685,7713,7714,7717,7718,7748,7749,2526,7821,727],{},[390,7653,829],{}," an array ",[457,7656,7658],{"className":7657},[460],[457,7659,7661],{"className":7660,"ariaHidden":465},[464],[457,7662,7664,7667,7670,7673,7676,7679],{"className":7663},[469],[457,7665],{"className":7666,"style":510},[473],[457,7668,1506],{"className":7669},[478,479],[457,7671,1510],{"className":7672},[542],[457,7674,2566],{"className":7675},[478],[457,7677,480],{"className":7678},[478,479],[457,7680,1526],{"className":7681},[589],".\n",[390,7684,1031],{},[457,7686,7688],{"className":7687},[460],[457,7689,7691],{"className":7690,"ariaHidden":465},[464],[457,7692,7694,7697,7704,7707,7710],{"className":7693},[469],[457,7695],{"className":7696,"style":510},[473],[457,7698,7700],{"className":7699},[5137],[457,7701,7703],{"className":7702,"style":5142},[478,5141],"ninv",[457,7705,543],{"className":7706},[542],[457,7708,1506],{"className":7709},[478,479],[457,7711,590],{"className":7712},[589],", the number of ",[390,7715,7716],{},"inversions"," — pairs\n",[457,7719,7721],{"className":7720},[460],[457,7722,7724],{"className":7723,"ariaHidden":465},[464],[457,7725,7727,7730,7733,7736,7739,7742,7745],{"className":7726},[469],[457,7728],{"className":7729,"style":510},[473],[457,7731,543],{"className":7732},[542],[457,7734,2785],{"className":7735},[478,479],[457,7737,903],{"className":7738},[902],[457,7740],{"className":7741,"style":647},[625],[457,7743,2803],{"className":7744,"style":2802},[478,479],[457,7746,590],{"className":7747},[589]," with ",[457,7750,7752],{"className":7751},[460],[457,7753,7755,7774,7794,7812],{"className":7754,"ariaHidden":465},[464],[457,7756,7758,7762,7765,7768,7771],{"className":7757},[469],[457,7759],{"className":7760,"style":7761},[473],"height:0.7804em;vertical-align:-0.136em;",[457,7763,440],{"className":7764},[478],[457,7766],{"className":7767,"style":626},[625],[457,7769,1261],{"className":7770},[630],[457,7772],{"className":7773,"style":626},[625],[457,7775,7777,7781,7784,7787,7791],{"className":7776},[469],[457,7778],{"className":7779,"style":7780},[473],"height:0.6986em;vertical-align:-0.0391em;",[457,7782,2785],{"className":7783},[478,479],[457,7785],{"className":7786,"style":626},[625],[457,7788,7790],{"className":7789},[630],"\u003C",[457,7792],{"className":7793,"style":626},[625],[457,7795,7797,7800,7803,7806,7809],{"className":7796},[469],[457,7798],{"className":7799,"style":2798},[473],[457,7801,2803],{"className":7802,"style":2802},[478,479],[457,7804],{"className":7805,"style":626},[625],[457,7807,1261],{"className":7808},[630],[457,7810],{"className":7811,"style":626},[625],[457,7813,7815,7818],{"className":7814},[469],[457,7816],{"className":7817,"style":474},[473],[457,7819,480],{"className":7820},[478,479],[457,7822,7824],{"className":7823},[460],[457,7825,7827,7855],{"className":7826,"ariaHidden":465},[464],[457,7828,7830,7833,7836,7839,7842,7845,7848,7852],{"className":7829},[469],[457,7831],{"className":7832,"style":510},[473],[457,7834,1506],{"className":7835},[478,479],[457,7837,1510],{"className":7838},[542],[457,7840,2785],{"className":7841},[478,479],[457,7843,1526],{"className":7844},[589],[457,7846],{"className":7847,"style":626},[625],[457,7849,7851],{"className":7850},[630],">",[457,7853],{"className":7854,"style":626},[625],[457,7856,7858,7861,7864,7867,7870],{"className":7857},[469],[457,7859],{"className":7860,"style":510},[473],[457,7862,1506],{"className":7863},[478,479],[457,7865,1510],{"className":7866},[542],[457,7868,2803],{"className":7869,"style":2802},[478,479],[457,7871,1526],{"className":7872},[589],[381,7874,7875,7876,7961,7962,7965,7966,8219],{},"A sorted array has zero inversions; a reverse-sorted one has the maximum,\n",[457,7877,7879],{"className":7878},[460],[457,7880,7882],{"className":7881,"ariaHidden":465},[464],[457,7883,7885,7889],{"className":7884},[469],[457,7886],{"className":7887,"style":7888},[473],"height:1.2em;vertical-align:-0.35em;",[457,7890,7892,7899,7955],{"className":7891},[478],[457,7893,7895],{"className":7894,"style":851},[542,850],[457,7896,543],{"className":7897},[7898,6644],"delimsizing",[457,7900,7903],{"className":7901},[7902],"mfrac",[457,7904,7906,7946],{"className":7905},[557,865],[457,7907,7909,7943],{"className":7908},[561],[457,7910,7913,7928],{"className":7911,"style":7912},[565],"height:0.7454em;",[457,7914,7916,7919],{"style":7915},"top:-2.355em;",[457,7917],{"className":7918,"style":574},[573],[457,7920,7922],{"className":7921},[578,579,580,581],[457,7923,7925],{"className":7924},[478,581],[457,7926,936],{"className":7927},[478,581],[457,7929,7931,7934],{"style":7930},"top:-3.144em;",[457,7932],{"className":7933,"style":574},[573],[457,7935,7937],{"className":7936},[578,579,580,581],[457,7938,7940],{"className":7939},[478,581],[457,7941,480],{"className":7942},[478,479,581],[457,7944,889],{"className":7945},[888],[457,7947,7949],{"className":7948},[561],[457,7950,7953],{"className":7951,"style":7952},[565],"height:0.345em;",[457,7954],{},[457,7956,7958],{"className":7957,"style":851},[589,850],[457,7959,590],{"className":7960},[7898,6644],". (Inversion counts also drive collaborative-filtering ",[427,7963,7964],{},"how similar are two rankings?"," scores.) The brute-force algorithm loops over all\npairs and counts the bad ones, costing exactly ",[457,7967,7969],{"className":7968},[460],[457,7970,7972,8053,8154,8175],{"className":7971,"ariaHidden":465},[464],[457,7973,7975,7978,8044,8047,8050],{"className":7974},[469],[457,7976],{"className":7977,"style":7888},[473],[457,7979,7981,7987,8038],{"className":7980},[478],[457,7982,7984],{"className":7983,"style":851},[542,850],[457,7985,543],{"className":7986},[7898,6644],[457,7988,7990],{"className":7989},[7902],[457,7991,7993,8030],{"className":7992},[557,865],[457,7994,7996,8027],{"className":7995},[561],[457,7997,7999,8013],{"className":7998,"style":7912},[565],[457,8000,8001,8004],{"style":7915},[457,8002],{"className":8003,"style":574},[573],[457,8005,8007],{"className":8006},[578,579,580,581],[457,8008,8010],{"className":8009},[478,581],[457,8011,936],{"className":8012},[478,581],[457,8014,8015,8018],{"style":7930},[457,8016],{"className":8017,"style":574},[573],[457,8019,8021],{"className":8020},[578,579,580,581],[457,8022,8024],{"className":8023},[478,581],[457,8025,480],{"className":8026},[478,479,581],[457,8028,889],{"className":8029},[888],[457,8031,8033],{"className":8032},[561],[457,8034,8036],{"className":8035,"style":7952},[565],[457,8037],{},[457,8039,8041],{"className":8040,"style":851},[589,850],[457,8042,590],{"className":8043},[7898,6644],[457,8045],{"className":8046,"style":626},[625],[457,8048,631],{"className":8049},[630],[457,8051],{"className":8052,"style":626},[625],[457,8054,8056,8060,8136,8139,8142,8145,8148,8151],{"className":8055},[469],[457,8057],{"className":8058,"style":8059},[473],"height:1.1901em;vertical-align:-0.345em;",[457,8061,8063,8067,8133],{"className":8062},[478],[457,8064],{"className":8065},[542,8066],"nulldelimiter",[457,8068,8070],{"className":8069},[7902],[457,8071,8073,8125],{"className":8072},[557,865],[457,8074,8076,8122],{"className":8075},[561],[457,8077,8080,8096,8107],{"className":8078,"style":8079},[565],"height:0.8451em;",[457,8081,8083,8087],{"style":8082},"top:-2.655em;",[457,8084],{"className":8085,"style":8086},[573],"height:3em;",[457,8088,8090],{"className":8089},[578,579,580,581],[457,8091,8093],{"className":8092},[478,581],[457,8094,936],{"className":8095},[478,581],[457,8097,8099,8102],{"style":8098},"top:-3.23em;",[457,8100],{"className":8101,"style":8086},[573],[457,8103],{"className":8104,"style":8106},[8105],"frac-line","border-bottom-width:0.04em;",[457,8108,8110,8113],{"style":8109},"top:-3.394em;",[457,8111],{"className":8112,"style":8086},[573],[457,8114,8116],{"className":8115},[578,579,580,581],[457,8117,8119],{"className":8118},[478,581],[457,8120,440],{"className":8121},[478,581],[457,8123,889],{"className":8124},[888],[457,8126,8128],{"className":8127},[561],[457,8129,8131],{"className":8130,"style":7952},[565],[457,8132],{},[457,8134],{"className":8135},[589,8066],[457,8137,480],{"className":8138},[478,479],[457,8140,543],{"className":8141},[542],[457,8143,480],{"className":8144},[478,479],[457,8146],{"className":8147,"style":669},[625],[457,8149,3573],{"className":8150},[673],[457,8152],{"className":8153,"style":669},[625],[457,8155,8157,8160,8163,8166,8169,8172],{"className":8156},[469],[457,8158],{"className":8159,"style":510},[473],[457,8161,440],{"className":8162},[478],[457,8164,590],{"className":8165},[589],[457,8167],{"className":8168,"style":626},[625],[457,8170,631],{"className":8171},[630],[457,8173],{"className":8174,"style":626},[625],[457,8176,8178,8181,8184,8187,8216],{"className":8177},[469],[457,8179],{"className":8180,"style":1430},[473],[457,8182,538],{"className":8183},[478],[457,8185,543],{"className":8186},[542],[457,8188,8190,8193],{"className":8189},[478],[457,8191,480],{"className":8192},[478,479],[457,8194,8196],{"className":8195},[553],[457,8197,8199],{"className":8198},[557],[457,8200,8202],{"className":8201},[561],[457,8203,8205],{"className":8204,"style":1455},[565],[457,8206,8207,8210],{"style":569},[457,8208],{"className":8209,"style":574},[573],[457,8211,8213],{"className":8212},[578,579,580,581],[457,8214,936],{"className":8215},[478,581],[457,8217,590],{"className":8218},[589]," comparisons. We can do far better.",[381,8221,8222,8225,8226,8241,8242,8259,8260,8277],{},[390,8223,8224],{},"Idea 0: divide and conquer, just like mergesort."," Split ",[457,8227,8229],{"className":8228},[460],[457,8230,8232],{"className":8231,"ariaHidden":465},[464],[457,8233,8235,8238],{"className":8234},[469],[457,8236],{"className":8237,"style":2746},[473],[457,8239,1506],{"className":8240},[478,479]," into a left half\n",[457,8243,8245],{"className":8244},[460],[457,8246,8248],{"className":8247,"ariaHidden":465},[464],[457,8249,8251,8254],{"className":8250},[469],[457,8252],{"className":8253,"style":2746},[473],[457,8255,8258],{"className":8256,"style":8257},[478,479],"margin-right:0.0502em;","B"," and a right half ",[457,8261,8263],{"className":8262},[460],[457,8264,8266],{"className":8265,"ariaHidden":465},[464],[457,8267,8269,8272],{"className":8268},[469],[457,8270],{"className":8271,"style":2746},[473],[457,8273,8276],{"className":8274,"style":8275},[478,479],"margin-right:0.0715em;","C",". Every inversion is one of three kinds:",[394,8279,8280,8315,8347],{},[397,8281,8282,8283,8298,8299,8314],{},"both endpoints in ",[457,8284,8286],{"className":8285},[460],[457,8287,8289],{"className":8288,"ariaHidden":465},[464],[457,8290,8292,8295],{"className":8291},[469],[457,8293],{"className":8294,"style":2746},[473],[457,8296,8258],{"className":8297,"style":8257},[478,479],", counted by recursing on ",[457,8300,8302],{"className":8301},[460],[457,8303,8305],{"className":8304,"ariaHidden":465},[464],[457,8306,8308,8311],{"className":8307},[469],[457,8309],{"className":8310,"style":2746},[473],[457,8312,8258],{"className":8313,"style":8257},[478,479],";",[397,8316,8282,8317,8298,8332,8314],{},[457,8318,8320],{"className":8319},[460],[457,8321,8323],{"className":8322,"ariaHidden":465},[464],[457,8324,8326,8329],{"className":8325},[469],[457,8327],{"className":8328,"style":2746},[473],[457,8330,8276],{"className":8331,"style":8275},[478,479],[457,8333,8335],{"className":8334},[460],[457,8336,8338],{"className":8337,"ariaHidden":465},[464],[457,8339,8341,8344],{"className":8340},[469],[457,8342],{"className":8343,"style":2746},[473],[457,8345,8276],{"className":8346,"style":8275},[478,479],[397,8348,8349,8350,746,8353,8368,8369,8384,8385,727],{},"one endpoint in each: a ",[390,8351,8352],{},"cross inversion",[457,8354,8356],{"className":8355},[460],[457,8357,8359],{"className":8358,"ariaHidden":465},[464],[457,8360,8362,8365],{"className":8361},[469],[457,8363],{"className":8364,"style":2781},[473],[457,8366,2785],{"className":8367},[478,479]," on the left and ",[457,8370,8372],{"className":8371},[460],[457,8373,8375],{"className":8374,"ariaHidden":465},[464],[457,8376,8378,8381],{"className":8377},[469],[457,8379],{"className":8380,"style":2798},[473],[457,8382,2803],{"className":8383,"style":2802},[478,479]," on the\nright with ",[457,8386,8388],{"className":8387},[460],[457,8389,8391,8418],{"className":8390,"ariaHidden":465},[464],[457,8392,8394,8397,8400,8403,8406,8409,8412,8415],{"className":8393},[469],[457,8395],{"className":8396,"style":510},[473],[457,8398,8258],{"className":8399,"style":8257},[478,479],[457,8401,1510],{"className":8402},[542],[457,8404,2785],{"className":8405},[478,479],[457,8407,1526],{"className":8408},[589],[457,8410],{"className":8411,"style":626},[625],[457,8413,7851],{"className":8414},[630],[457,8416],{"className":8417,"style":626},[625],[457,8419,8421,8424,8427,8430,8433],{"className":8420},[469],[457,8422],{"className":8423,"style":510},[473],[457,8425,8276],{"className":8426,"style":8275},[478,479],[457,8428,1510],{"className":8429},[542],[457,8431,2803],{"className":8432,"style":2802},[478,479],[457,8434,1526],{"className":8435},[589],[1634,8437,8439,8611],{"className":8438},[1637,1638],[1640,8440,8444],{"xmlns":1642,"width":8441,"height":8442,"viewBox":8443},"255.774","84.021","-75 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",[457,8616,8618],{"className":8617},[460],[457,8619,8621],{"className":8620,"ariaHidden":465},[464],[457,8622,8624,8627],{"className":8623},[469],[457,8625],{"className":8626,"style":2746},[473],[457,8628,8258],{"className":8629,"style":8257},[478,479]," to a smaller element in right half ",[457,8632,8634],{"className":8633},[460],[457,8635,8637],{"className":8636,"ariaHidden":465},[464],[457,8638,8640,8643],{"className":8639},[469],[457,8641],{"className":8642,"style":2746},[473],[457,8644,8276],{"className":8645,"style":8275},[478,479],[381,8647,8648,8649,8699,8700,8810,8811,8861],{},"Counting cross inversions with a double loop costs ",[457,8650,8652],{"className":8651},[460],[457,8653,8655],{"className":8654,"ariaHidden":465},[464],[457,8656,8658,8661,8664,8667,8696],{"className":8657},[469],[457,8659],{"className":8660,"style":1430},[473],[457,8662,538],{"className":8663},[478],[457,8665,543],{"className":8666},[542],[457,8668,8670,8673],{"className":8669},[478],[457,8671,480],{"className":8672},[478,479],[457,8674,8676],{"className":8675},[553],[457,8677,8679],{"className":8678},[557],[457,8680,8682],{"className":8681},[561],[457,8683,8685],{"className":8684,"style":1455},[565],[457,8686,8687,8690],{"style":569},[457,8688],{"className":8689,"style":574},[573],[457,8691,8693],{"className":8692},[578,579,580,581],[457,8694,936],{"className":8695},[478,581],[457,8697,590],{"className":8698},[589]," for the combine\nstep, giving ",[457,8701,8703],{"className":8702},[460],[457,8704,8706,8733,8766],{"className":8705,"ariaHidden":465},[464],[457,8707,8709,8712,8715,8718,8721,8724,8727,8730],{"className":8708},[469],[457,8710],{"className":8711,"style":510},[473],[457,8713,612],{"className":8714,"style":611},[478,479],[457,8716,543],{"className":8717},[542],[457,8719,480],{"className":8720},[478,479],[457,8722,590],{"className":8723},[589],[457,8725],{"className":8726,"style":626},[625],[457,8728,631],{"className":8729},[630],[457,8731],{"className":8732,"style":626},[625],[457,8734,8736,8739,8742,8745,8748,8751,8754,8757,8760,8763],{"className":8735},[469],[457,8737],{"className":8738,"style":510},[473],[457,8740,936],{"className":8741},[478],[457,8743,612],{"className":8744,"style":611},[478,479],[457,8746,543],{"className":8747},[542],[457,8749,480],{"className":8750},[478,479],[457,8752,1591],{"className":8753},[478],[457,8755,590],{"className":8756},[589],[457,8758],{"className":8759,"style":669},[625],[457,8761,674],{"className":8762},[673],[457,8764],{"className":8765,"style":669},[625],[457,8767,8769,8772,8775,8778,8807],{"className":8768},[469],[457,8770],{"className":8771,"style":1430},[473],[457,8773,538],{"className":8774},[478],[457,8776,543],{"className":8777},[542],[457,8779,8781,8784],{"className":8780},[478],[457,8782,480],{"className":8783},[478,479],[457,8785,8787],{"className":8786},[553],[457,8788,8790],{"className":8789},[557],[457,8791,8793],{"className":8792},[561],[457,8794,8796],{"className":8795,"style":1455},[565],[457,8797,8798,8801],{"style":569},[457,8799],{"className":8800,"style":574},[573],[457,8802,8804],{"className":8803},[578,579,580,581],[457,8805,936],{"className":8806},[478,581],[457,8808,590],{"className":8809},[589],", which the master theorem resolves\nto ",[457,8812,8814],{"className":8813},[460],[457,8815,8817],{"className":8816,"ariaHidden":465},[464],[457,8818,8820,8823,8826,8829,8858],{"className":8819},[469],[457,8821],{"className":8822,"style":1430},[473],[457,8824,538],{"className":8825},[478],[457,8827,543],{"className":8828},[542],[457,8830,8832,8835],{"className":8831},[478],[457,8833,480],{"className":8834},[478,479],[457,8836,8838],{"className":8837},[553],[457,8839,8841],{"className":8840},[557],[457,8842,8844],{"className":8843},[561],[457,8845,8847],{"className":8846,"style":1455},[565],[457,8848,8849,8852],{"style":569},[457,8850],{"className":8851,"style":574},[573],[457,8853,8855],{"className":8854},[578,579,580,581],[457,8856,936],{"className":8857},[478,581],[457,8859,590],{"className":8860},[589],", no gain. The combine step is the bottleneck.",[381,8863,8864,8867,8868,8871,8872,8893,8894,8945,8946,8970,8971,3691,8995,8998,8999,3418,9014,9038,9039,9091,9092,9107,9108,9132],{},[390,8865,8866],{},"Idea 1: count cross inversions during a merge."," Suppose the two halves\narrive ",[390,8869,8870],{},"already sorted",". Walk them with two cursors exactly as ",[457,8873,8875],{"className":8874},[460],[457,8876,8878],{"className":8877,"ariaHidden":465},[464],[457,8879,8881,8884],{"className":8880},[469],[457,8882],{"className":8883,"style":2481},[473],[457,8885,8887],{"className":8886},[2485,2486],[457,8888,8890],{"className":8889},[478,2490],[457,8891,2494],{"className":8892},[478],"\ndoes. When we are about to emit and ",[457,8895,8897],{"className":8896},[460],[457,8898,8900,8927],{"className":8899,"ariaHidden":465},[464],[457,8901,8903,8906,8909,8912,8915,8918,8921,8924],{"className":8902},[469],[457,8904],{"className":8905,"style":510},[473],[457,8907,8258],{"className":8908,"style":8257},[478,479],[457,8910,1510],{"className":8911},[542],[457,8913,2785],{"className":8914},[478,479],[457,8916,1526],{"className":8917},[589],[457,8919],{"className":8920,"style":626},[625],[457,8922,7851],{"className":8923},[630],[457,8925],{"className":8926,"style":626},[625],[457,8928,8930,8933,8936,8939,8942],{"className":8929},[469],[457,8931],{"className":8932,"style":510},[473],[457,8934,8276],{"className":8935,"style":8275},[478,479],[457,8937,1510],{"className":8938},[542],[457,8940,2803],{"className":8941,"style":2802},[478,479],[457,8943,1526],{"className":8944},[589],", the element ",[457,8947,8949],{"className":8948},[460],[457,8950,8952],{"className":8951,"ariaHidden":465},[464],[457,8953,8955,8958,8961,8964,8967],{"className":8954},[469],[457,8956],{"className":8957,"style":510},[473],[457,8959,8276],{"className":8960,"style":8275},[478,479],[457,8962,1510],{"className":8963},[542],[457,8965,2803],{"className":8966,"style":2802},[478,479],[457,8968,1526],{"className":8969},[589]," is smaller\nthan ",[457,8972,8974],{"className":8973},[460],[457,8975,8977],{"className":8976,"ariaHidden":465},[464],[457,8978,8980,8983,8986,8989,8992],{"className":8979},[469],[457,8981],{"className":8982,"style":510},[473],[457,8984,8258],{"className":8985,"style":8257},[478,479],[457,8987,1510],{"className":8988},[542],[457,8990,2785],{"className":8991},[478,479],[457,8993,1526],{"className":8994},[589],[385,8996,8997],{},"and"," than everything after it in ",[457,9000,9002],{"className":9001},[460],[457,9003,9005],{"className":9004,"ariaHidden":465},[464],[457,9006,9008,9011],{"className":9007},[469],[457,9009],{"className":9010,"style":2746},[473],[457,9012,8258],{"className":9013,"style":8257},[478,479],[457,9015,9017],{"className":9016},[460],[457,9018,9020],{"className":9019,"ariaHidden":465},[464],[457,9021,9023,9026,9029,9032,9035],{"className":9022},[469],[457,9024],{"className":9025,"style":510},[473],[457,9027,8276],{"className":9028,"style":8275},[478,479],[457,9030,1510],{"className":9031},[542],[457,9033,2803],{"className":9034,"style":2802},[478,479],[457,9036,1526],{"className":9037},[589]," forms an inversion\nwith all ",[457,9040,9042],{"className":9041},[460],[457,9043,9045,9063,9082],{"className":9044,"ariaHidden":465},[464],[457,9046,9048,9051,9054,9057,9060],{"className":9047},[469],[457,9049],{"className":9050,"style":3583},[473],[457,9052,381],{"className":9053},[478,479],[457,9055],{"className":9056,"style":669},[625],[457,9058,3573],{"className":9059},[673],[457,9061],{"className":9062,"style":669},[625],[457,9064,9066,9070,9073,9076,9079],{"className":9065},[469],[457,9067],{"className":9068,"style":9069},[473],"height:0.7429em;vertical-align:-0.0833em;",[457,9071,2785],{"className":9072},[478,479],[457,9074],{"className":9075,"style":669},[625],[457,9077,674],{"className":9078},[673],[457,9080],{"className":9081,"style":669},[625],[457,9083,9085,9088],{"className":9084},[469],[457,9086],{"className":9087,"style":3370},[473],[457,9089,440],{"className":9090},[478]," remaining