[{"data":1,"prerenderedAt":14347},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Ffoundations\u002Frecurrences":374,"course-wordcounts":14216,"ref-card-index":14272},[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":22,"blurb":376,"body":377,"description":14178,"extension":14179,"meta":14180,"module":5,"navigation":14182,"path":23,"practice":14183,"rawbody":14198,"readingTime":14199,"seo":14204,"sources":14205,"status":14212,"stem":14213,"summary":27,"topics":14214,"__hash__":14215},"course\u002F01.algorithms\u002F01.foundations\u002F03.recurrences.md","",{"type":378,"value":379,"toc":14166},"minimark",[380,443,448,495,545,618,637,766,769,1045,1056,1232,1324,1328,1363,1469,1816,2150,2664,2795,2818,2822,2848,3173,3437,3443,4221,4479,4877,5015,5018,5213,5217,5694,5772,5866,5945,6362,6831,7265,7480,7484,7491,7590,7748,8061,8491,9715,9918,10180,10183,10352,10357,10920,10992,10995,11459,11525,11953,12039,12084,12955,13038,13041,13045,13055,13260,13263,13267,13966,14162],[381,382,383,384,388,389,414,415,419,420,424,425,442],"p",{},"When an algorithm solves a problem by calling smaller copies of itself, its\nrunning time obeys an equation that refers to ",[385,386,387],"em",{},"itself",": the cost on an input of\nsize ",[390,391,394],"span",{"className":392},[393],"katex",[390,395,399],{"className":396,"ariaHidden":398},[397],"katex-html","true",[390,400,403,408],{"className":401},[402],"base",[390,404],{"className":405,"style":407},[406],"strut","height:0.4306em;",[390,409,413],{"className":410},[411,412],"mord","mathnormal","n"," is some local work plus the cost of the recursive calls on smaller\ninputs. Such an equation is a ",[416,417,418],"strong",{},"recurrence",". Counting loops, as in ",[421,422,423],"a",{"href":17},"the previous\nlesson",", no longer suffices; we need\ntechniques to turn a recurrence into a closed ",[390,426,428],{"className":427},[393],[390,429,431],{"className":430,"ariaHidden":398},[397],[390,432,434,438],{"className":433},[402],[390,435],{"className":436,"style":437},[406],"height:0.6833em;",[390,439,441],{"className":440},[411],"Θ","-bound. This lesson develops\nthree, in increasing order of power and precision.",[444,445,447],"h2",{"id":446},"from-a-recursive-algorithm-to-a-recurrence","From a recursive algorithm to a recurrence",[381,449,450,476,477,480,481,484,485,488,489,494],{},[390,451,453],{"className":452},[393],[390,454,456],{"className":455,"ariaHidden":398},[397],[390,457,459,463],{"className":458},[402],[390,460],{"className":461,"style":462},[406],"height:0.8889em;vertical-align:-0.1944em;",[390,464,468],{"className":465},[466,467],"enclosing","textsc",[390,469,472],{"className":470},[411,471],"text",[390,473,475],{"className":474},[411],"Divide-and-conquer"," is the paradigm all three books use to introduce\nrecurrences. It has three steps: ",[416,478,479],{},"divide"," the instance into subproblems,\n",[416,482,483],{},"conquer"," them by recursion, and ",[416,486,487],{},"combine"," their solutions. ",[421,490,491],{"href":34},[416,492,493],{},"Merge sort","\nsplits the array in half, sorts each half recursively, and merges the two sorted\nhalves.",[496,497,501],"pre",{"className":498,"code":499,"language":500,"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 $A[p..r]$\nnumber: 1\nif $p \u003C r$ then\n  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F midpoint\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\nreturn $A$\n","algorithm",[502,503,504,510,515,520,525,530,535,540],"code",{"__ignoreMap":376},[390,505,507],{"class":506,"line":6},"line",[390,508,509],{},"caption: $\\textsc{Merge-Sort}(A, p, r)$ — sort $A[p..r]$\n",[390,511,512],{"class":506,"line":18},[390,513,514],{},"number: 1\n",[390,516,517],{"class":506,"line":24},[390,518,519],{},"if $p \u003C r$ then\n",[390,521,522],{"class":506,"line":73},[390,523,524],{},"  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F midpoint\n",[390,526,527],{"class":506,"line":102},[390,528,529],{},"  call $\\textsc{Merge-Sort}(A, p, q)$ \u002F\u002F sort left half\n",[390,531,532],{"class":506,"line":108},[390,533,534],{},"  call $\\textsc{Merge-Sort}(A, q+1, r)$ \u002F\u002F sort right half\n",[390,536,537],{"class":506,"line":116},[390,538,539],{},"  call $\\textsc{Merge}(A, p, q, r)$ \u002F\u002F combine halves\n",[390,541,542],{"class":506,"line":196},[390,543,544],{},"return $A$\n",[381,546,547,548,571,572,587,588,617],{},"The ",[390,549,551],{"className":550},[393],[390,552,554],{"className":553,"ariaHidden":398},[397],[390,555,557,561],{"className":556},[402],[390,558],{"className":559,"style":560},[406],"height:0.8778em;vertical-align:-0.1944em;",[390,562,564],{"className":563},[466,467],[390,565,567],{"className":566},[411,471],[390,568,570],{"className":569},[411],"Merge"," subroutine walks the two sorted halves with two pointers, repeatedly\ncopying the smaller front element into the output. It touches each of the ",[390,573,575],{"className":574},[393],[390,576,578],{"className":577,"ariaHidden":398},[397],[390,579,581,584],{"className":580},[402],[390,582],{"className":583,"style":407},[406],[390,585,413],{"className":586},[411,412],"\nelements a constant number of times, so it costs ",[390,589,591],{"className":590},[393],[390,592,594],{"className":593,"ariaHidden":398},[397],[390,595,597,601,604,609,612],{"className":596},[402],[390,598],{"className":599,"style":600},[406],"height:1em;vertical-align:-0.25em;",[390,602,441],{"className":603},[411],[390,605,608],{"className":606},[607],"mopen","(",[390,610,413],{"className":611},[411,412],[390,613,616],{"className":614},[615],"mclose",")",".",[381,619,620,621,636],{},"Now read the cost off the structure. On an array of size ",[390,622,624],{"className":623},[393],[390,625,627],{"className":626,"ariaHidden":398},[397],[390,628,630,633],{"className":629},[402],[390,631],{"className":632,"style":407},[406],[390,634,413],{"className":635},[411,412],":",[638,639,640,672,736],"ul",{},[641,642,643,646,647,617],"li",{},[416,644,645],{},"Divide"," is computing the midpoint, ",[390,648,650],{"className":649},[393],[390,651,653],{"className":652,"ariaHidden":398},[397],[390,654,656,659,662,665,669],{"className":655},[402],[390,657],{"className":658,"style":600},[406],[390,660,441],{"className":661},[411],[390,663,608],{"className":664},[607],[390,666,668],{"className":667},[411],"1",[390,670,616],{"className":671},[615],[641,673,674,677,678,697,698,617],{},[416,675,676],{},"Conquer"," is two recursive calls, each on ",[390,679,681],{"className":680},[393],[390,682,684],{"className":683,"ariaHidden":398},[397],[390,685,687,690,693],{"className":686},[402],[390,688],{"className":689,"style":600},[406],[390,691,413],{"className":692},[411,412],[390,694,696],{"className":695},[411],"\u002F2"," elements, costing ",[390,699,701],{"className":700},[393],[390,702,704],{"className":703,"ariaHidden":398},[397],[390,705,707,710,714,719,724,727,730,733],{"className":706},[402],[390,708],{"className":709,"style":600},[406],[390,711,713],{"className":712},[411],"2",[390,715],{"className":716,"style":718},[717],"mspace","margin-right:0.1667em;",[390,720,723],{"className":721,"style":722},[411,412],"margin-right:0.1389em;","T",[390,725,608],{"className":726},[607],[390,728,413],{"className":729},[411,412],[390,731,696],{"className":732},[411],[390,734,616],{"className":735},[615],[641,737,738,741,742,617],{},[416,739,740],{},"Combine"," is the merge, ",[390,743,745],{"className":744},[393],[390,746,748],{"className":747,"ariaHidden":398},[397],[390,749,751,754,757,760,763],{"className":750},[402],[390,752],{"className":753,"style":600},[406],[390,755,441],{"className":756},[411],[390,758,608],{"className":759},[607],[390,761,413],{"className":762},[411,412],[390,764,616],{"className":765},[615],[381,767,768],{},"Adding these (and noting that a one-element array is sorted at no cost) gives the\nrecurrence",[390,770,773],{"className":771},[772],"katex-display",[390,774,776],{"className":775},[393],[390,777,779,809],{"className":778,"ariaHidden":398},[397],[390,780,782,785,788,791,794,797,801,806],{"className":781},[402],[390,783],{"className":784,"style":600},[406],[390,786,723],{"className":787,"style":722},[411,412],[390,789,608],{"className":790},[607],[390,792,413],{"className":793},[411,412],[390,795,616],{"className":796},[615],[390,798],{"className":799,"style":800},[717],"margin-right:0.2778em;",[390,802,805],{"className":803},[804],"mrel","=",[390,807],{"className":808,"style":800},[717],[390,810,812,816],{"className":811},[402],[390,813],{"className":814,"style":815},[406],"height:3.08em;vertical-align:-1.29em;",[390,817,820,831,1041],{"className":818},[819],"minner",[390,821,825],{"className":822,"style":824},[607,823],"delimcenter","top:0em;",[390,826,830],{"className":827},[828,829],"delimsizing","size4","{",[390,832,834],{"className":833},[411],[390,835,838,947,952],{"className":836},[837],"mtable",[390,839,842],{"className":840},[841],"col-align-l",[390,843,847,938],{"className":844},[845,846],"vlist-t","vlist-t2",[390,848,851,933],{"className":849},[850],"vlist-r",[390,852,856,879],{"className":853,"style":855},[854],"vlist","height:1.79em;",[390,857,859,864],{"style":858},"top:-3.79em;",[390,860],{"className":861,"style":863},[862],"pstrut","height:3.008em;",[390,865,867,870,873,876],{"className":866},[411],[390,868,441],{"className":869},[411],[390,871,608],{"className":872},[607],[390,874,668],{"className":875},[411],[390,877,616],{"className":878},[615],[390,880,882,885],{"style":881},"top:-2.15em;",[390,883],{"className":884,"style":863},[862],[390,886,888,891,894,897,900,903,906,909,913,918,921,924,927,930],{"className":887},[411],[390,889,713],{"className":890},[411],[390,892],{"className":893,"style":718},[717],[390,895,723],{"className":896,"style":722},[411,412],[390,898,608],{"className":899},[607],[390,901,413],{"className":902},[411,412],[390,904,696],{"className":905},[411],[390,907,616],{"className":908},[615],[390,910],{"className":911,"style":912},[717],"margin-right:0.2222em;",[390,914,917],{"className":915},[916],"mbin","+",[390,919],{"className":920,"style":912},[717],[390,922,441],{"className":923},[411],[390,925,608],{"className":926},[607],[390,928,413],{"className":929},[411,412],[390,931,616],{"className":932},[615],[390,934,937],{"className":935},[936],"vlist-s","​",[390,939,941],{"className":940},[850],[390,942,945],{"className":943,"style":944},[854],"height:1.29em;",[390,946],{},[390,948],{"className":949,"style":951},[950],"arraycolsep","width:1em;",[390,953,955],{"className":954},[841],[390,956,958,1033],{"className":957},[845,846],[390,959,961,1030],{"className":960},[850],[390,962,964,999],{"className":963,"style":855},[854],[390,965,966,969],{"style":858},[390,967],{"className":968,"style":863},[862],[390,970,972,979,982,985,988,991,994],{"className":971},[411],[390,973,975],{"className":974},[411,471],[390,976,978],{"className":977},[411],"if ",[390,980,413],{"className":981},[411,412],[390,983],{"className":984,"style":800},[717],[390,986,805],{"className":987},[804],[390,989],{"className":990,"style":800},[717],[390,992,668],{"className":993},[411],[390,995,998],{"className":996},[997],"mpunct",",",[390,1000,1001,1004],{"style":881},[390,1002],{"className":1003,"style":863},[862],[390,1005,1007,1013,1016,1019,1023,1026],{"className":1006},[411],[390,1008,1010],{"className":1009},[411,471],[390,1011,978],{"className":1012},[411],[390,1014,413],{"className":1015},[411,412],[390,1017],{"className":1018,"style":800},[717],[390,1020,1022],{"className":1021},[804],">",[390,1024],{"className":1025,"style":800},[717],[390,1027,1029],{"className":1028},[411],"1.",[390,1031,937],{"className":1032},[936],[390,1034,1036],{"className":1035},[850],[390,1037,1039],{"className":1038,"style":944},[854],[390,1040],{},[390,1042],{"className":1043},[615,1044],"nulldelimiter",[381,1046,1047,1048,1051,1052,1055],{},"We can write this compactly, and as an ",[385,1049,1050],{},"inequality"," (since the\ncombine step costs ",[385,1053,1054],{},"at most"," linear), as",[390,1057,1059],{"className":1058},[772],[390,1060,1062],{"className":1061},[393],[390,1063,1065,1093,1129,1165,1192],{"className":1064,"ariaHidden":398},[397],[390,1066,1068,1071,1074,1077,1080,1083,1086,1090],{"className":1067},[402],[390,1069],{"className":1070,"style":600},[406],[390,1072,723],{"className":1073,"style":722},[411,412],[390,1075,608],{"className":1076},[607],[390,1078,413],{"className":1079},[411,412],[390,1081,616],{"className":1082},[615],[390,1084],{"className":1085,"style":800},[717],[390,1087,1089],{"className":1088},[804],"≤",[390,1091],{"className":1092,"style":800},[717],[390,1094,1096,1099,1102,1105,1108,1111,1114,1117,1120,1123,1126],{"className":1095},[402],[390,1097],{"className":1098,"style":600},[406],[390,1100,713],{"className":1101},[411],[390,1103],{"className":1104,"style":718},[717],[390,1106,723],{"className":1107,"style":722},[411,412],[390,1109,608],{"className":1110},[607],[390,1112,413],{"className":1113},[411,412],[390,1115,696],{"className":1116},[411],[390,1118,616],{"className":1119},[615],[390,1121],{"className":1122,"style":912},[717],[390,1124,917],{"className":1125},[916],[390,1127],{"className":1128,"style":912},[717],[390,1130,1132,1135,1140,1143,1146,1149,1152,1155,1159,1162],{"className":1131},[402],[390,1133],{"className":1134,"style":600},[406],[390,1136,1139],{"className":1137,"style":1138},[411,412],"margin-right:0.0278em;","O",[390,1141,608],{"className":1142},[607],[390,1144,413],{"className":1145},[411,412],[390,1147,616],{"className":1148},[615],[390,1150],{"className":1151,"style":800},[717],[390,1153],{"className":1154,"style":800},[717],[390,1156,1158],{"className":1157},[804],"⟹",[390,1160],{"className":1161,"style":800},[717],[390,1163],{"className":1164,"style":800},[717],[390,1166,1168,1171,1174,1177,1180,1183,1186,1189],{"className":1167},[402],[390,1169],{"className":1170,"style":600},[406],[390,1172,723],{"className":1173,"style":722},[411,412],[390,1175,608],{"className":1176},[607],[390,1178,413],{"className":1179},[411,412],[390,1181,616],{"className":1182},[615],[390,1184],{"className":1185,"style":800},[717],[390,1187,805],{"className":1188},[804],[390,1190],{"className":1191,"style":800},[717],[390,1193,1195,1198,1201,1204,1207,1210,1220,1223,1226,1229],{"className":1194},[402],[390,1196],{"className":1197,"style":600},[406],[390,1199,1139],{"className":1200,"style":1138},[411,412],[390,1202,608],{"className":1203},[607],[390,1205,413],{"className":1206},[411,412],[390,1208],{"className":1209,"style":718},[717],[390,1211,1214],{"className":1212},[1213],"mop",[390,1215,1219],{"className":1216,"style":1218},[411,1217],"mathrm","margin-right:0.0139em;","log",[390,1221],{"className":1222,"style":718},[717],[390,1224,413],{"className":1225},[411,412],[390,1227,616],{"className":1228},[615],[390,1230,998],{"className":1231},[997],[381,1233,1234,1235,1253,1254,1283,1284,1313,1314,1323],{},"and the bulk of the work is justifying that implication. This is the equation we\nmust solve. (We freely write ",[390,1236,1238],{"className":1237},[393],[390,1239,1241],{"className":1240,"ariaHidden":398},[397],[390,1242,1244,1247,1250],{"className":1243},[402],[390,1245],{"className":1246,"style":600},[406],[390,1248,413],{"className":1249},[411,412],[390,1251,696],{"className":1252},[411]," rather than\n",[390,1255,1257],{"className":1256},[393],[390,1258,1260],{"className":1259,"ariaHidden":398},[397],[390,1261,1263,1266],{"className":1262},[402],[390,1264],{"className":1265,"style":600},[406],[390,1267,1269,1273,1276,1279],{"className":1268},[819],[390,1270,1272],{"className":1271,"style":824},[607,823],"⌊",[390,1274,413],{"className":1275},[411,412],[390,1277,696],{"className":1278},[411],[390,1280,1282],{"className":1281,"style":824},[615,823],"⌋"," and ",[390,1285,1287],{"className":1286},[393],[390,1288,1290],{"className":1289,"ariaHidden":398},[397],[390,1291,1293,1296],{"className":1292},[402],[390,1294],{"className":1295,"style":600},[406],[390,1297,1299,1303,1306,1309],{"className":1298},[819],[390,1300,1302],{"className":1301,"style":824},[607,823],"⌈",[390,1304,413],{"className":1305},[411,412],[390,1307,696],{"className":1308},[411],[390,1310,1312],{"className":1311,"style":824},[615,823],"⌉","; the floors and ceilings change the answer by\nlower-order amounts that the asymptotics absorb. Skiena and CLRS both justify\ndropping them.",[1315,1316,1317],"sup",{},[421,1318,668],{"href":1319,"ariaDescribedBy":1320,"dataFootnoteRef":376,"id":1322},"#user-content-fn-skiena-floors",[1321],"footnote-label","user-content-fnref-skiena-floors",") Throughout we also assume a constant base case, which lets us\nignore the boundary condition when finding the asymptotic order.",[444,1325,1327],{"id":1326},"method-1-the-recursion-tree","Method 1: the recursion tree",[381,1329,1330,1331,1334,1335,1338,1339,617],{},"The most intuitive method ",[385,1332,1333],{},"draws"," the recurrence. Each node is a subproblem\nlabeled with the ",[416,1336,1337],{},"non-recursive"," work it does; its children are the subproblems\nit spawns. Summing all node labels gives ",[390,1340,1342],{"className":1341},[393],[390,1343,1345],{"className":1344,"ariaHidden":398},[397],[390,1346,1348,1351,1354,1357,1360],{"className":1347},[402],[390,1349],{"className":1350,"style":600},[406],[390,1352,723],{"className":1353,"style":722},[411,412],[390,1355,608],{"className":1356},[607],[390,1358,413],{"className":1359},[411,412],[390,1361,616],{"className":1362},[615],[381,1364,1365,1366,1385,1386,1404,1405,1385,1432,1451,1452,1468],{},"For merge sort, the root does ",[390,1367,1369],{"className":1368},[393],[390,1370,1372],{"className":1371,"ariaHidden":398},[397],[390,1373,1375,1378,1382],{"className":1374},[402],[390,1376],{"className":1377,"style":407},[406],[390,1379,1381],{"className":1380},[411,412],"c",[390,1383,413],{"className":1384},[411,412]," work and has two children of size ",[390,1387,1389],{"className":1388},[393],[390,1390,1392],{"className":1391,"ariaHidden":398},[397],[390,1393,1395,1398,1401],{"className":1394},[402],[390,1396],{"className":1397,"style":600},[406],[390,1399,413],{"className":1400},[411,412],[390,1402,696],{"className":1403},[411],". Each\nof those does ",[390,1406,1408],{"className":1407},[393],[390,1409,1411],{"className":1410,"ariaHidden":398},[397],[390,1412,1414,1417,1420,1423,1426,1429],{"className":1413},[402],[390,1415],{"className":1416,"style":600},[406],[390,1418,1381],{"className":1419},[411,412],[390,1421,608],{"className":1422},[607],[390,1424,413],{"className":1425},[411,412],[390,1427,696],{"className":1428},[411],[390,1430,616],{"className":1431},[615],[390,1433,1435],{"className":1434},[393],[390,1436,1438],{"className":1437,"ariaHidden":398},[397],[390,1439,1441,1444,1447],{"className":1440},[402],[390,1442],{"className":1443,"style":600},[406],[390,1445,413],{"className":1446},[411,412],[390,1448,1450],{"className":1449},[411],"\u002F4",", and so on, until\nthe leaves are size-",[390,1453,1455],{"className":1454},[393],[390,1456,1458],{"className":1457,"ariaHidden":398},[397],[390,1459,1461,1465],{"className":1460},[402],[390,1462],{"className":1463,"style":1464},[406],"height:0.6444em;",[390,1466,668],{"className":1467},[411]," subproblems.",[1470,1471,1475,1792],"figure",{"className":1472},[1473,1474],"tikz-figure","tikz-diagram-rendered",[1476,1477,1482],"svg",{"xmlns":1478,"width":1479,"height":1480,"viewBox":1481},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","389.637","207.928","-75 -75 292.228 155.946",[1483,1484,1487,1492,1501,1504,1529,1532,1553,1556,1578,1581,1602,1605,1626,1629,1650,1653,1669,1722,1770],"g",{"stroke":1485,"style":1486},"currentColor","stroke-miterlimit:10;stroke-width:.4",[1488,1489],"path",{"fill":1490,"d":1491},"none","M36.184-72.07h-9.072a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h9.072a4 4 0 0 0 4-4v-9.072a4 4 0 0 0-4-4ZM23.112-54.998",[1483,1493,1495],{"transform":1494},"translate(-5.165 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32.488-72.086",[1499],[1483,1781,1782],{"transform":1775},[1488,1783],{"d":1784,"fill":1485,"stroke":1485,"className":1785,"style":1500},"M32.488-68.086Q32.488-68.310 32.654-68.472Q32.820-68.633 33.040-68.633Q33.176-68.633 33.308-68.559Q33.440-68.486 33.513-68.354Q33.586-68.222 33.586-68.086Q33.586-67.866 33.425-67.700Q33.264-67.534 33.040-67.534Q32.820-67.534 32.654-67.700Q32.488-67.866 32.488-68.086",[1499],[1483,1787,1788],{"transform":1775},[1488,1789],{"d":1790,"fill":1485,"stroke":1485,"className":1791,"style":1500},"M32.488-64.086Q32.488-64.310 32.654-64.471Q32.820-64.633 33.040-64.633Q33.176-64.633 33.308-64.559Q33.440-64.486 33.513-64.354Q33.586-64.222 33.586-64.086Q33.586-63.866 33.425-63.700Q33.264-63.534 33.040-63.534Q32.820-63.534 32.654-63.700Q32.488-63.866 32.488-64.086",[1499],[1793,1794,1797,1798,617],"figcaption",{"className":1795},[1796],"tikz-cap","Recursion tree for merge sort, with each level summing to ",[390,1799,1801],{"className":1800},[393],[390,1802,1804],{"className":1803,"ariaHidden":398},[397],[390,1805,1807,1810,1813],{"className":1806},[402],[390,1808],{"className":1809,"style":407},[406],[390,1811,1381],{"className":1812},[411,412],[390,1814,413],{"className":1815},[411,412],[381,1817,1818,1819,1840,1841,1859,1860,1927,1928,1995,1996,2085,2086,2101,2102,2117,2118,2133,2134,2149],{},"The key observation: ",[416,1820,1821,1822,617],{},"every level sums to ",[390,1823,1825],{"className":1824},[393],[390,1826,1828],{"className":1827,"ariaHidden":398},[397],[390,1829,1831,1834,1837],{"className":1830},[402],[390,1832],{"className":1833,"style":407},[406],[390,1835,1381],{"className":1836},[411,412],[390,1838,413],{"className":1839},[411,412]," The root level is ",[390,1842,1844],{"className":1843},[393],[390,1845,1847],{"className":1846,"ariaHidden":398},[397],[390,1848,1850,1853,1856],{"className":1849},[402],[390,1851],{"className":1852,"style":407},[406],[390,1854,1381],{"className":1855},[411,412],[390,1857,413],{"className":1858},[411,412],"; the\nnext level is ",[390,1861,1863],{"className":1862},[393],[390,1864,1866,1885,1915],{"className":1865,"ariaHidden":398},[397],[390,1867,1869,1872,1875,1878,1882],{"className":1868},[402],[390,1870],{"className":1871,"style":1464},[406],[390,1873,713],{"className":1874},[411],[390,1876],{"className":1877,"style":912},[717],[390,1879,1881],{"className":1880},[916],"⋅",[390,1883],{"className":1884,"style":912},[717],[390,1886,1888,1891,1894,1897,1900,1903,1906,1909,1912],{"className":1887},[402],[390,1889],{"className":1890,"style":600},[406],[390,1892,1381],{"className":1893},[411,412],[390,1895,608],{"className":1896},[607],[390,1898,413],{"className":1899},[411,412],[390,1901,696],{"className":1902},[411],[390,1904,616],{"className":1905},[615],[390,1907],{"className":1908,"style":800},[717],[390,1910,805],{"className":1911},[804],[390,1913],{"className":1914,"style":800},[717],[390,1916,1918,1921,1924],{"className":1917},[402],[390,1919],{"className":1920,"style":407},[406],[390,1922,1381],{"className":1923},[411,412],[390,1925,413],{"className":1926},[411,412],"; the level below is ",[390,1929,1931],{"className":1930},[393],[390,1932,1934,1953,1983],{"className":1933,"ariaHidden":398},[397],[390,1935,1937,1940,1944,1947,1950],{"className":1936},[402],[390,1938],{"className":1939,"style":1464},[406],[390,1941,1943],{"className":1942},[411],"4",[390,1945],{"className":1946,"style":912},[717],[390,1948,1881],{"className":1949},[916],[390,1951],{"className":1952,"style":912},[717],[390,1954,1956,1959,1962,1965,1968,1971,1974,1977,1980],{"className":1955},[402],[390,1957],{"className":1958,"style":600},[406],[390,1960,1381],{"className":1961},[411,412],[390,1963,608],{"className":1964},[607],[390,1966,413],{"className":1967},[411,412],[390,1969,1450],{"className":1970},[411],[390,1972,616],{"className":1973},[615],[390,1975],{"className":1976,"style":800},[717],[390,1978,805],{"className":1979},[804],[390,1981],{"className":1982,"style":800},[717],[390,1984,1986,1989,1992],{"className":1985},[402],[390,1987],{"className":1988,"style":407},[406],[390,1990,1381],{"className":1991},[411,412],[390,1993,413],{"className":1994},[411,412],". The\nsubproblem sizes shrink by half each level, so the tree has\n",[390,1997,1999],{"className":1998},[393],[390,2000,2002,2076],{"className":2001,"ariaHidden":398},[397],[390,2003,2005,2009,2061,2064,2067,2070,2073],{"className":2004},[402],[390,2006],{"className":2007,"style":2008},[406],"height:0.9386em;vertical-align:-0.2441em;",[390,2010,2012,2018],{"className":2011},[1213],[390,2013,2015],{"className":2014},[1213],[390,2016,1219],{"className":2017,"style":1218},[411,1217],[390,2019,2022],{"className":2020},[2021],"msupsub",[390,2023,2025,2052],{"className":2024},[845,846],[390,2026,2028,2049],{"className":2027},[850],[390,2029,2032],{"className":2030,"style":2031},[854],"height:0.207em;",[390,2033,2035,2039],{"style":2034},"top:-2.4559em;margin-right:0.05em;",[390,2036],{"className":2037,"style":2038},[862],"height:2.7em;",[390,2040,2046],{"className":2041},[2042,2043,2044,2045],"sizing","reset-size6","size3","mtight",[390,2047,713],{"className":2048},[411,2045],[390,2050,937],{"className":2051},[936],[390,2053,2055],{"className":2054},[850],[390,2056,2059],{"className":2057,"style":2058},[854],"height:0.2441em;",[390,2060],{},[390,2062],{"className":2063,"style":718},[717],[390,2065,413],{"className":2066},[411,412],[390,2068],{"className":2069,"style":912},[717],[390,2071,917],{"className":2072},[916],[390,2074],{"className":2075,"style":912},[717],[390,2077,2079,2082],{"className":2078},[402],[390,2080],{"className":2081,"style":1464},[406],[390,2083,668],{"className":2084},[411]," levels (from size ",[390,2087,2089],{"className":2088},[393],[390,2090,2092],{"className":2091,"ariaHidden":398},[397],[390,2093,2095,2098],{"className":2094},[402],[390,2096],{"className":2097,"style":407},[406],[390,2099,413],{"className":2100},[411,412]," down to size ",[390,2103,2105],{"className":2104},[393],[390,2106,2108],{"className":2107,"ariaHidden":398},[397],[390,2109,2111,2114],{"className":2110},[402],[390,2112],{"className":2113,"style":1464},[406],[390,2115,668],{"className":2116},[411],"), and the bottom level holds\nthe ",[390,2119,2121],{"className":2120},[393],[390,2122,2124],{"className":2123,"ariaHidden":398},[397],[390,2125,2127,2130],{"className":2126},[402],[390,2128],{"className":2129,"style":407},[406],[390,2131,413],{"className":2132},[411,412]," size-",[390,2135,2137],{"className":2136},[393],[390,2138,2140],{"className":2139,"ariaHidden":398},[397],[390,2141,2143,2146],{"className":2142},[402],[390,2144],{"className":2145,"style":1464},[406],[390,2147,668],{"className":2148},[411]," leaves. Therefore",[390,2151,2153],{"className":2152},[772],[390,2154,2156],{"className":2155},[393],[390,2157,2159,2186,2339,2534,2607,2628],{"className":2158,"ariaHidden":398},[397],[390,2160,2162,2165,2168,2171,2174,2177,2180,2183],{"className":2161},[402],[390,2163],{"className":2164,"style":600},[406],[390,2166,723],{"className":2167,"style":722},[411,412],[390,2169,608],{"className":2170},[607],[390,2172,413],{"className":2173},[411,412],[390,2175,616],{"className":2176},[615],[390,2178],{"className":2179,"style":800},[717],[390,2181,805],{"className":2182},[804],[390,2184],{"className":2185,"style":800},[717],[390,2187,2189,2193,2329,2332,2336],{"className":2188},[402],[390,2190],{"className":2191,"style":2192},[406],"height:2.0535em;vertical-align:-1.4702em;",[390,2194,2197],{"className":2195},[819,2196],"munder",[390,2198,2200,2320],{"className":2199},[845,846],[390,2201,2203,2317],{"className":2202},[850],[390,2204,2206,2226],{"className":2205,"style":407},[854],[390,2207,2209,2213],{"style":2208},"top:-1.6659em;",[390,2210],{"className":2211,"style":2212},[862],"height:3em;",[390,2214,2216],{"className":2215},[2042,2043,2044,2045],[390,2217,2219],{"className":2218},[411,2045],[390,2220,2222],{"className":2221},[411,471,2045],[390,2223,2225],{"className":2224},[411,2045],"per level",[390,2227,2229,2232],{"style":2228},"top:-3em;",[390,2230],{"className":2231,"style":2212},[862],[390,2233,2235],{"className":2234},[819,2196],[390,2236,2238,2308],{"className":2237},[845,846],[390,2239,2241,2305],{"className":2240},[850],[390,2242,2244,2291],{"className":2243,"style":407},[854],[390,2245,2249,2252],{"className":2246,"style":2248},[2247],"svg-align","top:-2.352em;",[390,2250],{"className":2251,"style":2212},[862],[390,2253,2257,2271,2281],{"className":2254,"style":2256},[2255],"stretchy","height:0.548em;min-width:1.6em;",[390,2258,2262],{"className":2259,"style":2261},[2260],"brace-left","height:0.548em;",[1476,2263,2268],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2267},"400em","0.548em","0 0 400000 548","xMinYMin slice",[1488,2269],{"d":2270},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[390,2272,2275],{"className":2273,"style":2261},[2274],"brace-center",[1476,2276,2278],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2277},"xMidYMin slice",[1488,2279],{"d":2280},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[390,2282,2285],{"className":2283,"style":2261},[2284],"brace-right",[1476,2286,2288],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2287},"xMaxYMin slice",[1488,2289],{"d":2290},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[390,2292,2293,2296],{"style":2228},[390,2294],{"className":2295,"style":2212},[862],[390,2297,2299,2302],{"className":2298},[411],[390,2300,1381],{"className":2301},[411,412],[390,2303,413],{"className":2304},[411,412],[390,2306,937],{"className":2307},[936],[390,2309,2311],{"className":2310},[850],[390,2312,2315],{"className":2313,"style":2314},[854],"height:0.648em;",[390,2316],{},[390,2318,937],{"className":2319},[936],[390,2321,2323],{"className":2322},[850],[390,2324,2327],{"className":2325,"style":2326},[854],"height:1.4702em;",[390,2328],{},[390,2330],{"className":2331,"style":912},[717],[390,2333,2335],{"className":2334},[916],"×",[390,2337],{"className":2338,"style":912},[717],[390,2340,2342,2346,2525,2528,2531],{"className":2341},[402],[390,2343],{"className":2344,"style":2345},[406],"height:2.3341em;vertical-align:-1.5841em;",[390,2347,2349],{"className":2348},[819,2196],[390,2350,2352,2516],{"className":2351},[845,846],[390,2353,2355,2513],{"className":2354},[850],[390,2356,2359,2378],{"className":2357,"style":2358},[854],"height:0.75em;",[390,2360,2362,2365],{"style":2361},"top:-1.4159em;",[390,2363],{"className":2364,"style":2212},[862],[390,2366,2368],{"className":2367},[2042,2043,2044,2045],[390,2369,2371],{"className":2370},[411,2045],[390,2372,2374],{"className":2373},[411,471,2045],[390,2375,2377],{"className":2376},[411,2045],"levels",[390,2379,2380,2383],{"style":2228},[390,2381],{"className":2382,"style":2212},[862],[390,2384,2386],{"className":2385},[819,2196],[390,2387,2389,2504],{"className":2388},[845,846],[390,2390,2392,2501],{"className":2391},[850],[390,2393,2395,2426],{"className":2394,"style":2358},[854],[390,2396,2399,2402],{"className":2397,"style":2398},[2247],"top:-2.102em;",[390,2400],{"className":2401,"style":2212},[862],[390,2403,2405,2412,2419],{"className":2404,"style":2256},[2255],[390,2406,2408],{"className":2407,"style":2261},[2260],[1476,2409,2410],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2267},[1488,2411],{"d":2270},[390,2413,2415],{"className":2414,"style":2261},[2274],[1476,2416,2417],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2277},[1488,2418],{"d":2280},[390,2420,2422],{"className":2421,"style":2261},[2284],[1476,2423,2424],{"xmlns":1478,"width":2264,"height":2265,"viewBox":2266,"preserveAspectRatio":2287},[1488,2425],{"d":2290},[390,2427,2428,2431],{"style":2228},[390,2429],{"className":2430,"style":2212},[862],[390,2432,2434,2437,2480,2483,2486,2489,2492,2495,2498],{"className":2433},[411],[390,2435,608],{"className":2436},[607],[390,2438,2440,2446],{"className":2439},[1213],[390,2441,2443],{"className":2442},[1213],[390,2444,1219],{"className":2445,"style":1218},[411,1217],[390,2447,2449],{"className":2448},[2021],[390,2450,2452,2472],{"className":2451},[845,846],[390,2453,2455,2469],{"className":2454},[850],[390,2456,2458],{"className":2457,"style":2031},[854],[390,2459,2460,2463],{"style":2034},[390,2461],{"className":2462,"style":2038},[862],[390,2464,2466],{"className":2465},[2042,2043,2044,2045],[390,2467,713],{"className":2468},[411,2045],[390,2470,937],{"className":2471},[936],[390,2473,2475],{"className":2474},[850],[390,2476,2478],{"className":2477,"style":2058},[854],[390,2479],{},[390,2481],{"className":2482,"style":718},[717],[390,2484,413],{"className":2485},[411,412],[390,2487],{"className":2488,"style":912},[717],[390,2490,917],{"className":2491},[916],[390,2493],{"className":2494,"style":912},[717],[390,2496,668],{"className":2497},[411],[390,2499,616],{"className":2500},[615],[390,2502,937],{"className":2503},[936],[390,2505,2507],{"className":2506},[850],[390,2508,2511],{"className":2509,"style":2510},[854],"height:0.898em;",[390,2512],{},[390,2514,937],{"className":2515},[936],[390,2517,2519],{"className":2518},[850],[390,2520,2523],{"className":2521,"style":2522},[854],"height:1.5841em;",[390,2524],{},[390,2526],{"className":2527,"style":800},[717],[390,2529,805],{"className":2530},[804],[390,2532],{"className":2533,"style":800},[717],[390,2535,2537,2540,2543,2546,2549,2592,2595,2598,2601,2604],{"className":2536},[402],[390,2538],{"className":2539,"style":2008},[406],[390,2541,1381],{"className":2542},[411,412],[390,2544,413],{"className":2545},[411,412],[390,2547],{"className":2548,"style":718},[717],[390,2550,2552,2558],{"className":2551},[1213],[390,2553,2555],{"className":2554},[1213],[390,2556,1219],{"className":2557,"style":1218},[411,1217],[390,2559,2561],{"className":2560},[2021],[390,2562,2564,2584],{"className":2563},[845,846],[390,2565,2567,2581],{"className":2566},[850],[390,2568,2570],{"className":2569,"style":2031},[854],[390,2571,2572,2575],{"style":2034},[390,2573],{"className":2574,"style":2038},[862],[390,2576,2578],{"className":2577},[2042,2043,2044,2045],[390,2579,713],{"className":2580},[411,2045],[390,2582,937],{"className":2583},[936],[390,2585,2587],{"className":2586},[850],[390,2588,2590],{"className":2589,"style":2058},[854],[390,2591],{},[390,2593],{"className":2594,"style":718},[717],[390,2596,413],{"className":2597},[411,412],[390,2599],{"className":2600,"style":912},[717],[390,2602,917],{"className":2603},[916],[390,2605],{"className":2606,"style":912},[717],[390,2608,2610,2613,2616,2619,2622,2625],{"className":2609},[402],[390,2611],{"className":2612,"style":407},[406],[390,2614,1381],{"className":2615},[411,412],[390,2617,413],{"className":2618},[411,412],[390,2620],{"className":2621,"style":800},[717],[390,2623,805],{"className":2624},[804],[390,2626],{"className":2627,"style":800},[717],[390,2629,2631,2634,2637,2640,2643,2646,2652,2655,2658,2661],{"className":2630},[402],[390,2632],{"className":2633,"style":600},[406],[390,2635,441],{"className":2636},[411],[390,2638,608],{"className":2639},[607],[390,2641,413],{"className":2642},[411,412],[390,2644],{"className":2645,"style":718},[717],[390,2647,2649],{"className":2648},[1213],[390,2650,1219],{"className":2651,"style":1218},[411,1217],[390,2653],{"className":2654,"style":718},[717],[390,2656,413],{"className":2657},[411,412],[390,2659,616],{"className":2660},[615],[390,2662,617],{"className":2663},[411],[381,2665,2666,2667,2710,2711,617,2764,2771,2772,2775,2776,2794],{},"This is the result: ",[416,2668,2669,2670,2709],{},"merge sort runs in ",[390,2671,2673],{"className":2672},[393],[390,2674,2676],{"className":2675,"ariaHidden":398},[397],[390,2677,2679,2682,2685,2688,2691,2694,2700,2703,2706],{"className":2678},[402],[390,2680],{"className":2681,"style":600},[406],[390,2683,441],{"className":2684},[411],[390,2686,608],{"className":2687},[607],[390,2689,413],{"className":2690},[411,412],[390,2692],{"className":2693,"style":718},[717],[390,2695,2697],{"className":2696},[1213],[390,2698,1219],{"className":2699,"style":1218},[411,1217],[390,2701],{"className":2702,"style":718},[717],[390,2704,413],{"className":2705},[411,412],[390,2707,616],{"className":2708},[615]," time",",\nstrictly better than insertion sort's ",[390,2712,2714],{"className":2713},[393],[390,2715,2717],{"className":2716,"ariaHidden":398},[397],[390,2718,2720,2724,2727,2730,2761],{"className":2719},[402],[390,2721],{"className":2722,"style":2723},[406],"height:1.0641em;vertical-align:-0.25em;",[390,2725,441],{"className":2726},[411],[390,2728,608],{"className":2729},[607],[390,2731,2733,2736],{"className":2732},[411],[390,2734,413],{"className":2735},[411,412],[390,2737,2739],{"className":2738},[2021],[390,2740,2742],{"className":2741},[845],[390,2743,2745],{"className":2744},[850],[390,2746,2749],{"className":2747,"style":2748},[854],"height:0.8141em;",[390,2750,2752,2755],{"style":2751},"top:-3.063em;margin-right:0.05em;",[390,2753],{"className":2754,"style":2038},[862],[390,2756,2758],{"className":2757},[2042,2043,2044,2045],[390,2759,713],{"className":2760},[411,2045],[390,2762,616],{"className":2763},[615],[1315,2765,2766],{},[421,2767,713],{"href":2768,"ariaDescribedBy":2769,"dataFootnoteRef":376,"id":2770},"#user-content-fn-clrs-tree",[1321],"user-content-fnref-clrs-tree"," The recursion tree also\nexposes ",[385,2773,2774],{},"why",": the per-level work stays flat at ",[390,2777,2779],{"className":2778},[393],[390,2780,2782],{"className":2781,"ariaHidden":398},[397],[390,2783,2785,2788,2791],{"className":2784},[402],[390,2786],{"className":2787,"style":407},[406],[390,2789,1381],{"className":2790},[411,412],[390,2792,413],{"className":2793},[411,412]," while the depth is only\nlogarithmic.",[381,2796,2797,2798,2813,2814,2817],{},"The tree is a derivation, not yet a proof — it asks us to trust the level sums\nand the level count. When a recurrence is irregular (unequal splits, work that\nisn't a clean power of ",[390,2799,2801],{"className":2800},[393],[390,2802,2804],{"className":2803,"ariaHidden":398},[397],[390,2805,2807,2810],{"className":2806},[402],[390,2808],{"className":2809,"style":407},[406],[390,2811,413],{"className":2812},[411,412],"), the