elements of ",[457,9093,9095],{"className":9094},[460],[457,9096,9098],{"className":9097,"ariaHidden":465},[464],[457,9099,9101,9104],{"className":9100},[469],[457,9102],{"className":9103,"style":2746},[473],[457,9105,8258],{"className":9106,"style":8257},[478,479]," at once. Add that count, emit\n",[457,9109,9111],{"className":9110},[460],[457,9112,9114],{"className":9113,"ariaHidden":465},[464],[457,9115,9117,9120,9123,9126,9129],{"className":9116},[469],[457,9118],{"className":9119,"style":510},[473],[457,9121,8276],{"className":9122,"style":8275},[478,479],[457,9124,1510],{"className":9125},[542],[457,9127,2803],{"className":9128,"style":2802},[478,479],[457,9130,1526],{"className":9131},[589],", and move on.",[381,9134,9135,9136,9187,9188,9203,9204,9228,9229,9253],{},"This batching is exactly why the count collapses to linear time: a single\ncomparison reveals ",[457,9137,9139],{"className":9138},[460],[457,9140,9142,9160,9178],{"className":9141,"ariaHidden":465},[464],[457,9143,9145,9148,9151,9154,9157],{"className":9144},[469],[457,9146],{"className":9147,"style":3583},[473],[457,9149,381],{"className":9150},[478,479],[457,9152],{"className":9153,"style":669},[625],[457,9155,3573],{"className":9156},[673],[457,9158],{"className":9159,"style":669},[625],[457,9161,9163,9166,9169,9172,9175],{"className":9162},[469],[457,9164],{"className":9165,"style":9069},[473],[457,9167,2785],{"className":9168},[478,479],[457,9170],{"className":9171,"style":669},[625],[457,9173,674],{"className":9174},[673],[457,9176],{"className":9177,"style":669},[625],[457,9179,9181,9184],{"className":9180},[469],[457,9182],{"className":9183,"style":3370},[473],[457,9185,440],{"className":9186},[478]," inversions, not one. Because ",[457,9189,9191],{"className":9190},[460],[457,9192,9194],{"className":9193,"ariaHidden":465},[464],[457,9195,9197,9200],{"className":9196},[469],[457,9198],{"className":9199,"style":2746},[473],[457,9201,8258],{"className":9202,"style":8257},[478,479]," is sorted, every\nelement from ",[457,9205,9207],{"className":9206},[460],[457,9208,9210],{"className":9209,"ariaHidden":465},[464],[457,9211,9213,9216,9219,9222,9225],{"className":9212},[469],[457,9214],{"className":9215,"style":510},[473],[457,9217,8258],{"className":9218,"style":8257},[478,479],[457,9220,1510],{"className":9221},[542],[457,9223,2785],{"className":9224},[478,479],[457,9226,1526],{"className":9227},[589]," onward exceeds ",[457,9230,9232],{"className":9231},[460],[457,9233,9235],{"className":9234,"ariaHidden":465},[464],[457,9236,9238,9241,9244,9247,9250],{"className":9237},[469],[457,9239],{"className":9240,"style":510},[473],[457,9242,8276],{"className":9243,"style":8275},[478,479],[457,9245,1510],{"className":9246},[542],[457,9248,2803],{"className":9249,"style":2802},[478,479],[457,9251,1526],{"className":9252},[589],", so each is inverted with it.",[1634,9255,9257,9612],{"className":9256},[1637,1638],[1640,9258,9262],{"xmlns":1642,"width":9259,"height":9260,"viewBox":9261},"376.292","198.056","-75 -75 282.219 148.542",[1647,9263,9264,9271,9274,9280,9283,9289,9301,9313,9325,9334,9343,9410,9417,9429,9432,9439,9448,9456,9464,9473,9481],{"stroke":1649,"style":1650},[1647,9265,9267],{"transform":9266},"translate(-37.335 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45.284-0.496Q45.284-0.322 45.161-0.199Q45.038-0.076 44.870-0.076Q44.703-0.076 44.576-0.199Q44.450-0.322 44.450-0.496M46.822-0.496Q46.822-0.664 46.948-0.787Q47.075-0.910 47.242-0.910Q47.410-0.910 47.533-0.787Q47.656-0.664 47.656-0.496Q47.656-0.322 47.533-0.199Q47.410-0.076 47.242-0.076Q47.075-0.076 46.948-0.199Q46.822-0.322 46.822-0.496M49.908 1.281L48.493 1.281Q48.462 1.281 48.433 1.245Q48.404 1.209 48.404 1.178L48.438 1.066Q48.466 1.007 48.517 1.001Q48.743 1.001 48.823 0.963Q48.903 0.925 48.951 0.758L49.727-2.366Q49.761-2.493 49.761-2.609Q49.761-2.749 49.708-2.845Q49.655-2.940 49.532-2.940Q49.310-2.940 49.209-2.713Q49.108-2.486 48.999-2.065Q48.989-2 48.927-2L48.818-2Q48.787-2 48.763-2.031Q48.739-2.062 48.739-2.086L48.739-2.113Q48.852-2.547 49.033-2.855Q49.214-3.162 49.546-3.162Q49.799-3.162 50.012-3.036Q50.226-2.909 50.288-2.687Q50.496-2.903 50.749-3.033Q51.002-3.162 51.265-3.162Q51.586-3.162 51.833-3.007Q52.079-2.851 52.208-2.585Q52.338-2.318 52.338-1.993Q52.338-1.641 52.193-1.289Q52.048-0.937 51.797-0.649Q51.545-0.360 51.210-0.184Q50.875-0.008 50.517-0.008Q50.065-0.008 49.812-0.397L49.508 0.806Q49.488 0.867 49.488 0.932Q49.488 1.001 49.929 1.001Q50.014 1.028 50.014 1.113L49.987 1.226Q49.959 1.274 49.908 1.281M50.288-2.301L49.915-0.825Q49.977-0.575 50.134-0.402Q50.291-0.230 50.530-0.230Q50.739-0.230 50.928-0.356Q51.118-0.483 51.255-0.672Q51.392-0.862 51.477-1.064Q51.583-1.327 51.667-1.694Q51.750-2.062 51.750-2.294Q51.750-2.448 51.699-2.600Q51.648-2.752 51.535-2.846Q51.422-2.940 51.251-2.940Q50.971-2.940 50.727-2.757Q50.482-2.575 50.288-2.301",[1667],[1647,9607,9608],{"transform":9487},[1652,9609],{"d":9610,"fill":3110,"stroke":3110,"className":9611,"style":1668},"M53.857 1.674L52.781 1.674L52.781 1.332L53.515 1.332L53.515-4.984L52.781-4.984L52.781-5.326L53.857-5.326",[1667],[2323,9613,9615,9616,9667,9668,9683,9684,9714,9715,9739,9740,9764,9765,9816],{"className":9614},[2326],"When the merge finds ",[457,9617,9619],{"className":9618},[460],[457,9620,9622,9649],{"className":9621,"ariaHidden":465},[464],[457,9623,9625,9628,9631,9634,9637,9640,9643,9646],{"className":9624},[469],[457,9626],{"className":9627,"style":510},[473],[457,9629,8258],{"className":9630,"style":8257},[478,479],[457,9632,1510],{"className":9633},[542],[457,9635,2785],{"className":9636},[478,479],[457,9638,1526],{"className":9639},[589],[457,9641],{"className":9642,"style":626},[625],[457,9644,7851],{"className":9645},[630],[457,9647],{"className":9648,"style":626},[625],[457,9650,9652,9655,9658,9661,9664],{"className":9651},[469],[457,9653],{"className":9654,"style":510},[473],[457,9656,8276],{"className":9657,"style":8275},[478,479],[457,9659,1510],{"className":9660},[542],[457,9662,2803],{"className":9663,"style":2802},[478,479],[457,9665,1526],{"className":9666},[589],", sortedness of ",[457,9669,9671],{"className":9670},[460],[457,9672,9674],{"className":9673,"ariaHidden":465},[464],[457,9675,9677,9680],{"className":9676},[469],[457,9678],{"className":9679,"style":2746},[473],[457,9681,8258],{"className":9682,"style":8257},[478,479]," means every remaining ",[457,9685,9687],{"className":9686},[460],[457,9688,9690],{"className":9689,"ariaHidden":465},[464],[457,9691,9693,9696,9699,9702,9705,9708,9711],{"className":9692},[469],[457,9694],{"className":9695,"style":510},[473],[457,9697,8258],{"className":9698,"style":8257},[478,479],[457,9700,1510],{"className":9701},[542],[457,9703,2785],{"className":9704},[478,479],[457,9706,1517],{"className":9707},[478],[457,9709,381],{"className":9710},[478,479],[457,9712,1526],{"className":9713},[589]," also exceeds ",[457,9716,9718],{"className":9717},[460],[457,9719,9721],{"className":9720,"ariaHidden":465},[464],[457,9722,9724,9727,9730,9733,9736],{"className":9723},[469],[457,9725],{"className":9726,"style":510},[473],[457,9728,8276],{"className":9729,"style":8275},[478,479],[457,9731,1510],{"className":9732},[542],[457,9734,2803],{"className":9735,"style":2802},[478,479],[457,9737,1526],{"className":9738},[589]," — so emitting ",[457,9741,9743],{"className":9742},[460],[457,9744,9746],{"className":9745,"ariaHidden":465},[464],[457,9747,9749,9752,9755,9758,9761],{"className":9748},[469],[457,9750],{"className":9751,"style":510},[473],[457,9753,8276],{"className":9754,"style":8275},[478,479],[457,9756,1510],{"className":9757},[542],[457,9759,2803],{"className":9760,"style":2802},[478,479],[457,9762,1526],{"className":9763},[589]," adds ",[457,9766,9768],{"className":9767},[460],[457,9769,9771,9789,9807],{"className":9770,"ariaHidden":465},[464],[457,9772,9774,9777,9780,9783,9786],{"className":9773},[469],[457,9775],{"className":9776,"style":3583},[473],[457,9778,381],{"className":9779},[478,479],[457,9781],{"className":9782,"style":669},[625],[457,9784,3573],{"className":9785},[673],[457,9787],{"className":9788,"style":669},[625],[457,9790,9792,9795,9798,9801,9804],{"className":9791},[469],[457,9793],{"className":9794,"style":9069},[473],[457,9796,2785],{"className":9797},[478,479],[457,9799],{"className":9800,"style":669},[625],[457,9802,674],{"className":9803},[673],[457,9805],{"className":9806,"style":669},[625],[457,9808,9810,9813],{"className":9809},[469],[457,9811],{"className":9812,"style":3370},[473],[457,9814,440],{"className":9815},[478]," cross inversions in one stroke.",[2418,9818,9820],{"className":2420,"code":9819,"language":2422,"meta":376,"style":376},"caption: $\\textsc{Count-Cross-Inv}(B[1..p], C[1..q])$ — cross inversions, $B, C$ sorted\nnumber: 3\n$\\mathit{ans} \\gets 0$\n$i \\gets 1$\n$j \\gets 1$\nwhile $i \\le p$ and $j \\le q$ do\n  if $B[i] \\le C[j]$ then\n    $i \\gets i + 1$ \u002F\u002F no inversion\n  else\n    $\\mathit{ans} \\gets \\mathit{ans} + (p - i + 1)$ \u002F\u002F $C[j]$ inverts with $B[i..p]$\n    $j \\gets j + 1$\nreturn $\\mathit{ans}$\n",[2424,9821,9822,9827,9832,9837,9841,9845,9850,9855,9860,9864,9869,9873],{"__ignoreMap":376},[457,9823,9824],{"class":2428,"line":6},[457,9825,9826],{},"caption: $\\textsc{Count-Cross-Inv}(B[1..p], C[1..q])$ — cross inversions, $B, C$ sorted\n",[457,9828,9829],{"class":2428,"line":18},[457,9830,9831],{},"number: 3\n",[457,9833,9834],{"class":2428,"line":24},[457,9835,9836],{},"$\\mathit{ans} \\gets 0$\n",[457,9838,9839],{"class":2428,"line":73},[457,9840,2638],{},[457,9842,9843],{"class":2428,"line":102},[457,9844,2644],{},[457,9846,9847],{"class":2428,"line":108},[457,9848,9849],{},"while $i \\le p$ and $j \\le q$ do\n",[457,9851,9852],{"class":2428,"line":116},[457,9853,9854],{},"  if $B[i] \\le C[j]$ then\n",[457,9856,9857],{"class":2428,"line":196},[457,9858,9859],{},"    $i \\gets i + 1$ \u002F\u002F no inversion\n",[457,9861,9862],{"class":2428,"line":202},[457,9863,2674],{},[457,9865,9866],{"class":2428,"line":283},[457,9867,9868],{},"    $\\mathit{ans} \\gets \\mathit{ans} + (p - i + 1)$ \u002F\u002F $C[j]$ inverts with $B[i..p]$\n",[457,9870,9871],{"class":2428,"line":333},[457,9872,2686],{},[457,9874,9875],{"class":2428,"line":354},[457,9876,9877],{},"return $\\mathit{ans}$\n",[381,9879,9880,9881,9950,9951,2526,9966,9981,9982,3691,10021,10024,10025,10049,10050,10220],{},"This runs in ",[457,9882,9884],{"className":9883},[460],[457,9885,9887,9911,9932],{"className":9886,"ariaHidden":465},[464],[457,9888,9890,9893,9896,9899,9902,9905,9908],{"className":9889},[469],[457,9891],{"className":9892,"style":510},[473],[457,9894,538],{"className":9895},[478],[457,9897,543],{"className":9898},[542],[457,9900,381],{"className":9901},[478,479],[457,9903],{"className":9904,"style":669},[625],[457,9906,674],{"className":9907},[673],[457,9909],{"className":9910,"style":669},[625],[457,9912,9914,9917,9920,9923,9926,9929],{"className":9913},[469],[457,9915],{"className":9916,"style":510},[473],[457,9918,427],{"className":9919,"style":1544},[478,479],[457,9921,590],{"className":9922},[589],[457,9924],{"className":9925,"style":626},[625],[457,9927,631],{"className":9928},[630],[457,9930],{"className":9931,"style":626},[625],[457,9933,9935,9938,9941,9944,9947],{"className":9934},[469],[457,9936],{"className":9937,"style":510},[473],[457,9939,538],{"className":9940},[478],[457,9942,543],{"className":9943},[542],[457,9945,480],{"className":9946},[478,479],[457,9948,590],{"className":9949},[589],", the linear merge pattern. But it\ndemands sorted halves, so we must sort them first: sorting ",[457,9952,9954],{"className":9953},[460],[457,9955,9957],{"className":9956,"ariaHidden":465},[464],[457,9958,9960,9963],{"className":9959},[469],[457,9961],{"className":9962,"style":2746},[473],[457,9964,8258],{"className":9965,"style":8257},[478,479],[457,9967,9969],{"className":9968},[460],[457,9970,9972],{"className":9971,"ariaHidden":465},[464],[457,9973,9975,9978],{"className":9974},[469],[457,9976],{"className":9977,"style":2746},[473],[457,9979,8276],{"className":9980,"style":8275},[478,479]," costs an\nextra ",[457,9983,9985],{"className":9984},[460],[457,9986,9988],{"className":9987,"ariaHidden":465},[464],[457,9989,9991,9994,9997,10000,10003,10006,10012,10015,10018],{"className":9990},[469],[457,9992],{"className":9993,"style":510},[473],[457,9995,538],{"className":9996},[478],[457,9998,543],{"className":9999},[542],[457,10001,480],{"className":10002},[478,479],[457,10004],{"className":10005,"style":647},[625],[457,10007,10009],{"className":10008},[5137],[457,10010,5143],{"className":10011,"style":5142},[478,5141],[457,10013],{"className":10014,"style":647},[625],[457,10016,480],{"className":10017},[478,479],[457,10019,590],{"className":10020},[589],[385,10022,10023],{},"per level",", and there are ",[457,10026,10028],{"className":10027},[460],[457,10029,10031],{"className":10030,"ariaHidden":465},[464],[457,10032,10034,10037,10043,10046],{"className":10033},[469],[457,10035],{"className":10036,"style":6698},[473],[457,10038,10040],{"className":10039},[5137],[457,10041,5143],{"className":10042,"style":5142},[478,5141],[457,10044],{"className":10045,"style":647},[625],[457,10047,480],{"className":10048},[478,479]," levels, giving\n",[457,10051,10053],{"className":10052},[460],[457,10054,10056,10083,10116,10158],{"className":10055,"ariaHidden":465},[464],[457,10057,10059,10062,10065,10068,10071,10074,10077,10080],{"className":10058},[469],[457,10060],{"className":10061,"style":510},[473],[457,10063,612],{"className":10064,"style":611},[478,479],[457,10066,543],{"className":10067},[542],[457,10069,480],{"className":10070},[478,479],[457,10072,590],{"className":10073},[589],[457,10075],{"className":10076,"style":626},[625],[457,10078,631],{"className":10079},[630],[457,10081],{"className":10082,"style":626},[625],[457,10084,10086,10089,10092,10095,10098,10101,10104,10107,10110,10113],{"className":10085},[469],[457,10087],{"className":10088,"style":510},[473],[457,10090,936],{"className":10091},[478],[457,10093,612],{"className":10094,"style":611},[478,479],[457,10096,543],{"className":10097},[542],[457,10099,480],{"className":10100},[478,479],[457,10102,1591],{"className":10103},[478],[457,10105,590],{"className":10106},[589],[457,10108],{"className":10109,"style":669},[625],[457,10111,674],{"className":10112},[673],[457,10114],{"className":10115,"style":669},[625],[457,10117,10119,10122,10125,10128,10131,10134,10140,10143,10146,10149,10152,10155],{"className":10118},[469],[457,10120],{"className":10121,"style":510},[473],[457,10123,538],{"className":10124},[478],[457,10126,543],{"className":10127},[542],[457,10129,480],{"className":10130},[478,479],[457,10132],{"className":10133,"style":647},[625],[457,10135,10137],{"className":10136},[5137],[457,10138,5143],{"className":10139,"style":5142},[478,5141],[457,10141],{"className":10142,"style":647},[625],[457,10144,480],{"className":10145},[478,479],[457,10147,590],{"className":10148},[589],[457,10150],{"className":10151,"style":626},[625],[457,10153,631],{"className":10154},[630],[457,10156],{"className":10157,"style":626},[625],[457,10159,10161,10165,10168,10171,10174,10177,10211,10214,10217],{"className":10160},[469],[457,10162],{"className":10163,"style":10164},[473],"height:1.1484em;vertical-align:-0.25em;",[457,10166,538],{"className":10167},[478],[457,10169,543],{"className":10170},[542],[457,10172,480],{"className":10173},[478,479],[457,10175],{"className":10176,"style":647},[625],[457,10178,10180,10186],{"className":10179},[5137],[457,10181,10183],{"className":10182},[5137],[457,10184,5143],{"className":10185,"style":5142},[478,5141],[457,10187,10189],{"className":10188},[553],[457,10190,10192],{"className":10191},[557],[457,10193,10195],{"className":10194},[561],[457,10196,10199],{"className":10197,"style":10198},[565],"height:0.8984em;",[457,10200,10202,10205],{"style":10201},"top:-3.1473em;margin-right:0.05em;",[457,10203],{"className":10204,"style":574},[573],[457,10206,10208],{"className":10207},[578,579,580,581],[457,10209,936],{"className":10210},[478,581],[457,10212],{"className":10213,"style":647},[625],[457,10215,480],{"className":10216},[478,479],[457,10218,590],{"className":10219},[589],". Better than\nquadratic, but the repeated sorting is wasteful.",[381,10222,10223,10229,10230,10232],{},[390,10224,10225,10226,10228],{},"Idea 2: sort ",[385,10227,8997],{}," count in one pass."," We are doing almost all of\nmergesort's work anyway, so let the recursion return both\nthe inversion count ",[385,10231,8997],{}," a sorted copy of its slice. Then the cross-counting\nmerge also produces the sorted output the parent needs, for free.",[2418,10234,10236],{"className":2420,"code":10235,"language":2422,"meta":376,"style":376},"caption: $\\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, \\mathit{hi})$ — sort $A[\\mathit{lo}..\\mathit{hi}]$, return its inversion count\nnumber: 4\nif $\\mathit{hi} \\le \\mathit{lo}$ then\n  return $0$ \u002F\u002F single element: no inversions\n$t \\gets \\floor{(\\mathit{lo} + \\mathit{hi}) \u002F 2}$\n$c \\gets \\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, t)$ \u002F\u002F left inversions + sort left\n$c \\gets c + \\textsc{Sort-And-Count-Inv}(A, t + 1, \\mathit{hi})$ \u002F\u002F right inversions + sort right\n$c \\gets c + \\textsc{Count-Cross-Inv-And-Merge}(A, \\mathit{lo}, t, \\mathit{hi})$ \u002F\u002F cross + merge\nreturn $c$\n",[2424,10237,10238,10243,10248,10253,10258,10263,10268,10273,10278],{"__ignoreMap":376},[457,10239,10240],{"class":2428,"line":6},[457,10241,10242],{},"caption: $\\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, \\mathit{hi})$ — sort $A[\\mathit{lo}..\\mathit{hi}]$, return its inversion count\n",[457,10244,10245],{"class":2428,"line":18},[457,10246,10247],{},"number: 4\n",[457,10249,10250],{"class":2428,"line":24},[457,10251,10252],{},"if $\\mathit{hi} \\le \\mathit{lo}$ then\n",[457,10254,10255],{"class":2428,"line":73},[457,10256,10257],{},"  return $0$ \u002F\u002F single element: no inversions\n",[457,10259,10260],{"class":2428,"line":102},[457,10261,10262],{},"$t \\gets \\floor{(\\mathit{lo} + \\mathit{hi}) \u002F 2}$\n",[457,10264,10265],{"class":2428,"line":108},[457,10266,10267],{},"$c \\gets \\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, t)$ \u002F\u002F left inversions + sort left\n",[457,10269,10270],{"class":2428,"line":116},[457,10271,10272],{},"$c \\gets c + \\textsc{Sort-And-Count-Inv}(A, t + 1, \\mathit{hi})$ \u002F\u002F right inversions + sort right\n",[457,10274,10275],{"class":2428,"line":196},[457,10276,10277],{},"$c \\gets c + \\textsc{Count-Cross-Inv-And-Merge}(A, \\mathit{lo}, t, \\mathit{hi})$ \u002F\u002F cross + merge\n",[457,10279,10280],{"class":2428,"line":202},[457,10281,10282],{},"return $c$\n",[381,10284,10285,10286,10308,10309,10330,10331,10353,10354,10428],{},"The helper ",[457,10287,10289],{"className":10288},[460],[457,10290,10292],{"className":10291,"ariaHidden":465},[464],[457,10293,10295,10298],{"className":10294},[469],[457,10296],{"className":10297,"style":6698},[473],[457,10299,10301],{"className":10300},[2485,2486],[457,10302,10304],{"className":10303},[478,2490],[457,10305,10307],{"className":10306},[478],"Count-Cross-Inv-And-Merge"," is just ",[457,10310,10312],{"className":10311},[460],[457,10313,10315],{"className":10314,"ariaHidden":465},[464],[457,10316,10318,10321],{"className":10317},[469],[457,10319],{"className":10320,"style":2481},[473],[457,10322,10324],{"className":10323},[2485,2486],[457,10325,10327],{"className":10326},[478,2490],[457,10328,2494],{"className":10329},[478]," with the\ncounting rule from ",[457,10332,10334],{"className":10333},[460],[457,10335,10337],{"className":10336,"ariaHidden":465},[464],[457,10338,10340,10343],{"className":10339},[469],[457,10341],{"className":10342,"style":2746},[473],[457,10344,10346],{"className":10345},[2485,2486],[457,10347,10349],{"className":10348},[478,2490],[457,10350,10352],{"className":10351},[478],"Count-Cross-Inv"," folded in: whenever it takes from\nthe right half because ",[457,10355,10357],{"className":10356},[460],[457,10358,10360,10389,10410],{"className":10359,"ariaHidden":465},[464],[457,10361,10363,10366,10369,10372,10380,10383,10386],{"className":10362},[469],[457,10364],{"className":10365,"style":510},[473],[457,10367,1506],{"className":10368},[478,479],[457,10370,1510],{"className":10371},[542],[457,10373,10375],{"className":10374},[478],[457,10376,10379],{"className":10377},[478,10378],"mathit","mid",[457,10381],{"className":10382,"style":669},[625],[457,10384,674],{"className":10385},[673],[457,10387],{"className":10388,"style":669},[625],[457,10390,10392,10395,10398,10401,10404,10407],{"className":10391},[469],[457,10393],{"className":10394,"style":510},[473],[457,10396,2803],{"className":10397,"style":2802},[478,479],[457,10399,1526],{"className":10400},[589],[457,10402],{"className":10403,"style":626},[625],[457,10405,7790],{"className":10406},[630],[457,10408],{"className":10409,"style":626},[625],[457,10411,10413,10416,10419,10422,10425],{"className":10412},[469],[457,10414],{"className":10415,"style":510},[473],[457,10417,1506],{"className":10418},[478,479],[457,10420,1510],{"className":10421},[542],[457,10423,2785],{"className":10424},[478,479],[457,10426,1526],{"className":10427},[589],", it