tree still gives a reliable ",[385,2815,2816],{},"guess",", which we\nthen certify with the next method.",[444,2819,2821],{"id":2820},"method-2-substitution-guess-and-verify","Method 2: substitution (guess and verify)",[381,2823,547,2824,2827,2828,2831,2832,2847],{},[416,2825,2826],{},"substitution method"," is the rigorous one: guess the form of the\nanswer, then prove it by ",[416,2829,2830],{},"induction"," on ",[390,2833,2835],{"className":2834},[393],[390,2836,2838],{"className":2837,"ariaHidden":398},[397],[390,2839,2841,2844],{"className":2840},[402],[390,2842],{"className":2843,"style":407},[406],[390,2845,413],{"className":2846},[411,412],". It is the only method that always\nworks, and the only one that produces a complete proof.",[381,2849,2850,2851,2917,2918,2999,3000,3036,3037,3137,3138,617],{},"We verify the guess ",[390,2852,2854],{"className":2853},[393],[390,2855,2857,2884],{"className":2856,"ariaHidden":398},[397],[390,2858,2860,2863,2866,2869,2872,2875,2878,2881],{"className":2859},[402],[390,2861],{"className":2862,"style":600},[406],[390,2864,723],{"className":2865,"style":722},[411,412],[390,2867,608],{"className":2868},[607],[390,2870,413],{"className":2871},[411,412],[390,2873,616],{"className":2874},[615],[390,2876],{"className":2877,"style":800},[717],[390,2879,805],{"className":2880},[804],[390,2882],{"className":2883,"style":800},[717],[390,2885,2887,2890,2893,2896,2899,2902,2908,2911,2914],{"className":2886},[402],[390,2888],{"className":2889,"style":600},[406],[390,2891,1139],{"className":2892,"style":1138},[411,412],[390,2894,608],{"className":2895},[607],[390,2897,413],{"className":2898},[411,412],[390,2900],{"className":2901,"style":718},[717],[390,2903,2905],{"className":2904},[1213],[390,2906,1219],{"className":2907,"style":1218},[411,1217],[390,2909],{"className":2910,"style":718},[717],[390,2912,413],{"className":2913},[411,412],[390,2915,616],{"className":2916},[615]," for the merge-sort recurrence\n",[390,2919,2921],{"className":2920},[393],[390,2922,2924,2951,2987],{"className":2923,"ariaHidden":398},[397],[390,2925,2927,2930,2933,2936,2939,2942,2945,2948],{"className":2926},[402],[390,2928],{"className":2929,"style":600},[406],[390,2931,723],{"className":2932,"style":722},[411,412],[390,2934,608],{"className":2935},[607],[390,2937,413],{"className":2938},[411,412],[390,2940,616],{"className":2941},[615],[390,2943],{"className":2944,"style":800},[717],[390,2946,805],{"className":2947},[804],[390,2949],{"className":2950,"style":800},[717],[390,2952,2954,2957,2960,2963,2966,2969,2972,2975,2978,2981,2984],{"className":2953},[402],[390,2955],{"className":2956,"style":600},[406],[390,2958,713],{"className":2959},[411],[390,2961],{"className":2962,"style":718},[717],[390,2964,723],{"className":2965,"style":722},[411,412],[390,2967,608],{"className":2968},[607],[390,2970,413],{"className":2971},[411,412],[390,2973,696],{"className":2974},[411],[390,2976,616],{"className":2977},[615],[390,2979],{"className":2980,"style":912},[717],[390,2982,917],{"className":2983},[916],[390,2985],{"className":2986,"style":912},[717],[390,2988,2990,2993,2996],{"className":2989},[402],[390,2991],{"className":2992,"style":407},[406],[390,2994,1381],{"className":2995},[411,412],[390,2997,413],{"className":2998},[411,412],". We claim there is a constant ",[390,3001,3003],{"className":3002},[393],[390,3004,3006,3026],{"className":3005,"ariaHidden":398},[397],[390,3007,3009,3013,3017,3020,3023],{"className":3008},[402],[390,3010],{"className":3011,"style":3012},[406],"height:0.7335em;vertical-align:-0.0391em;",[390,3014,3016],{"className":3015},[411,412],"d",[390,3018],{"className":3019,"style":800},[717],[390,3021,1022],{"className":3022},[804],[390,3024],{"className":3025,"style":800},[717],[390,3027,3029,3032],{"className":3028},[402],[390,3030],{"className":3031,"style":1464},[406],[390,3033,3035],{"className":3034},[411],"0"," with\n",[390,3038,3040],{"className":3039},[393],[390,3041,3043,3070],{"className":3042,"ariaHidden":398},[397],[390,3044,3046,3049,3052,3055,3058,3061,3064,3067],{"className":3045},[402],[390,3047],{"className":3048,"style":600},[406],[390,3050,723],{"className":3051,"style":722},[411,412],[390,3053,608],{"className":3054},[607],[390,3056,413],{"className":3057},[411,412],[390,3059,616],{"className":3060},[615],[390,3062],{"className":3063,"style":800},[717],[390,3065,1089],{"className":3066},[804],[390,3068],{"className":3069,"style":800},[717],[390,3071,3073,3076,3079,3082,3085,3088,3131,3134],{"className":3072},[402],[390,3074],{"className":3075,"style":2008},[406],[390,3077,3016],{"className":3078},[411,412],[390,3080],{"className":3081,"style":718},[717],[390,3083,413],{"className":3084},[411,412],[390,3086],{"className":3087,"style":718},[717],[390,3089,3091,3097],{"className":3090},[1213],[390,3092,3094],{"className":3093},[1213],[390,3095,1219],{"className":3096,"style":1218},[411,1217],[390,3098,3100],{"className":3099},[2021],[390,3101,3103,3123],{"className":3102},[845,846],[390,3104,3106,3120],{"className":3105},[850],[390,3107,3109],{"className":3108,"style":2031},[854],[390,3110,3111,3114],{"style":2034},[390,3112],{"className":3113,"style":2038},[862],[390,3115,3117],{"className":3116},[2042,2043,2044,2045],[390,3118,713],{"className":3119},[411,2045],[390,3121,937],{"className":3122},[936],[390,3124,3126],{"className":3125},[850],[390,3127,3129],{"className":3128,"style":2058},[854],[390,3130],{},[390,3132],{"className":3133,"style":718},[717],[390,3135,413],{"className":3136},[411,412]," for all ",[390,3139,3141],{"className":3140},[393],[390,3142,3144,3164],{"className":3143,"ariaHidden":398},[397],[390,3145,3147,3151,3154,3157,3161],{"className":3146},[402],[390,3148],{"className":3149,"style":3150},[406],"height:0.7719em;vertical-align:-0.136em;",[390,3152,413],{"className":3153},[411,412],[390,3155],{"className":3156,"style":800},[717],[390,3158,3160],{"className":3159},[804],"≥",[390,3162],{"className":3163,"style":800},[717],[390,3165,3167,3170],{"className":3166},[402],[390,3168],{"className":3169,"style":1464},[406],[390,3171,713],{"className":3172},[411],[381,3174,3175,3178,3179,3194,3195,617],{},[416,3176,3177],{},"Inductive hypothesis."," Assume the bound holds for all sizes smaller than ",[390,3180,3182],{"className":3181},[393],[390,3183,3185],{"className":3184,"ariaHidden":398},[397],[390,3186,3188,3191],{"className":3187},[402],[390,3189],{"className":3190,"style":407},[406],[390,3192,413],{"className":3193},[411,412],";\nin particular 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step."," Substitute into the 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exactly when ",[390,4311,4313],{"className":4312},[393],[390,4314,4316,4335],{"className":4315,"ariaHidden":398},[397],[390,4317,4319,4323,4326,4329,4332],{"className":4318},[402],[390,4320],{"className":4321,"style":4322},[406],"height:0.8304em;vertical-align:-0.136em;",[390,4324,3016],{"className":4325},[411,412],[390,4327],{"className":4328,"style":800},[717],[390,4330,3160],{"className":4331},[804],[390,4333],{"className":4334,"style":800},[717],[390,4336,4338,4341],{"className":4337},[402],[390,4339],{"className":4340,"style":407},[406],[390,4342,1381],{"className":4343},[411,412],". So choosing any ",[390,4346,4348],{"className":4347},[393],[390,4349,4351,4369],{"className":4350,"ariaHidden":398},[397],[390,4352,4354,4357,4360,4363,4366],{"className":4353},[402],[390,4355],{"className":4356,"style":4322},[406],[390,4358,3016],{"className":4359},[411,412],[390,4361],{"className":4362,"style":800},[717],[390,4364,3160],{"className":4365},[804],[390,4367],{"className":4368,"style":800},[717],[390,4370,4372,4375],{"className":4371},[402],[390,4373],{"className":4374,"style":407},[406],[390,4376,1381],{"className":4377},[411,412],"\n(and picking the base-case constant large enough to cover ",[390,4380,4382],{"className":4381},[393],[390,4383,4385,4403],{"className":4384,"ariaHidden":398},[397],[390,4386,4388,4391,4394,4397,4400],{"className":4387},[402],[390,4389],{"className":4390,"style":407},[406],[390,4392,413],{"className":4393},[411,412],[390,4395],{"className":4396,"style":800},[717],[390,4398,805],{"className":4399},[804],[390,4401],{"className":4402,"style":800},[717],[390,4404,4406,4409],{"className":4405},[402],[390,4407],{"className":4408,"style":1464},[406],[390,4410,713],{"className":4411},[411],") completes the\ninduction. Hence 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19.265-57.656L18.898-57.260Q18.898-57.236 18.845-57.236L18.786-57.236Q18.698-57.236 18.698-57.319",[1499],[1483,4822,4823],{"transform":4756},[1488,4824],{"d":4825,"fill":1485,"stroke":1485,"className":4826,"style":1500},"M28.151-57.348L25.969-57.348L25.969-57.700Q26.310-57.700 26.530-57.753Q26.750-57.807 26.750-58.017L26.750-60.746Q26.750-61.132 26.601-61.220Q26.452-61.308 26.013-61.308L26.013-61.660L27.453-61.767L27.453-58.017Q27.453-57.807 27.643-57.753Q27.834-57.700 28.151-57.700L28.151-57.348M26.393-63.486Q26.393-63.705 26.559-63.871Q26.725-64.037 26.940-64.037Q27.082-64.037 27.214-63.964Q27.346-63.891 27.419-63.759Q27.492-63.627 27.492-63.486Q27.492-63.271 27.326-63.105Q27.160-62.939 26.940-62.939Q26.725-62.939 26.559-63.105Q26.393-63.271 26.393-63.486M30.949-57.348L28.688-57.348L28.688-57.700Q29.030-57.700 29.250-57.753Q29.470-57.807 29.470-58.017L29.470-61.308L28.698-61.308L28.698-61.660L29.470-61.660L29.470-62.856Q29.470-63.232 29.682-63.527Q29.894-63.823 30.224-64.013Q30.554-64.203 30.939-64.301Q31.325-64.399 31.682-64.399Q31.994-64.399 32.285-64.296Q32.575-64.194 32.741-63.969Q33.215-64.399 33.869-64.399Q34.206-64.399 34.477-64.199Q34.748-63.998 34.748-63.666Q34.748-63.476 34.619-63.347Q34.489-63.217 34.299-63.217Q34.108-63.217 33.974-63.347Q33.840-63.476 33.840-63.666Q33.840-63.979 34.108-64.086Q33.947-64.140 33.820-64.140Q33.527-64.140 33.317-63.935Q33.107-63.730 33.002-63.430Q32.897-63.129 32.897-62.836L32.897-61.660L34.059-61.660L34.059-61.308L32.932-61.308L32.932-58.017Q32.932-57.812 33.224-57.756Q33.517-57.700 33.898-57.700L33.898-57.348L31.447-57.348L31.447-57.700Q31.789-57.700 32.009-57.753Q32.228-57.807 32.228-58.017L32.228-61.308L30.168-61.308L30.168-58.017Q30.168-57.807 30.390-57.753Q30.612-57.700 30.949-57.700L30.949-57.348M30.139-62.836L30.139-61.660L32.228-61.660L32.228-62.856Q32.228-62.944 32.248-63.110Q31.979-63.237 31.979-63.530Q31.979-63.691 32.072-63.810Q32.165-63.930 32.321-63.969Q32.038-64.140 31.599-64.140Q31.232-64.140 30.893-63.972Q30.554-63.803 30.346-63.503Q30.139-63.202 30.139-62.836",[1499],[1483,4828,4829],{"transform":4756},[1488,4830],{"d":4831,"fill":1485,"stroke":1485,"className":4832,"style":1500},"M39.343-57.236Q38.713-57.236 38.350-57.712Q37.986-58.188 37.986-58.837Q37.986-59.477 38.318-60.165Q38.650-60.854 39.214-61.310Q39.778-61.767 40.427-61.767Q40.720-61.767 40.957-61.603Q41.194-61.440 41.326-61.166L41.887-63.398Q41.926-63.569 41.936-63.666Q41.936-63.827 41.287-63.827Q41.184-63.827 41.184-63.959Q41.189-63.984 41.206-64.047Q41.223-64.111 41.250-64.145Q41.277-64.179 41.326-64.179L42.673-64.286Q42.795-64.286 42.795-64.160L41.355-58.408Q41.287-58.242 41.287-57.929Q41.287-57.499 41.575-57.499Q41.887-57.499 42.051-57.897Q42.214-58.295 42.327-58.818Q42.346-58.876 42.405-58.876L42.527-58.876Q42.566-58.876 42.590-58.842Q42.615-58.808 42.615-58.779Q42.439-58.080 42.231-57.658Q42.024-57.236 41.555-57.236Q41.218-57.236 40.959-57.433Q40.701-57.631 40.637-57.958Q39.993-57.236 39.343-57.236M39.353-57.499Q39.714-57.499 40.054-57.770Q40.393-58.041 40.637-58.408Q40.647-58.417 40.647-58.447L41.194-60.649Q41.135-61 40.935-61.254Q40.735-61.508 40.403-61.508Q40.066-61.508 39.775-61.232Q39.485-60.956 39.285-60.580Q39.089-60.180 38.911-59.482Q38.733-58.784 38.733-58.398Q38.733-58.051 38.882-57.775Q39.031-57.499 39.353-57.499",[1499],[1483,4834,4835],{"transform":4756},[1488,4836],{"d":4837,"fill":1485,"stroke":1485,"className":4838,"style":1500},"M52.346-55.976L46.609-55.976Q46.526-55.976 46.472-56.039Q46.418-56.103 46.418-56.176Q46.418-56.259 46.472-56.318Q46.526-56.376 46.609-56.376L52.346-56.376Q52.424-56.376 52.475-56.318Q52.526-56.259 52.526-56.176Q52.526-56.103 52.475-56.039Q52.424-55.976 52.346-55.976M46.418-58.119Q46.418-58.242 46.550-58.310L51.858-60.820L46.516-63.339Q46.418-63.383 46.418-63.510Q46.418-63.583 46.477-63.647Q46.535-63.710 46.618-63.710Q46.667-63.710 46.696-63.696L52.439-60.986Q52.526-60.942 52.526-60.820Q52.526-60.698 52.409-60.629L46.696-57.939Q46.667-57.919 46.618-57.919Q46.535-57.919 46.477-57.978Q46.418-58.036 46.418-58.119",[1499],[1483,4840,4841],{"transform":4756},[1488,4842],{"d":4843,"fill":1485,"stroke":1485,"className":4844,"style":1500},"M57.316-58.539Q57.316-58.105 57.538-57.802Q57.760-57.499 58.175-57.499Q58.771-57.499 59.320-57.773Q59.870-58.046 60.216-58.530Q60.246-58.559 60.294-58.559Q60.343-58.559 60.394-58.503Q60.446-58.447 60.446-58.398Q60.446-58.359 60.426-58.339Q60.060-57.827 59.440-57.531Q58.820-57.236 58.156-57.236Q57.677-57.236 57.311-57.463Q56.945-57.690 56.745-58.071Q56.544-58.452 56.544-58.930Q56.544-59.604 56.920-60.268Q57.296-60.932 57.921-61.349Q58.546-61.767 59.235-61.767Q59.684-61.767 60.043-61.550Q60.402-61.332 60.402-60.908Q60.402-60.634 60.243-60.441Q60.084-60.248 59.816-60.248Q59.655-60.248 59.545-60.348Q59.435-60.449 59.435-60.610Q59.435-60.844 59.606-61.010Q59.777-61.176 60.006-61.176L60.026-61.176Q59.909-61.347 59.686-61.428Q59.464-61.508 59.225-61.508Q58.639-61.508 58.200-61.008Q57.760-60.507 57.538-59.811Q57.316-59.116 57.316-58.539",[1499],[1488,4846],{"fill":1490,"d":4847},"M115.44-42.426v23.816",[1488,4849],{"d":4850},"m115.44-14.542 2.218-5.898-2.218 1.93-2.218-1.93Z",[1488,4852],{"fill":1490,"d":4853},"m148.088-13.973 59.275-26.557",[1488,4855],{"d":4856,"style":4857},"m211.075-42.193-6.29.387 2.67 1.235-.856 2.814Z","stroke-width:.399988",[1793,4859,4861,4862,617],{"className":4860},[1796],"The substitution method: assume the bound on smaller inputs, substitute into the recurrence, and check that the leftover residual term lets the same bound re-emerge for ",[390,4863,4865],{"className":4864},[393],[390,4866,4868],{"className":4867,"ariaHidden":398},[397],[390,4869,4871,4874],{"className":4870},[402],[390,4872],{"className":4873,"style":407},[406],[390,4875,413],{"className":4876},[411,412],[381,4878,4879,4880,4947,4948,5014],{},"A symmetric argument with the inequality reversed gives ",[390,4881,4883],{"className":4882},[393],[390,4884,4886,4913],{"className":4885,"ariaHidden":398},[397],[390,4887,4889,4892,4895,4898,4901,4904,4907,4910],{"className":4888},[402],[390,4890],{"className":4891,"style":600},[406],[390,4893,723],{"className":4894,"style":722},[411,412],[390,4896,608],{"className":4897},[607],[390,4899,413],{"className":4900},[411,412],[390,4902,616],{"className":4903},[615],[390,4905],{"className":4906,"style":800},[717],[390,4908,805],{"className":4909},[804],[390,4911],{"className":4912,"style":800},[717],[390,4914,4916,4919,4923,4926,4929,4932,4938,4941,4944],{"className":4915},[402],[390,4917],{"className":4918,"style":600},[406],[390,4920,4922],{"className":4921},[411],"Ω",[390,4924,608],{"className":4925},[607],[390,4927,413],{"className":4928},[411,412],[390,4930],{"className":4931,"style":718},[717],[390,4933,4935],{"className":4934},[1213],[390,4936,1219],{"className":4937,"style":1218},[411,1217],[390,4939],{"className":4940,"style":718},[717],[390,4942,413],{"className":4943},[411,412],[390,4945,616],{"className":4946},[615],",\nand together they yield ",[390,4949,4951],{"className":4950},[393],[390,4952,4954,4981],{"className":4953,"ariaHidden":398},[397],[390,4955,4957,4960,4963,4966,4969,4972,4975,4978],{"className":4956},[402],[390,4958],{"className":4959,"style":600},[406],[390,4961,723],{"className":4962,"style":722},[411,412],[390,4964,608],{"className":4965},[607],[390,4967,413],{"className":4968},[411,412],[390,4970,616],{"className":4971},[615],[390,4973],{"className":4974,"style":800},[717],[390,4976,805],{"className":4977},[804],[390,4979],{"className":4980,"style":800},[717],[390,4982,4984,4987,4990,4993,4996,4999,5005,5008,5011],{"className":4983},[402],[390,4985],{"className":4986,"style":600},[406],[390,4988,441],{"className":4989},[411],[390,4991,608],{"className":4992},[607],[390,4994,413],{"className":4995},[411,412],[390,4997],{"className":4998,"style":718},[717],[390,5000,5002],{"className":5001},[1213],[390,5003,1219],{"className":5004,"style":1218},[411,1217],[390,5006],{"className":5007,"style":718},[717],[390,5009,413],{"className":5010},[411,412],[390,5012,616],{"className":5013},[615],", confirming the tree.",[381,5016,5017],{},"Two warnings the books repeat:",[638,5019,5020,5030],{},[641,5021,5022,5025,5026,5029],{},[416,5023,5024],{},"Guess the right form."," Substitution verifies a guess; it cannot invent one.