adds the number of\nelements still waiting in the left half. The combine step is now plain linear,\nso",[457,10430,10432],{"className":10431},[595],[457,10433,10435],{"className":10434},[460],[457,10436,10438,10465,10501,10528],{"className":10437,"ariaHidden":465},[464],[457,10439,10441,10444,10447,10450,10453,10456,10459,10462],{"className":10440},[469],[457,10442],{"className":10443,"style":510},[473],[457,10445,612],{"className":10446,"style":611},[478,479],[457,10448,543],{"className":10449},[542],[457,10451,480],{"className":10452},[478,479],[457,10454,590],{"className":10455},[589],[457,10457],{"className":10458,"style":626},[625],[457,10460,631],{"className":10461},[630],[457,10463],{"className":10464,"style":626},[625],[457,10466,10468,10471,10474,10477,10480,10483,10486,10489,10492,10495,10498],{"className":10467},[469],[457,10469],{"className":10470,"style":510},[473],[457,10472,936],{"className":10473},[478],[457,10475],{"className":10476,"style":647},[625],[457,10478,612],{"className":10479,"style":611},[478,479],[457,10481,543],{"className":10482},[542],[457,10484,480],{"className":10485},[478,479],[457,10487,1591],{"className":10488},[478],[457,10490,590],{"className":10491},[589],[457,10493],{"className":10494,"style":669},[625],[457,10496,674],{"className":10497},[673],[457,10499],{"className":10500,"style":669},[625],[457,10502,10504,10507,10510,10513,10516,10519,10522,10525],{"className":10503},[469],[457,10505],{"className":10506,"style":510},[473],[457,10508,538],{"className":10509},[478],[457,10511,543],{"className":10512},[542],[457,10514,480],{"className":10515},[478,479],[457,10517,590],{"className":10518},[589],[457,10520],{"className":10521,"style":626},[625],[457,10523,631],{"className":10524},[630],[457,10526],{"className":10527,"style":626},[625],[457,10529,10531,10534,10537,10540,10543,10546,10552,10555,10558,10561],{"className":10530},[469],[457,10532],{"className":10533,"style":510},[473],[457,10535,538],{"className":10536},[478],[457,10538,543],{"className":10539},[542],[457,10541,480],{"className":10542},[478,479],[457,10544],{"className":10545,"style":647},[625],[457,10547,10549],{"className":10548},[5137],[457,10550,5143],{"className":10551,"style":5142},[478,5141],[457,10553],{"className":10554,"style":647},[625],[457,10556,480],{"className":10557},[478,479],[457,10559,590],{"className":10560},[589],[457,10562,727],{"className":10563},[478],[381,10565,10566,10567,6742,10591,10615,10616,10619],{},"This is the same recurrence as mergesort, and the same recursion tree explains\nit: ",[457,10568,10570],{"className":10569},[460],[457,10571,10573],{"className":10572,"ariaHidden":465},[464],[457,10574,10576,10579,10585,10588],{"className":10575},[469],[457,10577],{"className":10578,"style":6698},[473],[457,10580,10582],{"className":10581},[5137],[457,10583,5143],{"className":10584,"style":5142},[478,5141],[457,10586],{"className":10587,"style":647},[625],[457,10589,480],{"className":10590},[478,479],[457,10592,10594],{"className":10593},[460],[457,10595,10597],{"className":10596,"ariaHidden":465},[464],[457,10598,10600,10603,10606,10609,10612],{"className":10599},[469],[457,10601],{"className":10602,"style":510},[473],[457,10604,538],{"className":10605},[478],[457,10607,543],{"className":10608},[542],[457,10610,480],{"className":10611},[478,479],[457,10613,590],{"className":10614},[589]," work each. Counting how disordered a list is\ncosts no more, ",[434,10617,10618],{"href":17},"asymptotically",", than sorting it.",[814,10621,10623],{"id":10622},"karatsuba-multiplication-the-same-idea-elsewhere","Karatsuba multiplication: the same idea elsewhere",[381,10625,10626,10627,10642,10643,10667,10668,10683,10684,10734],{},"Sorting is not the only home for divide and conquer. Take a problem that feels\nunrelated: multiplying two large integers. Adding two ",[457,10628,10630],{"className":10629},[460],[457,10631,10633],{"className":10632,"ariaHidden":465},[464],[457,10634,10636,10639],{"className":10635},[469],[457,10637],{"className":10638,"style":474},[473],[457,10640,480],{"className":10641},[478,479],"-bit numbers is easy.\nThe grade-school ripple-carry method is ",[457,10644,10646],{"className":10645},[460],[457,10647,10649],{"className":10648,"ariaHidden":465},[464],[457,10650,10652,10655,10658,10661,10664],{"className":10651},[469],[457,10653],{"className":10654,"style":510},[473],[457,10656,538],{"className":10657},[478],[457,10659,543],{"className":10660},[542],[457,10662,480],{"className":10663},[478,479],[457,10665,590],{"className":10666},[589],", and you cannot beat linear\nsince you must at least read the input. Multiplication is the interesting one.\nThe grade-school algorithm forms ",[457,10669,10671],{"className":10670},[460],[457,10672,10674],{"className":10673,"ariaHidden":465},[464],[457,10675,10677,10680],{"className":10676},[469],[457,10678],{"className":10679,"style":474},[473],[457,10681,480],{"className":10682},[478,479]," shifted partial products and adds them,\ncosting ",[457,10685,10687],{"className":10686},[460],[457,10688,10690],{"className":10689,"ariaHidden":465},[464],[457,10691,10693,10696,10699,10702,10731],{"className":10692},[469],[457,10694],{"className":10695,"style":1430},[473],[457,10697,538],{"className":10698},[478],[457,10700,543],{"className":10701},[542],[457,10703,10705,10708],{"className":10704},[478],[457,10706,480],{"className":10707},[478,479],[457,10709,10711],{"className":10710},[553],[457,10712,10714],{"className":10713},[557],[457,10715,10717],{"className":10716},[561],[457,10718,10720],{"className":10719,"style":1455},[565],[457,10721,10722,10725],{"style":569},[457,10723],{"className":10724,"style":574},[573],[457,10726,10728],{"className":10727},[578,579,580,581],[457,10729,936],{"className":10730},[478,581],[457,10732,590],{"className":10733},[589],". Can divide and conquer do better?",[381,10736,10737,10738,10753,10754,10791,10792,10808,10809,10825,10826,10841,10842,10825,10857,10872,10873,10889,10890,2526,10905,10921],{},"Write each ",[457,10739,10741],{"className":10740},[460],[457,10742,10744],{"className":10743,"ariaHidden":465},[464],[457,10745,10747,10750],{"className":10746},[469],[457,10748],{"className":10749,"style":474},[473],[457,10751,480],{"className":10752},[478,479],"-bit integer as a high half and a low half. With ",[457,10755,10757],{"className":10756},[460],[457,10758,10760,10778],{"className":10759,"ariaHidden":465},[464],[457,10761,10763,10766,10769,10772,10775],{"className":10762},[469],[457,10764],{"className":10765,"style":474},[473],[457,10767,480],{"className":10768},[478,479],[457,10770],{"className":10771,"style":626},[625],[457,10773,631],{"className":10774},[630],[457,10776],{"className":10777,"style":626},[625],[457,10779,10781,10784,10787],{"className":10780},[469],[457,10782],{"className":10783,"style":3370},[473],[457,10785,936],{"className":10786},[478],[457,10788,10790],{"className":10789},[478,479],"t",", split\n",[457,10793,10795],{"className":10794},[460],[457,10796,10798],{"className":10797,"ariaHidden":465},[464],[457,10799,10801,10804],{"className":10800},[469],[457,10802],{"className":10803,"style":474},[473],[457,10805,10807],{"className":10806},[478,479],"x"," into its least-significant ",[457,10810,10812],{"className":10811},[460],[457,10813,10815],{"className":10814,"ariaHidden":465},[464],[457,10816,10818,10822],{"className":10817},[469],[457,10819],{"className":10820,"style":10821},[473],"height:0.6151em;",[457,10823,10790],{"className":10824},[478,479]," bits ",[457,10827,10829],{"className":10828},[460],[457,10830,10832],{"className":10831,"ariaHidden":465},[464],[457,10833,10835,10838],{"className":10834},[469],[457,10836],{"className":10837,"style":474},[473],[457,10839,434],{"className":10840},[478,479]," and its top 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Unrolling shows this is no improvement at 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which\ndominates everything: ",[457,12768,12770],{"className":12769},[460],[457,12771,12773,12800],{"className":12772,"ariaHidden":465},[464],[457,12774,12776,12779,12782,12785,12788,12791,12794,12797],{"className":12775},[469],[457,12777],{"className":12778,"style":510},[473],[457,12780,612],{"className":12781,"style":611},[478,479],[457,12783,543],{"className":12784},[542],[457,12786,480],{"className":12787},[478,479],[457,12789,590],{"className":12790},[589],[457,12792],{"className":12793,"style":626},[625],[457,12795,631],{"className":12796},[630],[457,12798],{"className":12799,"style":626},[625],[457,12801,12803,12806,12809,12812,12841],{"className":12802},[469],[457,12804],{"className":12805,"style":1430},[473],[457,12807,538],{"className":12808},[478],[457,12810,543],{"className":12811},[542],[457,12813,12815,12818],{"className":12814},[478],[457,12816,480],{"className":12817},[478,479],[457,12819,12821],{"className":12820},[553],[457,12822,12824],{"className":12823},[557],[457,12825,12827],{"className":12826},[561],[457,12828,12830],{"className":12829,"style":1455},[565],[457,12831,12832,12835],{"style":569},[457,12833],{"className":12834,"style":574},[573],[457,12836,12838],{"className":12837},[578,579,580,581],[457,12839,936],{"className":12840},[478,581],[457,12842,590],{"className":12843},[589],". Splitting bought us nothing, because\n",[457,12846,12848],{"className":12847},[460],[457,12849,12851,12869],{"className":12850,"ariaHidden":465},[464],[457,12852,12854,12857,12860,12863,12866],{"className":12853},[469],[457,12855],{"className":12856,"style":474},[473],[457,12858,434],{"className":12859},[478,479],[457,12861],{"className":12862,"style":626},[625],[457,12864,631],{"className":12865},[630],[457,12867],{"className":12868,"style":626},[625],[457,12870,12872,12875],{"className":12871},[469],[457,12873],{"className":12874,"style":3370},[473],[457,12876,2365],{"className":12877},[478]," subproblems of half size is exactly the quadratic regime.",[381,12880,12881,12884,12885,3691,12924,12927],{},[390,12882,12883],{},"Karatsuba's trick"," is to compute the middle coefficient ",[457,12886,12888],{"className":12887},[460],[457,12889,12891,12912],{"className":12890,"ariaHidden":465},[464],[457,12892,12894,12897,12900,12903,12906,12909],{"className":12893},[469],[457,12895],{"className":12896,"style":3759},[473],[457,12898,434],{"className":12899},[478,479],[457,12901,10920],{"className":12902},[478,479],[457,12904],{"className":12905,"style":669},[625],[457,12907,674],{"className":12908},[673],[457,12910],{"className":12911,"style":669},[625],[457,12913,12915,12918,12921],{"className":12914},[469],[457,12916],{"className":12917,"style":759},[473],[457,12919,521],{"className":12920},[478,479],[457,12922,585],{"className":12923},[478,479],[385,12925,12926],{},"without"," a\nthird and fourth multiplication. Notice the algebraic identity",[457,12929,12931],{"className":12930},[595],[457,12932,12934],{"className":12933},[460],[457,12935,12937,12958,12985,13006,13027,13048,13069,13115,13136,13157,13184,13205,13226],{"className":12936,"ariaHidden":465},[464],[457,12938,12940,12943,12946,12949,12952,12955],{"className":12939},[469],[457,12941],{"className":12942,"style":510},[473],[457,12944,543],{"className":12945},[542],[457,12947,434],{"className":12948},[478,479],[457,12950],{"className":12951,"style":669},[625],[457,12953,3573],{"className":12954},[673],[457,12956],{"className":12957,"style":669},[625],[457,12959,12961,12964,12967,12970,12973,12976,12979,12982],{"className":12960},[469],[457,12962],{"className":12963,"style":510},[473],[457,12965,521],{"className":12966},[478,479],[457,12968,590],{"className":12969},[589],[457,12971,543],{"className":12972},[542],[457,12974,10920],{"className":12975},[478,479],[457,12977],{"className":12978,"style":669},[625],[457,12980,3573],{"className":12981},[673],[457,12983],{"className":12984,"style":669},[625],[457,12986,12988,12991,12994,12997,13000,13003],{"className":12987},[469],[457,12989],{"className":12990,"style":510},[473],[457,12992,585],{"className":12993},[478,479],[457,12995,590],{"className":12996},[589],[457,12998],{"className":12999,"style":626},[625],[457,13001,631],{"className":13002},[630],[457,13004],{"className":13005,"style":626},[625],[457,13007,13009,13012,13015,13018,13021,13024],{"className":13008},[469],[457,13010],{"className":13011,"style":3759},[473],[457,13013,434],{"className":13014},[478,479],[457,13016,10920],{"className":13017},[478,479],[457,13019],{"className":13020,"style":669},[625],[457,13022,3573],{"className":13023},[673],[457,13025],{"className":13026,"style":669},[625],[457,13028,13030,13033,13036,13039,13042,13045],{"className":13029},[469],[457,13031],{"className":13032,"style":3563},[473],[457,13034,434],{"className":13035},[478,479],[457,13037,585],{"className":13038},[478,479],[457,13040],{"className":13041,"style":669},[625],[457,13043,3573],{"className":13044},[673],[457,13046],{"className":13047,"style":669},[625],[457,13049,13051,13054,13057,13060,13063,13066],{"className":13050},[469],[457,13052],{"className":13053,"style":3759},[473],[457,13055,521],{"className":13056},[478,479],[457,13058,10920],{"className":13059},[478,479],[457,13061],{"className":13062,"style":669},[625],[457,13064,674],{"className":13065},[673],[457,13067],{"className":13068,"style":669},[625],[457,13070,13072,13075,13078,13081,13084,13087,13090,13097,13100,13103,13106,13109,13112],{"className":13071},[469],[457,13073],{"className":13074,"style":6698},[473],[457,13076,521],{"className":13077},[478,479],[457,13079,585],{"className":13080},[478,479],[457,13082,903],{"className":13083},[902],[457,13085],{"className":13086,"style":5061},[625],[457,13088],{"className":13089,"style":647},[625],[457,13091,13093],{"className":13092},[478,2490],[457,13094,13096],{"className":13095},[478],"so",[457,13098],{"className":13099,"style":5061},[625],[457,13101,434],{"className":13102},[478,479],[457,13104,10920],{"className":13105},[478,479],[457,13107],{"className":13108,"style":669},[625],[457,13110,674],{"className":13111},[673],[457,13113],{"className":13114,"style":669},[625],[457,13116,13118,13121,13124,13127,13130,13133],{"className":13117},[469],[457,13119],{"className":13120,"style":759},[473],[457,13122,521],{"className":13123},[478,479],[457,13125,585],{"className":13126},[478,479],[457,13128],{"className":13129,"style":626},[625],[457,13131,631],{"className":13132},[630],[457,13134],{"className":13135,"style":626},[625],[457,13137,13139,13142,13145,13148,13151,13154],{"className":13138},[469],[457,13140],{"className":13141,"style":510},[473],[457,13143,543],{"className":13144},[542],[457,13146,434],{"className":13147},[478,479],[457,13149],{"className":13150,"style":669},[625],[457,13152,3573],{"className":13153},[673],[457,13155],{"className":13156,"style":669},[625],[457,13158,13160,13163,13166,13169,13172,13175,13178,13181],{"className":13159},[469],[457,13161],{"className":13162,"style":510},[473],[457,13164,521],{"className":13165},[478,479],[457,13167,590],{"className":13168},[589],[457,13170,543],{"className":13171},[542],[457,13173,10920],{"className":13174},[478,479],[457,13176],{"className":13177,"style":669},[625],[457,13179,3573],{"className":13180},[673],[457,13182],{"className":13183,"style":669},[625],[457,13185,13187,13190,13193,13196,13199,13202],{"className":13186},[469],[457,13188],{"className":13189,"style":510},[473],[457,13191,585],{"className":13192},[478,479],[457,13194,590],{"className":13195},[589],[457,13197],{"className":13198,"style":669},[625],[457,13200,674],{"className":13201},[673],[457,13203],{"className":13204,"style":669},[625],[457,13206,13208,13211,13214,13217,13220,13223],{"className":13207},[469],[457,13209],{"className":13210,"style":3563},[473],[457,13212,434],{"className":13213},[478,479],[457,13215,585],{"className":13216},[478,479],[457,13218],{"className":13219,"style":669},[625],[457,13221,674],{"className":13222},[673],[457,13224],{"className":13225,"style":669},[625],[457,13227,13229,13232,13235,13238],{"className":13228},[469],[457,13230],{"className":13231,"style":759},[473],[457,13233,521],{"className":13234},[478,479],[457,13236,10920],{"className":13237},[478,479],[457,13239,727],{"className":13240},[478],[381,13242,13243,13244,2526,13262,13280,13281,13284,13285,13351],{},"We were going to compute ",[457,13245,13247],{"className":13246},[460],[457,13248,13250],{"className":13249,"ariaHidden":465},[464],[457,13251,13253,13256,13259],{"className":13252},[469],[457,13254],{"className":13255,"style":474},[473],[457,13257,434],{"className":13258},[478,479],[457,13260,585],{"className":13261},[478,479],[457,13263,13265],{"className":13264},[460],[457,13266,13268],{"className":13267,"ariaHidden":465},[464],[457,13269,13271,13274,13277],{"className":13270},[469],[457,13272],{"className":13273,"style":759},[473],[457,13275,521],{"className":13276},[478,479],[457,13278,10920],{"className":13279},[478,479]," anyway. So once we have those two\nproducts, the middle term needs only ",[390,13282,13283],{},"one"," more multiplication,\n",[457,13286,13288],{"className":13287},[460],[457,13289,13291,13312,13339],{"className":13290,"ariaHidden":465},[464],[457,13292,13294,13297,13300,13303,13306,13309],{"className":13293},[469],[457,13295],{"className":13296,"style":510},[473],[457,13298,543],{"className":13299},[542],[457,13301,434],{"className":13302},[478,479],[457,13304],{"className":13305,"style":669},[625],[457,13307,3573],{"className":13308},[673],[457,13310],{"className":13311,"style":669},[625],[457,13313,13315,13318,13321,13324,13327,13330,13333,13336],{"className":13314},[469],[457,13316],{"className":13317,"style":510},[473],[457,13319,521],{"className":13320},[478,479],[457,13322,590],{"className":13323},[589],[457,13325,543],{"className":13326},[542],[457,13328,10920],{"className":13329},[478,479],[457,13331],{"className":13332,"style":669},[625],[457,13334,3573],{"className":13335},[673],[457,13337],{"className":13338,"style":669},[625],[457,13340,13342,13345,13348],{"className":13341},[469],[457,13343],{"className":13344,"style":510},[473],[457,13346,585],{"className":13347},[478,479],[457,13349,590],{"className":13350},[589],", plus a few additions. Three half-size products suffice:",[2418,13353,13355],{"className":2420,"code":13354,"language":2422,"meta":376,"style":376},"caption: $\\textsc{Karatsuba}(x, y)$ — multiply two $n$-bit integers\nnumber: 5\nif $n = 1$ then\n  return $x \\cdot y$ \u002F\u002F base case\n$t \\gets \\floor{n \u002F 2}$\nsplit $x = a + 2^{t} b$ and $y = c + 2^{t} d$ \u002F\u002F low and high halves\n$\\mathit{ac} \\gets \\textsc{Karatsuba}(a, c)$ \u002F\u002F recursive call 1\n$\\mathit{bd} \\gets \\textsc{Karatsuba}(b, d)$ \u002F\u002F recursive call 2\n$m \\gets \\textsc{Karatsuba}(a - b, d - c)$ \u002F\u002F recursive call 3 — the trick\n$\\mathit{mid} \\gets m + \\mathit{ac} + \\mathit{bd}$ \u002F\u002F recovers $ad + bc$\nreturn $\\mathit{ac} + 2^{t}\\,\\mathit{mid} + 2^{2t}\\,\\mathit{bd}$ \u002F\u002F shift and add\n",[2424,13356,13357,13362,13367,13372,13377,13382,13387,13392,13397,13402,13407],{"__ignoreMap":376},[457,13358,13359],{"class":2428,"line":6},[457,13360,13361],{},"caption: $\\textsc{Karatsuba}(x, y)$ — multiply two $n$-bit integers\n",[457,13363,13364],{"class":2428,"line":18},[457,13365,13366],{},"number: 5\n",[457,13368,13369],{"class":2428,"line":24},[457,13370,13371],{},"if $n = 1$ then\n",[457,13373,13374],{"class":2428,"line":73},[457,13375,13376],{},"  return $x \\cdot y$ \u002F\u002F base case\n",[457,13378,13379],{"class":2428,"line":102},[457,13380,13381],{},"$t \\gets \\floor{n \u002F 2}$\n",[457,13383,13384],{"class":2428,"line":108},[457,13385,13386],{},"split $x = a + 2^{t} b$ and $y = c + 2^{t} d$ \u002F\u002F low and high halves\n",[457,13388,13389],{"class":2428,"line":116},[457,13390,13391],{},"$\\mathit{ac} \\gets \\textsc{Karatsuba}(a, c)$ \u002F\u002F recursive call 1\n",[457,13393,13394],{"class":2428,"line":196},[457,13395,13396],{},"$\\mathit{bd} \\gets \\textsc{Karatsuba}(b, d)$ \u002F\u002F recursive call 2\n",[457,13398,13399],{"class":2428,"line":202},[457,13400,13401],{},"$m \\gets \\textsc{Karatsuba}(a - b, d - c)$ \u002F\u002F recursive call 3 — the trick\n",[457,13403,13404],{"class":2428,"line":283},[457,13405,13406],{},"$\\mathit{mid} \\gets m + \\mathit{ac} + \\mathit{bd}$ \u002F\u002F recovers $ad + bc$\n",[457,13408,13409],{"class":2428,"line":333},[457,13410,13411],{},"return $\\mathit{ac} + 2^{t}\\,\\mathit{mid} + 2^{2t}\\,\\mathit{bd}$ \u002F\u002F shift and add\n",[381,13413,13414,13415,13448,13449,13473],{},"Now ",[457,13416,13418],{"className":13417},[460],[457,13419,13421,13439],{"className":13420,"ariaHidden":465},[464],[457,13422,13424,13427,13430,13433,13436],{"className":13423},[469],[457,13425],{"className":13426,"style":474},[473],[457,13428,434],{"className":13429},[478,479],[457,13431],{"className":13432,"style":626},[625],[457,13434,631],{"className":13435},[630],[457,13437],{"className":13438,"style":626},[625],[457,13440,13442,13445],{"className":13441},[469],[457,13443],{"className":13444,"style":3370},[473],[457,13446,2393],{"className":13447},[478]," recursive calls, each on half-size inputs, with ",[457,13450,13452],{"className":13451},[460],[457,13453,13455],{"className":13454,"ariaHidden":465},[464],[457,13456,13458,13461,13464,13467,13470],{"className":13457},[469],[457,13459],{"className":13460,"style":510},[473],[457,13462,538],{"className":13463},[478],[457,13465,543],{"className":13466},[542],[457,13468,480],{"className":13469},[478,479],[457,13471,590],{"className":13472},[589]," work to\nsplit, shift, and add:",[457,13475,13477],{"className":13476},[595],[457,13478,13480],{"className":13479},[460],[457,13481,13483,13510,13546],{"className":13482,"ariaHidden":465},[464],[457,13484,13486,13489,13492,13495,13498,13501,13504,13507],{"className":13485},[469],[457,13487],{"className":13488,"style":510},[473],[457,13490,612],{"className":13491,"style":611},[478,479],[457,13493,543],{"className":13494},[542],[457,13496,480],{"className":13497},[478,479],[457,13499,590],{"className":13500},[589],[457,13502],{"className":13503,"style":626},[625],[457,13505,631],{"className":13506},[630],[457,13508],{"className":13509,"style":626},[625],[457,13511,13513,13516,13519,13522,13525,13528,13531,13534,13537,13540,13543],{"className":13512},[469],[457,13514],{"className":13515,"style":510},[473],[457,13517,2393],{"className":13518},[478],[457,13520],{"className":13521,"style":647},[625],[457,13523,612],{"className":13524,"style":611},[478,479],[457,13526,543],{"className":13527},[542],[457,13529,480],{"className":13530},[478,479],[457,13532,1591],{"className":13533},[478],[457,13535,590],{"className":13536},[589],[457,13538],{"className":13539,"style":669},[625],[457,13541,674],{"className":13542},[673],[457,13544],{"className":13545,"style":669},[625],[457,13547,13549,13552,13555,13558,13561,13564],{"className":13548},[469],[457,13550],{"className":13551,"style":510},[473],[457,13553,538],{"className":13554},[478],[457,13556,543],{"className":13557},[542],[457,13559,480],{"className":13560},[478,479],[457,13562,590],{"className":13563},[589],[457,13565,727],{"className":13566},[478],[381,13568,13569,13570,5547,13585,13629,13630,13680,13681,13738,13739,13742],{},"The recursion tree makes the cost legible. Level ",[457,13571,13573],{"className":13572},[460],[457,13574,13576],{"className":13575,"ariaHidden":465},[464],[457,13577,13579,13582],{"className":13578},[469],[457,13580],{"className":13581,"style":2781},[473],[457,13583,2785],{"className":13584},[478,479],[457,13586,13588],{"className":13587},[460],[457,13589,13591],{"className":13590,"ariaHidden":465},[464],[457,13592,13594,13597],{"className":13593},[469],[457,13595],{"className":13596,"style":5560},[473],[457,13598,13600,13603],{"className":13599},[478],[457,13601,2393],{"className":13602},[478],[457,13604,13606],{"className":13605},[553],[457,13607,13609],{"className":13608},[557],[457,13610,13612],{"className":13611},[561],[457,13613,13615],{"className":13614,"style":5560},[565],[457,13616,13617,13620],{"style":569},[457,13618],{"className":13619,"style":574},[573],[457,13621,13623],{"className":13622},[578,579,580,581],[457,13624,13626],{"className":13625},[478,581],[457,13627,2785],{"className":13628},[478,479,581]," nodes, each\ndoing ",[457,13631,13633],{"className":13632},[460],[457,13634,13636],{"className":13635,"ariaHidden":465},[464],[457,13637,13639,13642,13645,13648],{"className":13638},[469],[457,13640],{"className":13641,"style":5603},[473],[457,13643,11761],{"className":13644},[478,479],[457,13646,517],{"className":13647},[478],[457,13649,13651,13654],{"className":13650},[478],[457,13652,936],{"className":13653},[478],[457,13655,13657],{"className":13656},[553],[457,13658,13660],{"className":13659},[557],[457,13661,13663],{"className":13662},[561],[457,13664,13666],{"className":13665,"style":5560},[565],[457,13667,13668,13671],{"style":569},[457,13669],{"className":13670,"style":574},[573],[457,13672,13674],{"className":13673},[578,579,580,581],[457,13675,13677],{"className":13676},[478,581],[457,13678,2785],{"className":13679},[478,479,581]," work, for a row total of ",[457,13682,13684],{"className":13683},[460],[457,13685,13687],{"className":13686,"ariaHidden":465},[464],[457,13688,13690,13693,13696,13700,13732,13735],{"className":13689},[469],[457,13691],{"className":13692,"style":5603},[473],[457,13694,543],{"className":13695},[542],[457,13697,13699],{"className":13698},[478],"3\u002F2",[457,13701,13703,13706],{"className":13702},[589],[457,13704,590],{"className":13705},[589],[457,13707,13709],{"className":13708},[553],[457,13710,13712],{"className":13711},[557],[457,13713,13715],{"className":13714},[561],[457,13716,13718],{"className":13717,"style":5560},[565],[457,13719,13720,13723],{"style":569},[457,13721],{"className":13722,"style":574},[573],[457,13724,13726],{"className":13725},[578,579,580,581],[457,13727,13729],{"className":13728},[478,581],[457,13730,2785],{"className":13731},[478,479,581],[457,13733],{"className":13734,"style":647},[625],[457,13736,11761],{"className":13737},[478,479],". The rows ",[385,13740,13741],{},"grow","\ngeometrically downward, so the leaves dominate:",[1634,13744,13746,13911],{"className":13745},[1637,1638],[1640,13747,13751],{"xmlns":1642,"width":13748,"height":13749,"viewBox":13750},"289.051","113.050","-75 -75 216.788 84.787",[1647,13752,13753,13765,13788,13791,13796,13799,13804,13807,13812,13815,13836,13839,13844,13847,13852,13855,13860,13863,13884,13887,13892,13895,13900,13903,13908],{"stroke":1649,"style":1650},[1647,13754,13755,13758],{"fill":1837},[1652,13756],{"d":13757},"M42.86-65.668a6.402 6.402 0 1 0-12.804 0 6.402 6.402 0 0 0 12.803 0Zm-6.403 0",[1647,13759,13761],{"transform":13760},"translate(-2.472 1.507)",[1652,13762],{"d":13763,"fill":1649,"stroke":1649,"className":13764,"style":1668},"M37.240-65.822Q37.240-65.870 37.247-65.894L37.759-67.958Q37.793-68.085 37.793-68.201Q37.793-68.341 37.740-68.437Q37.687-68.532 37.564-68.532Q37.342-68.532 37.241-68.305Q37.141-68.078 37.031-67.657Q37.021-67.592 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88.503-.562",[1647,13896,13897],{"fill":1837},[1652,13898],{"fill":1837,"d":13899},"M114.133 2.618a3.699 3.699 0 1 0-7.397 0 3.699 3.699 0 0 0 7.397 0Zm-3.698 0",[1652,13901],{"fill":1654,"d":13902},"M110.435-24.095V-1.28",[1647,13904,13905],{"fill":1837},[1652,13906],{"fill":1837,"d":13907},"M138.318 2.618a3.699 3.699 0 1 0-7.397 0 3.699 3.699 0 0 0 7.397 0Zm-3.699 0",[1652,13909],{"fill":1654,"d":13910},"m114.729-25.462 17.637 24.9",[2323,13912,13914,13915,13930,13931,13946,13947,14046],{"className":13913},[2326],"Each Karatsuba node spawns ",[457,13916,13918],{"className":13917},[460],[457,13919,13921],{"className":13920,"ariaHidden":465},[464],[457,13922,13924,13927],{"className":13923},[469],[457,13925],{"className":13926,"style":3370},[473],[457,13928,2393],{"className":13929},[478]," half-size children, so the rows grow by ",[457,13932,13934],{"className":13933},[460],[457,13935,13937],{"className":13936,"ariaHidden":465},[464],[457,13938,13940,13943],{"className":13939},[469],[457,13941],{"className":13942,"style":510},[473],[457,13944,13699],{"className":13945},[478]," downward and the 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at face value, the four output blocks need ",[390,16979,16980],{},"eight"," block multiplications,\ngiving ",[457,16983,16985],{"className":16984},[460],[457,16986,16988,17015,17051],{"className":16987,"ariaHidden":465},[464],[457,16989,16991,16994,16997,17000,17003,17006,17009,17012],{"className":16990},[469],[457,16992],{"className":16993,"style":510},[473],[457,16995,612],{"className":16996,"style":611},[478,479],[457,16998,543],{"className":16999},[542],[457,17001,480],{"className":17002},[478,479],[457,17004,590],{"className":17005},[589],[457,17007],{"className":17008,"style":626},[625],[457,17010,631],{"className":17011},[630],[457,17013],{"className":17014,"style":626},[625],[457,17016,17018,17021,17024,17027,17030,17033,17036,17039,17042,17045,17048],{"className":17017},[469],[457,17019],{"className":17020,"style":510},[473],[457,17022,3417],{"className":17023},[478],[457,17025],{"className":17026,"style":647},[625],[457,17028,612],{"className":17029,"style":611},[478,479],[457,17031,543],{"className":17032},[542],[457,17034,480],{"className":17035},[478,479],[457,17037,1591],{"className":17038},[478],[457,17040,590],{"className":17041},[589],[457,17043],{"className":17044,"style":669},[625],[457,17046,674],{"className":17047},[673],[457,17049],{"className":17050,"style":669},[625],[457,17052,17054,17057,17060,17063,17092],{"className":17053},[469],[457,17055],{"className":17056,"style":1430},[473],[457,17058,538],{"className":17059},[478],[457,17061,543],{"className":17062},[542],[457,17064,17066,17069],{"className":17065},[478],[457,17067,480],{"className":17068},[478,479],[457,17070,17072],{"className":17071},[553],[457,17073,17075],{"className":17074},[557],[457,17076,17078],{"className":17077},[561],[457,17079,17081],{"className":17080,"style":1455},[565],[457,17082,17083,17086],{"style":569},[457,17084],{"className":17085,"style":574},[573],[457,17087,17089],{"className":17088},[578,579,580,581],[457,17090,936],{"className":17091},[478,581],[457,17093,590],{"className":17094},[589],". The master theorem (next section) returns\n",[457,17097,17099],{"className":17098},[460],[457,17100,17102,17204],{"className":17101,"ariaHidden":465},[464],[457,17103,17105,17108,17111,17114,17192,17195,17198,17201],{"className":17104},[469],[457,17106],{"className":17107,"style":12609},[473],[457,17109,538],{"className":17110},[478],[457,17112,543],{"className":17113},[542],[457,17115,17117,17120],{"className":17116},[478],[457,17118,480],{"className":17119},[478,479],[457,17121,17123],{"className":17122},[553],[457,17124,17126],{"className":17125},[557],[457,17127,17129],{"className":17128},[561],[457,17130,17132],{"className":17131,"style":6581},[565],[457,17133,17134,17137],{"style":569},[457,17135],{"className":17136,"style":574},[573],[457,17138,17140],{"className":17139},[578,579,580,581],[457,17141,17143,17186,17189],{"className":17142},[478,581],[457,17144,17146,17152],{"className":17145},[5137,581],[457,17147,17149],{"className":17148},[5137,581],[457,17150,5143],{"className":17151,"style":5142},[478,5141,581],[457,17153,17155],{"className":17154},[553],[457,17156,17158,17178],{"className":17157},[557,865],[457,17159,17161,17175],{"className":17160},[561],[457,17162,17164],{"className":17163,"style":12660},[565],[457,17165,17166,17169],{"style":6635},[457,17167],{"className":17168,"style":6639},[573],[457,17170,17172],{"className":17171},[578,6643,6644,581],[457,17173,936],{"className":17174},[478,581],[457,17176,889],{"className":17177},[888],[457,17179,17181],{"className":17180},[561],[457,17182,17184],{"className":17183,"style":6657},[565],[457,17185],{},[457,17187],{"className":17188,"style":6663},[625,581],[457,17190,3417],{"className":17191},[478,581],[457,17193,590],{"className":17194},[589],[457,17196],{"className":17197,"style":626},[625],[457,17199,631],{"className":17200},[630],[457,17202],{"className":17203,"style":626},[625],[457,17205,17207,17210,17213,17216,17245],{"className":17206},[469],[457,17208],{"className":17209,"style":1430},[473],[457,17211,538],{"className":17212},[478],[457,17214,543],{"className":17215},[542],[457,17217,17219,17222],{"className":17218},[478],[457,17220,480],{"className":17221},[478,479],[457,17223,17225],{"className":17224},[553],[457,17226,17228],{"className":17227},[557],[457,17229,17231],{"className":17230},[561],[457,17232,17234],{"className":17233,"style":1455},[565],[457,17235,17236,17239],{"style":569},[457,17237],{"className":17238,"style":574},[573],[457,17240,17242],{"className":17241},[578,579,580,581],[457,17243,2393],{"className":17244},[478,581],[457,17246,590],{"className":17247},[589],": more subproblems exactly cancel their smaller\nsize, so blocking alone buys nothing.",[381,17250,17251,17252,17255,17256,17263,17264,17288],{},"Strassen's 1969 insight is that ",[390,17253,17254],{},"seven"," products suffice.",[431,17257,17258],{},[434,17259,2365],{"href":17260,"ariaDescribedBy":17261,"dataFootnoteRef":376,"id":17262},"#user-content-fn-strassen",[438],"user-content-fnref-strassen"," Form seven\nrecursive ",[457,17265,17267],{"className":17266},[460],[457,17268,17270],{"className":17269,"ariaHidden":465},[464],[457,17271,17273,17276,17279,17282,17285],{"className":17272},[469],[457,17274],{"className":17275,"style":510},[473],[457,17277,543],{"className":17278},[542],[457,17280,480],{"className":17281},[478,479],[457,17283,1591],{"className":17284},[478],[457,17286,590],{"className":17287},[589],"-size 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recover the output blocks by ",[390,19130,19131],{},"addition only",[457,19133,19135],{"className":19134},[595],[457,19136,19138],{"className":19137},[460],[457,19139,19141],{"className":19140,"ariaHidden":465},[464],[457,19142,19144,19148],{"className":19143},[469],[457,19145],{"className":19146,"style":19147},[473],"height:2.7em;vertical-align:-1.1em;",[457,19149,19151],{"className":19150},[478],[457,19152,19154,19282,19627,19630,19755],{"className":19153},[11785],[457,19155,19157],{"className":19156},[11789],[457,19158,19160,19273],{"className":19159},[557,865],[457,19161,19163,19270],{"className":19162},[561],[457,19164,19167,19219],{"className":19165,"style":19166},[565],"height:1.6em;",[457,19168,19169,19172],{"style":17378},[457,19170],{"className":19171,"style":8086},[573],[457,19173,19175],{"className":19174},[478],[457,19176,19178,19181],{"className":19177},[478],[457,19179,8276],{"className":19180,"style":8275},[478,479],[457,19182,19184],{"className":19183},[553],[457,19185,19187,19211],{"className":19186},[557,865],[457,19188,19190,19208],{"className":19189},[561],[457,19191,19193],{"className":19192,"style":872},[565],[457,19194,19196,19199],{"style":19195},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[457,19197],{"className":19198,"style":574},[573],[457,19200,19202],{"className":19201},[578,579,580,581],[457,19203,19205],{"className":19204},[478,581],[457,19206,3374],{"className":19207},[478,581],[457,19209,889],{"className":19210},[888],[457,19212,19214],{"className":19213},[561],[457,19215,19217],{"className":19216,"style":896},[565],[457,19218],{},[457,19220,19221,19224],{"style":17427},[457,19222],{"className":19223,"style":8086},[573],[457,19225,19227],{"className":19226},[478],[457,19228,19230,19233],{"className":19229},[478],[457,19231,8276],{"className":19232,"style":8275},[478,479],[457,19234,19236],{"className":19235},[553],[457,19237,19239,19262],{"className":19238},[557,865],[457,19240,19242,19259],{"className":19241},[561],[457,19243,19245],{"className":19244,"style":872},[565],[457,19246,19247,19250],{"style":19195},[457,19248],{"className":19249,"style":574},[573],[457,19251,19253],{"className":19252},[578,579,580,581],[457,19254,19256],{"className":19255},[478,581],[457,19257,15636],{"className":19258},[478,581],[457,19260,889],{"className":19261},[888],[457,19263,19265],{"className":19264},[561],[457,19266,19268],{"className":19267,"style":896},[565],[457,19269],{},[457,19271,889],{"className":19272},[888],[457,19274,19276],{"className":19275},[561],[457,19277,19280],{"className":19278,"style":19279},[565],"height:1.1em;",[457,19281],{},[457,19283,19285],{"className":19284},[11845],[457,19286,19288,19619],{"className":19287},[557,865],[457,19289,19291,19616],{"className":19290},[561],[457,19292,19294,19504],{"className":19293,"style":19166},[565],[457,19295,19296,19299],{"style":17378},[457,19297],{"className":19298,"style":8086},[573],[457,19300,19302,19305,19308,19311,19314,19354,19357,19360,19363,19403,19406,19409,19412,19452,19455,19458,19461,19501],{"className":19301},[478],[457,19303],{"className":19304},[478],[457,19306],{"className":19307,"style":626},[625],[457,19309,631],{"className":19310},[630],[457,19312],{"className":19313,"style":626},[625],[457,19315,19317,19320],{"className":19316},[478],[457,19318,17340],{"className":19319,"style":17339},[478,479],[457,19321,19323],{"className":19322},[553],[457,19324,19326,19346],{"className":19325},[557,865],[457,19327,19329,19343],{"className":19328},[561],[457,19330,19332],{"className":19331,"style":872},[565],[457,19333,19334,19337],{"style":17355},[457,19335],{"className":19336,"style":574},[573],[457,19338,19340],{"className":19339},[578,579,580,581],[457,19341,440],{"className":19342},[478,581],[457,19344,889],{"className":19345},[888],[457,19347,19349],{"className":19348},[561],[457,19350,19352],{"className":19351,"style":896},[565],[457,19353],{},[457,19355],{"className":19356,"style":669},[625],[457,19358,674],{"className":19359},[673],[457,19361],{"className":19362,"style":669},[625],[457,19364,19366,19369],{"className":19365},[478],[457,19367,17340],{"className":19368,"style":17339},[478,479],[457,19370,19372],{"className":19371},[553],[457,19373,19375,19395],{"className":19374},[557,865],[457,19376,19378,19392],{"className":19377},[561],[457,19379,19381],{"className":19380,"style":872},[565],[457,19382,19383,19386],{"style":17355},[457,19384],{"className":19385,"style":574},[573],[457,19387,19389],{"className":19388},[578,579,580,581],[457,19390,2365],{"className":19391},[478,581],[457,19393,889],{"className":19394},[888],[457,19396,19398],{"className":19397},[561],[457,19399,19401],{"className":19400,"style":896},[565],[457,19402],{},[457,19404],{"className":19405,"style":669},[625],[457,19407,3573],{"className":19408},[673],[457,19410],{"className":19411,"style":669},[625],[457,19413,19415,19418],{"className":19414},[478],[457,19416,17340],{"className":19417,"style":17339},[478,479],[457,19419,19421],{"className":19420},[553],[457,19422,19424,19444],{"className":19423},[557,865],[457,19425,19427,19441],{"className":19426},[561],[457,19428,19430],{"className":19429,"style":872},[565],[457,19431,19432,19435],{"style":17355},[457,19433],{"className":19434,"style":574},[573],[457,19436,19438],{"className":19437},[578,579,580,581],[457,19439,2346],{"className":19440},[478,581],[457,19442,889],{"className":19443},[888],[457,19445,19447],{"className":19446},[561],[457,19448,19450],{"className":19449,"style":896},[565],[457,19451],{},[457,19453],{"className":19454,"style":669},[625],[457,19456,674],{"className":19457},[673],[457,19459],{"className":19460,"style":669},[625],[457,19462,19464,19467],{"className":19463},[478],[457,19465,17340],{"className":19466,"style":17339},[478,479],[457,19468,19470],{"className":19469},[553],[457,19471,19473,19493],{"className":19472},[557,865],[457,19474,19476,19490],{"className":19475},[561],[457,19477,19479],{"className":19478,"style":872},[565],[457,19480,19481,19484],{"style":17355},[457,19482],{"className":19483,"style":574},[573],[457,19485,19487],{"className":19486},[578,579,580,581],[457,19488,1659],{"className":19489},[478,581],[457,19491,889],{"className":19492},[888],[457,19494,19496],{"className":19495},[561],[457,19497,19499],{"className":19498,"style":896},[565],[457,19500],{},[457,19502,903],{"className":19503},[902],[457,19505,19506,19509],{"style":17427},[457,19507],{"className":19508,"style":8086},[573],[457,19510,19512,19515,19518,19521,19524,19564,19567,19570,19573,19613],{"className":19511},[478],[457,19513],{"className":19514},[478],[457,19516],{"className":19517,"style":626},[625],[457,19519,631],{"className":19520},[630],[457,19522],{"className":19523,"style":626},[625],[457,19525,19527,19530],{"className":19526},[478],[457,19528,17340],{"className":19529,"style":17339},[478,479],[457,19531,19533],{"className":19532},[553],[457,19534,19536,19556],{"className":19535},[557,865],[457,19537,19539,19553],{"className":19538},[561],[457,19540,19542],{"className":19541,"style":872},[565],[457,19543,19544,19547],{"style":17355},[457,19545],{"className":19546,"style":574},[573],[457,19548,19550],{"className":19549},[578,579,580,581],[457,19551,936],{"className":19552},[478,581],[457,19554,889],{"className":19555},[888],[457,19557,19559],{"className":19558},[561],[457,19560,19562],{"className":19561,"style":896},[565],[457,19563],{},[457,19565],{"className":19566,"style":669},[625],[457,19568,674],{"className":19569},[673],[457,19571],{"className":19572,"style":669},[625],[457,19574,19576,19579],{"className":19575},[478],[457,19577,17340],{"className":19578,"style":17339},[478,479],[457,19580,19582],{"className":19581},[553],[457,19583,19585,19605],{"className":19584},[557,865],[457,19586,19588,19602],{"className":19587},[561],[457,19589,19591],{"className":19590,"style":872},[565],[457,19592,19593,19596],{"style":17355},[457,19594],{"className":19595,"style":574},[573],[457,19597,19599],{"className":19598},[578,579,580,581],[457,19600,2365],{"className":19601},[478,581],[457,19603,889],{"className":19604},[888],[457,19606,19608],{"className":19607},[561],[457,19609,19611],{"className":19610,"style":896},[565],[457,19612],{},[457,19614,903],{"className":19615},[902],[457,19617,889],{"className":19618},[888],[457,19620,19622],{"className":19621},[561],[457,19623,19625],{"className":19624,"style":19279},[565],[457,19626],{},[457,19628],{"className":19629,"style":18296},[15663],[457,19631,19633],{"className":19632},[11789],[457,19634,19636,19747],{"className":19635},[557,865],[457,19637,19639,19744],{"className":19638},[561],[457,19640,19642,19693],{"className":19641,"style":19166},[565],[457,19643,19644,19647],{"style":17378},[457,19645],{"className":19646,"style":8086},[573],[457,19648,19650],{"className":19649},[478],[457,19651,19653,19656],{"className":19652},[478],[457,19654,8276],{"className":19655,"style":8275},[478,479],[457,19657,19659],{"className":19658},[553],[457,19660,19662,19685],{"className":19661},[557,865],[457,19663,19665,19682],{"className":19664},[561],[457,19666,19668],{"className":19667,"style":872},[565],[457,19669,19670,19673],{"style":19195},[457,19671],{"className":19672,"style":574},[573],[457,19674,19676],{"className":19675},[578,579,580,581],[457,19677,19679],{"className":19678},[478,581],[457,19680,15720],{"className":19681},[478,581],[457,19683,889],{"className":19684},[888],[457,19686,19688],{"className":19687},[561],[457,19689,19691],{"className":19690,"style":896},[565],[457,19692],{},[457,19694,19695,19698],{"style":17427},[457,19696],{"className":19697,"style":8086},[573],[457,19699,19701],{"className":19700},[478],[457,19702,19704,19707],{"className":19703},[478],[457,19705,8276],{"className":19706,"style":8275},[478,479],[457,19708,19710],{"className":19709},[553],[457,19711,19713,19736],{"className":19712},[557,865],[457,19714,19716,19733],{"className":19715},[561],[457,19717,19719],{"className":19718,"style":872},[565],[457,19720,19721,19724],{"style":19195},[457,19722],{"className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sName":20019,"style":574},[573],[457,20021,20023],{"className":20022},[578,579,580,581],[457,20024,2393],{"className":20025},[478,581],[457,20027,889],{"className":20028},[888],[457,20030,20032],{"className":20031},[561],[457,20033,20035],{"className":20034,"style":896},[565],[457,20036],{},[457,20038],{"className":20039,"style":669},[625],[457,20041,674],{"className":20042},[673],[457,20044],{"className":20045,"style":669},[625],[457,20047,20049,20052],{"className":20048},[478],[457,20050,17340],{"className":20051,"style":17339},[478,479],[457,20053,20055],{"className":20054},[553],[457,20056,20058,20078],{"className":20057},[557,865],[457,20059,20061,20075],{"className":20060},[561],[457,20062,20064],{"className":20063,"style":872},[565],[457,20065,20066,20069],{"style":17355},[457,20067],{"className":20068,"style":574},[573],[457,20070,20072],{"className":20071},[578,579,580,581],[457,20073,2412],{"className":20074},[478,581],[457,20076,889],{"className":20077},[888],[457,20079,20081],{"className":20080},[561],[457,20082,20084],{"className":20083,"style":896},[565],[457,20085],{},[457,20087,727],{"className":20088},[478],[457,20090,889],{"className":20091},[888],[457,20093,20095],{"className":20094},[561],[457,20096,20098],{"className":20097,"style":19279},[565],[457,20099],{},[381,20101,20102,20103,20153,20154,20267],{},"The eighth multiplication is gone, traded for a handful of extra block additions\nthat cost only ",[457,20104,20106],{"className":20105},[460],[457,20107,20109],{"className":20108,"ariaHidden":465},[464],[457,20110,20112,20115,20118,20121,20150],{"className":20111},[469],[457,20113],{"className":20114,"style":1430},[473],[457,20116,538],{"className":20117},[478],[457,20119,543],{"className":20120},[542],[457,20122,20124,20127],{"className":20123},[478],[457,20125,480],{"className":20126},[478,479],[457,20128,20130],{"className":20129},[553],[457,20131,20133],{"className":20132},[557],[457,20134,20136],{"className":20135},[561],[457,20137,20139],{"className":20138,"style":1455},[565],[457,20140,20141,20144],{"style":569},[457,20142],{"className":20143,"style":574},[573],[457,20145,20147],{"className":20146},[578,579,580,581],[457,20148,936],{"className":20149},[478,581],[457,20151,590],{"className":20152},[589],". The recurrence becomes 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71.249-19.021Q71.450-18.888 71.696-18.845L72.321-18.732Q72.751-18.642 73.059-18.345Q73.368-18.048 73.368-17.634Q73.368-17.064 72.970-16.786Q72.571-16.509 71.977-16.509Q71.427-16.509 71.075-16.845L70.778-16.532Q70.755-16.509 70.720-16.509L70.673-16.509Q70.649-16.509 70.618-16.540Q70.587-16.571 70.587-16.595",[1667],[2323,20716,20718,20719,20734,20735,20750,20751,2863,20801,727],{"className":20717},[2326],"Each Strassen node makes ",[457,20720,20722],{"className":20721},[460],[457,20723,20725],{"className":20724,"ariaHidden":465},[464],[457,20726,20728,20731],{"className":20727},[469],[457,20729],{"className":20730,"style":3370},[473],[457,20732,1659],{"className":20733},[478]," recursive half-size multiplications instead of the naive ",[457,20736,20738],{"className":20737},[460],[457,20739,20741],{"className":20740,"ariaHidden":465},[464],[457,20742,20744,20747],{"className":20743},[469],[457,20745],{"className":20746,"style":3370},[473],[457,20748,3417],{"className":20749},[478]," — the fan-out, not the block size, is what drops the cost from ",[457,20752,20754],{"className":20753},[460],[457,20755,20757],{"className":20756,"ariaHidden":465},[464],[457,20758,20760,20763,20766,20769,20798],{"className":20759},[469],[457,20761],{"className":20762,"style":1430},[473],[457,20764,538],{"className":20765},[478],[457,20767,543],{"className":20768},[542],[457,20770,20772,20775],{"className":20771},[478],[457,20773,480],{"className":20774},[478,479],[457,20776,20778],{"className":20777},[553],[457,20779,20781],{"className":20780},[557],[457,20782,20784],{"className":20783},[561],[457,20785,20787],{"className":20786,"style":1455},[565],[457,20788,20789,20792],{"style":569},[457,20790],{"className":20791,"style":574},[573],[457,20793,20795],{"className":20794},[578,579,580,581],[457,20796,2393],{"className":20797},[478,581],[457,20799,590],{"className":20800},[589],[457,20802,20804],{"className":20803},[460],[457,20805,20807],{"className":20806,"ariaHidden":465},[464],[457,20808,20810,20813,20816,20819,20897],{"className":20809},[469],[457,20811],{"className":20812,"style":12609},[473],[457,20814,538],{"className":20815},[478],[457,20817,543],{"className":20818},[542],[457,20820,20822,20825],{"className":20821},[478],[457,20823,480],{"className":20824},[478,479],[457,20826,20828],{"className":20827},[553],[457,20829,20831],{"className":20830},[557],[457,20832,20834],{"className":20833},[561],[457,20835,20837],{"className":20836,"style":6581},[565],[457,20838,20839,20842],{"style":569},[457,20840],{"className":20841,"style":574},[573],[457,20843,20845],{"className":20844},[578,579,580,581],[457,20846,20848,20891,20894],{"className":20847},[478,581],[457,20849,20851,20857],{"className":20850},[5137,581],[457,20852,20854],{"className":20853},[5137,581],[457,20855,5143],{"className":20856,"style":5142},[478,5141,581],[457,20858,20860],{"className":20859},[553],[457,20861,20863,20883],{"className":20862},[557,865],[457,20864,20866,20880],{"className":20865},[561],[457,20867,20869],{"className":20868,"style":12660},[565],[457,20870,20871,20874],{"style":6635},[457,20872],{"className":20873,"style":6639},[573],[457,20875,20877],{"className":20876},[578,6643,6644,581],[457,20878,936],{"className":20879},[478,581],[457,20881,889],{"className":20882},[888],[457,20884,20886],{"className":20885},[561],[457,20887,20889],{"className":20888,"style":6657},[565],[457,20890],{},[457,20892],{"className":20893,"style":6663},[625,581],[457,20895,1659],{"className":20896},[478,581],[457,20898,590],{"className":20899},[589],[381,20901,20902,20903,20918,20919,20969,20970,727],{},"The constant is large, so the crossover with the cubic method only pays off for\nsizable ",[457,20904,20906],{"className":20905},[460],[457,20907,20909],{"className":20908,"ariaHidden":465},[464],[457,20910,20912,20915],{"className":20911},[469],[457,20913],{"className":20914,"style":474},[473],[457,20916,480],{"className":20917},[478,479],"; in practice Strassen is switched in above a threshold and the base\ncase falls back to the cache-friendly schoolbook multiply. Theoretically, though,\nit was the first crack in the ",[457,20920,20922],{"className":20921},[460],[457,20923,20925],{"className":20924,"ariaHidden":465},[464],[457,20926,20928,20931,20934,20937,20966],{"className":20927},[469],[457,20929],{"className":20930,"style":1430},[473],[457,20932,538],{"className":20933},[478],[457,20935,543],{"className":20936},[542],[457,20938,20940,20943],{"className":20939},[478],[457,20941,480],{"className":20942},[478,479],[457,20944,20946],{"className":20945},[553],[457,20947,20949],{"className":20948},[557],[457,20950,20952],{"className":20951},[561],[457,20953,20955],{"className":20954,"style":1455},[565],[457,20956,20957,20960],{"style":569},[457,20958],{"className":20959,"style":574},[573],[457,20961,20963],{"className":20962},[578,579,580,581],[457,20964,2393],{"className":20965},[478,581],[457,20967,590],{"className":20968},[589]," wall, and a long line of refinements\nhas pushed the exponent down toward ",[457,20971,20973],{"className":20972},[460],[457,20974,20976],{"className":20975,"ariaHidden":465},[464],[457,20977,20979,20982],{"className":20978},[469],[457,20980],{"className":20981,"style":3370},[473],[457,20983,20985],{"className":20984},[478],"2.37",[814,20987,20989],{"id":20988},"the-master-theorem","The master theorem",[381,20991,20992,20993,21112,21113,3691,21116,21178,21179,3691,21182,10921],{},"Every recurrence in this lesson has the form ",[457,20994,20996],{"className":20995},[460],[457,20997,20999,21026,21065],{"className":20998,"ariaHidden":465},[464],[457,21000,21002,21005,21008,21011,21014,21017,21020,21023],{"className":21001},[469],[457,21003],{"className":21004,"style":510},[473],[457,21006,612],{"className":21007,"style":611},[478,479],[457,21009,543],{"className":21010},[542],[457,21012,480],{"className":21013},[478,479],[457,21015,590],{"className":21016},[589],[457,21018],{"className":21019,"style":626},[625],[457,21021,631],{"className":21022},[630],[457,21024],{"className":21025,"style":626},[625],[457,21027,21029,21032,21035,21038,21041,21044,21047,21050,21053,21056,21059,21062],{"className":21028},[469],[457,21030],{"className":21031,"style":510},[473],[457,21033,434],{"className":21034},[478,479],[457,21036],{"className":21037,"style":647},[625],[457,21039,612],{"className":21040,"style":611},[478,479],[457,21042,543],{"className":21043},[542],[457,21045,480],{"className":21046},[478,479],[457,21048,517],{"className":21049},[478],[457,21051,521],{"className":21052},[478,479],[457,21054,590],{"className":21055},[589],[457,21057],{"className":21058,"style":669},[625],[457,21060,674],{"className":21061},[673],[457,21063],{"className":21064,"style":669},[625],[457,21066,21068,21071,21074,21077,21109],{"className":21067},[469],[457,21069],{"className":21070,"style":510},[473],[457,21072,538],{"className":21073},[478],[457,21075,543],{"className":21076},[542],[457,21078,21080,21083],{"className":21079},[478],[457,21081,480],{"className":21082},[478,479],[457,21084,21086],{"className":21085},[553],[457,21087,21089],{"className":21088},[557],[457,21090,21092],{"className":21091},[561],[457,21093,21095],{"className":21094,"style":566},[565],[457,21096,21097,21100],{"style":569},[457,21098],{"className":21099,"style":574},[573],[457,21101,21103],{"className":21102},[578,579,580,581],[457,21104,21106],{"className":21105},[478,581],[457,21107,585],{"className":21108},[478,479,581],[457,21110,590],{"className":21111},[589],".\nThe recursion-tree analysis we did by hand each time generalizes to a single\nrule. Compare the ",[390,21114,21115],{},"branching 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the rate at which leaves\nproliferate, against the ",[390,21180,21181],{},"work 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the root's work\ndominates. Reading off our examples:",[6885,22249,22250,22416],{},[6888,22251,22252],{},[6891,22253,22254,22257,22260,22277,22294,22311,22390],{},[6894,22255,22256],{},"Algorithm",[6894,22258,22259],{},"Recurrence",[6894,22261,22262],{},[457,22263,22265],{"className":22264},[460],[457,22266,22268],{"className":22267,"ariaHidden":465},[464],[457,22269,22271,22274],{"className":22270},[469],[457,22272],{"className":22273,"style":474},[473],[457,22275,434],{"className":22276},[478,479],[6894,22278,22279],{},[457,22280,22282],{"className":22281},[460],[457,22283,22285],{"className":22284,"ariaHidden":465},[464],[457,22286,22288,22291],{"className":22287},[469],[457,22289],{"className":22290,"style":759},[473],[457,22292,521],{"className":22293},[478,479],[6894,22295,22296],{},[457,22297,22299],{"className":22298},[460],[457,22300,22302],{"className":22301,"ariaHidden":465},[464],[457,22303,22305,22308],{"className":22304},[469],[457,22306],{"className":22307,"style":474},[473],[457,22309,585],{"className":22310},[478,479],[6894,22312,22313,22374,22375],{},[457,22314,22316],{"className":22315},[460],[457,22317,22319],{"className":22318,"ariaHidden":465},[464],[457,22320,22322,22325,22368,22371],{"className":22321},[469],[457,22323],{"className":22324,"style":5973},[473],[457,22326,22328,22334],{"className":22327},[5137],[457,22329,22331],{"className":22330},[5137],[457,22332,5143],{"className":22333,"style":5142},[478,5141],[457,22335,22337],{"className":22336},[553],[457,22338,22340,22360],{"className":22339},[557,865],[457,22341,22343,22357],{"className":22342},[561],[457,22344,22346],{"className":22345,"style":21149},[565],[457,22347,22348,22351],{"style":5998},[457,22349],{"className":22350,"style":574},[573],[457,22352,22354],{"className":22353},[578,579,580,581],[457,22355,521],{"className":22356},[478,479,581],[457,22358,889],{"className":22359},[888],[457,22361,22363],{"className":22362},[561],[457,22364,22366],{"className":22365,"style":6017},[565],[457,22367],{},[457,22369],{"className":22370,"style":647},[625],[457,22372,434],{"className":22373},[478,479]," 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naive combine",[6913,22805,22806],{},[457,22807,22809],{"className":22808},[460],[457,22810,22812,22845],{"className":22811,"ariaHidden":465},[464],[457,22813,22815,22818,22821,22824,22827,22830,22833,22836,22839,22842],{"className":22814},[469],[457,22816],{"className":22817,"style":510},[473],[457,22819,936],{"className":22820},[478],[457,22822,612],{"className":22823,"style":611},[478,479],[457,22825,543],{"className":22826},[542],[457,22828,480],{"className":22829},[478,479],[457,22831,1591],{"className":22832},[478],[457,22834,590],{"className":22835},[589],[457,22837],{"className":22838,"style":669},[625],[457,22840,674],{"className":22841},[673],[457,22843],{"className":22844,"style":669},[625],[457,22846,22848,22851,22854,22857,22886],{"className":22847},[469],[457,22849],{"className":22850,"style":1430},[473],[457,22852,538],{"className":22853},[478],[457,22855,543],{"className":22856},[542],[457,22858,22860,22863],{"className":22859},[478],[457,22861,480],{"className":22862},[478,479],[457,22864,22866],{"className":22865},[553],[457,22867,22869],{"className":22868},[557],[457,22870,22872],{"className":22871},[561],[457,22873,22875],{"className":22874,"style":1455},[565],[457,22876,22877,22880],{"style":569},[457,22878],{"className":22879,"style":574},[573],[457,22881,22883],{"className":22882},[578,579,580,581],[457,22884,936],{"className":22885},[478,581],[457,22887,590],{"className":22888},[589],[6913,22890,22891],{},[457,22892,22894],{"className":22893},[460],[457,22895,22897],{"className":22896,"ariaHidden":465},[464],[457,22898,22900,22903],{"className":22899},[469],[457,22901],{"className":22902,"style":3370},[473],[457,22904,936],{"className":22905},[478],[6913,22907,22908],{},[457,22909,22911],{"className":22910},[460],[457,22912,22914],{"className":22913,"ariaHidden":465},[464],[457,22915,22917,22920],{"className":22916},[469],[457,22918],{"className":22919,"style":3370},[473],[457,22921,936],{"className":22922},[478],[6913,22924,22925],{},[457,22926,22928],{"className":22927},[460],[457,22929,22931],{"className":22930,"ariaHidden":465},[464],[457,22932,22934,22937],{"className":22933},[469],[457,22935],{"className":22936,"style":3370},[473],[457,22938,936],{"className":22939},[478],[6913,22941,22942,22976],{},[457,22943,22945],{"className":22944},[460],[457,22946,22948,22967],{"className":22947,"ariaHidden":465},[464],[457,22949,22951,22955,22958,22961,22964],{"className":22950},[469],[457,22952],{"className":22953,"style":22954},[473],"height:0.6835em;vertical-align:-0.0391em;",[457,22956,440],{"className":22957},[478],[457,22959],{"className":22960,"style":626},[625],[457,22962,7790],{"className":22963},[630],[457,22965],{"className":22966,"style":626},[625],[457,22968,22970,22973],{"className":22969},[469],[457,22971],{"className":22972,"style":3370},[473],[457,22974,936],{"className":22975},[478],", 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leaf-heavy",[6913,23227,23228],{},[457,23229,23231],{"className":23230},[460],[457,23232,23234],{"className":23233,"ariaHidden":465},[464],[457,23235,23237,23240,23243,23246,23324],{"className":23236},[469],[457,23238],{"className":23239,"style":12609},[473],[457,23241,538],{"className":23242},[478],[457,23244,543],{"className":23245},[542],[457,23247,23249,23252],{"className":23248},[478],[457,23250,480],{"className":23251},[478,479],[457,23253,23255],{"className":23254},[553],[457,23256,23258],{"className":23257},[557],[457,23259,23261],{"className":23260},[561],[457,23262,23264],{"className":23263,"style":6581},[565],[457,23265,23266,23269],{"style":569},[457,23267],{"className":23268,"style":574},[573],[457,23270,23272],{"className":23271},[578,579,580,581],[457,23273,23275,23318,23321],{"className":23274},[478,581],[457,23276,23278,23284],{"className":23277},[5137,581],[457,23279,23281],{"className":23280},[5137,581],[457,23282,5143],{"className":23283,"style":5142},[478,5141,581],[457,23285,23287],{"className":23286},[553],[457,23288,23290,23310],{"className":23289},[557,865],[457,23291,23293,23307],{"className":23292},[561],[457,23294,23296],{"className":23295,"style":12660},[565],[457,23297,23298,23301],{"style":6635},[457,23299],{"className":23300,"style":6639},[573],[457,23302,23304],{"className":23303},[578,6643,6644,581],[457,23305,936],{"className":23306},[478,581],[457,23308,889],{"className":23309},[888],[457,23311,23313],{"className":23312},[561],[457,23314,23316],{"className":23315,"style":6657},[565],[457,23317],{},[457,23319],{"className":23320,"style":6663},[625,581],[457,23322,2393],{"className":23323},[478,581],[457,23325,590],{"className":23326},[589],[381,23328,23329,23330,23354,23355,23379,23380,2526,23464,23548,23549,746,23564,23579,23580,727],{},"One last sanity check worth stressing: it makes no difference whether the\ncombine cost is written ",[457,23331,23333],{"className":23332},[460],[457,23334,23336],{"className":23335,"ariaHidden":465},[464],[457,23337,23339,23342,23345,23348,23351],{"className":23338},[469],[457,23340],{"className":23341,"style":510},[473],[457,23343,538],{"className":23344},[478],[457,23346,543],{"className":23347},[542],[457,23349,480],{"className":23350},[478,479],[457,23352,590],{"className":23353},[589]," or bounded above by ",[457,23356,23358],{"className":23357},[460],[457,23359,23361],{"className":23360,"ariaHidden":465},[464],[457,23362,23364,23367,23370,23373,23376],{"className":23363},[469],[457,23365],{"className":23366,"style":510},[473],[457,23368,3641],{"className":23369,"style":1521},[478,479],[457,23371,543],{"className":23372},[542],[457,23374,480],{"className":23375},[478,479],[457,23377,590],{"className":23378},[589],". The