\nUse the recursion tree (or the Master Theorem below) to ",[385,5027,5028],{},"find"," the candidate.",[641,5031,5032,5035,5036,5085,5086,5125,5126,5129,5130,5212],{},[416,5033,5034],{},"Beware the off-by-a-term trap."," A guess like ",[390,5037,5039],{"className":5038},[393],[390,5040,5042,5069],{"className":5041,"ariaHidden":398},[397],[390,5043,5045,5048,5051,5054,5057,5060,5063,5066],{"className":5044},[402],[390,5046],{"className":5047,"style":600},[406],[390,5049,723],{"className":5050,"style":722},[411,412],[390,5052,608],{"className":5053},[607],[390,5055,413],{"className":5056},[411,412],[390,5058,616],{"className":5059},[615],[390,5061],{"className":5062,"style":800},[717],[390,5064,1089],{"className":5065},[804],[390,5067],{"className":5068,"style":800},[717],[390,5070,5072,5076,5079,5082],{"className":5071},[402],[390,5073],{"className":5074,"style":5075},[406],"height:0.6944em;",[390,5077,3016],{"className":5078},[411,412],[390,5080],{"className":5081,"style":718},[717],[390,5083,413],{"className":5084},[411,412]," for a\n",[390,5087,5089],{"className":5088},[393],[390,5090,5092],{"className":5091,"ariaHidden":398},[397],[390,5093,5095,5098,5101,5104,5107,5110,5116,5119,5122],{"className":5094},[402],[390,5096],{"className":5097,"style":600},[406],[390,5099,441],{"className":5100},[411],[390,5102,608],{"className":5103},[607],[390,5105,413],{"className":5106},[411,412],[390,5108],{"className":5109,"style":718},[717],[390,5111,5113],{"className":5112},[1213],[390,5114,1219],{"className":5115,"style":1218},[411,1217],[390,5117],{"className":5118,"style":718},[717],[390,5120,413],{"className":5121},[411,412],[390,5123,616],{"className":5124},[615]," recurrence will ",[385,5127,5128],{},"seem"," to almost work but leave an\nunkillable residual term. Strengthening the hypothesis (subtracting a\nlower-order term, as in ",[390,5131,5133],{"className":5132},[393],[390,5134,5136,5163,5202],{"className":5135,"ariaHidden":398},[397],[390,5137,5139,5142,5145,5148,5151,5154,5157,5160],{"className":5138},[402],[390,5140],{"className":5141,"style":600},[406],[390,5143,723],{"className":5144,"style":722},[411,412],[390,5146,608],{"className":5147},[607],[390,5149,413],{"className":5150},[411,412],[390,5152,616],{"className":5153},[615],[390,5155],{"className":5156,"style":800},[717],[390,5158,1089],{"className":5159},[804],[390,5161],{"className":5162,"style":800},[717],[390,5164,5166,5169,5172,5175,5178,5181,5187,5190,5193,5196,5199],{"className":5165},[402],[390,5167],{"className":5168,"style":462},[406],[390,5170,3016],{"className":5171},[411,412],[390,5173],{"className":5174,"style":718},[717],[390,5176,413],{"className":5177},[411,412],[390,5179],{"className":5180,"style":718},[717],[390,5182,5184],{"className":5183},[1213],[390,5185,1219],{"className":5186,"style":1218},[411,1217],[390,5188],{"className":5189,"style":718},[717],[390,5191,413],{"className":5192},[411,412],[390,5194],{"className":5195,"style":912},[717],[390,5197,3954],{"className":5198},[916],[390,5200],{"className":5201,"style":912},[717],[390,5203,5205,5208],{"className":5204},[402],[390,5206],{"className":5207,"style":5075},[406],[390,5209,5211],{"className":5210},[411,412],"bn",") is the standard fix.",[444,5214,5216],{"id":5215},"a-second-example-counting-inversions","A second example: counting inversions",[381,5218,5219,5220,5223,5224,5227,5228,5351,5352,5385,5386,5422,5423,5534,5535,5550,5551,5633,5634,5665,5666,617],{},"A second divide-and-conquer problem makes the point sharply. Its\nrecurrence has the ",[385,5221,5222],{},"same shape"," as merge sort but a different combine cost, and\nthe combine cost is exactly the thing you must get right. An ",[416,5225,5226],{},"inversion","\nof a list ",[390,5229,5231],{"className":5230},[393],[390,5232,5234],{"className":5233,"ariaHidden":398},[397],[390,5235,5237,5240,5244,5287,5290,5293,5297,5300,5303,5306,5347],{"className":5236},[402],[390,5238],{"className":5239,"style":600},[406],[390,5241,5243],{"className":5242},[607],"⟨",[390,5245,5247,5250],{"className":5246},[411],[390,5248,421],{"className":5249},[411,412],[390,5251,5253],{"className":5252},[2021],[390,5254,5256,5278],{"className":5255},[845,846],[390,5257,5259,5275],{"className":5258},[850],[390,5260,5263],{"className":5261,"style":5262},[854],"height:0.3011em;",[390,5264,5266,5269],{"style":5265},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[390,5267],{"className":5268,"style":2038},[862],[390,5270,5272],{"className":5271},[2042,2043,2044,2045],[390,5273,668],{"className":5274},[411,2045],[390,5276,937],{"className":5277},[936],[390,5279,5281],{"className":5280},[850],[390,5282,5285],{"className":5283,"style":5284},[854],"height:0.15em;",[390,5286],{},[390,5288,998],{"className":5289},[997],[390,5291],{"className":5292,"style":718},[717],[390,5294,5296],{"className":5295},[819],"…",[390,5298],{"className":5299,"style":718},[717],[390,5301,998],{"className":5302},[997],[390,5304],{"className":5305,"style":718},[717],[390,5307,5309,5312],{"className":5308},[411],[390,5310,421],{"className":5311},[411,412],[390,5313,5315],{"className":5314},[2021],[390,5316,5318,5339],{"className":5317},[845,846],[390,5319,5321,5336],{"className":5320},[850],[390,5322,5325],{"className":5323,"style":5324},[854],"height:0.1514em;",[390,5326,5327,5330],{"style":5265},[390,5328],{"className":5329,"style":2038},[862],[390,5331,5333],{"className":5332},[2042,2043,2044,2045],[390,5334,413],{"className":5335},[411,412,2045],[390,5337,937],{"className":5338},[936],[390,5340,5342],{"className":5341},[850],[390,5343,5345],{"className":5344,"style":5284},[854],[390,5346],{},[390,5348,5350],{"className":5349},[615],"⟩"," is a pair ",[390,5353,5355],{"className":5354},[393],[390,5356,5358],{"className":5357,"ariaHidden":398},[397],[390,5359,5361,5364,5367,5371,5374,5377,5382],{"className":5360},[402],[390,5362],{"className":5363,"style":600},[406],[390,5365,608],{"className":5366},[607],[390,5368,5370],{"className":5369},[411,412],"i",[390,5372,998],{"className":5373},[997],[390,5375],{"className":5376,"style":718},[717],[390,5378,5381],{"className":5379,"style":5380},[411,412],"margin-right:0.0572em;","j",[390,5383,616],{"className":5384},[615]," with ",[390,5387,5389],{"className":5388},[393],[390,5390,5392,5412],{"className":5391,"ariaHidden":398},[397],[390,5393,5395,5399,5402,5405,5409],{"className":5394},[402],[390,5396],{"className":5397,"style":5398},[406],"height:0.6986em;vertical-align:-0.0391em;",[390,5400,5370],{"className":5401},[411,412],[390,5403],{"className":5404,"style":800},[717],[390,5406,5408],{"className":5407},[804],"\u003C",[390,5410],{"className":5411,"style":800},[717],[390,5413,5415,5419],{"className":5414},[402],[390,5416],{"className":5417,"style":5418},[406],"height:0.854em;vertical-align:-0.1944em;",[390,5420,5381],{"className":5421,"style":5380},[411,412]," but\n",[390,5424,5426],{"className":5425},[393],[390,5427,5429,5486],{"className":5428,"ariaHidden":398},[397],[390,5430,5432,5436,5477,5480,5483],{"className":5431},[402],[390,5433],{"className":5434,"style":5435},[406],"height:0.6891em;vertical-align:-0.15em;",[390,5437,5439,5442],{"className":5438},[411],[390,5440,421],{"className":5441},[411,412],[390,5443,5445],{"className":5444},[2021],[390,5446,5448,5469],{"className":5447},[845,846],[390,5449,5451,5466],{"className":5450},[850],[390,5452,5455],{"className":5453,"style":5454},[854],"height:0.3117em;",[390,5456,5457,5460],{"style":5265},[390,5458],{"className":5459,"style":2038},[862],[390,5461,5463],{"className":5462},[2042,2043,2044,2045],[390,5464,5370],{"className":5465},[411,412,2045],[390,5467,937],{"className":5468},[936],[390,5470,5472],{"className":5471},[850],[390,5473,5475],{"className":5474,"style":5284},[854],[390,5476],{},[390,5478],{"className":5479,"style":800},[717],[390,5481,1022],{"className":5482},[804],[390,5484],{"className":5485,"style":800},[717],[390,5487,5489,5493],{"className":5488},[402],[390,5490],{"className":5491,"style":5492},[406],"height:0.7167em;vertical-align:-0.2861em;",[390,5494,5496,5499],{"className":5495},[411],[390,5497,421],{"className":5498},[411,412],[390,5500,5502],{"className":5501},[2021],[390,5503,5505,5525],{"className":5504},[845,846],[390,5506,5508,5522],{"className":5507},[850],[390,5509,5511],{"className":5510,"style":5454},[854],[390,5512,5513,5516],{"style":5265},[390,5514],{"className":5515,"style":2038},[862],[390,5517,5519],{"className":5518},[2042,2043,2044,2045],[390,5520,5381],{"className":5521,"style":5380},[411,412,2045],[390,5523,937],{"className":5524},[936],[390,5526,5528],{"className":5527},[850],[390,5529,5532],{"className":5530,"style":5531},[854],"height:0.2861em;",[390,5533],{},"; the number of inversions measures how far from sorted the list is\n(a sorted list has ",[390,5536,5538],{"className":5537},[393],[390,5539,5541],{"className":5540,"ariaHidden":398},[397],[390,5542,5544,5547],{"className":5543},[402],[390,5545],{"className":5546,"style":1464},[406],[390,5548,3035],{"className":5549},[411],", a reversed list has ",[390,5552,5554],{"className":5553},[393],[390,5555,5557],{"className":5556,"ariaHidden":398},[397],[390,5558,5560,5564],{"className":5559},[402],[390,5561],{"className":5562,"style":5563},[406],"height:1.2em;vertical-align:-0.35em;",[390,5565,5567,5573,5627],{"className":5566},[411],[390,5568,5570],{"className":5569,"style":824},[607,823],[390,5571,608],{"className":5572},[828,3651],[390,5574,5576],{"className":5575},[3253],[390,5577,5579,5619],{"className":5578},[845,846],[390,5580,5582,5616],{"className":5581},[850],[390,5583,5586,5601],{"className":5584,"style":5585},[854],"height:0.7454em;",[390,5587,5589,5592],{"style":5588},"top:-2.355em;",[390,5590],{"className":5591,"style":2038},[862],[390,5593,5595],{"className":5594},[2042,2043,2044,2045],[390,5596,5598],{"className":5597},[411,2045],[390,5599,713],{"className":5600},[411,2045],[390,5602,5604,5607],{"style":5603},"top:-3.144em;",[390,5605],{"className":5606,"style":2038},[862],[390,5608,5610],{"className":5609},[2042,2043,2044,2045],[390,5611,5613],{"className":5612},[411,2045],[390,5614,413],{"className":5615},[411,412,2045],[390,5617,937],{"className":5618},[936],[390,5620,5622],{"className":5621},[850],[390,5623,5625],{"className":5624,"style":3314},[854],[390,5626],{},[390,5628,5630],{"className":5629,"style":824},[615,823],[390,5631,616],{"className":5632},[828,3651],"). The task: given\n",[390,5635,5637],{"className":5636},[393],[390,5638,5640],{"className":5639,"ariaHidden":398},[397],[390,5641,5643,5646,5650,5654,5658,5661],{"className":5642},[402],[390,5644],{"className":5645,"style":600},[406],[390,5647,5649],{"className":5648},[411,412],"A",[390,5651,5653],{"className":5652},[607],"[",[390,5655,5657],{"className":5656},[411],"1..",[390,5659,413],{"className":5660},[411,412],[390,5662,5664],{"className":5663},[615],"]",", return ",[390,5667,5669],{"className":5668},[393],[390,5670,5672],{"className":5671,"ariaHidden":398},[397],[390,5673,5675,5678,5685,5688,5691],{"className":5674},[402],[390,5676],{"className":5677,"style":600},[406],[390,5679,5681],{"className":5680},[1213],[390,5682,5684],{"className":5683,"style":1218},[411,1217],"ninv",[390,5686,608],{"className":5687},[607],[390,5689,5649],{"className":5690},[411,412],[390,5692,616],{"className":5693},[615],[381,5695,5696,5697,5747,5748,5751,5752,5767,5768,5771],{},"The brute-force algorithm compares every pair and runs in ",[390,5698,5700],{"className":5699},[393],[390,5701,5703],{"className":5702,"ariaHidden":398},[397],[390,5704,5706,5709,5712,5715,5744],{"className":5705},[402],[390,5707],{"className":5708,"style":2723},[406],[390,5710,441],{"className":5711},[411],[390,5713,608],{"className":5714},[607],[390,5716,5718,5721],{"className":5717},[411],[390,5719,413],{"className":5720},[411,412],[390,5722,5724],{"className":5723},[2021],[390,5725,5727],{"className":5726},[845],[390,5728,5730],{"className":5729},[850],[390,5731,5733],{"className":5732,"style":2748},[854],[390,5734,5735,5738],{"style":2751},[390,5736],{"className":5737,"style":2038},[862],[390,5739,5741],{"className":5740},[2042,2043,2044,2045],[390,5742,713],{"className":5743},[411,2045],[390,5745,616],{"className":5746},[615],". To beat\nit, ",[416,5749,5750],{},"mimic merge sort",": split ",[390,5753,5755],{"className":5754},[393],[390,5756,5758],{"className":5757,"ariaHidden":398},[397],[390,5759,5761,5764],{"className":5760},[402],[390,5762],{"className":5763,"style":437},[406],[390,5765,5649],{"className":5766},[411,412]," in half, recursively count inversions inside\neach half, then count the ",[416,5769,5770],{},"cross inversions",", the pairs with one element in the\nleft half and one in the right. That gives a recurrence of the merge-sort form,",[390,5773,5775],{"className":5774},[772],[390,5776,5778],{"className":5777},[393],[390,5779,5781,5808,5844],{"className":5780,"ariaHidden":398},[397],[390,5782,5784,5787,5790,5793,5796,5799,5802,5805],{"className":5783},[402],[390,5785],{"className":5786,"style":600},[406],[390,5788,723],{"className":5789,"style":722},[411,412],[390,5791,608],{"className":5792},[607],[390,5794,413],{"className":5795},[411,412],[390,5797,616],{"className":5798},[615],[390,5800],{"className":5801,"style":800},[717],[390,5803,1089],{"className":5804},[804],[390,5806],{"className":5807,"style":800},[717],[390,5809,5811,5814,5817,5820,5823,5826,5829,5832,5835,5838,5841],{"className":5810},[402],[390,5812],{"className":5813,"style":600},[406],[390,5815,713],{"className":5816},[411],[390,5818],{"className":5819,"style":718},[717],[390,5821,723],{"className":5822,"style":722},[411,412],[390,5824,608],{"className":5825},[607],[390,5827,413],{"className":5828},[411,412],[390,5830,696],{"className":5831},[411],[390,5833,616],{"className":5834},[615],[390,5836],{"className":5837,"style":912},[717],[390,5839,917],{"className":5840},[916],[390,5842],{"className":5843,"style":912},[717],[390,5845,5847,5850,5853,5860,5863],{"className":5846},[402],[390,5848],{"className":5849,"style":600},[406],[390,5851,608],{"className":5852},[607],[390,5854,5856],{"className":5855},[411,471],[390,5857,5859],{"className":5858},[411],"cost of counting cross inversions",[390,5861,616],{"className":5862},[615],[390,5864,617],{"className":5865},[411],[381,5867,5868,5869,5936,5937,5940,5941,5944],{},"The three kinds of inversion partition cleanly along the split. On\n",[390,5870,5872],{"className":5871},[393],[390,5873,5875,5893],{"className":5874,"ariaHidden":398},[397],[390,5876,5878,5881,5884,5887,5890],{"className":5877},[402],[390,5879],{"className":5880,"style":437},[406],[390,5882,5649],{"className":5883},[411,412],[390,5885],{"className":5886,"style":800},[717],[390,5888,805],{"className":5889},[804],[390,5891],{"className":5892,"style":800},[717],[390,5894,5896,5899,5902,5905,5908,5911,5914,5917,5920,5923,5926,5929,5933],{"className":5895},[402],[390,5897],{"className":5898,"style":600},[406],[390,5900,5243],{"className":5901},[607],[390,5903,713],{"className":5904},[411],[390,5906,998],{"className":5907},[997],[390,5909],{"className":5910,"style":718},[717],[390,5912,1943],{"className":5913},[411],[390,5915,998],{"className":5916},[997],[390,5918],{"className":5919,"style":718},[717],[390,5921,668],{"className":5922},[411],[390,5924,998],{"className":5925},[997],[390,5927],{"className":5928,"style":718},[717],[390,5930,5932],{"className":5931},[411],"3",[390,5934,5350],{"className":5935},[615],", 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91.073-35.248Q90.681-35.383 90.415-35.668Q90.150-35.953 90.150-36.371Q90.150-36.566 90.282-36.695Q90.415-36.824 90.607-36.824Q90.732-36.824 90.835-36.765Q90.939-36.707 91.001-36.601Q91.064-36.496 91.064-36.371Q91.064-36.176 90.929-36.045Q90.794-35.914 90.591-35.914M94.685-33.281L94.603-33.281Q94.568-33.281 94.542-33.310Q94.517-33.340 94.517-33.379Q94.517-33.429 94.548-33.449Q94.935-33.785 95.218-34.234Q95.501-34.683 95.667-35.183Q95.833-35.683 95.907-36.201Q95.982-36.718 95.982-37.281Q95.982-37.851 95.907-38.367Q95.833-38.883 95.667-39.379Q95.501-39.875 95.222-40.322Q94.943-40.769 94.548-41.113Q94.517-41.133 94.517-41.183Q94.517-41.222 94.542-41.252Q94.568-41.281 94.603-41.281L94.685-41.281Q94.696-41.281 94.706-41.279Q94.716-41.277 94.724-41.273Q95.337-40.816 95.739-40.181Q96.142-39.547 96.337-38.801Q96.532-38.054 96.532-37.281Q96.532-36.508 96.337-35.761Q96.142-35.015 95.739-34.381Q95.337-33.746 94.724-33.289Q94.712-33.289 94.704-33.287Q94.696-33.285 94.685-33.281",[1499],[1793,6170,6172,6173,6221,6222,1283,6252,6282,6283,6361],{"className":6171},[1796],"Cross inversions on ",[390,6174,6176],{"className":6175},[393],[390,6177,6179],{"className":6178,"ariaHidden":398},[397],[390,6180,6182,6185,6188,6191,6194,6197,6200,6203,6206,6209,6212,6215,6218],{"className":6181},[402],[390,6183],{"className":6184,"style":600},[406],[390,6186,5243],{"className":6187},[607],[390,6189,713],{"className":6190},[411],[390,6192,998],{"className":6193},[997],[390,6195],{"className":6196,"style":718},[717],[390,6198,1943],{"className":6199},[411],[390,6201,998],{"className":6202},[997],[390,6204],{"className":6205,"style":718},[717],[390,6207,668],{"className":6208},[411],[390,6210,998],{"className":6211},[997],[390,6213],{"className":6214,"style":718},[717],[390,6216,5932],{"className":6217},[411],[390,6219,5350],{"className":6220},[615]," split into halves ",[390,6223,6225],{"className":6224},[393],[390,6226,6228],{"className":6227,"ariaHidden":398},[397],[390,6229,6231,6234,6237,6240,6243,6246,6249],{"className":6230},[402],[390,6232],{"className":6233,"style":600},[406],[390,6235,5243],{"className":6236},[607],[390,6238,713],{"className":6239},[411],[390,6241,998],{"className":6242},[997],[390,6244],{"className":6245,"style":718},[717],[390,6247,1943],{"className":6248},[411],[390,6250,5350],{"className":6251},[615],[390,6253,6255],{"className":6254},[393],[390,6256,6258],{"className":6257,"ariaHidden":398},[397],[390,6259,6261,6264,6267,6270,6273,6276,6279],{"className":6260},[402],[390,6262],{"className":6263,"style":600},[406],[390,6265,5243],{"className":6266},[607],[390,6268,668],{"className":6269},[411],[390,6271,998],{"className":6272},[997],[390,6274],{"className":6275,"style":718},[717],[390,6277,5932],{"className":6278},[411],[390,6280,5350],{"className":6281},[615],". Each red arc is a cross inversion (a left element bigger than a right one): ",[390,6284,6286],{"className":6285},[393],[390,6287,6289],{"className":6288,"ariaHidden":398},[397],[390,6290,6292,6295,6298,6301,6304,6307,6310,6313,6316,6319,6322,6325,6328,6331,6334,6337,6340,6343,6346,6349,6352,6355,6358],{"className":6291},[402],[390,6293],{"className":6294,"style":600},[406],[390,6296,608],{"className":6297},[607],[390,6299,713],{"className":6300},[411],[390,6302,998],{"className":6303},[997],[390,6305],{"className":6306,"style":718},[717],[390,6308,668],{"className":6309},[411],[390,6311,616],{"className":6312},[615],[390,6314,998],{"className":6315},[997],[390,6317],{"className":6318,"style":718},[717],[390,6320,608],{"className":6321},[607],[390,6323,1943],{"className":6324},[411],[390,6326,998],{"className":6327},[997],[390,6329],{"className":6330,"style":718},[717],[390,6332,668],{"className":6333},[411],[390,6335,616],{"className":6336},[615],[390,6338,998],{"className":6339},[997],[390,6341],{"className":6342,"style":718},[717],[390,6344,608],{"className":6345},[607],[390,6347,1943],{"className":6348},[411],[390,6350,998],{"className":6351},[997],[390,6353],{"className":6354,"style":718},[717],[390,6356,5932],{"className":6357},[411],[390,6359,616],{"className":6360},[615],". Within-half inversions are handled by recursion; the combine step counts only these crossing pairs.",[381,6363,6364,6365,6368,6369,6419,6420,6528,6529,6775,6776,6779,6780,6830],{},"Counting cross inversions ",[385,6366,6367],{},"naively",", with a double loop over the two halves, costs\n",[390,6370,6372],{"className":6371},[393],[390,6373,6375],{"className":6374,"ariaHidden":398},[397],[390,6376,6378,6381,6384,6387,6416],{"className":6377},[402],[390,6379],{"className":6380,"style":2723},[406],[390,6382,441],{"className":6383},[411],[390,6385,608],{"className":6386},[607],[390,6388,6390,6393],{"className":6389},[411],[390,6391,413],{"className":6392},[411,412],[390,6394,6396],{"className":6395},[2021],[390,6397,6399],{"className":6398},[845],[390,6400,6402],{"className":6401},[850],[390,6403,6405],{"className":6404,"style":2748},[854],[390,6406,6407,6410],{"style":2751},[390,6408],{"className":6409,"style":2038},[862],[390,6411,6413],{"className":6412},[2042,2043,2044,2045],[390,6414,713],{"className":6415},[411,2045],[390,6417,616],{"className":6418},[615],", so the recurrence becomes ",[390,6421,6423],{"className":6422},[393],[390,6424,6426,6453,6489],{"className":6425,"ariaHidden":398},[397],[390,6427,6429,6432,6435,6438,6441,6444,6447,6450],{"className":6428},[402],[390,6430],{"className":6431,"style":600},[406],[390,6433,723],{"className":6434,"style":722},[411,412],[390,6436,608],{"className":6437},[607],[390,6439,413],{"className":6440},[411,412],[390,6442,616],{"className":6443},[615],[390,6445],{"className":6446,"style":800},[717],[390,6448,1089],{"className":6449},[804],[390,6451],{"className":6452,"style":800},[717],[390,6454,6456,6459,6462,6465,6468,6471,6474,6477,6480,6483,6486],{"className":6455},[402],[390,6457],{"className":6458,"style":600},[406],[390,6460,713],{"className":6461},[411],[390,6463],{"className":6464,"style":718},[717],[390,6466,723],{"className":6467,"style":722},[411,412],[390,6469,608],{"className":6470},[607],[390,6472,413],{"className":6473},[411,412],[390,6475,696],{"className":6476},[411],[390,6478,616],{"className":6479},[615],[390,6481],{"className":6482,"style":912},[717],[390,6484,917],{"className":6485},[916],[390,6487],{"className":6488,"style":912},[717],[390,6490,6492,6495,6499],{"className":6491},[402],[390,6493],{"className":6494,"style":2748},[406],[390,6496,6498],{"className":6497},[411,412],"b",[390,6500,6502,6505],{"className":6501},[411],[390,6503,413],{"className":6504},[411,412],[390,6506,6508],{"className":6507},[2021],[390,6509,6511],{"className":6510},[845],[390,6512,6514],{"className":6513},[850],[390,6515,6517],{"className":6516,"style":2748},[854],[390,6518,6519,6522],{"style":2751},[390,6520],{"className":6521,"style":2038},[862],[390,6523,6525],{"className":6524},[2042,2043,2044,2045],[390,6526,713],{"className":6527},[411,2045],". Feed that to\nthe recursion tree: the per-level work is now ",[390,6530,6532],{"className":6531},[393],[390,6533,6535,6599,6655],{"className":6534,"ariaHidden":398},[397],[390,6536,6538,6542,6545,6548,6577,6580,6584,6587,6590,6593,6596],{"className":6537},[402],[390,6539],{"className":6540,"style":6541},[406],"height:1.0085em;vertical-align:-0.1944em;",[390,6543,6498],{"className":6544},[411,412],[390,6546],{"className":6547,"style":718},[717],[390,6549,6551,6554],{"className":6550},[411],[390,6552,413],{"className":6553},[411,412],[390,6555,6557],{"className":6556},[2021],[390,6558,6560],{"className":6559},[845],[390,6561,6563],{"className":6562},[850],[390,6564,6566],{"className":6565,"style":2748},[854],[390,6567,6568,6571],{"style":2751},[390,6569],{"className":6570,"style":2038},[862],[390,6572,6574],{"className":6573},[2042,2043,2044,2045],[390,6575,713],{"className":6576},[411,2045],[390,6578,998],{"className":6579},[997],[390,6581,6583],{"className":6582},[717]," ",[390,6585],{"className":6586,"style":718},[717],[390,6588,713],{"className":6589},[411],[390,6591],{"className":6592,"style":912},[717],[390,6594,1881],{"className":6595},[916],[390,6597],{"className":6598,"style":912},[717],[390,6600,6602,6605,6608,6611,6614,6617,6646,6649,6652],{"className":6601},[402],[390,6603],{"className":6604,"style":2723},[406],[390,6606,6498],{"className":6607},[411,412],[390,6609,608],{"className":6610},[607],[390,6612,413],{"className":6613},[411,412],[390,6615,696],{"className":6616},[411],[390,6618,6620,6623],{"className":6619},[615],[390,6621,616],{"className":6622},[615],[390,6624,6626],{"className":6625},[2021],[390,6627,6629],{"className":6628},[845],[390,6630,6632],{"className":6631},[850],[390,6633,6635],{"className":6634,"style":2748},[854],[390,6636,6637,6640],{"style":2751},[390,6638],{"className":6639,"style":2038},[862],[390,6641,6643],{"className":6642},[2042,2043,2044,2045],[390,6644,713],{"className":6645},[411,2045],[390,6647],{"className":6648,"style":800},[717],[390,6650,805],{"className":6651},[804],[390,6653],{"className":6654,"style":800},[717],[390,6656,6658,6662,6731,6734,6763,6766,6769,6772],{"className":6657},[402],[390,6659],{"className":6660,"style":6661},[406],"height:1.1901em;vertical-align:-0.345em;",[390,6663,6665,6668,6728],{"className":6664},[411],[390,6666],{"className":6667},[607,1044],[390,6669,6671],{"className":6670},[3253],[390,6672,6674,6720],{"className":6673},[845,846],[390,6675,6677,6717],{"className":6676},[850],[390,6678,6681,6695,6703],{"className":6679,"style":6680},[854],"height:0.8451em;",[390,6682,6683,6686],{"style":3266},[390,6684],{"className":6685,"style":2212},[862],[390,6687,6689],{"className":6688},[2042,2043,2044,2045],[390,6690,6692],{"className":6691},[411,2045],[390,6693,713],{"className":6694},[411,2045],[390,6696,6697,6700],{"style":3281},[390,6698],{"className":6699,"style":2212},[862],[390,6701],{"className":6702,"style":3289},[3288],[390,6704,6705,6708],{"style":3292},[390,6706],{"className":6707,"style":2212},[862],[390,6709,6711],{"className":6710},[2042,2043,2044,2045],[390,6712,6714],{"className":6713},[411,2045],[390,6715,668],{"className":6716},[411,2045],[390,6718,937],{"className":6719},[936],[390,6721,6723],{"className":6722},[850],[390,6724,6726],{"className":6725,"style":3314},[854],[390,6727],{},[390,6729],{"className":6730},[615,1044],[390,6732,6498],{"className":6733},[411,412],[390,6735,6737,6740],{"className":6736},[411],[390,6738,413],{"className":6739},[411,412],[390,6741,6743],{"className":6742},[2021],[390,6744,6746],{"className":6745},[845],[390,6747,6749],{"className":6748},[850],[390,6750,6752],{"className":6751,"style":2748},[854],[390,6753,6754,6757],{"style":2751},[390,6755],{"className":6756,"style":2038},[862],[390,6758,6760],{"className":6759},[2042,2043,2044,2045],[390,6761,713],{"className":6762},[411,2045],[390,6764,998],{"className":6765},[997],[390,6767,6583],{"className":6768},[717],[390,6770],{"className":6771,"style":718},[717],[390,6773,5296],{"className":6774},[819],", which ",[385,6777,6778],{},"shrinks geometrically",", so the root dominates and the\ntree sums to ",[390,6781,6783],{"className":6782},[393],[390,6784,6786],{"className":6785,"ariaHidden":398},[397],[390,6787,6789,6792,6795,6798,6827],{"className":6788},[402],[390,6790],{"className":6791,"style":2723},[406],[390,6793,441],{"className":6794},[411],[390,6796,608],{"className":6797},[607],[390,6799,6801,6804],{"className":6800},[411],[390,6802,413],{"className":6803},[411,412],[390,6805,6807],{"className":6806},[2021],[390,6808,6810],{"className":6809},[845],[390,6811,6813],{"className":6812},[850],[390,6814,6816],{"className":6815,"style":2748},[854],[390,6817,6818,6821],{"style":2751},[390,6819],{"className":6820,"style":2038},[862],[390,6822,6824],{"className":6823},[2042,2043,2044,2045],[390,6825,713],{"className":6826},[411,2045],[390,6828,616],{"className":6829},[615],". That is no improvement. The split bought us nothing because\nthe combine step is as expensive as the brute force.",[1470,6832,6834,7211],{"className":6833},[1473,1474],[1476,6835,6838],{"xmlns":1478,"width":6836,"height":1480,"viewBox":6837},"447.612","-75 -75 335.709 155.946",[1483,6839,6840,6843,6858,6861,6903,6906,6942,6945,6982,6985,7021,7024,7060,7063,7099,7102,7123,7158,7190],{"stroke":1485,"style":1486},[1488,6841],{"fill":1490,"d":6842},"M59.547-72.07h-10.78a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h10.78a4 4 0 0 0 4-4v-9.072a4 4 0 0 0-4-4Zm-14.78 17.072",[1483,6844,6845,6852],{"stroke":1490},[1483,6846,6848],{"transform":6847},"translate(-7.39 4.07)",[1488,6849],{"d":6850,"fill":1485,"stroke":1485,"className":6851,"style":1500},"M55.886-63.422Q55.280-63.422 54.943-63.895Q54.606-64.369 54.606-65.004Q54.606-65.096 54.653-65.367Q54.699-65.638 54.699-65.702L55.666-69.584Q55.705-69.755 55.715-69.852Q55.715-70.013 55.065-70.013Q54.968-70.013 54.968-70.145Q54.972-70.170 54.990-70.233Q55.007-70.297 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",[390,7216,7218],{"className":7217},[393],[390,7219,7221],{"className":7220,"ariaHidden":398},[397],[390,7222,7224,7227,7230,7233,7262],{"className":7223},[402],[390,7225],{"className":7226,"style":2723},[406],[390,7228,441],{"className":7229},[411],[390,7231,608],{"className":7232},[607],[390,7234,7236,7239],{"className":7235},[411],[390,7237,413],{"className":7238},[411,412],[390,7240,7242],{"className":7241},[2021],[390,7243,7245],{"className":7244},[845],[390,7246,7248],{"className":7247},[850],[390,7249,7251],{"className":7250,"style":2748},[854],[390,7252,7253,7256],{"style":2751},[390,7254],{"className":7255,"style":2038},[862],[390,7257,7259],{"className":7258},[2042,2043,2044,2045],[390,7260,713],{"className":7261},[411,2045],[390,7263,616],{"className":7264},[615],[381,7266,7267,7268,7271,7272,7296,7297,7347,7348,7351,7352,7439,7440,7479],{},"The lesson the recurrence teaches is therefore a ",[385,7269,7270],{},"demand",": the combine step must\nrun in ",[390,7273,7275],{"className":7274},[393],[390,7276,7278],{"className":7277,"ariaHidden":398},[397],[390,7279,7281,7284,7287,7290,7293],{"className":7280},[402],[390,7282],{"className":7283,"style":600},[406],[390,7285,1139],{"className":7286,"style":1138},[411,412],[390,7288,608],{"className":7289},[607],[390,7291,413],{"className":7292},[411,412],[390,7294,616],{"className":7295},[615],", not ",[390,7298,7300],{"className":7299},[393],[390,7301,7303],{"className":7302,"ariaHidden":398},[397],[390,7304,7306,7309,7312,7315,7344],{"className":7305},[402],[390,7307],{"className":7308,"style":2723},[406],[390,7310,1139],{"className":7311,"style":1138},[411,412],[390,7313,608],{"className":7314},[607],[390,7316,7318,7321],{"className":7317},[411],[390,7319,413],{"className":7320},[411,412],[390,7322,7324],{"className":7323},[2021],[390,7325,7327],{"className":7326},[845],[390,7328,7330],{"className":7329},[850],[390,7331,7333],{"className":7332,"style":2748},[854],[390,7334,7335,7338],{"style":2751},[390,7336],{"className":7337,"style":2038},[862],[390,7339,7341],{"className":7340},[2042,2043,2044,2045],[390,7342,713],{"className":7343},[411,2045],[390,7345,616],{"className":7346},[615],". If we can count cross inversions in linear time,\nwhich one can, by counting them ",[385,7349,7350],{},"while merging"," the two sorted halves, the\nrecurrence collapses to ",[390,7353,7355],{"className":7354},[393],[390,7356,7358,7385,7421],{"className":7357,"ariaHidden":398},[397],[390,7359,7361,7364,7367,7370,7373,7376,7379,7382],{"className":7360},[402],[390,7362],{"className":7363,"style":600},[406],[390,7365,723],{"className":7366,"style":722},[411,412],[390,7368,608],{"className":7369},[607],[390,7371,413],{"className":7372},[411,412],[390,7374,616],{"className":7375},[615],[390,7377],{"className":7378,"style":800},[717],[390,7380,1089],{"className":7381},[804],[390,7383],{"className":7384,"style":800},[717],[390,7386,7388,7391,7394,7397,7400,7403,7406,7409,7412,7415,7418],{"className":7387},[402],[390,7389],{"className":7390,"style":600},[406],[390,7392,713],{"className":7393},[411],[390,7395],{"className":7396,"style":718},[717],[390,7398,723],{"className":7399,"style":722},[411,412],[390,7401,608],{"className":7402},[607],[390,7404,413],{"className":7405},[411,412],[390,7407,696],{"className":7408},[411],[390,7410,616],{"className":7411},[615],[390,7413],{"className":7414,"style":912},[717],[390,7416,917],{"className":7417},[916],[390,7419],{"className":7420,"style":912},[717],[390,7422,7424,7427,7430,7433,7436],{"className":7423},[402],[390,7425],{"className":7426,"style":600},[406],[390,7428,1139],{"className":7429,"style":1138},[411,412],[390,7431,608],{"className":7432},[607],[390,7434,413],{"className":7435},[411,412],[390,7437,616],{"className":7438},[615],", the merge-sort recurrence,\nand we get ",[390,7441,7443],{"className":7442},[393],[390,7444,7446],{"className":7445,"ariaHidden":398},[397],[390,7447,7449,7452,7455,7458,7461,7464,7470,7473,7476],{"className":7448},[402],[390,7450],{"className":7451,"style":600},[406],[390,7453,441],{"className":7454},[411],[390,7456,608],{"className":7457},[607],[390,7459,413],{"className":7460},[411,412],[390,7462],{"className":7463,"style":718},[717],[390,7465,7467],{"className":7466},[1213],[390,7468,1219],{"className":7469,"style":1218},[411,1217],[390,7471],{"className":7472,"style":718},[717],[390,7474,413],{"className":7475},[411,412],[390,7477,616],{"className":7478},[615],". The recurrence both predicts\nthe running time and tells you precisely how fast the combine step has to be for\ndivide-and-conquer to pay off.",[444,7481,7483],{"id":7482},"method-3-the-master-theorem","Method 3: the Master Theorem",[381,7485,7486,7487,7490],{},"Merge sort's recurrence is one instance of a common pattern. The ",[416,7488,7489],{},"Master\nTheorem"," solves every recurrence of the form",[390,7492,7494],{"className":7493},[772],[390,7495,7497],{"className":7496},[393],[390,7498,7500,7527,7567],{"className":7499,"ariaHidden":398},[397],[390,7501,7503,7506,7509,7512,7515,7518,7521,7524],{"className":7502},[402],[390,7504],{"className":7505,"style":600},[406],[390,7507,723],{"className":7508,"style":722},[411,412],[390,7510,608],{"className":7511},[607],[390,7513,413],{"className":7514},[411,412],[390,7516,616],{"className":7517},[615],[390,7519],{"className":7520,"style":800},[717],[390,7522,805],{"className":7523},[804],[390,7525],{"className":7526,"style":800},[717],[390,7528,7530,7533,7536,7539,7542,7545,7548,7552,7555,7558,7561,7564],{"className":7529},[402],[390,7531],{"className":7532,"style":600},[406],[390,7534,421],{"className":7535},[411,412],[390,7537],{"className":7538,"style":718},[717],[390,7540,723],{"className":7541,"style":722},[411,412],[390,7543,608],{"className":7544},[607],[390,7546,413],{"className":7547},[411,412],[390,7549,7551],{"className":7550},[411],"\u002F",[390,7553,6498],{"className":7554},[411,412],[390,7556,616],{"className":7557},[615],[390,7559],{"className":7560,"style":912},[717],[390,7562,917],{"className":7563},[916],[390,7565],{"className":7566,"style":912},[717],[390,7568,7570,7573,7578,7581,7584,7587],{"className":7569},[402],[390,7571],{"className":7572,"style":600},[406],[390,7574,7577],{"className":7575,"style":7576},[411,412],"margin-right:0.1076em;","f",[390,7579,608],{"className":7580},[607],[390,7582,413],{"className":7583},[411,412],[390,7585,616],{"className":7586},[615],[390,7588,998],{"className":7589},[997],[381,7591,7592,7593,1283,7626,7659,7660,7684,7685,7700,7701,7722,7723,7747],{},"where ",[390,7594,7596],{"className":7595},[393],[390,7597,7599,7617],{"className":7598,"ariaHidden":398},[397],[390,7600,7602,7605,7608,7611,7614],{"className":7601},[402],[390,7603],{"className":7604,"style":3150},[406],[390,7606,421],{"className":7607},[411,412],[390,7609],{"className":7610,"style":800},[717],[390,7612,3160],{"className":7613},[804],[390,7615],{"className":7616,"style":800},[717],[390,7618,7620,7623],{"className":7619},[402],[390,7621],{"className":7622,"style":1464},[406],[390,7624,668],{"className":7625},[411],[390,7627,7629],{"className":7628},[393],[390,7630,7632,7650],{"className":7631,"ariaHidden":398},[397],[390,7633,7635,7638,7641,7644,7647],{"className":7634},[402],[390,7636],{"className":7637,"style":3012},[406],[390,7639,6498],{"className":7640},[411,412],[390,7642],{"className":7643,"style":800},[717],[390,7645,1022],{"className":7646},[804],[390,7648],{"className":7649,"style":800},[717],[390,7651,7653,7656],{"className":7652},[402],[390,7654],{"className":7655,"style":1464},[406],[390,7657,668],{"className":7658},[411]," are constants and ",[390,7661,7663],{"className":7662},[393],[390,7664,7666],{"className":7665,"ariaHidden":398},[397],[390,7667,7669,7672,7675,7678,7681],{"className":7668},[402],[390,7670],{"className":7671,"style":600},[406],[390,7673,7577],{"className":7674,"style":7576},[411,412],[390,7676,608],{"className":7677},[607],[390,7679,413],{"className":7680},[411,412],[390,7682,616],{"className":7683},[615]," is the divide-and-combine\nwork. Here ",[390,7686,7688],{"className":7687},[393],[390,7689,7691],{"className":7690,"ariaHidden":398},[397],[390,7692,7694,7697],{"className":7693},[402],[390,7695],{"className":7696,"style":407},[406],[390,7698,421],{"className":7699},[411,412]," is the number of subproblems, ",[390,7702,7704],{"className":7703},[393],[390,7705,7707],{"className":7706,"ariaHidden":398},[397],[390,7708,7710,7713,7716,7719],{"className":7709},[402],[390,7711],{"className":7712,"style":600},[406],[390,7714,413],{"className":7715},[411,412],[390,7717,7551],{"className":7718},[411],[390,7720,6498],{"className":7721},[411,412]," is each subproblem's size, and\n",[390,7724,7726],{"className":7725},[393],[390,7727,7729],{"className":7728,"ariaHidden":398},[397],[390,7730,7732,7735,7738,7741,7744],{"className":7731},[402],[390,7733],{"className":7734,"style":600},[406],[390,7736,7577],{"className":7737,"style":7576},[411,412],[390,7739,608],{"className":7740},[607],[390,7742,413],{"className":7743},[411,412],[390,7745,616],{"className":7746},[615]," is the work done outside the recursion.",[381,7749,7750,7751,7775,7776,7779,7876,7877,8060],{},"The theorem compares ",[390,7752,7754],{"className":7753},[393],[390,7755,7757],{"className":7756,"ariaHidden":398},[397],[390,7758,7760,7763,7766,7769,7772],{"className":7759},[402],[390,7761],{"className":7762,"style":600},[406],[390,7764,7577],{"className":7765,"style":7576},[411,412],[390,7767,608],{"className":7768},[607],[390,7770,413],{"className":7771},[411,412],[390,7773,616],{"className":7774},[615]," against the ",[416,7777,7778],{},"watershed function",[390,7780,7782],{"className":7781},[393],[390,7783,7785],{"className":7784,"ariaHidden":398},[397],[390,7786,7788,7792],{"className":7787},[402],[390,7789],{"className":7790,"style":7791},[406],"height:0.8491em;",[390,7793,7795,7798],{"className":7794},[411],[390,7796,413],{"className":7797},[411,412],[390,7799,7801],{"className":7800},[2021],[390,7802,7804],{"className":7803},[845],[390,7805,7807],{"className":7806},[850],[390,7808,7810],{"className":7809,"style":7791},[854],[390,7811,7812,7815],{"style":2751},[390,7813],{"className":7814,"style":2038},[862],[390,7816,7818],{"className":7817},[2042,2043,2044,2045],[390,7819,7821,7869,7873],{"className":7820},[411,2045],[390,7822,7824,7830],{"className":7823},[1213,2045],[390,7825,7827],{"className":7826},[1213,2045],[390,7828,1219],{"className":7829,"style":1218},[411,1217,2045],[390,7831,7833],{"className":7832},[2021],[390,7834,7836,7860],{"className":7835},[845,846],[390,7837,7839,7857],{"className":7838},[850],[390,7840,7843],{"className":7841,"style":7842},[854],"height:0.2302em;",[390,7844,7846,7850],{"style":7845},"top:-2.2341em;margin-right:0.0714em;",[390,7847],{"className":7848,"style":7849},[862],"height:2.5em;",[390,7851,7854],{"className":7852},[2042,7853,3651,2045],"reset-size3",[390,7855,6498],{"className":7856},[411,412,2045],[390,7858,937],{"className":7859},[936],[390,7861,7863],{"className":7862},[850],[390,7864,7867],{"className":7865,"style":7866},[854],"height:0.2659em;",[390,7868],{},[390,7870],{"className":7871,"style":7872},[717,2045],"margin-right:0.1952em;",[390,7874,421],{"className":7875},[411,412,2045],", the total cost of the leaves, which equals the number of leaves\n",[390,7878,7880],{"className":7879},[393],[390,7881,7883,7976],{"className":7882,"ariaHidden":398},[397],[390,7884,7886,7889,7967,7970,7973],{"className":7885},[402],[390,7887],{"className":7888,"style":7791},[406],[390,7890,7892,7895],{"className":7891},[411],[390,7893,421],{"className":7894},[411,412],[390,7896,7898],{"className":7897},[2021],[390,7899,7901],{"className":7900},[845],[390,7902,7904],{"className":7903},[850],[390,7905,7907],{"className":7906,"style":7791},[854],[390,7908,7909,7912],{"style":2751},[390,7910],{"className":7911,"style":2038},[862],[390,7913,7915],{"className":7914},[2042,2043,2044,2045],[390,7916,7918,7961,7964],{"className":7917},[411,2045],[390,7919,7921,7927],{"className":7920},[1213,2045],[390,7922,7924],{"className":7923},[1213,2045],[390,7925,1219],{"className":7926,"style":1218},[411,1217,2045],[390,7928,7930],{"className":7929},[2021],[390,7931,7933,7953],{"className":7932},[845,846],[390,7934,7936,7950],{"className":7935},[850],[390,7937,7939],{"className":7938,"style":7842},[854],[390,7940,7941,7944],{"style":7845},[390,7942],{"className":7943,"style":7849},[862],[390,7945,7947],{"className":7946},[2042,7853,3651,2045],[390,7948,6498],{"className":7949},[411,412,2045],[390,7951,937],{"className":7952},[936],[390,7954,7956],{"className":7955},[850],[390,7957,7959],{"className":7958,"style":7866},[854],[390,7960],{},[390,7962],{"className":7963,"style":7872},[717,2045],[390,7965,413],{"className":7966},[411,412,2045],[390,7968],{"className":7969,"style":800},[717],[390,7971,805],{"className":7972},[804],[390,7974],{"className":7975,"style":800},[717],[390,7977,7979,7982],{"className":7978},[402],[390,7980],{"className":7981,"style":7791},[406],[390,7983,7985,7988],{"className":7984},[411],[390,7986,413],{"className":7987},[411,412],[390,7989,7991],{"className":7990},[2021],[390,7992,7994],{"className":7993},[845],[390,7995,7997],{"className":7996},[850],[390,7998,8000],{"className":7999,"style":7791},[854],[390,8001,8002,8005],{"style":2751},[390,8003],{"className":8004,"style":2038},[862],[390,8006,8008],{"className":8007},[2042,2043,2044,2045],[390,8009,8011,8054,8057],{"className":8010},[411,2045],[390,8012,8014,8020],{"className":8013},[1213,2045],[390,8015,8017],{"className":8016},[1213,2045],[390,8018,1219],{"className":8019,"style":1218},[411,1217,2045],[390,8021,8023],{"className":8022},[2021],[390,8024,8026,8046],{"className":8025},[845,846],[390,8027,8029,8043],{"className":8028},[850],[390,8030,8032],{"className":8031,"style":7842},[854],[390,8033,8034,8037],{"style":7845},[390,8035],{"className":8036,"style":7849},[862],[390,8038,8040],{"className":8039},[2042,7853,3651,2045],[390,8041,6498],{"className":8042},[411,412,2045],[390,8044,937],{"className":8045},[936],[390,8047,8049],{"className":8048},[850],[390,8050,8052],{"className":8051,"style":7866},[854],[390,8053],{},[390,8055],{"className":8056,"style":7872},[717,2045],[390,8058,421],{"className":8059},[411,412,2045]," times the constant base-case cost. Which of the two\ndominates determines the answer.",[1470,8062,8064,8372],{"className":8063},[1473,1474],[1476,8065,8069],{"xmlns":1478,"width":8066,"height":8067,"viewBox":8068},"265.373","233.695","-75 -75 199.030 175.271",[1483,8070,8071,8103,8106,8131,8134,8157,8179,8192,8203,8221,8224,8227,8284],{"stroke":1485,"style":1486},[1483,8072,8073,8076],{"fill":4491},[1488,8074],{"d":8075},"M27.627-72.07H11.875a4 4 0 0 0-4 4v9.072a4 4 0 0 0 4 4h15.752a4 4 0 0 0 4-4v-9.072a4 4 0 0 0-4-4ZM7.875-54.998",[1483,8077,8078,8085,8091,8097],{"fill":1485,"stroke":1490,"fontSize":1655},[1483,8079,8081],{"transform":8080},"translate(-9.876 2.5)",[1488,8082],{"d":8083,"fill":1485,"stroke":1485,"className":8084,"style":1500},"M20.723-61.884Q20.908-61.742 21.172-61.742Q21.528-61.742 21.753-62.533Q21.846-62.914 22.261-65.043L22.720-67.494L21.860-67.494Q21.763-67.494 21.763-67.626Q21.802-67.846 21.890-67.846L22.783-67.846L22.900-68.495Q22.959-68.798 23.008-69.015Q23.057-69.232 23.113-69.418Q23.169-69.603 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166.392-67.510Q166.520-67.317 166.766-67.317",[1499],[1793,8373,8375,8376,8400,8401,617],{"className":8374},[1796],"The Master Theorem weighs the root's combine work ",[390,8377,8379],{"className":8378},[393],[390,8380,8382],{"className":8381,"ariaHidden":398},[397],[390,8383,8385,8388,8391,8394,8397],{"className":8384},[402],[390,8386],{"className":8387,"style":600},[406],[390,8389,7577],{"className":8390,"style":7576},[411,412],[390,8392,608],{"className":8393},[607],[390,8395,413],{"className":8396},[411,412],[390,8398,616],{"className":8399},[615]," against the leaves' total cost, the watershed ",[390,8402,8404],{"className":8403},[393],[390,8405,8407],{"className":8406,"ariaHidden":398},[397],[390,8408,8410,8413],{"className":8409},[402],[390,8411],{"className":8412,"style":7791},[406],[390,8414,8416,8419],{"className":8415},[411],[390,8417,413],{"className":8418},[411,412],[390,8420,8422],{"className":8421},[2021],[390,8423,8425],{"className":8424},[845],[390,8426,8428],{"className":8427},[850],[390,8429,8431],{"className":8430,"style":7791},[854],[390,8432,8433,8436],{"style":2751},[390,8434],{"className":8435,"style":2038},[862],[390,8437,8439],{"className":8438},[2042,2043,2044,2045],[390,8440,8442,8485,8488],{"className":8441},[411,2045],[390,8443,8445,8451],{"className":8444},[1213,2045],[390,8446,8448],{"className":8447},[1213,2045],[390,8449,1219],{"className":8450,"style":1218},[411,1217,2045],[390,8452,8454],{"className":8453},[2021],[390,8455,8457,8477],{"className":8456},[845,846],[390,8458,8460,8474],{"className":8459},[850],[390,8461,8463],{"className":8462,"style":7842},[854],[390,8464,8465,8468],{"style":7845},[390,8466],{"className":8467,"style":7849},[862],[390,8469,8471],{"className":8470},[2042,7853,3651,2045],[390,8472,6498],{"className":8473},[411,412,2045],[390,8475,937],{"className":8476},[936],[390,8478,8480],{"className":8479},[850],[390,8481,8483],{"className":8482,"style":7866},[854],[390,8484],{},[390,8486],{"className":8487,"style":7872},[717,2045],[390,8489,421],{"className":8490},[411,412,2045],[8492,8493,8495,8695,9017,9345],"callout",{"type":8494},"theorem",[381,8496,8497,8500,8501,5385,8591,8624,8625,8658,8659,8694],{},[416,8498,8499],{},"Theorem (Master)."," Let ",[390,8502,8504],{"className":8503},[393],[390,8505,8507,8534,8573],{"className":8506,"ariaHidden":398},[397],[390,8508,8510,8513,8516,8519,8522,8525,8528,8531],{"className":8509},[402],[390,8511],{"className":8512,"style":600},[406],[390,8514,723],{"className":8515,"style":722},[411,412],[390,8517,608],{"className":8518},[607],[390,8520,413],{"className":8521},[411,412],[390,8523,616],{"className":8524},[615],[390,8526],{"className":8527,"style":800},[717],[390,8529,805],{"className":8530},[804],[390,8532],{"className":8533,"style":800},[717],[390,8535,8537,8540,8543,8546,8549,8552,8555,8558,8561,8564,8567,8570],{"className":8536},[402],[390,8538],{"className":8539,"style":600},[406],[390,8541,421],{"className":8542},[411,412],[390,8544],{"className":8545,"style":718},[717],[390,8547,723],{"className":8548,"style":722},[411,412],[390,8550,608],{"className":8551},[607],[390,8553,413],{"className":8554},[411,412],[390,8556,7551],{"className":8557},[411],[390,8559,6498],{"className":8560},[411,412],[390,8562,616],{"className":8563},[615],[390,8565],{"className":8566,"style":912},[717],[390,8568,917],{"className":8569},[916],[390,8571],{"className":8572,"style":912},[717],[390,8574,8576,8579,8582,8585,8588],{"className":8575},[402],[390,8577],{"className":8578,"style":600},[406],[390,8580,7577],{"className":8581,"style":7576},[411,412],[390,8583,608],{"className":8584},[607],[390,8586,413],{"className":8587},[411,412],[390,8589,616],{"className":8590},[615],[390,8592,8594],{"className":8593},[393],[390,8595,8597,8615],{"className":8596,"ariaHidden":398},[397],[390,8598,8600,8603,8606,8609,8612],{"className":8599},[402],[390,8601],{"className":8602,"style":3150},[406],[390,8604,421],{"className":8605},[411,412],[390,8607],{"className":8608,"style":800},[717],[390,8610,3160],{"className":8611},[804],[390,8613],{"className":8614,"style":800},[717],[390,8616,8618,8621],{"className":8617},[402],[390,8619],{"className":8620,"style":1464},[406],[390,8622,668],{"className":8623},[411],", ",[390,8626,8628],{"className":8627},[393],[390,8629,8631,8649],{"className":8630,"ariaHidden":398},[397],[390,8632,8634,8637,8640,8643,8646],{"className":8633},[402],[390,8635],{"className":8636,"style":3012},[406],[390,8638,6498],{"className":8639},[411,412],[390,8641],{"className":8642,"style":800},[717],[390,8644,1022],{"className":8645},[804],[390,8647],{"className":8648,"style":800},[717],[390,8650,8652,8655],{"className":8651},[402],[390,8653],{"className":8654,"style":1464},[406],[390,8656,668],{"className":8657},[411],".\nLet ",[390,8660,8662],{"className":8661},[393],[390,8663,8665,8685],{"className":8664,"ariaHidden":398},[397],[390,8666,8668,8672,8676,8679,8682],{"className":8667},[402],[390,8669],{"className":8670,"style":8671},[406],"height:0.5782em;vertical-align:-0.0391em;",[390,8673,8675],{"className":8674},[411,412],"ϵ",[390,8677],{"className":8678,"style":800},[717],[390,8680,1022],{"className":8681},[804],[390,8683],{"className":8684,"style":800},[717],[390,8686,8688,8691],{"className":8687},[402],[390,8689],{"className":8690,"style":1464},[406],[390,8692,3035],{"className":8693},[411]," be a constant. Then:",[381,8696,8697,8700,8701,8849,8850,8865,8866,8869,8870],{},[416,8698,8699],{},"Case 1 (leaves dominate)."," If ",[390,8702,8704],{"className":8703},[393],[390,8705,8707,8734],{"className":8706,"ariaHidden":398},[397],[390,8708,8710,8713,8716,8719,8722,8725,8728,8731],{"className":8709},[402],[390,8711],{"className":8712,"style":600},[406],[390,8714,7577],{"className":8715,"style":7576},[411,412],[390,8717,608],{"className":8718},[607],[390,8720,413],{"className":8721},[411,412],[390,8723,616],{"className":8724},[615],[390,8726],{"className":8727,"style":800},[717],[390,8729,805],{"className":8730},[804],[390,8732],{"className":8733,"style":800},[717],[390,8735,8737,8740,8743,8747,8750],{"className":8736},[402],[390,8738],{"className":8739,"style":5563},[406],[390,8741,1139],{"className":8742,"style":1138},[411,412],[390,8744],{"className":8745,"style":8746},[717],"margin-right:-0.1667em;",[390,8748],{"className":8749,"style":718},[717],[390,8751,8753,8759,8843],{"className":8752},[819],[390,8754,8756],{"className":8755,"style":824},[607,823],[390,8757,608],{"className":8758},[828,3651],[390,8760,8762,8765],{"className":8761},[411],[390,8763,413],{"className":8764},[411,412],[390,8766,8768],{"className":8767},[2021],[390,8769,8771],{"className":8770},[845],[390,8772,8774],{"className":8773},[850],[390,8775,8777],{"className":8776,"style":7791},[854],[390,8778,8779,8782],{"style":2751},[390,8780],{"className":8781,"style":2038},[862],[390,8783,8785],{"className":8784},[2042,2043,2044,2045],[390,8786,8788,8831,8834,8837,8840],{"className":8787},[411,2045],[390,8789,8791,8797],{"className":8790},[1213,2045],[390,8792,8794],{"className":8793},[1213,2045],[390,8795,1219],{"className":8796,"style":1218},[411,1217,2045],[390,8798,8800],{"className":8799},[2021],[390,8801,8803,8823],{"className":8802},[845,846],[390,8804,8806,8820],{"className":8805},[850],[390,8807,8809],{"className":8808,"style":7842},[854],[390,8810,8811,8814],{"style":7845},[390,8812],{"className":8813,"style":7849},[862],[390,8815,8817],{"className":8816},[2042,7853,3651,2045],[390,8818,6498],{"className":8819},[411,412,2045],[390,8821,937],{"className":8822},[936],[390,8824,8826],{"className":8825},[850],[390,8827,8829],{"className":8828,"style":7866},[854],[390,8830],{},[390,8832],{"className":8833,"style":7872},[717,2045],[390,8835,421],{"className":8836},[411,412,2045],[390,8838,3954],{"className":8839},[916,2045],[390,8841,8675],{"className":8842},[411,412,2045],[390,8844,8846],{"className":8845,"style":824},[615,823],[390,8847,616],{"className":8848},[828,3651],"\n(that is, ",[390,8851,8853],{"className":8852},[393],[390,8854,8856],{"className":8855,"ariaHidden":398},[397],[390,8857,8859,8862],{"className":8858},[402],[390,8860],{"className":8861,"style":462},[406],[390,8863,7577],{"className":8864,"style":7576},[411,412]," is ",[385,8867,8868],{},"polynomially smaller"," than the watershed), then\n",[390,8871,8873],{"className":8872},[393],[390,8874,8876,8903],{"className":8875,"ariaHidden":398},[397],[390,8877,8879,8882,8885,8888,8891,8894,8897,8900],{"className":8878},[402],[390,8880],{"className":8881,"style":600},[406],[390,8883,723],{"className":8884,"style":722},[411,412],[390,8886,608],{"className":8887},[607],[390,8889,413],{"className":8890},[411,412],[390,8892,616],{"className":8893},[615],[390,8895],{"className":8896,"style":800},[717],[390,8898,805],{"className":8899},[804],[390,8901],{"className":8902,"style":800},[717],[390,8904,8906,8909,8912,8915,8918,9011,9014],{"className":8905},[402],[390,8907],{"className":8908,"style":5563},[406],[390,8910,441],{"className":8911},[411],[390,8913],{"className":8914,"style":8746},[717],[390,8916],{"className":8917,"style":718},[717],[390,8919,8921,8927,9005],{"className":8920},[819],[390,8922,8924],{"className":8923,"style":824},[607,823],[390,8925,608],{"className":8926},[828,3651],[390,8928,8930,8933],{"className":8929},[411],[390,8931,413],{"className":8932},[411,412],[390,8934,8936],{"className":8935},[2021],[390,8937,8939],{"className":8938},[845],[390,8940,8942],{"className":8941},[850],[390,8943,8945],{"className":8944,"style":7791},[854],[390,8946,8947,8950],{"style":2751},[390,8948],{"className":8949,"style":2038},[862],[390,8951,8953],{"className":8952},[2042,2043,2044,2045],[390,8954,8956,8999,9002],{"className":8955},[411,2045],[390,8957,8959,8965],{"className":8958},[1213,2045],[390,8960,8962],{"className":8961},[1213,2045],[390,8963,1219],{"className":8964,"style":1218},[411,1217,2045],[390,8966,8968],{"className":8967},[2021],[390,8969,8971,8991],{"className":8970},[845,846],[390,8972,8974,8988],{"className":8973},[850],[390,8975,8977],{"className":8976,"style":7842},[854],[390,8978,8979,8982],{"style":7845},[390,8980],{"className":8981,"style":7849},[862],[390,8983,8985],{"className":8984},[2042,7853,3651,2045],[390,8986,6498],{"className":8987},[411,412,2045],[390,8989,937],{"className":8990},[936],[390,8992,8994],{"className":8993},[850],[390,8995,8997],{"className":8996,"style":7866},[854],[390,8998],{},[390,9000],{"className":9001,"style":7872},[717,2045],[390,9003,421],{"className":9004},[411,412,2045],[390,9006,9008],{"className":9007,"style":824},[615,823],[390,9009,616],{"className":9010},[828,3651],[390,9012],{"className":9013,"style":718},[717],[390,9015,617],{"className":9016},[411],[381,9018,9019,8700,9022,9163,9164,9179,9180],{},[416,9020,9021],{},"Case 2 (balanced).",[390,9023,9025],{"className":9024},[393],[390,9026,9028,9055],{"className":9027,"ariaHidden":398},[397],[390,9029,9031,9034,9037,9040,9043,9046,9049,9052],{"className":9030},[402],[390,9032],{"className":9033,"style":600},[406],[390,9035,7577],{"className":9036,"style":7576},[411,412],[390,9038,608],{"className":9039},[607],[390,9041,413],{"className":9042},[411,412],[390,9044,616],{"className":9045},[615],[390,9047],{"className":9048,"style":800},[717],[390,9050,805],{"className":9051},[804],[390,9053],{"className":9054,"style":800},[717],[390,9056,9058,9061,9064,9067,9070],{"className":9057},[402],[390,9059],{"className":9060,"style":5563},[406],[390,9062,441],{"className":9063},[411],[390,9065],{"className":9066,"style":8746},[717],[390,9068],{"className":9069,"style":718},[717],[390,9071,9073,9079,9157],{"className":9072},[819],[390,9074,9076],{"className":9075,"style":824},[607,823],[390,9077,608],{"className":9078},[828,3651],[390,9080,9082,9085],{"className":9081},[411],[390,9083,413],{"className":9084},[411,412],[390,9086,9088],{"className":9087},[2021],[390,9089,9091],{"className":9090},[845],[390,9092,9094],{"className":9093},[850],[390,9095,9097],{"className":9096,"style":7791},[854],[390,9098,9099,9102],{"style":2751},[390,9100],{"className":9101,"style":2038},[862],[390,9103,9105],{"className":9104},[2042,2043,2044,2045],[390,9106,9108,9151,9154],{"className":9107},[411,2045],[390,9109,9111,9117],{"className":9110},[1213,2045],[390,9112,9114],{"className":9113},[1213,2045],[390,9115,1219],{"className":9116,"style":1218},[411,1217,2045],[390,9118,9120],{"className":9119},[2021],[390,9121,9123,9143],{"className":9122},[845,846],[390,9124,9126,9140],{"className":9125},[850],[390,9127,9129],{"className":9128,"style":7842},[854],[390,9130,9131,9134],{"style":7845},[390,9132],{"className":9133,"style":7849},[862],[390,9135,9137],{"className":9136},[2042,7853,3651,2045],[390,9138,6498],{"className":9139},[411,412,2045],[390,9141,937],{"className":9142},[936],[390,9144,9146],{"className":9145},[850],[390,9147,9149],{"className":9148,"style":7866},[854],[390,9150],{},[390,9152],{"className":9153,"style":7872},[717,2045],[390,9155,421],{"className":9156},[411,412,2045],[390,9158,9160],{"className":9159,"style":824},[615,823],[390,9161,616],{"className":9162},[828,3651]," (",[390,9165,9167],{"className":9166},[393],[390,9168,9170],{"className":9169,"ariaHidden":398},[397],[390,9171,9173,9176],{"className":9172},[402],[390,9174],{"className":9175,"style":462},[406],[390,9177,7577],{"className":9178,"style":7576},[411,412],"\nmatches the watershed), then\n",[390,9181,9183],{"className":9182},[393],[390,9184,9186,9213],{"className":9185,"ariaHidden":398},[397],[390,9187,9189,9192,9195,9198,9201,9204,9207,9210],{"className":9188},[402],[390,9190],{"className":9191,"style":600},[406],[390,9193,723],{"className":9194,"style":722},[411,412],[390,9196,608],{"className":9197},[607],[390,9199,413],{"className":9200},[411,412],[390,9202,616],{"className":9203},[615],[390,9205],{"className":9206,"style":800},[717],[390,9208,805],{"className":9209},[804],[390,9211],{"className":9212,"style":800},[717],[390,9214,9216,9219,9222,9225,9228,9339,9342],{"className":9215},[402],[390,9217],{"className":9218,"style":5563},[406],[390,9220,441],{"className":9221},[411],[390,9223],{"className":9224,"style":8746},[717],[390,9226],{"className":9227,"style":718},[717],[390,9229,9231,9237,9315,9318,9321,9327,9330,9333],{"className":9230},[819],[390,9232,9234],{"className":9233,"style":824},[607,823],[390,9235,608],{"className":9236},[828,3651],[390,9238,9240,9243],{"className":9239},[411],[390,9241,413],{"className":9242},[411,412],[390,9244,9246],{"className":9245},[2021],[390,9247,9249],{"className":9248},[845],[390,9250,9252],{"className":9251},[850],[390,9253,9255],{"className":9254,"style":7791},[854],[390,9256,9257,9260],{"style":2751},[390,9258],{"className":9259,"style":2038},[862],[390,9261,9263],{"className":9262},[2042,2043,2044,2045],[390,9264,9266,9309,9312],{"className":9265},[411,2045],[390,9267,9269,9275],{"className":9268},[1213,2045],[390,9270,9272],{"className":9271},[1213,2045],[390,9273,1219],{"className":9274,"style":1218},[411,1217,2045],[390,9276,9278],{"className":9277},[2021],[390,9279,9281,9301],{"className":9280},[845,846],[390,9282,9284,9298],{"className":9283},[850],[390,9285,9287],{"className":9286,"style":7842},[854],[390,9288,9289,9292],{"style":7845},[390,9290],{"className":9291,"style":7849},[862],[390,9293,9295],{"className":9294},[2042,7853,3651,2045],[390,9296,6498],{"className":9297},[411,412,2045],[390,9299,937],{"className":9300},[936],[390,9302,9304],{"className":9303},[850],[390,9305,9307],{"className":9306,"style":7866},[854],[390,9308],{},[390,9310],{"className":9311,"style":7872},[717,2045],[390,9313,421],{"className":9314},[411,412,2045],[390,9316],{"className":9317,"style":718},[717],[390,9319],{"className":9320,"style":718},[717],[390,9322,9324],{"className":9323},[1213],[390,9325,1219],{"className":9326,"style":1218},[411,1217],[390,9328],{"className":9329,"style":718},[717],[390,9331,413],{"className":9332},[411,412],[390,9334,9336],{"className":9335,"style":824},[615,823],[390,9337,616],{"className":9338},[828,3651],[390,9340],{"className":9341,"style":718},[717],[390,9343,617],{"className":9344},[411],[381,9346,9347,8700,9350,9497,9498,8865,9513,9516,9517,9520,9521,9592,9593,9626,9627,9642,9643],{},[416,9348,9349],{},"Case 