recurrences\n",[457,23381,23383],{"className":23382},[460],[457,23384,23386,23413,23446],{"className":23385,"ariaHidden":465},[464],[457,23387,23389,23392,23395,23398,23401,23404,23407,23410],{"className":23388},[469],[457,23390],{"className":23391,"style":510},[473],[457,23393,612],{"className":23394,"style":611},[478,479],[457,23396,543],{"className":23397},[542],[457,23399,480],{"className":23400},[478,479],[457,23402,590],{"className":23403},[589],[457,23405],{"className":23406,"style":626},[625],[457,23408,631],{"className":23409},[630],[457,23411],{"className":23412,"style":626},[625],[457,23414,23416,23419,23422,23425,23428,23431,23434,23437,23440,23443],{"className":23415},[469],[457,23417],{"className":23418,"style":510},[473],[457,23420,936],{"className":23421},[478],[457,23423,612],{"className":23424,"style":611},[478,479],[457,23426,543],{"className":23427},[542],[457,23429,480],{"className":23430},[478,479],[457,23432,1591],{"className":23433},[478],[457,23435,590],{"className":23436},[589],[457,23438],{"className":23439,"style":669},[625],[457,23441,674],{"className":23442},[673],[457,23444],{"className":23445,"style":669},[625],[457,23447,23449,23452,23455,23458,23461],{"className":23448},[469],[457,23450],{"className":23451,"style":510},[473],[457,23453,538],{"className":23454},[478],[457,23456,543],{"className":23457},[542],[457,23459,480],{"className":23460},[478,479],[457,23462,590],{"className":23463},[589],[457,23465,23467],{"className":23466},[460],[457,23468,23470,23497,23530],{"className":23469,"ariaHidden":465},[464],[457,23471,23473,23476,23479,23482,23485,23488,23491,23494],{"className":23472},[469],[457,23474],{"className":23475,"style":510},[473],[457,23477,612],{"className":23478,"style":611},[478,479],[457,23480,543],{"className":23481},[542],[457,23483,480],{"className":23484},[478,479],[457,23486,590],{"className":23487},[589],[457,23489],{"className":23490,"style":626},[625],[457,23492,1261],{"className":23493},[630],[457,23495],{"className":23496,"style":626},[625],[457,23498,23500,23503,23506,23509,23512,23515,23518,23521,23524,23527],{"className":23499},[469],[457,23501],{"className":23502,"style":510},[473],[457,23504,936],{"className":23505},[478],[457,23507,612],{"className":23508,"style":611},[478,479],[457,23510,543],{"className":23511},[542],[457,23513,480],{"className":23514},[478,479],[457,23516,1591],{"className":23517},[478],[457,23519,590],{"className":23520},[589],[457,23522],{"className":23523,"style":669},[625],[457,23525,674],{"className":23526},[673],[457,23528],{"className":23529,"style":669},[625],[457,23531,23533,23536,23539,23542,23545],{"className":23532},[469],[457,23534],{"className":23535,"style":510},[473],[457,23537,3641],{"className":23538,"style":1521},[478,479],[457,23540,543],{"className":23541},[542],[457,23543,480],{"className":23544},[478,479],[457,23546,590],{"className":23547},[589]," have the same\nsolution. The master theorem cares only about ",[457,23550,23552],{"className":23551},[460],[457,23553,23555],{"className":23554,"ariaHidden":465},[464],[457,23556,23558,23561],{"className":23557},[469],[457,23559],{"className":23560,"style":474},[473],[457,23562,434],{"className":23563},[478,479],[457,23565,23567],{"className":23566},[460],[457,23568,23570],{"className":23569,"ariaHidden":465},[464],[457,23571,23573,23576],{"className":23572},[469],[457,23574],{"className":23575,"style":759},[473],[457,23577,521],{"className":23578},[478,479],", and the exponent ",[457,23581,23583],{"className":23582},[460],[457,23584,23586],{"className":23585,"ariaHidden":465},[464],[457,23587,23589,23592],{"className":23588},[469],[457,23590],{"className":23591,"style":474},[473],[457,23593,585],{"className":23594},[478,479],[814,23596,23598],{"id":23597},"takeaways","Takeaways",[394,23600,23601,23723,23769,23946,23980,24149,24454],{},[397,23602,23603,23606,23607,727],{},[390,23604,23605],{},"Divide and conquer"," = divide into smaller copies, conquer recursively,\ncombine. Trust the recursion; focus on the split and the merge. The cost is\nalways a recurrence ",[457,23608,23610],{"className":23609},[460],[457,23611,23613,23640,23679],{"className":23612,"ariaHidden":465},[464],[457,23614,23616,23619,23622,23625,23628,23631,23634,23637],{"className":23615},[469],[457,23617],{"className":23618,"style":510},[473],[457,23620,612],{"className":23621,"style":611},[478,479],[457,23623,543],{"className":23624},[542],[457,23626,480],{"className":23627},[478,479],[457,23629,590],{"className":23630},[589],[457,23632],{"className":23633,"style":626},[625],[457,23635,631],{"className":23636},[630],[457,23638],{"className":23639,"style":626},[625],[457,23641,23643,23646,23649,23652,23655,23658,23661,23664,23667,23670,23673,23676],{"className":23642},[469],[457,23644],{"className":23645,"style":510},[473],[457,23647,434],{"className":23648},[478,479],[457,23650],{"className":23651,"style":647},[625],[457,23653,612],{"className":23654,"style":611},[478,479],[457,23656,543],{"className":23657},[542],[457,23659,480],{"className":23660},[478,479],[457,23662,517],{"className":23663},[478],[457,23665,521],{"className":23666},[478,479],[457,23668,590],{"className":23669},[589],[457,23671],{"className":23672,"style":669},[625],[457,23674,674],{"className":23675},[673],[457,23677],{"className":23678,"style":669},[625],[457,23680,23682,23685,23688,23691,23720],{"className":23681},[469],[457,23683],{"className":23684,"style":510},[473],[457,23686,538],{"className":23687},[478],[457,23689,543],{"className":23690},[542],[457,23692,23694,23697],{"className":23693},[478],[457,23695,480],{"className":23696},[478,479],[457,23698,23700],{"className":23699},[553],[457,23701,23703],{"className":23702},[557],[457,23704,23706],{"className":23705},[561],[457,23707,23709],{"className":23708,"style":566},[565],[457,23710,23711,23714],{"style":569},[457,23712],{"className":23713,"style":574},[573],[457,23715,23717],{"className":23716},[578,579,580,581],[457,23718,585],{"className":23719},[478,479,581],[457,23721,590],{"className":23722},[589],[397,23724,23725,23746,23747,23768],{},[457,23726,23728],{"className":23727},[460],[457,23729,23731],{"className":23730,"ariaHidden":465},[464],[457,23732,23734,23737],{"className":23733},[469],[457,23735],{"className":23736,"style":2481},[473],[457,23738,23740],{"className":23739},[2485,2486],[457,23741,23743],{"className":23742},[478,2490],[457,23744,1487],{"className":23745},[478]," divides at the midpoint and combines with a linear-time\n",[457,23748,23750],{"className":23749},[460],[457,23751,23753],{"className":23752,"ariaHidden":465},[464],[457,23754,23756,23759],{"className":23755},[469],[457,23757],{"className":23758,"style":2481},[473],[457,23760,23762],{"className":23761},[2485,2486],[457,23763,23765],{"className":23764},[478,2490],[457,23766,2494],{"className":23767},[478]," whose correctness is a clean loop-invariant argument.",[397,23770,23771,23772,23856,23857,23881,23882,23906,23907,727],{},"The recurrence ",[457,23773,23775],{"className":23774},[460],[457,23776,23778,23805,23838],{"className":23777,"ariaHidden":465},[464],[457,23779,23781,23784,23787,23790,23793,23796,23799,23802],{"className":23780},[469],[457,23782],{"className":23783,"style":510},[473],[457,23785,612],{"className":23786,"style":611},[478,479],[457,23788,543],{"className":23789},[542],[457,23791,480],{"className":23792},[478,479],[457,23794,590],{"className":23795},[589],[457,23797],{"className":23798,"style":626},[625],[457,23800,631],{"className":23801},[630],[457,23803],{"className":23804,"style":626},[625],[457,23806,23808,23811,23814,23817,23820,23823,23826,23829,23832,23835],{"className":23807},[469],[457,23809],{"className":23810,"style":510},[473],[457,23812,936],{"className":23813},[478],[457,23815,612],{"className":23816,"style":611},[478,479],[457,23818,543],{"className":23819},[542],[457,23821,480],{"className":23822},[478,479],[457,23824,1591],{"className":23825},[478],[457,23827,590],{"className":23828},[589],[457,23830],{"className":23831,"style":669},[625],[457,23833,674],{"className":23834},[673],[457,23836],{"className":23837,"style":669},[625],[457,23839,23841,23844,23847,23850,23853],{"className":23840},[469],[457,23842],{"className":23843,"style":510},[473],[457,23845,538],{"className":23846},[478],[457,23848,543],{"className":23849},[542],[457,23851,480],{"className":23852},[478,479],[457,23854,590],{"className":23855},[589]," unfolds into a recursion tree\nwith ",[457,23858,23860],{"className":23859},[460],[457,23861,23863],{"className":23862,"ariaHidden":465},[464],[457,23864,23866,23869,23875,23878],{"className":23865},[469],[457,23867],{"className":23868,"style":6698},[473],[457,23870,23872],{"className":23871},[5137],[457,23873,5143],{"className":23874,"style":5142},[478,5141],[457,23876],{"className":23877,"style":647},[625],[457,23879,480],{"className":23880},[478,479]," levels of ",[457,23883,23885],{"className":23884},[460],[457,23886,23888],{"className":23887,"ariaHidden":465},[464],[457,23889,23891,23894,23897,23900,23903],{"className":23890},[469],[457,23892],{"className":23893,"style":510},[473],[457,23895,538],{"className":23896},[478],[457,23898,543],{"className":23899},[542],[457,23901,480],{"className":23902},[478,479],[457,23904,590],{"className":23905},[589]," work each, giving ",[457,23908,23910],{"className":23909},[460],[457,23911,23913],{"className":23912,"ariaHidden":465},[464],[457,23914,23916,23919,23922,23925,23928,23931,23937,23940,23943],{"className":23915},[469],[457,23917],{"className":23918,"style":510},[473],[457,23920,538],{"className":23921},[478],[457,23923,543],{"className":23924},[542],[457,23926,480],{"className":23927},[478,479],[457,23929],{"className":23930,"style":647},[625],[457,23932,23934],{"className":23933},[5137],[457,23935,5143],{"className":23936,"style":5142},[478,5141],[457,23938],{"className":23939,"style":647},[625],[457,23941,480],{"className":23942},[478,479],[457,23944,590],{"className":23945},[589],[397,23947,6765,23948,23950,23951,23954,23955,23979],{},[390,23949,6768],{}," and worst-case optimal among ",[434,23952,23953],{"href":62},"comparison sorts",", at the\ncost of ",[457,23956,23958],{"className":23957},[460],[457,23959,23961],{"className":23960,"ariaHidden":465},[464],[457,23962,23964,23967,23970,23973,23976],{"className":23963},[469],[457,23965],{"className":23966,"style":510},[473],[457,23968,538],{"className":23969},[478],[457,23971,543],{"className":23972},[542],[457,23974,480],{"className":23975},[478,479],[457,23977,590],{"className":23978},[589]," extra space, ideal for linked lists and external sorting.",[397,23981,23982,23984,23985,24006,24007,24058,24059,24098,24099,727],{},[390,23983,7474],{}," reuses the merge: fold a cross-inversion count into\n",[457,23986,23988],{"className":23987},[460],[457,23989,23991],{"className":23990,"ariaHidden":465},[464],[457,23992,23994,23997],{"className":23993},[469],[457,23995],{"className":23996,"style":2481},[473],[457,23998,24000],{"className":23999},[2485,2486],[457,24001,24003],{"className":24002},[478,2490],[457,24004,2494],{"className":24005},[478]," so each step adds ",[457,24008,24010],{"className":24009},[460],[457,24011,24013,24031,24049],{"className":24012,"ariaHidden":465},[464],[457,24014,24016,24019,24022,24025,24028],{"className":24015},[469],[457,24017],{"className":24018,"style":3583},[473],[457,24020,381],{"className":24021},[478,479],[457,24023],{"className":24024,"style":669},[625],[457,24026,3573],{"className":24027},[673],[457,24029],{"className":24030,"style":669},[625],[457,24032,24034,24037,24040,24043,24046],{"className":24033},[469],[457,24035],{"className":24036,"style":9069},[473],[457,24038,2785],{"className":24039},[478,479],[457,24041],{"className":24042,"style":669},[625],[457,24044,674],{"className":24045},[673],[457,24047],{"className":24048,"style":669},[625],[457,24050,24052,24055],{"className":24051},[469],[457,24053],{"className":24054,"style":3370},[473],[457,24056,440],{"className":24057},[478],", sorting and counting together\nin ",[457,24060,24062],{"className":24061},[460],[457,24063,24065],{"className":24064,"ariaHidden":465},[464],[457,24066,24068,24071,24074,24077,24080,24083,24089,24092,24095],{"className":24067},[469],[457,24069],{"className":24070,"style":510},[473],[457,24072,538],{"className":24073},[478],[457,24075,543],{"className":24076},[542],[457,24078,480],{"className":24079},[478,479],[457,24081],{"className":24082,"style":647},[625],[457,24084,24086],{"className":24085},[5137],[457,24087,5143],{"className":24088,"style":5142},[478,5141],[457,24090],{"className":24091,"style":647},[625],[457,24093,480],{"className":24094},[478,479],[457,24096,590],{"className":24097},[589]," instead of the brute-force ",[457,24100,24102],{"className":24101},[460],[457,24103,24105],{"className":24104,"ariaHidden":465},[464],[457,24106,24108,24111,24114,24117,24146],{"className":24107},[469],[457,24109],{"className":24110,"style":1430},[473],[457,24112,538],{"className":24113},[478],[457,24115,543],{"className":24116},[542],[457,24118,24120,24123],{"className":24119},[478],[457,24121,480],{"className":24122},[478,479],[457,24124,24126],{"className":24125},[553],[457,24127,24129],{"className":24128},[557],[457,24130,24132],{"className":24131},[561],[457,24133,24135],{"className":24134,"style":1455},[565],[457,24136,24137,24140],{"style":569},[457,24138],{"className":24139,"style":574},[573],[457,24141,24143],{"className":24142},[578,579,580,581],[457,24144,936],{"className":24145},[478,581],[457,24147,590],{"className":24148},[589],[397,24150,24151,24153,24154,24304,24305,2863,24355,727],{},[390,24152,23033],{}," multiplication shows the paradigm's reach: the identity\n",[457,24155,24157],{"className":24156},[460],[457,24158,24160,24181,24202,24223,24250,24271,24292],{"className":24159,"ariaHidden":465},[464],[457,24161,24163,24166,24169,24172,24175,24178],{"className":24162},[469],[457,24164],{"className":24165,"style":3759},[473],[457,24167,434],{"className":24168},[478,479],[457,24170,10920],{"className":24171},[478,479],[457,24173],{"className":24174,"style":669},[625],[457,24176,674],{"className":24177},[673],[457,24179],{"className":24180,"style":669},[625],[457,24182,24184,24187,24190,24193,24196,24199],{"className":24183},[469],[457,24185],{"className":24186,"style":759},[473],[457,24188,521],{"className":24189},[478,479],[457,24191,585],{"className":24192},[478,479],[457,24194],{"className":24195,"style":626},[625],[457,24197,631],{"className":24198},[630],[457,24200],{"className":24201,"style":626},[625],[457,24203,24205,24208,24211,24214,24217,24220],{"className":24204},[469],[457,24206],{"className":24207,"style":510},[473],[457,24209,543],{"className":24210},[542],[457,24212,434],{"className":24213},[478,479],[457,24215],{"className":24216,"style":669},[625],[457,24218,3573],{"className":24219},[673],[457,24221],{"className":24222,"style":669},[625],[457,24224,24226,24229,24232,24235,24238,24241,24244,24247],{"className":24225},[469],[457,24227],{"className":24228,"style":510},[473],[457,24230,521],{"className":24231},[478,479],[457,24233,590],{"className":24234},[589],[457,24236,543],{"className":24237},[542],[457,24239,10920],{"className":24240},[478,479],[457,24242],{"className":24243,"style":669},[625],[457,24245,3573],{"className":24246},[673],[457,24248],{"className":24249,"style":669},[625],[457,24251,24253,24256,24259,24262,24265,24268],{"className":24252},[469],[457,24254],{"className":24255,"style":510},[473],[457,24257,585],{"className":24258},[478,479],[457,24260,590],{"className":24261},[589],[457,24263],{"className":24264,"style":669},[625],[457,24266,674],{"className":24267},[673],[457,24269],{"className":24270,"style":669},[625],[457,24272,24274,24277,24280,24283,24286,24289],{"className":24273},[469],[457,24275],{"className":24276,"style":3563},[473],[457,24278,434],{"className":24279},[478,479],[457,24281,585],{"className":24282},[478,479],[457,24284],{"className":24285,"style":669},[625],[457,24287,674],{"className":24288},[673],[457,24290],{"className":24291,"style":669},[625],[457,24293,24295,24298,24301],{"className":24294},[469],[457,24296],{"className":24297,"style":759},[473],[457,24299,521],{"className":24300},[478,479],[457,24302,10920],{"className":24303},[478,479]," cuts four subproblems to three, taking\ninteger multiplication from ",[457,24306,24308],{"className":24307},[460],[457,24309,24311],{"className":24310,"ariaHidden":465},[464],[457,24312,24314,24317,24320,24323,24352],{"className":24313},[469],[457,24315],{"className":24316,"style":1430},[473],[457,24318,538],{"className":24319},[478],[457,24321,543],{"className":24322},[542],[457,24324,24326,24329],{"className":24325},[478],[457,24327,480],{"className":24328},[478,479],[457,24330,24332],{"className":24331},[553],[457,24333,24335],{"className":24334},[557],[457,24336,24338],{"className":24337},[561],[457,24339,24341],{"className":24340,"style":1455},[565],[457,24342,24343,24346],{"style":569},[457,24344],{"className":24345,"style":574},[573],[457,24347,24349],{"className":24348},[578,579,580,581],[457,24350,936],{"className":24351},[478,581],[457,24353,590],{"className":24354},[589],[457,24356,24358],{"className":24357},[460],[457,24359,24361],{"className":24360,"ariaHidden":465},[464],[457,24362,24364,24367,24370,24373,24451],{"className":24363},[469],[457,24365],{"className":24366,"style":12609},[473],[457,24368,538],{"className":24369},[478],[457,24371,543],{"className":24372},[542],[457,24374,24376,24379],{"className":24375},[478],[457,24377,480],{"className":24378},[478,479],[457,24380,24382],{"className":24381},[553],[457,24383,24385],{"className":24384},[557],[457,24386,24388],{"className":24387},[561],[457,24389,24391],{"className":24390,"style":6581},[565],[457,24392,24393,24396],{"style":569},[457,24394],{"className":24395,"style":574},[573],[457,24397,24399],{"className":24398},[578,579,580,581],[457,24400,24402,24445,24448],{"className":24401},[478,581],[457,24403,24405,24411],{"className":24404},[5137,581],[457,24406,24408],{"className":24407},[5137,581],[457,24409,5143],{"className":24410,"style":5142},[478,5141,581],[457,24412,24414],{"className":24413},[553],[457,24415,24417,24437],{"className":24416},[557,865],[457,24418,24420,24434],{"className":24419},[561],[457,24421,24423],{"className":24422,"style":12660},[565],[457,24424,24425,24428],{"style":6635},[457,24426],{"className":24427,"style":6639},[573],[457,24429,24431],{"className":24430},[578,6643,6644,581],[457,24432,936],{"className":24433},[478,581],[457,24435,889],{"className":24436},[888],[457,24438,24440],{"className":24439},[561],[457,24441,24443],{"className":24442,"style":6657},[565],[457,24444],{},[457,24446],{"className":24447,"style":6663},[625,581],[457,24449,2393],{"className":24450},[478,581],[457,24452,590],{"className":24453},[589],[397,24455,24456,24457,24459,24460,2863,24521,24536,24537],{},"The ",[390,24458,807],{}," turns the tree into a rule: compare ",[457,24461,24463],{"className":24462},[460],[457,24464,24466],{"className":24465,"ariaHidden":465},[464],[457,24467,24469,24472,24515,24518],{"className":24468},[469],[457,24470],{"className":24471,"style":5973},[473],[457,24473,24475,24481],{"className":24474},[5137],[457,24476,24478],{"className":24477},[5137],[457,24479,5143],{"className":24480,"style":5142},[478,5141],[457,24482,24484],{"className":24483},[553],[457,24485,24487,24507],{"className":24486},[557,865],[457,24488,24490,24504],{"className":24489},[561],[457,24491,24493],{"className":24492,"style":21149},[565],[457,24494,24495,24498],{"style":5998},[457,24496],{"className":24497,"style":574},[573],[457,24499,24501],{"className":24500},[578,579,580,581],[457,24502,521],{"className":24503},[478,479,581],[457,24505,889],{"className":24506},[888],[457,24508,24510],{"className":24509},[561],[457,24511,24513],{"className":24512,"style":6017},[565],[457,24514],{},[457,24516],{"className":24517,"style":647},[625],[457,24519,434],{"className":24520},[478,479],[457,24522,24524],{"className":24523},[460],[457,24525,24527],{"className":24526,"ariaHidden":465},[464],[457,24528,24530,24533],{"className":24529},[469],[457,24531],{"className":24532,"style":474},[473],[457,24534,585],{"className":24535},[478,479],"\nfor leaf-heavy, balanced, or root-heavy behavior.",[431,24538,24539],{},[434,24540,2346],{"href":24541,"ariaDescribedBy":24542,"dataFootnoteRef":376,"id":24543},"#user-content-fn-clrs-master",[438],"user-content-fnref-clrs-master",[24545,24546,24549,24554],"section",{"className":24547,"dataFootnotes":376},[24548],"footnotes",[814,24550,24553],{"className":24551,"id":438},[24552],"sr-only","Footnotes",[24555,24556,24557,24577,24589,24604,24722],"ol",{},[397,24558,24560,746,24563,24566,24567,24569,24570],{"id":24559},"user-content-fn-erickson-rec",[390,24561,24562],{},"Erickson",[385,24564,24565],{},"Algorithms",", Ch. 1 — Recursion: the ",[427,24568,429],{}," stance of assuming recursive calls already work and focusing on divide and combine. ",[434,24571,24576],{"href":24572,"ariaLabel":24573,"className":24574,"dataFootnoteBackref":376},"#user-content-fnref-erickson-rec","Back to reference 1",[24575],"data-footnote-backref","↩",[397,24578,24580,24583,24584],{"id":24579},"user-content-fn-clrs-merge",[390,24581,24582],{},"CLRS",", Ch. 2 (§2.3) — Designing algorithms: mergesort as the canonical divide-and-conquer sort built on a linear-time merge. ",[434,24585,24576],{"href":24586,"ariaLabel":24587,"className":24588,"dataFootnoteBackref":376},"#user-content-fnref-clrs-merge","Back to reference 2",[24575],[397,24590,24592,746,24595,24598,24599],{"id":24591},"user-content-fn-skiena-sort",[390,24593,24594],{},"Skiena",[385,24596,24597],{},"The Algorithm Design Manual",", §4 — Sorting and Searching: mergesort's stability and suitability for linked lists and external sorting. ",[434,24600,24576],{"href":24601,"ariaLabel":24602,"className":24603,"dataFootnoteBackref":376},"#user-content-fnref-skiena-sort","Back to reference 3",[24575],[397,24605,24607,24609,24610,24709,24710,3691,24713,24716,24717],{"id":24606},"user-content-fn-strassen",[390,24608,24582],{},", Ch. 4 (§4.2) — Strassen's algorithm for matrix multiplication and its ",[457,24611,24613],{"className":24612},[460],[457,24614,24616],{"className":24615,"ariaHidden":465},[464],[457,24617,24619,24622,24625,24628,24706],{"className":24618},[469],[457,24620],{"className":24621,"style":12609},[473],[457,24623,538],{"className":24624},[478],[457,24626,543],{"className":24627},[542],[457,24629,24631,24634],{"className":24630},[478],[457,24632,480],{"className":24633},[478,479],[457,24635,24637],{"className":24636},[553],[457,24638,24640],{"className":24639},[557],[457,24641,24643],{"className":24642},[561],[457,24644,24646],{"className":24645,"style":6581},[565],[457,24647,24648,24651],{"style":569},[457,24649],{"className":24650,"style":574},[573],[457,24652,24654],{"className":24653},[578,579,580,581],[457,24655,24657,24700,24703],{"className":24656},[478,581],[457,24658,24660,24666],{"className":24659},[5137,581],[457,24661,24663],{"className":24662},[5137,581],[457,24664,5143],{"className":24665,"style":5142},[478,5141,581],[457,24667,24669],{"className":24668},[553],[457,24670,24672,24692],{"className":24671},[557,865],[457,24673,24675,24689],{"className":24674},[561],[457,24676,24678],{"className":24677,"style":12660},[565],[457,24679,24680,24683],{"style":6635},[457,24681],{"className":24682,"style":6639},[573],[457,24684,24686],{"className":24685},[578,6643,6644,581],[457,24687,936],{"className":24688},[478,581],[457,24690,889],{"className":24691},[888],[457,24693,24695],{"className":24694},[561],[457,24696,24698],{"className":24697,"style":6657},[565],[457,24699],{},[457,24701],{"className":24702,"style":6663},[625,581],[457,24704,1659],{"className":24705},[478,581],[457,24707,590],{"className":24708},[589]," recurrence; the original result is V. Strassen (1969), ",[427,24711,24712],{},"Gaussian elimination is not optimal,",[385,24714,24715],{},"Numerische Mathematik"," 13, 354–356. ",[434,24718,24576],{"href":24719,"ariaLabel":24720,"className":24721,"dataFootnoteBackref":376},"#user-content-fnref-strassen","Back to reference 4",[24575],[397,24723,24725,24727,24728,24789,24790,24805,24806],{"id":24724},"user-content-fn-clrs-master",[390,24726,24582],{},", Ch. 4 — Divide-and-Conquer: the master theorem comparing ",[457,24729,24731],{"className":24730},[460],[457,24732,24734],{"className":24733,"ariaHidden":465},[464],[457,24735,24737,24740,24783,24786],{"className":24736},[469],[457,24738],{"className":24739,"style":5973},[473],[457,24741,24743,24749],{"className":24742},[5137],[457,24744,24746],{"className":24745},[5137],[457,24747,5143],{"className":24748,"style":5142},[478,5141],[457,24750,24752],{"className":24751},[553],[457,24753,24755,24775],{"className":24754},[557,865],[457,24756,24758,24772],{"className":24757},[561],[457,24759,24761],{"className":24760,"style":21149},[565],[457,24762,24763,24766],{"style":5998},[457,24764],{"className":24765,"style":574},[573],[457,24767,24769],{"className":24768},[578,579,580,581],[457,24770,521],{"className":24771},[478,479,581],[457,24773,889],{"className":24774},[888],[457,24776,24778],{"className":24777},[561],[457,24779,24781],{"className":24780,"style":6017},[565],[457,24782],{},[457,24784],{"className":24785,"style":647},[625],[457,24787,434],{"className":24788},[478,479]," against the work exponent ",[457,24791,24793],{"className":24792},[460],[457,24794,24796],{"className":24795,"ariaHidden":465},[464],[457,24797,24799,24802],{"className":24798},[469],[457,24800],{"className":24801,"style":474},[473],[457,24803,585],{"className":24804},[478,479]," to classify leaf-heavy, balanced, and root-heavy recurrences. ",[434,24807,24576],{"href":24808,"ariaLabel":24809,"className":24810,"dataFootnoteBackref":376},"#user-content-fnref-clrs-master","Back to reference 5",[24575],[24812,24813,24814],"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":24816},[24817,24818,24821,24822,24823,24824,24825,24826,24827,24828,24829],{"id":816,"depth":18,"text":817},{"id":1477,"depth":18,"text":1487,"children":24819},[24820],{"id":3503,"depth":24,"text":3504},{"id":4835,"depth":18,"text":4836},{"id":6761,"depth":18,"text":6762},{"id":6882,"depth":18,"text":6883},{"id":7473,"depth":18,"text":7474},{"id":10622,"depth":18,"text":10623},{"id":15200,"depth":18,"text":15201},{"id":20988,"depth":18,"text":20989},{"id":23597,"depth":18,"text":23598},{"id":438,"depth":18,"text":24553},"Some problems are easiest to solve by reducing them to smaller versions of\nthemselves. This is the divide-and-conquer paradigm, and it is one of the\nmost productive ideas in all of algorithm design. Every divide-and-conquer\nalgorithm has the same three-part skeleton:","md",{"moduleNumber":18,"lessonNumber":6,"order":24833},201,true,[24836,24840,24844,24848,24851],{"title":24837,"slug":24838,"difficulty":24839},"Merge Sorted Array","merge-sorted-array","Easy",{"title":24841,"slug":24842,"difficulty":24843},"Sort an Array","sort-an-array","Medium",{"title":24845,"slug":24846,"difficulty":24847},"Merge k Sorted Lists","merge-k-sorted-lists","Hard",{"title":24849,"slug":24850,"difficulty":24847},"Count of Smaller Numbers After Self","count-of-smaller-numbers-after-self",{"title":24852,"slug":24853,"difficulty":24847},"Reverse Pairs","reverse-pairs","---\ntitle: Divide and Conquer & Mergesort\nmodule: Divide & Conquer\nmoduleNumber: 2\nlessonNumber: 1\norder: 201\nsummary: >-\n  Divide and conquer breaks a problem into smaller copies of itself, solves\n  them recursively, and stitches the answers together. We meet the paradigm\n  through mergesort — its merge step, its loop-invariant proof, and the\n  recursion tree that pins its cost at $\\Theta(n\\log n)$ — then glimpse\n  Karatsuba multiplication as a second example of the same idea.\ntopics: [Divide & Conquer, Comparison Sorting]\nsources:\n  - book: CLRS\n    ref: \"Ch. 2 & Ch. 4 — Getting Started; Divide-and-Conquer\"\n  - book: Skiena\n    ref: \"§4 — Sorting and Searching\"\n  - book: Erickson\n    ref: \"Ch. 1 — Recursion\"\npractice:\n  - title: 'Merge Sorted Array'\n    slug: merge-sorted-array\n    difficulty: Easy\n  - title: 'Sort an Array'\n    slug: sort-an-array\n    difficulty: Medium\n  - title: 'Merge k Sorted Lists'\n    slug: merge-k-sorted-lists\n    difficulty: Hard\n  - title: 'Count of Smaller Numbers After Self'\n    slug: count-of-smaller-numbers-after-self\n    difficulty: Hard\n  - title: 'Reverse Pairs'\n    slug: reverse-pairs\n    difficulty: Hard\n---\n\nSome problems are easiest to solve by reducing them to _smaller versions of\nthemselves_. This is the **divide-and-conquer** paradigm, and it is one of the\nmost productive ideas in all of algorithm design. Every divide-and-conquer\nalgorithm has the same three-part skeleton:\n\n- **Divide** the problem into one or more subproblems that are smaller\n  instances of the _same_ problem.\n- **Conquer** the subproblems by solving them recursively. When a subproblem is\n  small enough (the **base case**), solve it directly without recursing.\n- **Combine** the subproblem solutions into a solution for the original\n  problem.\n\nErickson's advice captures the mindset: assume the recursion already\nworks, so that the recursive calls correctly solve the smaller instances, and\nfocus your energy on the divide and combine steps. This \"recursion fairy\"[^erickson-rec]\nstance turns a single hard problem into two manageable questions: _how do I split?_ and\n_how do I merge?_\n\nThe payoff is always a **recurrence**. If an instance of size $n$ spawns $a$\nsubproblems each of size $n\u002Fb$, and the divide-plus-combine work costs\n$\\Theta(n^c)$, then the total cost obeys\n\n$$\nT(n) = a\\,T(n\u002Fb) + \\Theta(n^c).\n$$\n\nAlmost every algorithm in this module is an exercise in choosing $a$, $b$, and\n$c$ wisely and then reading off $T(n)$. The **master theorem** (stated at the\nend of this lesson) turns that reading-off into a mechanical three-case rule;\nthe [recursion tree](\u002Falgorithms\u002Ffoundations\u002Frecurrences) is the picture behind it. Mergesort is the cleanest first\nexample, so we start there.\n\n## The sorting problem, revisited\n\nRecall the specification from the previous module:\n\n> **Input:** a sequence $\\vector{a_1, a_2, \\dots, a_n}$ of $n$ numbers.\n> **Output:** a permutation $\\vector{a'_1, \\dots, a'_n}$ with\n> $a'_1 \\le a'_2 \\le \\cdots \\le a'_n$.\n\nInsertion sort grew a sorted prefix one element at a time, costing\n$\\Theta(n^2)$ in the worst case. Divide and conquer does much better.\nThe trick is to ask: _if I already had two sorted halves, could I finish the\njob cheaply?_ The answer, yes, by merging, gives us **mergesort**.[^clrs-merge]\n\n## Mergesort\n\nTo sort the subarray $A[p..r]$, split it at the midpoint\n$q = \\floor{(p + r)\u002F2}$, recursively sort the two halves, and merge them back\ntogether. A single element ($p \\ge r$) is already sorted, so it is the base\ncase.\n\n$$\n% caption: Mergesort on $\\langle 5,2,4,7,1,3,2,6\\rangle$. The top half divides (black\n%          arrows) down to singletons — the base cases; the bottom half merges (blue\n%          arrows) those sorted runs back up, pair by pair, to the final sorted list.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]},\n  every node\u002F.style={draw, minimum height=5mm, inner sep=2pt, font=\\scriptsize},\n  divide\u002F.style={->}, merge\u002F.style={->, acc, thick}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % --- divide (top half) ---\n  \\node (a) at (0,3) {$5\\ 2\\ 4\\ 7\\ 1\\ 3\\ 2\\ 6$};\n  \\node (b) at (-2.6,2) {$5\\ 2\\ 4\\ 7$};\n  \\node (c) at (2.6,2) {$1\\ 3\\ 2\\ 6$};\n  \\node (d) at (-3.9,1) {$5\\ 2$};\n  \\node (e) at (-1.3,1) {$4\\ 7$};\n  \\node (f) at (1.3,1) {$1\\ 3$};\n  \\node (g) at (3.9,1) {$2\\ 6$};\n  \\node[fill=acc!12] (h) at (-4.5,0) {$5$};\n  \\node[fill=acc!12] (i) at (-3.3,0) {$2$};\n  \\node[fill=acc!12] (j) at (-1.9,0) {$4$};\n  \\node[fill=acc!12] (k) at (-0.7,0) {$7$};\n  \\node[fill=acc!12] (l) at (0.7,0) {$1$};\n  \\node[fill=acc!12] (m) at (1.9,0) {$3$};\n  \\node[fill=acc!12] (n) at (3.3,0) {$2$};\n  \\node[fill=acc!12] (o) at (4.5,0) {$6$};\n  \\draw[divide] (a) -- (b); \\draw[divide] (a) -- (c);\n  \\draw[divide] (b) -- (d); \\draw[divide] (b) -- (e);\n  \\draw[divide] (c) -- (f); \\draw[divide] (c) -- (g);\n  \\draw[divide] (d) -- (h); \\draw[divide] (d) -- (i);\n  \\draw[divide] (e) -- (j); \\draw[divide] (e) -- (k);\n  \\draw[divide] (f) -- (l); \\draw[divide] (f) -- (m);\n  \\draw[divide] (g) -- (n); \\draw[divide] (g) -- (o);\n  % --- merge (bottom half) ---\n  \\node (p1) at (-3.9,-1.4) {$2\\ 5$};\n  \\node (p2) at (-1.3,-1.4) {$4\\ 7$};\n  \\node (p3) at (1.3,-1.4) {$1\\ 3$};\n  \\node (p4) at (3.9,-1.4) {$2\\ 6$};\n  \\node (q1) at (-2.6,-2.6) {$2\\ 4\\ 5\\ 7$};\n  \\node (q2) at (2.6,-2.6) {$1\\ 2\\ 3\\ 6$};\n  \\node[fill=acc!18, draw=acc, very thick] (fin) at (0,-3.8) {$1\\ 2\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7$};\n  \\draw[merge] (h) -- (p1); \\draw[merge] (i) -- (p1);\n  \\draw[merge] (j) -- (p2); \\draw[merge] (k) -- (p2);\n  \\draw[merge] (l) -- (p3); \\draw[merge] (m) -- (p3);\n  \\draw[merge] (n) -- (p4); \\draw[merge] (o) -- (p4);\n  \\draw[merge] (p1) -- (q1); \\draw[merge] (p2) -- (q1);\n  \\draw[merge] (p3) -- (q2); \\draw[merge] (p4) -- (q2);\n  \\draw[merge] (q1) -- (fin); \\draw[merge] (q2) -- (fin);\n  \\node[draw=none, black!55, font=\\scriptsize] at (-5.7,1) {divide};\n  \\node[draw=none, acc, font=\\scriptsize] at (-5.7,-2.6) {merge};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{Merge-Sort}(A, p, r)$ — sort $A[p..r]$ in increasing order\nnumber: 1\nif $p \u003C r$ then\n  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F split point\n  call $\\textsc{Merge-Sort}(A, p, q)$ \u002F\u002F sort left half\n  call $\\textsc{Merge-Sort}(A, q + 1, r)$ \u002F\u002F sort right half\n  call $\\textsc{Merge}(A, p, q, r)$ \u002F\u002F combine halves\n```\n\nAll the real work lives in the **combine** step. $\\textsc{Merge}$ takes two adjacent\nsorted runs, $A[p..q]$ and $A[q+1..r]$, and interleaves them into a single\nsorted run in place. It copies each half into a scratch array, then repeatedly\ntakes the smaller of the two front elements and writes it back.\n\n```algorithm\ncaption: $\\textsc{Merge}(A, p, q, r)$ — merge sorted $A[p..q]$ and $A[q+1..r]$\nnumber: 2\n$n_1 \\gets q - p + 1$\n$n_2 \\gets r - q$\nlet $L[1..n_1 + 1]$ and $R[1..n_2 + 1]$ be new arrays\nfor $i \\gets 1$ to $n_1$ do\n  $L[i] \\gets A[p + i - 1]$ \u002F\u002F copy left half\nfor $j \\gets 1$ to $n_2$ do\n  $R[j] \\gets A[q + j]$ \u002F\u002F copy right half\n$L[n_1 + 1] \\gets \\infty$ \u002F\u002F sentinel guards the run end\n$R[n_2 + 1] \\gets \\infty$\n$i \\gets 1$\n$j \\gets 1$\nfor $k \\gets p$ to $r$ do\n  if $L[i] \\le R[j]$ then\n    $A[k] \\gets L[i]$\n    $i \\gets i + 1$\n  else\n    $A[k] \\gets R[j]$\n    $j \\gets j + 1$\n```\n\nThe two **sentinel** values $\\infty$ are a small but useful device: once one\nhalf is used up, its front element is forever $\\infty$, so the comparison\nalways picks from the other half. This removes the need to test \"have we run\nout?\" on every iteration.\n\nPicture the merge in flight. Two sorted runs $L$ and $R$ sit above the output;\nthe cursors $i$ and $j$ point at their smallest uncopied elements, and $k$\nmarks where the next winner lands in $A$. Each step compares $L[i]$ to $R[j]$,\nwrites the smaller, and advances that one cursor.\n\n$$\n% caption: Merge step with cursors $i$ and $j$ on sorted runs $L$ and $R$ writing the\n%          smaller value into $A$ at $k$.\n\\begin{tikzpicture}[\n  font=\\small,\n  cell\u002F.style={draw, minimum width=8mm, minimum height=8mm},\n  lbl\u002F.style={font=\\scriptsize\\itshape},\n  >={Stealth[round]}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % left run L (cursor at index 3, value 11)\n  \\node[lbl] at (-1.1, 1.4) {$L$};\n  \\node[cell] at (0,1.4) {$2$};\n  \\node[cell] at (1,1.4) {$5$};\n  \\node[cell, fill=acc!12] at (2,1.4) {$11$};\n  \\node[cell] at (3,1.4) {$17$};\n  \\node[cell] at (4,1.4) {$\\infty$};\n  \\node[lbl, acc] at (2,2.3) {$i$};\n  \\draw[->, acc] (2,2.05) -- (2,1.85);\n  % right run R (cursor at index 2, value 8) -- the smaller, so the winner\n  \\node[lbl] at (-1.1, 0) {$R$};\n  \\node[cell] at (0,0) {$3$};\n  \\node[cell, fill=acc!15, draw=acc, very thick] at (1,0) {$8$};\n  \\node[cell] at (2,0) {$15$};\n  \\node[cell] at (3,0) {$\\infty$};\n  \\node[lbl, acc] at (1,-0.9) {$j$};\n  \\draw[->, acc] (1,-0.65) -- (1,-0.2);\n  % output A (filled through index 2)\n  \\node[lbl] at (-1.1, -1.8) {$A$};\n  \\node[cell, fill=black!7] at (0,-1.8) {$2$};\n  \\node[cell, fill=black!7] at (1,-1.8) {$3$};\n  \\node[cell, fill=black!7] at (2,-1.8) {$5$};\n  \\node[cell, fill=acc!15, draw=acc, very thick] (Ak) at (3,-1.8) {};\n  \\node[cell] at (4,-1.8) {};\n  \\node[lbl, acc] at (3,-2.7) {$k$};\n  \\draw[->, acc] (3,-2.45) -- (3,-2.0);\n  % the smaller of L[i]=11, R[j]=8 is 8 -> written to A[k]: clean arc from R[j] east down to A[k] top, kept left of the right margin so the caption sits clear of every cell and label\n  \\draw[->, red!75!black, thick] (1.42,0) to[out=-35, in=125] (3,-1.38);\n  \\node[lbl, red!75!black, anchor=west, align=left] at (4.5,-0.9)\n    {smaller of $L[i],R[j]$\\\\ written to $A[k]$};\n\\end{tikzpicture}\n$$\n\nHere $L[i] = 11$ and $R[j] = 8$, so $R[j]$ wins: it is written to $A[k]$ and $j$\nadvances. The two $\\infty$ sentinels guard the right ends so the comparison is\nalways well-defined.\n\n### Why merge is correct\n\nMerge runs in $\\Theta(n)$ time on $n = r - p + 1$ elements: each of the $n$\niterations of the final **for** loop does $O(1)$ work and advances exactly one\nof $i$, $j$. Correctness rests on a loop invariant:\n\n> **Invariant (Merge loop invariant).** _At the start of each iteration of the **for** loop, the subarray\n> $A[p..k-1]$ contains the $k - p$ smallest elements of $L$ and $R$, in sorted\n> order. Also $L[i]$ and $R[j]$ are the smallest elements of their arrays\n> not yet copied back._\n\n- **Initialization.** Before the first iteration $k = p$, so $A[p..k-1]$ is\n  empty: it holds the $0$ smallest elements, vacuously sorted. Since nothing\n  has been copied, $L[1]$ and $R[1]$ are indeed the smallest uncopied elements.\n- **Maintenance.** Suppose $L[i] \\le R[j]$ (the other case is symmetric). Then\n  $L[i]$ is the smallest uncopied element. Appending it to the sorted\n  $A[p..k-1]$ keeps $A[p..k]$ sorted and now containing the $k - p + 1$\n  smallest elements. Incrementing $i$ and $k$ restores the invariant.\n- **Termination.** The loop ends with $k = r + 1$, so $A[p..r]$ holds all\n  $r - p + 1$ elements in sorted order. The sentinels guarantee we never read\n  past the real data.\n\nRun the loop to completion on the two halves $\\langle 2,4,5,7\\rangle$ and\n$\\langle 1,2,3,6\\rangle$ and every element lands in its sorted slot below:\n\n$$\n% caption: The completed merge of sorted halves $\\langle 2,4,5,7\\rangle$ and\n%          $\\langle 1,2,3,6\\rangle$ interleaves into one sorted run.\n\\begin{tikzpicture}[font=\\small, >={Stealth[round]},\n  cell\u002F.style={draw, minimum width=7mm, minimum height=7mm, font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[font=\\scriptsize\\itshape] at (-1.4,1) {sorted halves};\n  \\foreach \\v\u002F\\x in {2\u002F0,4\u002F1,5\u002F2,7\u002F3} \\node[cell] at (\\x,1) {$\\v$};\n  \\foreach \\v\u002F\\x in {1\u002F4.4,2\u002F5.4,3\u002F6.4,6\u002F7.4} \\node[cell, fill=acc!12] at (\\x,1) {$\\v$};\n  \\node[font=\\scriptsize\\itshape] at (-1.4,-0.4) {merged};\n  \\foreach \\v\u002F\\x in {1\u002F0,2\u002F1,2\u002F2,3\u002F3,4\u002F4,5\u002F5,6\u002F6,7\u002F7} \\node[cell, fill=black!7] at (\\x,-0.4) {$\\v$};\n\\end{tikzpicture}\n$$\n\n## Analyzing the cost\n\nLet $T(n)$ be the worst-case running time of mergesort on $n$ elements.\nSplitting costs $\\Theta(1)$, the two recursive calls cost $2\\,T(n\u002F2)$, and the\nmerge costs $\\Theta(n)$. So\n\n$$\nT(n) = 2\\,T(n\u002F2) + \\Theta(n), \\qquad T(1) = \\Theta(1).\n$$\n\nTo see why this resolves to $\\Theta(n\\log n)$, draw the **recursion tree**.\nEach node is labeled with the _non-recursive_ work it does, the cost of its\nown merge. The root merges $n$ elements; its two children each merge $n\u002F2$; the\nnext level has four nodes each merging $n\u002F4$; and so on.\n\n$$\n% caption: Recursion tree for mergesort with _each_ node showing its merge cost, summing\n%          to $cn$ per level.\n\\begin{tikzpicture}[level distance=14mm,\n  level 1\u002F.style={sibling distance=46mm},\n  level 2\u002F.style={sibling distance=23mm},\n  every node\u002F.style={draw, rounded corners, minimum size=7mm, font=\\small}]\n  \\node {$cn$}\n    child {node {$\\tfrac{cn}{2}$}\n      child {node {$\\tfrac{cn}{4}$}}\n      child {node {$\\tfrac{cn}{4}$}}}\n    child {node {$\\tfrac{cn}{2}$}\n      child {node {$\\tfrac{cn}{4}$}}\n      child {node {$\\tfrac{cn}{4}$}}};\n\\end{tikzpicture}\n$$\n\nThe key observation: **each level sums to the same amount.** The root level is\n$cn$; the next is $2 \\cdot cn\u002F2 = cn$; the next is $4 \\cdot cn\u002F4 = cn$; in\ngeneral level $i$ has $2^i$ nodes each doing $cn\u002F2^i$ work, for a row total of\n$cn$.\n\n$$\n% caption: Doubling the node count while halving each node's work keeps every level's\n%          total fixed at $cn$; the $\\log_2 n + 1$ rows give $\\Theta(n\\log n)$.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]}, x=1cm, y=1cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[draw, fill=acc!10, minimum width=10mm] (a) at (0,2) {$cn$};\n  \\node[draw, minimum width=8mm] (b) at (-1.6,1) {$\\tfrac{cn}{2}$};\n  \\node[draw, minimum width=8mm] (c) at (1.6,1) {$\\tfrac{cn}{2}$};\n  \\node[draw, minimum width=6mm] (d) at (-2.6,0) {$\\tfrac{cn}{4}$};\n  \\node[draw, minimum width=6mm] (e) at (-0.7,0) {$\\tfrac{cn}{4}$};\n  \\node[draw, minimum width=6mm] (f) at (0.7,0) {$\\tfrac{cn}{4}$};\n  \\node[draw, minimum width=6mm] (g) at (2.6,0) {$\\tfrac{cn}{4}$};\n  \\draw[->] (a)--(b); \\draw[->] (a)--(c);\n  \\draw[->] (b)--(d); \\draw[->] (b)--(e);\n  \\draw[->] (c)--(f); \\draw[->] (c)--(g);\n  \\draw[densely dashed, acc] (3.6,2) -- (5.3,2);\n  \\draw[densely dashed, acc] (3.6,1) -- (5.3,1);\n  \\draw[densely dashed, acc] (3.6,0) -- (5.3,0);\n  \\node[anchor=west, acc] at (3.7,2.28) {row sum $=cn$};\n  \\node[anchor=west, acc] at (3.7,1.28) {row sum $=cn$};\n  \\node[anchor=west, acc] at (3.7,0.28) {row sum $=cn$};\n\\end{tikzpicture}\n$$\n\nHalving from $n$ down to the base case of $1$ takes $\\log_2 n$ steps, so\nthere are $\\log_2 n + 1$ levels. Multiplying the per-level cost by the number\nof levels:\n\n$$\nT(n) = cn \\cdot (\\log_2 n + 1) = \\Theta(n\\log n).\n$$\n\nThis is the canonical application of the **master theorem** ($a = 2$, $b = 2$,\n$f(n) = \\Theta(n)$, so $n^{\\log_b a} = n$ and we land in the balanced case),\nbut the recursion tree makes the $n\\log n$ concrete: $\\log n$ levels, $n$ work\napiece.\n\n## Stability\n\nMergesort is **stable**: equal elements keep their original relative order.\nThis falls out of the $\\le$ in $\\textsc{Merge}$: when $L[i] = R[j]$ we take from\n$L$, the _left_ (earlier) half, first. Stability matters when records are\nsorted on one key but carry others: a stable sort lets you sort by secondary\nkey, then primary key, and trust that ties on the primary preserve the\nsecondary ordering.\n\n## Mergesort versus other sorts\n\n| Property | Mergesort | Insertion sort | Heapsort | Quicksort |\n| --- | --- | --- | --- | --- |\n| Worst case | $\\Theta(n\\log n)$ | $\\Theta(n^2)$ | $\\Theta(n\\log n)$ | $\\Theta(n^2)$ |\n| Average case | $\\Theta(n\\log n)$ | $\\Theta(n^2)$ | $\\Theta(n\\log n)$ | $\\Theta(n\\log n)$ |\n| Extra space | $\\Theta(n)$ | $\\Theta(1)$ | $\\Theta(1)$ | $\\Theta(\\log n)$ |\n| Stable | yes | yes | no | no |\n| In place | no | yes | yes | yes |\n\nMergesort's worst-case guarantee and stability make it the sort of choice when\npredictability matters or when data does not fit in memory. Its sequential,\nmerge-based access pattern is ideal for sorting linked lists and for\n**external sorting** of data streamed from disk.[^skiena-sort] Its cost is the $\\Theta(n)$\nauxiliary array. [Quicksort](\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort), the subject of the next lesson, trades that\nguarantee for better constants and in-place operation.\n\n## Counting inversions\n\nHere is a problem that has nothing to do with sorting on its surface, yet falls\nto the very machinery we just built. Given a list\n$\\vector{a_1, a_2, \\dots, a_n}$, how _close to sorted_ is it? A natural measure\ncounts the pairs that are out of order.\n\n> **Input:** an array $A[1..n]$.\n> **Output:** $\\operatorname{ninv}(A)$, the number of **inversions** — pairs\n> $(i, j)$ with $1 \\le i \u003C j \\le n$ and $A[i] > A[j]$.\n\nA sorted array has zero inversions; a reverse-sorted one has the maximum,\n$\\binom{n}{2}$. (Inversion counts also drive collaborative-filtering \"how\nsimilar are two rankings?\" scores.) The brute-force algorithm loops over all\npairs and counts the bad ones, costing exactly $\\binom{n}{2} = \\tfrac12 n(n-1)\n= \\Theta(n^2)$ comparisons. We can do far better.\n\n**Idea 0: divide and conquer, just like mergesort.** Split $A$ into a left half\n$B$ and a right half $C$. Every inversion is one of three kinds:\n\n- both endpoints in $B$, counted by recursing on $B$;\n- both endpoints in $C$, counted by recursing on $C$;\n- one endpoint in each: a **cross inversion**, $i$ on the left and $j$ on the\n  right with $B[i] > C[j]$.\n\n$$\n% caption: A cross inversion linking an element in left half $B$ to a smaller element in\n%          right half $C$.\n\\begin{tikzpicture}[font=\\small, >={Stealth[round]}]\n  \\draw (0,0) rectangle (3.2,0.9); \\node at (1.6,0.6) {$B$ (left half)};\n  \\draw (3.2,0) rectangle (6.4,0.9); \\node at (4.8,0.6) {$C$ (right half)};\n  \\node[font=\\scriptsize] (b) at (1.0,0.2) {$\\bullet$};\n  \\node[font=\\scriptsize] (c) at (4.4,0.2) {$\\bullet$};\n  \\draw[->, red!70!black, thick] (b) to[bend right=30] (c);\n  \\node[red!70!black, font=\\scriptsize\\itshape] at (3.2,1.25)\n    {cross inversion: $B[i] > C[j]$};\n\\end{tikzpicture}\n$$\n\nCounting cross inversions with a double loop costs $\\Theta(n^2)$ for the combine\nstep, giving $T(n) = 2T(n\u002F2) + \\Theta(n^2)$, which the master theorem resolves\nto $\\Theta(n^2)$, no gain. The combine step is the bottleneck.\n\n**Idea 1: count cross inversions during a merge.** Suppose the two halves\narrive **already sorted**. Walk them with two cursors exactly as $\\textsc{Merge}$\ndoes. When we are about to emit and $B[i] > C[j]$, the element $C[j]$ is smaller\nthan $B[i]$ _and_ than everything after it in $B$, so $C[j]$ forms an inversion\nwith all $p - i + 1$ remaining elements of $B$ at once. Add that count, emit\n$C[j]$, and move on.\n\nThis batching is exactly why the count collapses to linear time: a single\ncomparison reveals $p - i + 1$ inversions, not one. Because $B$ is sorted, every\nelement from $B[i]$ onward exceeds $C[j]$, so each is inverted with it.\n\n$$\n% caption: When the merge finds $B[i] > C[j]$, sortedness of $B$ means every remaining\n%          $B[i..p]$ also exceeds $C[j]$ — so emitting $C[j]$ adds $p-i+1$ cross\n%          inversions in one stroke.\n\\begin{tikzpicture}[font=\\small, >={Stealth[round]},\n  cell\u002F.style={draw, minimum width=8mm, minimum height=8mm, font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % left run B (sorted), cursor i at value 6; B[i..p] = 6,8,9 all > C[j]\n  \\node[font=\\scriptsize\\itshape] at (-1.2,1.4) {$B$};\n  \\node[cell] at (0,1.4) {$2$};\n  \\node[cell] at (1,1.4) {$5$};\n  \\node[cell, fill=acc!15] at (2,1.4) {$6$};\n  \\node[cell, fill=acc!15] at (3,1.4) {$8$};\n  \\node[cell, fill=acc!15] at (4,1.4) {$9$};\n  \\node[font=\\scriptsize\\itshape, acc] at (1.4,2.15) {$i$};\n  \\draw[->, acc] (1.55,2.0) -- (1.85,1.85);\n  \\draw[acc] (1.55,2.05) -- (4.45,2.05) node[midway, above, font=\\scriptsize] {$B[i..p]$, all $> C[j]$};\n  % right run C, cursor j at value 4 (the small one being emitted)\n  \\node[font=\\scriptsize\\itshape] at (-1.2,-1.4) {$C$};\n  \\node[cell, fill=acc!15, draw=acc, very thick] at (2,-1.4) {$4$};\n  \\node[cell] at (3,-1.4) {$7$};\n  \\node[font=\\scriptsize\\itshape, acc] at (2,-2.25) {$j$};\n  \\draw[->, acc] (2,-2.0) -- (2,-1.85);\n  % red fan from C[j] up to each of the three B elements: p-i+1 = 3 inversions at once\n  \\foreach \\tx in {2,3,4} \\draw[->, red!75!black, thick] (2,-1.05) -- (\\tx,1.05);\n  \\node[font=\\scriptsize, red!75!black, anchor=west, align=left] at (5.0,-0.0)\n    {$C[j]$ inverts with all\\\\ $p-i+1=3$ of $B[i..p]$};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{Count-Cross-Inv}(B[1..p], C[1..q])$ — cross inversions, $B, C$ sorted\nnumber: 3\n$\\mathit{ans} \\gets 0$\n$i \\gets 1$\n$j \\gets 1$\nwhile $i \\le p$ and $j \\le q$ do\n  if $B[i] \\le C[j]$ then\n    $i \\gets i + 1$ \u002F\u002F no inversion\n  else\n    $\\mathit{ans} \\gets \\mathit{ans} + (p - i + 1)$ \u002F\u002F $C[j]$ inverts with $B[i..p]$\n    $j \\gets j + 1$\nreturn $\\mathit{ans}$\n```\n\nThis runs in $\\Theta(p + q) = \\Theta(n)$, the linear merge pattern. But it\ndemands sorted halves, so we must sort them first: sorting $B$ and $C$ costs an\nextra $\\Theta(n \\log n)$ _per level_, and there are $\\log n$ levels, giving\n$T(n) = 2T(n\u002F2) + \\Theta(n \\log n) = \\Theta(n \\log^2 n)$. Better than\nquadratic, but the repeated sorting is wasteful.\n\n**Idea 2: sort *and* count in one pass.** We are doing almost all of\nmergesort's work anyway, so let the recursion return both\nthe inversion count _and_ a sorted copy of its slice. Then the cross-counting\nmerge also produces the sorted output the parent needs, for free.\n\n```algorithm\ncaption: $\\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, \\mathit{hi})$ — sort $A[\\mathit{lo}..\\mathit{hi}]$, return its inversion count\nnumber: 4\nif $\\mathit{hi} \\le \\mathit{lo}$ then\n  return $0$ \u002F\u002F single element: no inversions\n$t \\gets \\floor{(\\mathit{lo} + \\mathit{hi}) \u002F 2}$\n$c \\gets \\textsc{Sort-And-Count-Inv}(A, \\mathit{lo}, t)$ \u002F\u002F left inversions + sort left\n$c \\gets c + \\textsc{Sort-And-Count-Inv}(A, t + 1, \\mathit{hi})$ \u002F\u002F right inversions + sort right\n$c \\gets c + \\textsc{Count-Cross-Inv-And-Merge}(A, \\mathit{lo}, t, \\mathit{hi})$ \u002F\u002F cross + merge\nreturn $c$\n```\n\nThe helper $\\textsc{Count-Cross-Inv-And-Merge}$ is just $\\textsc{Merge}$ with the\ncounting rule from $\\textsc{Count-Cross-Inv}$ folded in: whenever it takes from\nthe right half because $A[\\mathit{mid} + j] \u003C A[i]$, it adds the number of\nelements still waiting in the left half. The combine step is now plain linear,\nso\n\n$$\nT(n) = 2\\,T(n\u002F2) + \\Theta(n) = \\Theta(n \\log n).\n$$\n\nThis is the same recurrence as mergesort, and the same recursion tree explains\nit: $\\log n$ levels, $\\Theta(n)$ work each. Counting how disordered a list is\ncosts no more, [asymptotically](\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis), than sorting it.\n\n## Karatsuba multiplication: the same idea elsewhere\n\nSorting is not the only home for divide and conquer. Take a problem that feels\nunrelated: multiplying two large integers. Adding two $n$-bit numbers is easy.\nThe grade-school ripple-carry method is $\\Theta(n)$, and you cannot beat linear\nsince you must at least read the input. Multiplication is the interesting one.\nThe grade-school algorithm forms $n$ shifted partial products and adds them,\ncosting $\\Theta(n^2)$. Can divide and conquer do better?\n\nWrite each $n$-bit integer as a high half and a low half. With $n = 2t$, split\n$x$ into its least-significant $t$ bits $a$ and its top $t$ bits $b$, and\nlikewise split $y$ into $c$ and $d$:\n\n$$\n% caption: Splitting each $n$-bit integer $x$ and $y$ into high and low $t$-bit halves for\n%          Karatsuba.\n\\begin{tikzpicture}[font=\\small, >={Stealth[round]},\n  half\u002F.style={draw, minimum width=24mm, minimum height=8mm, inner xsep=6pt, outer sep=0}]\n  % x\n  \\node at (-1.4,0.4) {$x =$};\n  \\node[half, anchor=west] (xb) at (0,0.4) {$b$ (high $t$ bits)};\n  \\node[half, anchor=west] (xa) at (xb.east) {$a$ (low $t$ bits)};\n  % y\n  \\node at (-1.4,-1.0) {$y =$};\n  \\node[half, anchor=west] (yd) at (0,-1.0) {$d$ (high $t$ bits)};\n  \\node[half, anchor=west] (yc) at (yd.east) {$c$ (low $t$ bits)};\n  \\node[font=\\scriptsize\\itshape, anchor=west] at ($(xa.east)+(8mm,0)$) {$x = a + 2^{t} b$};\n  \\node[font=\\scriptsize\\itshape, anchor=west] at ($(yc.east)+(8mm,0)$) {$y = c + 2^{t} d$};\n\\end{tikzpicture}\n$$\n\nMultiplying out $x = a + 2^t b$ and $y = c + 2^t d$ gives\n\n$$\nxy = ac + 2^{t}(ad + bc) + 2^{2t}\\,bd .\n$$\n\nThe powers of two are just bit-shifts (free, up to the additions), so the cost\nis dominated by the **four** half-size products $ac$, $ad$, $bc$, $bd$. That\nyields $T(n) \\le 4\\,T(n\u002F2) + bn$. Unrolling shows this is no improvement at all:\n\n$$\n\\begin{aligned}\nT(n) &\\le 4\\,T(n\u002F2) + bn \\le 4\\big(4\\,T(n\u002F4) + b\\tfrac n2\\big) + bn\n   = 16\\,T(n\u002F4) + 2bn + bn \\\\\n&\\le \\cdots \\le 4^{i}\\,T(n\u002F2^{i}) + \\underbrace{\\big(bn + 2bn + \\cdots +\n   2^{\\,i-1}bn\\big)}_{(2^{i}-1)\\,bn} .\n\\end{aligned}\n$$\n\nAt $i = \\log_2 n$ the leaf term is $4^{\\log_2 n}\\,T(1) = n^{2}\\,T(1)$, which\ndominates everything: $T(n) = \\Theta(n^2)$. Splitting bought us nothing, because\n$a = 4$ subproblems of half size is exactly the quadratic regime.\n\n**Karatsuba's trick** is to compute the middle coefficient $ad + bc$ _without_ a\nthird and fourth multiplication. Notice the algebraic identity\n\n$$\n(a - b)(d - c) = ad - ac - bd + bc,\n\\qquad\\text{so}\\qquad\nad + bc = (a - b)(d - c) + ac + bd .\n$$\n\nWe were going to compute $ac$ and $bd$ anyway. So once we have those two\nproducts, the middle term needs only **one** more multiplication,\n$(a - b)(d - c)$, plus a few additions. Three half-size products suffice:\n\n```algorithm\ncaption: $\\textsc{Karatsuba}(x, y)$ — multiply two $n$-bit integers\nnumber: 5\nif $n = 1$ then\n  return $x \\cdot y$ \u002F\u002F base case\n$t \\gets \\floor{n \u002F 2}$\nsplit $x = a + 2^{t} b$ and $y = c + 2^{t} d$ \u002F\u002F low and high halves\n$\\mathit{ac} \\gets \\textsc{Karatsuba}(a, c)$ \u002F\u002F recursive call 1\n$\\mathit{bd} \\gets \\textsc{Karatsuba}(b, d)$ \u002F\u002F recursive call 2\n$m \\gets \\textsc{Karatsuba}(a - b, d - c)$ \u002F\u002F recursive call 3 — the trick\n$\\mathit{mid} \\gets m + \\mathit{ac} + \\mathit{bd}$ \u002F\u002F recovers $ad + bc$\nreturn $\\mathit{ac} + 2^{t}\\,\\mathit{mid} + 2^{2t}\\,\\mathit{bd}$ \u002F\u002F shift and add\n```\n\nNow $a = 3$ recursive calls, each on half-size inputs, with $\\Theta(n)$ work to\nsplit, shift, and add:\n\n$$\nT(n) = 3\\,T(n\u002F2) + \\Theta(n).\n$$\n\nThe recursion tree makes the cost legible. Level $i$ has $3^{i}$ nodes, each\ndoing $bn\u002F2^{i}$ work, for a row total of $(3\u002F2)^{i}\\,bn$. The rows _grow_\ngeometrically downward, so the leaves dominate:\n\n$$\n% caption: Each Karatsuba node spawns $3$ half-size children, so the rows grow by $3\u002F2$\n%          downward and the $\\Theta(n^{\\log_2 3})$ leaf level dominates.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]}, level distance=12mm,\n  level 1\u002F.style={sibling distance=26mm},\n  level 2\u002F.style={sibling distance=8.5mm}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\tikzstyle{every node}=[draw, circle, inner sep=1.6pt, fill=acc!12, minimum size=4.5mm]\n  \\tikzstyle{lf}=[draw, circle, fill=acc!30, minimum size=2.6mm, inner sep=0pt]\n  \\node {$n$}\n    child {node {$\\tfrac n2$} child {node[lf]{}} child {node[lf]{}} child {node[lf]{}}}\n    child {node {$\\tfrac n2$} child {node[lf]{}} child {node[lf]{}} child {node[lf]{}}}\n    child {node {$\\tfrac n2$} child {node[lf]{}} child {node[lf]{}} child {node[lf]{}}};\n\\end{tikzpicture}\n$$\n\n$$\nT(n) \\le bn\\sum_{i=0}^{\\log_2 n} \\Big(\\tfrac32\\Big)^{i}\n   = \\Theta\\!\\left(\\Big(\\tfrac32\\Big)^{\\log_2 n}\\right) \\cdot bn\n   = \\Theta\\!\\left(n^{\\log_2 3}\\right) \\approx \\Theta(n^{1.585}),\n$$\n\nusing $n^{\\log_2 3} = (3\u002F2)^{\\log_2 n}\\cdot n$ (the identity\n$x^{\\log_a y} = y^{\\log_a x}$). Trading one multiplication for a handful of\nadditions drops integer multiplication below quadratic. The lesson generalizes:\nthe _number_ of subproblems, not just their size, drives the asymptotics, which\nis exactly what the master theorem makes precise. (Pushing the same idea to\nmore than two chunks leads to the **Fast Fourier Transform** and\n$O(n \\log n \\log\\log n)$ multiplication.)\n\n## Strassen's matrix multiplication\n\nThe same \"spend additions to save a multiplication\" idea cracks the cubic barrier\nfor **matrix multiplication**. The schoolbook method costs $\\Theta(n^3)$: each of\nthe $n^2$ entries of $C = AB$ is a length-$n$ dot product. Divide and conquer\nsplits each $n \\times n$ matrix into four $(n\u002F2) \\times (n\u002F2)$ blocks, and the\nproduct is read off block by block exactly as for $2 \\times 2$ scalars:\n\n$$\nA = \\begin{pmatrix} A_{11} & A_{12} \\\\ A_{21} & A_{22} \\end{pmatrix},\\quad\nB = \\begin{pmatrix} B_{11} & B_{12} \\\\ B_{21} & B_{22} \\end{pmatrix},\\quad\nC = \\begin{pmatrix}\nA_{11}B_{11}+A_{12}B_{21} & A_{11}B_{12}+A_{12}B_{22} \\\\\nA_{21}B_{11}+A_{22}B_{21} & A_{21}B_{12}+A_{22}B_{22}\n\\end{pmatrix}.\n$$\n\nTaken at face value, the four output blocks need **eight** block multiplications,\ngiving $T(n) = 8\\,T(n\u002F2) + \\Theta(n^2)$. The master theorem (next section) returns\n$\\Theta(n^{\\log_2 8}) = \\Theta(n^3)$: more subproblems exactly cancel their smaller\nsize, so blocking alone buys nothing.\n\nStrassen's 1969 insight is that **seven** products suffice.[^strassen] Form seven\nrecursive $(n\u002F2)$-size multiplications,\n\n$$\n\\begin{aligned}\nM_1 &= (A_{11}+A_{22})(B_{11}+B_{22}), & M_5 &= (A_{11}+A_{12})\\,B_{22}, \\\\\nM_2 &= (A_{21}+A_{22})\\,B_{11},        & M_6 &= (A_{21}-A_{11})(B_{11}+B_{12}), \\\\\nM_3 &= A_{11}\\,(B_{12}-B_{22}),        & M_7 &= (A_{12}-A_{22})(B_{21}+B_{22}), \\\\\nM_4 &= A_{22}\\,(B_{21}-B_{11}),        &     &\n\\end{aligned}\n$$\n\nand recover the output blocks by **addition only**:\n\n$$\n\\begin{aligned}\nC_{11} &= M_1+M_4-M_5+M_7, & C_{12} &= M_3+M_5, \\\\\nC_{21} &= M_2+M_4,        & C_{22} &= M_1-M_2+M_3+M_6.\n\\end{aligned}\n$$\n\nThe eighth multiplication is gone, traded for a handful of extra block additions\nthat cost only $\\Theta(n^2)$. The recurrence becomes $T(n) = 7\\,T(n\u002F2) +\n\\Theta(n^2)$, and now the leaves win:\n\n$$\nT(n) = \\Theta\\!\\left(n^{\\log_2 7}\\right) \\approx \\Theta\\!\\left(n^{2.807}\\right).\n$$\n\n$$\n% caption: Each Strassen node makes $7$ recursive half-size multiplications instead of the\n%          naive $8$ — the fan-out, not the block size, is what drops the cost from\n%          $\\Theta(n^3)$ to $\\Theta(n^{\\log_2 7})$.\n\\begin{tikzpicture}[every node\u002F.style={circle, draw, minimum size=7mm, font=\\scriptsize}, >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[fill=acc!15] (r) at (0,1.6) {$n$};\n  \\foreach \\i in {0,...,6} {\n    \\node (c\\i) at ({(\\i-3)*1.05},0) {$\\tfrac n2$};\n    \\draw[->] (r) -- (c\\i);\n  }\n  \\node[draw=none, font=\\footnotesize, acc, anchor=west] at (3.7,0.8) {$7$ products};\n\\end{tikzpicture}\n$$\n\nThe constant is large, so the crossover with the cubic method only pays off for\nsizable $n$; in practice Strassen is switched in above a threshold and the base\ncase falls back to the cache-friendly schoolbook multiply. Theoretically, though,\nit was the first crack in the $\\Theta(n^3)$ wall, and a long line of refinements\nhas pushed the exponent down toward $2.37$.\n\n## The master theorem\n\nEvery recurrence in this lesson has the form $T(n) = a\\,T(n\u002Fb) + \\Theta(n^{c})$.\nThe recursion-tree analysis we did by hand each time generalizes to a single\nrule. Compare the **branching exponent** $\\log_b a$, the rate at which leaves\nproliferate, against the **work exponent** $c$:\n\n$$\nT(n) = a\\,T(n\u002Fb) + \\Theta(n^{c}) \\quad\\Longrightarrow\\quad\nT(n) =\n\\begin{cases}\n\\Theta\\!\\left(n^{\\log_b a}\\right) & \\text{if } \\log_b a > c\n  & \\text{(leaf-heavy)} \\\\[2pt]\n\\Theta\\!\\left(n^{c}\\log n\\right) & \\text{if } \\log_b a = c\n  & \\text{(balanced)} \\\\[2pt]\n\\Theta\\!\\left(n^{c}\\right) & \\text{if } \\log_b a \u003C c\n  & \\text{(root-heavy)} .\n\\end{cases}\n$$\n\nThe three cases are exactly the three shapes of recursion tree: when\n$\\log_b a > c$ the rows grow toward the leaves (Karatsuba), when they are equal\nevery row costs the same (mergesort), and when $\\log_b a \u003C c$ the root's work\ndominates. Reading off our examples:\n\n| Algorithm | Recurrence | $a$ | $b$ | $c$ | $\\log_b a$ vs $c$ | $T(n)$ |\n| --- | --- | --- | --- | --- | --- | --- |\n| Mergesort | $2T(n\u002F2) + \\Theta(n)$ | $2$ | $2$ | $1$ | $1 = 1$, balanced | $\\Theta(n\\log n)$ |\n| Counting inversions | $2T(n\u002F2) + \\Theta(n)$ | $2$ | $2$ | $1$ | $1 = 1$, balanced | $\\Theta(n\\log n)$ |\n| Inversions, naive combine | $2T(n\u002F2) + \\Theta(n^2)$ | $2$ | $2$ | $2$ | $1 \u003C 2$, root-heavy | $\\Theta(n^2)$ |\n| Karatsuba | $3T(n\u002F2) + \\Theta(n)$ | $3$ | $2$ | $1$ | $\\log_2 3 > 1$, leaf-heavy | $\\Theta(n^{\\log_2 3})$ |\n\nOne last sanity check worth stressing: it makes no difference whether the\ncombine cost is written $\\Theta(n)$ or bounded above by $O(n)$. The recurrences\n$T(n) = 2T(n\u002F2) + \\Theta(n)$ and $T(n) \\le 2T(n\u002F2) + O(n)$ have the same\nsolution. The master theorem cares only about $a$, $b$, and the exponent $c$.\n\n## Takeaways\n\n- **Divide and conquer** = divide into smaller copies, conquer recursively,\n  combine. Trust the recursion; focus on the split and the merge. The cost is\n  always a recurrence $T(n) = a\\,T(n\u002Fb) + \\Theta(n^c)$.\n- $\\textsc{Mergesort}$ divides at the midpoint and combines with a linear-time\n  $\\textsc{Merge}$ whose correctness is a clean loop-invariant argument.\n- The recurrence $T(n) = 2T(n\u002F2) + \\Theta(n)$ unfolds into a recursion tree\n  with $\\log n$ levels of $\\Theta(n)$ work each, giving $\\Theta(n\\log n)$.\n- Mergesort is **stable** and worst-case optimal among [comparison sorts](\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds), at the\n  cost of $\\Theta(n)$ extra space, ideal for linked lists and external sorting.\n- **Counting inversions** reuses the merge: fold a cross-inversion count into\n  $\\textsc{Merge}$ so each step adds $p - i + 1$, sorting and counting together\n  in $\\Theta(n\\log n)$ instead of the brute-force $\\Theta(n^2)$.\n- **Karatsuba** multiplication shows the paradigm's reach: the identity\n  $ad + bc = (a-b)(d-c) + ac + bd$ cuts four subproblems to three, taking\n  integer multiplication from $\\Theta(n^2)$ to $\\Theta(n^{\\log_2 3})$.\n- The **master theorem** turns the tree into a rule: compare $\\log_b a$ to $c$\n  for leaf-heavy, balanced, or root-heavy behavior.[^clrs-master]\n\n[^erickson-rec]: **Erickson**, _Algorithms_, Ch. 1 — Recursion: the \"recursion fairy\" stance of assuming recursive calls already work and focusing on divide and combine.\n[^clrs-merge]: **CLRS**, Ch. 2 (§2.3) — Designing algorithms: mergesort as the canonical divide-and-conquer sort built on a linear-time merge.\n[^skiena-sort]: **Skiena**, _The Algorithm Design Manual_, §4 — Sorting and Searching: mergesort's stability and suitability for linked lists and external sorting.\n[^clrs-master]: **CLRS**, Ch. 4 — Divide-and-Conquer: the master theorem comparing $\\log_b a$ against the work exponent $c$ to classify leaf-heavy, balanced, and root-heavy recurrences.\n[^strassen]: **CLRS**, Ch. 4 (§4.2) — Strassen's algorithm for matrix multiplication and its $\\Theta(n^{\\log_2 7})$ recurrence; the original result is V. Strassen (1969), \"Gaussian elimination is not optimal,\" _Numerische Mathematik_ 13, 354–356.\n",{"text":24856,"minutes":24857,"time":24858,"words":24859},"14 min read",13.14,788400,2628,{"title":33,"description":24830},[24862,24864,24866],{"book":24582,"ref":24863},"Ch. 2 & Ch. 4 — Getting Started; Divide-and-Conquer",{"book":24594,"ref":24865},"§4 — Sorting and Searching",{"book":24562,"ref":24867},"Ch. 1 — 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