3 (root dominates).",[390,9351,9353],{"className":9352},[393],[390,9354,9356,9383],{"className":9355,"ariaHidden":398},[397],[390,9357,9359,9362,9365,9368,9371,9374,9377,9380],{"className":9358},[402],[390,9360],{"className":9361,"style":600},[406],[390,9363,7577],{"className":9364,"style":7576},[411,412],[390,9366,608],{"className":9367},[607],[390,9369,413],{"className":9370},[411,412],[390,9372,616],{"className":9373},[615],[390,9375],{"className":9376,"style":800},[717],[390,9378,805],{"className":9379},[804],[390,9381],{"className":9382,"style":800},[717],[390,9384,9386,9389,9392,9395,9398],{"className":9385},[402],[390,9387],{"className":9388,"style":5563},[406],[390,9390,4922],{"className":9391},[411],[390,9393],{"className":9394,"style":8746},[717],[390,9396],{"className":9397,"style":718},[717],[390,9399,9401,9407,9491],{"className":9400},[819],[390,9402,9404],{"className":9403,"style":824},[607,823],[390,9405,608],{"className":9406},[828,3651],[390,9408,9410,9413],{"className":9409},[411],[390,9411,413],{"className":9412},[411,412],[390,9414,9416],{"className":9415},[2021],[390,9417,9419],{"className":9418},[845],[390,9420,9422],{"className":9421},[850],[390,9423,9425],{"className":9424,"style":7791},[854],[390,9426,9427,9430],{"style":2751},[390,9428],{"className":9429,"style":2038},[862],[390,9431,9433],{"className":9432},[2042,2043,2044,2045],[390,9434,9436,9479,9482,9485,9488],{"className":9435},[411,2045],[390,9437,9439,9445],{"className":9438},[1213,2045],[390,9440,9442],{"className":9441},[1213,2045],[390,9443,1219],{"className":9444,"style":1218},[411,1217,2045],[390,9446,9448],{"className":9447},[2021],[390,9449,9451,9471],{"className":9450},[845,846],[390,9452,9454,9468],{"className":9453},[850],[390,9455,9457],{"className":9456,"style":7842},[854],[390,9458,9459,9462],{"style":7845},[390,9460],{"className":9461,"style":7849},[862],[390,9463,9465],{"className":9464},[2042,7853,3651,2045],[390,9466,6498],{"className":9467},[411,412,2045],[390,9469,937],{"className":9470},[936],[390,9472,9474],{"className":9473},[850],[390,9475,9477],{"className":9476,"style":7866},[854],[390,9478],{},[390,9480],{"className":9481,"style":7872},[717,2045],[390,9483,421],{"className":9484},[411,412,2045],[390,9486,917],{"className":9487},[916,2045],[390,9489,8675],{"className":9490},[411,412,2045],[390,9492,9494],{"className":9493,"style":824},[615,823],[390,9495,616],{"className":9496},[828,3651],"\n(",[390,9499,9501],{"className":9500},[393],[390,9502,9504],{"className":9503,"ariaHidden":398},[397],[390,9505,9507,9510],{"className":9506},[402],[390,9508],{"className":9509,"style":462},[406],[390,9511,7577],{"className":9512,"style":7576},[411,412],[385,9514,9515],{},"polynomially larger"," than the watershed) and the ",[416,9518,9519],{},"regularity\ncondition"," ",[390,9522,9524],{"className":9523},[393],[390,9525,9527,9566],{"className":9526,"ariaHidden":398},[397],[390,9528,9530,9533,9536,9539,9542,9545,9548,9551,9554,9557,9560,9563],{"className":9529},[402],[390,9531],{"className":9532,"style":600},[406],[390,9534,421],{"className":9535},[411,412],[390,9537],{"className":9538,"style":718},[717],[390,9540,7577],{"className":9541,"style":7576},[411,412],[390,9543,608],{"className":9544},[607],[390,9546,413],{"className":9547},[411,412],[390,9549,7551],{"className":9550},[411],[390,9552,6498],{"className":9553},[411,412],[390,9555,616],{"className":9556},[615],[390,9558],{"className":9559,"style":800},[717],[390,9561,1089],{"className":9562},[804],[390,9564],{"className":9565,"style":800},[717],[390,9567,9569,9572,9577,9580,9583,9586,9589],{"className":9568},[402],[390,9570],{"className":9571,"style":600},[406],[390,9573,9576],{"className":9574,"style":9575},[411,412],"margin-right:0.0315em;","k",[390,9578],{"className":9579,"style":718},[717],[390,9581,7577],{"className":9582,"style":7576},[411,412],[390,9584,608],{"className":9585},[607],[390,9587,413],{"className":9588},[411,412],[390,9590,616],{"className":9591},[615]," holds for some constant ",[390,9594,9596],{"className":9595},[393],[390,9597,9599,9617],{"className":9598,"ariaHidden":398},[397],[390,9600,9602,9605,9608,9611,9614],{"className":9601},[402],[390,9603],{"className":9604,"style":3012},[406],[390,9606,9576],{"className":9607,"style":9575},[411,412],[390,9609],{"className":9610,"style":800},[717],[390,9612,5408],{"className":9613},[804],[390,9615],{"className":9616,"style":800},[717],[390,9618,9620,9623],{"className":9619},[402],[390,9621],{"className":9622,"style":1464},[406],[390,9624,668],{"className":9625},[411]," and all\nlarge ",[390,9628,9630],{"className":9629},[393],[390,9631,9633],{"className":9632,"ariaHidden":398},[397],[390,9634,9636,9639],{"className":9635},[402],[390,9637],{"className":9638,"style":407},[406],[390,9640,413],{"className":9641},[411,412],", then\n",[390,9644,9646],{"className":9645},[393],[390,9647,9649,9676],{"className":9648,"ariaHidden":398},[397],[390,9650,9652,9655,9658,9661,9664,9667,9670,9673],{"className":9651},[402],[390,9653],{"className":9654,"style":600},[406],[390,9656,723],{"className":9657,"style":722},[411,412],[390,9659,608],{"className":9660},[607],[390,9662,413],{"className":9663},[411,412],[390,9665,616],{"className":9666},[615],[390,9668],{"className":9669,"style":800},[717],[390,9671,805],{"className":9672},[804],[390,9674],{"className":9675,"style":800},[717],[390,9677,9679,9682,9685,9688,9694,9697,9700,9703,9706,9712],{"className":9678},[402],[390,9680],{"className":9681,"style":5563},[406],[390,9683,441],{"className":9684},[411],[390,9686],{"className":9687,"style":8746},[717],[390,9689,9691],{"className":9690},[607],[390,9692,608],{"className":9693},[828,3651],[390,9695,7577],{"className":9696,"style":7576},[411,412],[390,9698,608],{"className":9699},[607],[390,9701,413],{"className":9702},[411,412],[390,9704,616],{"className":9705},[615],[390,9707,9709],{"className":9708},[615],[390,9710,616],{"className":9711},[828,3651],[390,9713,617],{"className":9714},[411],[381,9716,9717,9718,9742,9743,9833,9834,9858,9859,9892,9893,9917],{},"The intuition matches the recursion tree. Compare the work at the root, ",[390,9719,9721],{"className":9720},[393],[390,9722,9724],{"className":9723,"ariaHidden":398},[397],[390,9725,9727,9730,9733,9736,9739],{"className":9726},[402],[390,9728],{"className":9729,"style":600},[406],[390,9731,7577],{"className":9732,"style":7576},[411,412],[390,9734,608],{"className":9735},[607],[390,9737,413],{"className":9738},[411,412],[390,9740,616],{"className":9741},[615],",\nto the work at the leaves, ",[390,9744,9746],{"className":9745},[393],[390,9747,9749],{"className":9748,"ariaHidden":398},[397],[390,9750,9752,9755],{"className":9751},[402],[390,9753],{"className":9754,"style":7791},[406],[390,9756,9758,9761],{"className":9757},[411],[390,9759,413],{"className":9760},[411,412],[390,9762,9764],{"className":9763},[2021],[390,9765,9767],{"className":9766},[845],[390,9768,9770],{"className":9769},[850],[390,9771,9773],{"className":9772,"style":7791},[854],[390,9774,9775,9778],{"style":2751},[390,9776],{"className":9777,"style":2038},[862],[390,9779,9781],{"className":9780},[2042,2043,2044,2045],[390,9782,9784,9827,9830],{"className":9783},[411,2045],[390,9785,9787,9793],{"className":9786},[1213,2045],[390,9788,9790],{"className":9789},[1213,2045],[390,9791,1219],{"className":9792,"style":1218},[411,1217,2045],[390,9794,9796],{"className":9795},[2021],[390,9797,9799,9819],{"className":9798},[845,846],[390,9800,9802,9816],{"className":9801},[850],[390,9803,9805],{"className":9804,"style":7842},[854],[390,9806,9807,9810],{"style":7845},[390,9808],{"className":9809,"style":7849},[862],[390,9811,9813],{"className":9812},[2042,7853,3651,2045],[390,9814,6498],{"className":9815},[411,412,2045],[390,9817,937],{"className":9818},[936],[390,9820,9822],{"className":9821},[850],[390,9823,9825],{"className":9824,"style":7866},[854],[390,9826],{},[390,9828],{"className":9829,"style":7872},[717,2045],[390,9831,421],{"className":9832},[411,412,2045],". In Case 1 the tree is leaf-heavy and\nthe answer is the leaf count. In Case 3 the root work dwarfs everything below it\nand the answer is ",[390,9835,9837],{"className":9836},[393],[390,9838,9840],{"className":9839,"ariaHidden":398},[397],[390,9841,9843,9846,9849,9852,9855],{"className":9842},[402],[390,9844],{"className":9845,"style":600},[406],[390,9847,7577],{"className":9848,"style":7576},[411,412],[390,9850,608],{"className":9851},[607],[390,9853,413],{"className":9854},[411,412],[390,9856,616],{"className":9857},[615],". In Case 2 the work is spread evenly across all\n",[390,9860,9862],{"className":9861},[393],[390,9863,9865],{"className":9864,"ariaHidden":398},[397],[390,9866,9868,9871,9874,9877,9883,9886,9889],{"className":9867},[402],[390,9869],{"className":9870,"style":600},[406],[390,9872,441],{"className":9873},[411],[390,9875,608],{"className":9876},[607],[390,9878,9880],{"className":9879},[1213],[390,9881,1219],{"className":9882,"style":1218},[411,1217],[390,9884],{"className":9885,"style":718},[717],[390,9887,413],{"className":9888},[411,412],[390,9890,616],{"className":9891},[615]," levels, as we saw for merge sort, giving the extra ",[390,9894,9896],{"className":9895},[393],[390,9897,9899],{"className":9898,"ariaHidden":398},[397],[390,9900,9902,9905,9911,9914],{"className":9901},[402],[390,9903],{"className":9904,"style":462},[406],[390,9906,9908],{"className":9907},[1213],[390,9909,1219],{"className":9910,"style":1218},[411,1217],[390,9912],{"className":9913,"style":718},[717],[390,9915,413],{"className":9916},[411,412],"\nfactor.",[1470,9919,9921,10176],{"className":9920},[1473,1474],[1476,9922,9926],{"xmlns":1478,"width":9923,"height":9924,"viewBox":9925},"434.729","171.159","-75 -75 326.047 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the work concentrates in the Master Theorem's three cases.",[381,10181,10182],{},"Each panel stacks the per-level work from root (top) to leaves (bottom); the bar\nwidth is the work at that level. The case is decided by which end is heavier.",[381,10184,10185,10186,10193,10194,10197,10198,10213,10214,10217,10218,10351],{},"A caution from CLRS: the cases do not cover every recurrence.",[1315,10187,10188],{},[421,10189,5932],{"href":10190,"ariaDescribedBy":10191,"dataFootnoteRef":376,"id":10192},"#user-content-fn-clrs-master",[1321],"user-content-fnref-clrs-master"," There are ",[416,10195,10196],{},"gaps",".\nWhen ",[390,10199,10201],{"className":10200},[393],[390,10202,10204],{"className":10203,"ariaHidden":398},[397],[390,10205,10207,10210],{"className":10206},[402],[390,10208],{"className":10209,"style":462},[406],[390,10211,7577],{"className":10212,"style":7576},[411,412]," is larger than the watershed but not ",[385,10215,10216],{},"polynomially"," larger (e.g.\n",[390,10219,10221],{"className":10220},[393],[390,10222,10224,10251],{"className":10223,"ariaHidden":398},[397],[390,10225,10227,10230,10233,10236,10239,10242,10245,10248],{"className":10226},[402],[390,10228],{"className":10229,"style":600},[406],[390,10231,7577],{"className":10232,"style":7576},[411,412],[390,10234,608],{"className":10235},[607],[390,10237,413],{"className":10238},[411,412],[390,10240,616],{"className":10241},[615],[390,10243],{"className":10244,"style":800},[717],[390,10246,805],{"className":10247},[804],[390,10249],{"className":10250,"style":800},[717],[390,10252,10254,10258,10336,10339,10345,10348],{"className":10253},[402],[390,10255],{"className":10256,"style":10257},[406],"height:1.0435em;vertical-align:-0.1944em;",[390,10259,10261,10264],{"className":10260},[411],[390,10262,413],{"className":10263},[411,412],[390,10265,10267],{"className":10266},[2021],[390,10268,10270],{"className":10269},[845],[390,10271,10273],{"className":10272},[850],[390,10274,10276],{"className":10275,"style":7791},[854],[390,10277,10278,10281],{"style":2751},[390,10279],{"className":10280,"style":2038},[862],[390,10282,10284],{"className":10283},[2042,2043,2044,2045],[390,10285,10287,10330,10333],{"className":10286},[411,2045],[390,10288,10290,10296],{"className":10289},[1213,2045],[390,10291,10293],{"className":10292},[1213,2045],[390,10294,1219],{"className":10295,"style":1218},[411,1217,2045],[390,10297,10299],{"className":10298},[2021],[390,10300,10302,10322],{"className":10301},[845,846],[390,10303,10305,10319],{"className":10304},[850],[390,10306,10308],{"className":10307,"style":7842},[854],[390,10309,10310,10313],{"style":7845},[390,10311],{"className":10312,"style":7849},[862],[390,10314,10316],{"className":10315},[2042,7853,3651,2045],[390,10317,6498],{"className":10318},[411,412,2045],[390,10320,937],{"className":10321},[936],[390,10323,10325],{"className":10324},[850],[390,10326,10328],{"className":10327,"style":7866},[854],[390,10329],{},[390,10331],{"className":10332,"style":7872},[717,2045],[390,10334,421],{"className":10335},[411,412,2045],[390,10337],{"className":10338,"style":718},[717],[390,10340,10342],{"className":10341},[1213],[390,10343,1219],{"className":10344,"style":1218},[411,1217],[390,10346],{"className":10347,"style":718},[717],[390,10349,413],{"className":10350},[411,412],"), or when the regularity condition fails,\nthe Master Theorem simply does not apply. Fall back to the recursion tree or\nsubstitution.",[10353,10354,10356],"h3",{"id":10355},"worked-examples","Worked examples",[381,10358,10359,9520,10362,10449,10450,10483,10484,10517,10518,10764,10765,10919],{},[416,10360,10361],{},"Example 1, merge sort.",[390,10363,10365],{"className":10364},[393],[390,10366,10368,10395,10431],{"className":10367,"ariaHidden":398},[397],[390,10369,10371,10374,10377,10380,10383,10386,10389,10392],{"className":10370},[402],[390,10372],{"className":10373,"style":600},[406],[390,10375,723],{"className":10376,"style":722},[411,412],[390,10378,608],{"className":10379},[607],[390,10381,413],{"className":10382},[411,412],[390,10384,616],{"className":10385},[615],[390,10387],{"className":10388,"style":800},[717],[390,10390,805],{"className":10391},[804],[390,10393],{"className":10394,"style":800},[717],[390,10396,10398,10401,10404,10407,10410,10413,10416,10419,10422,10425,10428],{"className":10397},[402],[390,10399],{"className":10400,"style":600},[406],[390,10402,713],{"className":10403},[411],[390,10405],{"className":10406,"style":718},[717],[390,10408,723],{"className":10409,"style":722},[411,412],[390,10411,608],{"className":10412},[607],[390,10414,413],{"className":10415},[411,412],[390,10417,696],{"className":10418},[411],[390,10420,616],{"className":10421},[615],[390,10423],{"className":10424,"style":912},[717],[390,10426,917],{"className":10427},[916],[390,10429],{"className":10430,"style":912},[717],[390,10432,10434,10437,10440,10443,10446],{"className":10433},[402],[390,10435],{"className":10436,"style":600},[406],[390,10438,441],{"className":10439},[411],[390,10441,608],{"className":10442},[607],[390,10444,413],{"className":10445},[411,412],[390,10447,616],{"className":10448},[615],". Here ",[390,10451,10453],{"className":10452},[393],[390,10454,10456,10474],{"className":10455,"ariaHidden":398},[397],[390,10457,10459,10462,10465,10468,10471],{"className":10458},[402],[390,10460],{"className":10461,"style":407},[406],[390,10463,421],{"className":10464},[411,412],[390,10466],{"className":10467,"style":800},[717],[390,10469,805],{"className":10470},[804],[390,10472],{"className":10473,"style":800},[717],[390,10475,10477,10480],{"className":10476},[402],[390,10478],{"className":10479,"style":1464},[406],[390,10481,713],{"className":10482},[411],",\n",[390,10485,10487],{"className":10486},[393],[390,10488,10490,10508],{"className":10489,"ariaHidden":398},[397],[390,10491,10493,10496,10499,10502,10505],{"className":10492},[402],[390,10494],{"className":10495,"style":5075},[406],[390,10497,6498],{"className":10498},[411,412],[390,10500],{"className":10501,"style":800},[717],[390,10503,805],{"className":10504},[804],[390,10506],{"className":10507,"style":800},[717],[390,10509,10511,10514],{"className":10510},[402],[390,10512],{"className":10513,"style":1464},[406],[390,10515,713],{"className":10516},[411],", so ",[390,10519,10521],{"className":10520},[393],[390,10522,10524,10617,10711,10755],{"className":10523,"ariaHidden":398},[397],[390,10525,10527,10530,10608,10611,10614],{"className":10526},[402],[390,10528],{"className":10529,"style":7791},[406],[390,10531,10533,10536],{"className":10532},[411],[390,10534,413],{"className":10535},[411,412],[390,10537,10539],{"className":10538},[2021],[390,10540,10542],{"className":10541},[845],[390,10543,10545],{"className":10544},[850],[390,10546,10548],{"className":10547,"style":7791},[854],[390,10549,10550,10553],{"style":2751},[390,10551],{"className":10552,"style":2038},[862],[390,10554,10556],{"className":10555},[2042,2043,2044,2045],[390,10557,10559,10602,10605],{"className":10558},[411,2045],[390,10560,10562,10568],{"className":10561},[1213,2045],[390,10563,10565],{"className":10564},[1213,2045],[390,10566,1219],{"className":10567,"style":1218},[411,1217,2045],[390,10569,10571],{"className":10570},[2021],[390,10572,10574,10594],{"className":10573},[845,846],[390,10575,10577,10591],{"className":10576},[850],[390,10578,10580],{"className":10579,"style":7842},[854],[390,10581,10582,10585],{"style":7845},[390,10583],{"className":10584,"style":7849},[862],[390,10586,10588],{"className":10587},[2042,7853,3651,2045],[390,10589,6498],{"className":10590},[411,412,2045],[390,10592,937],{"className":10593},[936],[390,10595,10597],{"className":10596},[850],[390,10598,10600],{"className":10599,"style":7866},[854],[390,10601],{},[390,10603],{"className":10604,"style":7872},[717,2045],[390,10606,421],{"className":10607},[411,412,2045],[390,10609],{"className":10610,"style":800},[717],[390,10612,805],{"className":10613},[804],[390,10615],{"className":10616,"style":800},[717],[390,10618,10620,10623,10702,10705,10708],{"className":10619},[402],[390,10621],{"className":10622,"style":7791},[406],[390,10624,10626,10629],{"className":10625},[411],[390,10627,413],{"className":10628},[411,412],[390,10630,10632],{"className":10631},[2021],[390,10633,10635],{"className":10634},[845],[390,10636,10638],{"className":10637},[850],[390,10639,10641],{"className":10640,"style":7791},[854],[390,10642,10643,10646],{"style":2751},[390,10644],{"className":10645,"style":2038},[862],[390,10647,10649],{"className":10648},[2042,2043,2044,2045],[390,10650,10652,10696,10699],{"className":10651},[411,2045],[390,10653,10655,10661],{"className":10654},[1213,2045],[390,10656,10658],{"className":10657},[1213,2045],[390,10659,1219],{"className":10660,"style":1218},[411,1217,2045],[390,10662,10664],{"className":10663},[2021],[390,10665,10667,10688],{"className":10666},[845,846],[390,10668,10670,10685],{"className":10669},[850],[390,10671,10674],{"className":10672,"style":10673},[854],"height:0.1944em;",[390,10675,10676,10679],{"style":7845},[390,10677],{"className":10678,"style":7849},[862],[390,10680,10682],{"className":10681},[2042,7853,3651,2045],[390,10683,713],{"className":10684},[411,2045],[390,10686,937],{"className":10687},[936],[390,10689,10691],{"className":10690},[850],[390,10692,10694],{"className":10693,"style":7866},[854],[390,10695],{},[390,10697],{"className":10698,"style":7872},[717,2045],[390,10700,713],{"className":10701},[411,2045],[390,10703],{"className":10704,"style":800},[717],[390,10706,805],{"className":10707},[804],[390,10709],{"className":10710,"style":800},[717],[390,10712,10714,10717,10746,10749,10752],{"className":10713},[402],[390,10715],{"className":10716,"style":2748},[406],[390,10718,10720,10723],{"className":10719},[411],[390,10721,413],{"className":10722},[411,412],[390,10724,10726],{"className":10725},[2021],[390,10727,10729],{"className":10728},[845],[390,10730,10732],{"className":10731},[850],[390,10733,10735],{"className":10734,"style":2748},[854],[390,10736,10737,10740],{"style":2751},[390,10738],{"className":10739,"style":2038},[862],[390,10741,10743],{"className":10742},[2042,2043,2044,2045],[390,10744,668],{"className":10745},[411,2045],[390,10747],{"className":10748,"style":800},[717],[390,10750,805],{"className":10751},[804],[390,10753],{"className":10754,"style":800},[717],[390,10756,10758,10761],{"className":10757},[402],[390,10759],{"className":10760,"style":407},[406],[390,10762,413],{"className":10763},[411,412],". And ",[390,10766,10768],{"className":10767},[393],[390,10769,10771,10798,10825],{"className":10770,"ariaHidden":398},[397],[390,10772,10774,10777,10780,10783,10786,10789,10792,10795],{"className":10773},[402],[390,10775],{"className":10776,"style":600},[406],[390,10778,7577],{"className":10779,"style":7576},[411,412],[390,10781,608],{"className":10782},[607],[390,10784,413],{"className":10785},[411,412],[390,10787,616],{"className":10788},[615],[390,10790],{"className":10791,"style":800},[717],[390,10793,805],{"className":10794},[804],[390,10796],{"className":10797,"style":800},[717],[390,10799,10801,10804,10807,10810,10813,10816,10819,10822],{"className":10800},[402],[390,10802],{"className":10803,"style":600},[406],[390,10805,441],{"className":10806},[411],[390,10808,608],{"className":10809},[607],[390,10811,413],{"className":10812},[411,412],[390,10814,616],{"className":10815},[615],[390,10817],{"className":10818,"style":800},[717],[390,10820,805],{"className":10821},[804],[390,10823],{"className":10824,"style":800},[717],[390,10826,10828,10832,10835,10838,10916],{"className":10827},[402],[390,10829],{"className":10830,"style":10831},[406],"height:1.0991em;vertical-align:-0.25em;",[390,10833,441],{"className":10834},[411],[390,10836,608],{"className":10837},[607],[390,10839,10841,10844],{"className":10840},[411],[390,10842,413],{"className":10843},[411,412],[390,10845,10847],{"className":10846},[2021],[390,10848,10850],{"className":10849},[845],[390,10851,10853],{"className":10852},[850],[390,10854,10856],{"className":10855,"style":7791},[854],[390,10857,10858,10861],{"style":2751},[390,10859],{"className":10860,"style":2038},[862],[390,10862,10864],{"className":10863},[2042,2043,2044,2045],[390,10865,10867,10910,10913],{"className":10866},[411,2045],[390,10868,10870,10876],{"className":10869},[1213,2045],[390,10871,10873],{"className":10872},[1213,2045],[390,10874,1219],{"className":10875,"style":1218},[411,1217,2045],[390,10877,10879],{"className":10878},[2021],[390,10880,10882,10902],{"className":10881},[845,846],[390,10883,10885,10899],{"className":10884},[850],[390,10886,10888],{"className":10887,"style":7842},[854],[390,10889,10890,10893],{"style":7845},[390,10891],{"className":10892,"style":7849},[862],[390,10894,10896],{"className":10895},[2042,7853,3651,2045],[390,10897,6498],{"className":10898},[411,412,2045],[390,10900,937],{"className":10901},[936],[390,10903,10905],{"className":10904},[850],[390,10906,10908],{"className":10907,"style":7866},[854],[390,10909],{},[390,10911],{"className":10912,"style":7872},[717,2045],[390,10914,421],{"className":10915},[411,412,2045],[390,10917,616],{"className":10918},[615],", which is Case 2. Therefore",[390,10921,10923],{"className":10922},[772],[390,10924,10926],{"className":10925},[393],[390,10927,10929,10956],{"className":10928,"ariaHidden":398},[397],[390,10930,10932,10935,10938,10941,10944,10947,10950,10953],{"className":10931},[402],[390,10933],{"className":10934,"style":600},[406],[390,10936,723],{"className":10937,"style":722},[411,412],[390,10939,608],{"className":10940},[607],[390,10942,413],{"className":10943},[411,412],[390,10945,616],{"className":10946},[615],[390,10948],{"className":10949,"style":800},[717],[390,10951,805],{"className":10952},[804],[390,10954],{"className":10955,"style":800},[717],[390,10957,10959,10962,10965,10968,10971,10974,10980,10983,10986,10989],{"className":10958},[402],[390,10960],{"className":10961,"style":600},[406],[390,10963,441],{"className":10964},[411],[390,10966,608],{"className":10967},[607],[390,10969,413],{"className":10970},[411,412],[390,10972],{"className":10973,"style":718},[717],[390,10975,10977],{"className":10976},[1213],[390,10978,1219],{"className":10979,"style":1218},[411,1217],[390,10981],{"className":10982,"style":718},[717],[390,10984,413],{"className":10985},[411,412],[390,10987,616],{"className":10988},[615],[390,10990,998],{"className":10991},[997],[381,10993,10994],{},"recovering exactly what the tree and substitution gave.",[381,10996,10997,11003,11084,11085,8624,11118,11151,11152,11304,11305,11458],{},[416,10998,10999,11000,617],{},"Example 2, ",[421,11001,11002],{"href":137},"binary search",[390,11004,11006],{"className":11005},[393],[390,11007,11009,11036,11066],{"className":11008,"ariaHidden":398},[397],[390,11010,11012,11015,11018,11021,11024,11027,11030,11033],{"className":11011},[402],[390,11013],{"className":11014,"style":600},[406],[390,11016,723],{"className":11017,"style":722},[411,412],[390,11019,608],{"className":11020},[607],[390,11022,413],{"className":11023},[411,412],[390,11025,616],{"className":11026},[615],[390,11028],{"className":11029,"style":800},[717],[390,11031,805],{"className":11032},[804],[390,11034],{"className":11035,"style":800},[717],[390,11037,11039,11042,11045,11048,11051,11054,11057,11060,11063],{"className":11038},[402],[390,11040],{"className":11041,"style":600},[406],[390,11043,723],{"className":11044,"style":722},[411,412],[390,11046,608],{"className":11047},[607],[390,11049,413],{"className":11050},[411,412],[390,11052,696],{"className":11053},[411],[390,11055,616],{"className":11056},[615],[390,11058],{"className":11059,"style":912},[717],[390,11061,917],{"className":11062},[916],[390,11064],{"className":11065,"style":912},[717],[390,11067,11069,11072,11075,11078,11081],{"className":11068},[402],[390,11070],{"className":11071,"style":600},[406],[390,11073,441],{"className":11074},[411],[390,11076,608],{"className":11077},[607],[390,11079,668],{"className":11080},[411],[390,11082,616],{"className":11083},[615],": one subproblem of\nhalf size, constant work to pick the side. Here ",[390,11086,11088],{"className":11087},[393],[390,11089,11091,11109],{"className":11090,"ariaHidden":398},[397],[390,11092,11094,11097,11100,11103,11106],{"className":11093},[402],[390,11095],{"className":11096,"style":407},[406],[390,11098,421],{"className":11099},[411,412],[390,11101],{"className":11102,"style":800},[717],[390,11104,805],{"className":11105},[804],[390,11107],{"className":11108,"style":800},[717],[390,11110,11112,11115],{"className":11111},[402],[390,11113],{"className":11114,"style":1464},[406],[390,11116,668],{"className":11117},[411],[390,11119,11121],{"className":11120},[393],[390,11122,11124,11142],{"className":11123,"ariaHidden":398},[397],[390,11125,11127,11130,11133,11136,11139],{"className":11126},[402],[390,11128],{"className":11129,"style":5075},[406],[390,11131,6498],{"className":11132},[411,412],[390,11134],{"className":11135,"style":800},[717],[390,11137,805],{"className":11138},[804],[390,11140],{"className":11141,"style":800},[717],[390,11143,11145,11148],{"className":11144},[402],[390,11146],{"className":11147,"style":1464},[406],[390,11149,713],{"className":11150},[411],", so\n",[390,11153,11155],{"className":11154},[393],[390,11156,11158,11251,11295],{"className":11157,"ariaHidden":398},[397],[390,11159,11161,11164,11242,11245,11248],{"className":11160},[402],[390,11162],{"className":11163,"style":7791},[406],[390,11165,11167,11170],{"className":11166},[411],[390,11168,413],{"className":11169},[411,412],[390,11171,11173],{"className":11172},[2021],[390,11174,11176],{"className":11175},[845],[390,11177,11179],{"className":11178},[850],[390,11180,11182],{"className":11181,"style":7791},[854],[390,11183,11184,11187],{"style":2751},[390,11185],{"className":11186,"style":2038},[862],[390,11188,11190],{"className":11189},[2042,2043,2044,2045],[390,11191,11193,11236,11239],{"className":11192},[411,2045],[390,11194,11196,11202],{"className":11195},[1213,2045],[390,11197,11199],{"className":11198},[1213,2045],[390,11200,1219],{"className":11201,"style":1218},[411,1217,2045],[390,11203,11205],{"className":11204},[2021],[390,11206,11208,11228],{"className":11207},[845,846],[390,11209,11211,11225],{"className":11210},[850],[390,11212,11214],{"className":11213,"style":10673},[854],[390,11215,11216,11219],{"style":7845},[390,11217],{"className":11218,"style":7849},[862],[390,11220,11222],{"className":11221},[2042,7853,3651,2045],[390,11223,713],{"className":11224},[411,2045],[390,11226,937],{"className":11227},[936],[390,11229,11231],{"className":11230},[850],[390,11232,11234],{"className":11233,"style":7866},[854],[390,11235],{},[390,11237],{"className":11238,"style":7872},[717,2045],[390,11240,668],{"className":11241},[411,2045],[390,11243],{"className":11244,"style":800},[717],[390,11246,805],{"className":11247},[804],[390,11249],{"className":11250,"style":800},[717],[390,11252,11254,11257,11286,11289,11292],{"className":11253},[402],[390,11255],{"className":11256,"style":2748},[406],[390,11258,11260,11263],{"className":11259},[411],[390,11261,413],{"className":11262},[411,412],[390,11264,11266],{"className":11265},[2021],[390,11267,11269],{"className":11268},[845],[390,11270,11272],{"className":11271},[850],[390,11273,11275],{"className":11274,"style":2748},[854],[390,11276,11277,11280],{"style":2751},[390,11278],{"className":11279,"style":2038},[862],[390,11281,11283],{"className":11282},[2042,2043,2044,2045],[390,11284,3035],{"className":11285},[411,2045],[390,11287],{"className":11288,"style":800},[717],[390,11290,805],{"className":11291},[804],[390,11293],{"className":11294,"style":800},[717],[390,11296,11298,11301],{"className":11297},[402],[390,11299],{"className":11300,"style":1464},[406],[390,11302,668],{"className":11303},[411],". Then ",[390,11306,11308],{"className":11307},[393],[390,11309,11311,11338,11365],{"className":11310,"ariaHidden":398},[397],[390,11312,11314,11317,11320,11323,11326,11329,11332,11335],{"className":11313},[402],[390,11315],{"className":11316,"style":600},[406],[390,11318,7577],{"className":11319,"style":7576},[411,412],[390,11321,608],{"className":11322},[607],[390,11324,413],{"className":11325},[411,412],[390,11327,616],{"className":11328},[615],[390,11330],{"className":11331,"style":800},[717],[390,11333,805],{"className":11334},[804],[390,11336],{"className":11337,"style":800},[717],[390,11339,11341,11344,11347,11350,11353,11356,11359,11362],{"className":11340},[402],[390,11342],{"className":11343,"style":600},[406],[390,11345,441],{"className":11346},[411],[390,11348,608],{"className":11349},[607],[390,11351,668],{"className":11352},[411],[390,11354,616],{"className":11355},[615],[390,11357],{"className":11358,"style":800},[717],[390,11360,805],{"className":11361},[804],[390,11363],{"className":11364,"style":800},[717],[390,11366,11368,11371,11374,11377,11455],{"className":11367},[402],[390,11369],{"className":11370,"style":10831},[406],[390,11372,441],{"className":11373},[411],[390,11375,608],{"className":11376},[607],[390,11378,11380,11383],{"className":11379},[411],[390,11381,413],{"className":11382},[411,412],[390,11384,11386],{"className":11385},[2021],[390,11387,11389],{"className":11388},[845],[390,11390,11392],{"className":11391},[850],[390,11393,11395],{"className":11394,"style":7791},[854],[390,11396,11397,11400],{"style":2751},[390,11398],{"className":11399,"style":2038},[862],[390,11401,11403],{"className":11402},[2042,2043,2044,2045],[390,11404,11406,11449,11452],{"className":11405},[411,2045],[390,11407,11409,11415],{"className":11408},[1213,2045],[390,11410,11412],{"className":11411},[1213,2045],[390,11413,1219],{"className":11414,"style":1218},[411,1217,2045],[390,11416,11418],{"className":11417},[2021],[390,11419,11421,11441],{"className":11420},[845,846],[390,11422,11424,11438],{"className":11423},[850],[390,11425,11427],{"className":11426,"style":7842},[854],[390,11428,11429,11432],{"style":7845},[390,11430],{"className":11431,"style":7849},[862],[390,11433,11435],{"className":11434},[2042,7853,3651,2045],[390,11436,6498],{"className":11437},[411,412,2045],[390,11439,937],{"className":11440},[936],[390,11442,11444],{"className":11443},[850],[390,11445,11447],{"className":11446,"style":7866},[854],[390,11448],{},[390,11450],{"className":11451,"style":7872},[717,2045],[390,11453,421],{"className":11454},[411,412,2045],[390,11456,616],{"className":11457},[615],", Case 2\nagain, and",[390,11460,11462],{"className":11461},[772],[390,11463,11465],{"className":11464},[393],[390,11466,11468,11495],{"className":11467,"ariaHidden":398},[397],[390,11469,11471,11474,11477,11480,11483,11486,11489,11492],{"className":11470},[402],[390,11472],{"className":11473,"style":600},[406],[390,11475,723],{"className":11476,"style":722},[411,412],[390,11478,608],{"className":11479},[607],[390,11481,413],{"className":11482},[411,412],[390,11484,616],{"className":11485},[615],[390,11487],{"className":11488,"style":800},[717],[390,11490,805],{"className":11491},[804],[390,11493],{"className":11494,"style":800},[717],[390,11496,11498,11501,11504,11507,11513,11516,11519,11522],{"className":11497},[402],[390,11499],{"className":11500,"style":600},[406],[390,11502,441],{"className":11503},[411],[390,11505,608],{"className":11506},[607],[390,11508,11510],{"className":11509},[1213],[390,11511,1219],{"className":11512,"style":1218},[411,1217],[390,11514],{"className":11515,"style":718},[717],[390,11517,413],{"className":11518},[411,412],[390,11520,616],{"className":11521},[615],[390,11523,617],{"className":11524},[411],[381,11526,11527,9520,11530,11608,11609,8624,11642,11151,11675,11809,11810,11914,11915,11948,11949,11952],{},[416,11528,11529],{},"Example 3, leaf-dominated.",[390,11531,11533],{"className":11532},[393],[390,11534,11536,11563,11599],{"className":11535,"ariaHidden":398},[397],[390,11537,11539,11542,11545,11548,11551,11554,11557,11560],{"className":11538},[402],[390,11540],{"className":11541,"style":600},[406],[390,11543,723],{"className":11544,"style":722},[411,412],[390,11546,608],{"className":11547},[607],[390,11549,413],{"className":11550},[411,412],[390,11552,616],{"className":11553},[615],[390,11555],{"className":11556,"style":800},[717],[390,11558,805],{"className":11559},[804],[390,11561],{"className":11562,"style":800},[717],[390,11564,11566,11569,11572,11575,11578,11581,11584,11587,11590,11593,11596],{"className":11565},[402],[390,11567],{"className":11568,"style":600},[406],[390,11570,1943],{"className":11571},[411],[390,11573],{"className":11574,"style":718},[717],[390,11576,723],{"className":11577,"style":722},[411,412],[390,11579,608],{"className":11580},[607],[390,11582,413],{"className":11583},[411,412],[390,11585,696],{"className":11586},[411],[390,11588,616],{"className":11589},[615],[390,11591],{"className":11592,"style":912},[717],[390,11594,917],{"className":11595},[916],[390,11597],{"className":11598,"style":912},[717],[390,11600,11602,11605],{"className":11601},[402],[390,11603],{"className":11604,"style":407},[406],[390,11606,413],{"className":11607},[411,412],". Now ",[390,11610,11612],{"className":11611},[393],[390,11613,11615,11633],{"className":11614,"ariaHidden":398},[397],[390,11616,11618,11621,11624,11627,11630],{"className":11617},[402],[390,11619],{"className":11620,"style":407},[406],[390,11622,421],{"className":11623},[411,412],[390,11625],{"className":11626,"style":800},[717],[390,11628,805],{"className":11629},[804],[390,11631],{"className":11632,"style":800},[717],[390,11634,11636,11639],{"className":11635},[402],[390,11637],{"className":11638,"style":1464},[406],[390,11640,1943],{"className":11641},[411],[390,11643,11645],{"className":11644},[393],[390,11646,11648,11666],{"className":11647,"ariaHidden":398},[397],[390,11649,11651,11654,11657,11660,11663],{"className":11650},[402],[390,11652],{"className":11653,"style":5075},[406],[390,11655,6498],{"className":11656},[411,412],[390,11658],{"className":11659,"style":800},[717],[390,11661,805],{"className":11662},[804],[390,11664],{"className":11665,"style":800},[717],[390,11667,11669,11672],{"className":11668},[402],[390,11670],{"className":11671,"style":1464},[406],[390,11673,713],{"className":11674},[411],[390,11676,11678],{"className":11677},[393],[390,11679,11681,11774],{"className":11680,"ariaHidden":398},[397],[390,11682,11684,11687,11765,11768,11771],{"className":11683},[402],[390,11685],{"className":11686,"style":7791},[406],[390,11688,11690,11693],{"className":11689},[411],[390,11691,413],{"className":11692},[411,412],[390,11694,11696],{"className":11695},[2021],[390,11697,11699],{"className":11698},[845],[390,11700,11702],{"className":11701},[850],[390,11703,11705],{"className":11704,"style":7791},[854],[390,11706,11707,11710],{"style":2751},[390,11708],{"className":11709,"style":2038},[862],[390,11711,11713],{"className":11712},[2042,2043,2044,2045],[390,11714,11716,11759,11762],{"className":11715},[411,2045],[390,11717,11719,11725],{"className":11718},[1213,2045],[390,11720,11722],{"className":11721},[1213,2045],[390,11723,1219],{"className":11724,"style":1218},[411,1217,2045],[390,11726,11728],{"className":11727},[2021],[390,11729,11731,11751],{"className":11730},[845,846],[390,11732,11734,11748],{"className":11733},[850],[390,11735,11737],{"className":11736,"style":10673},[854],[390,11738,11739,11742],{"style":7845},[390,11740],{"className":11741,"style":7849},[862],[390,11743,11745],{"className":11744},[2042,7853,3651,2045],[390,11746,713],{"className":11747},[411,2045],[390,11749,937],{"className":11750},[936],[390,11752,11754],{"className":11753},[850],[390,11755,11757],{"className":11756,"style":7866},[854],[390,11758],{},[390,11760],{"className":11761,"style":7872},[717,2045],[390,11763,1943],{"className":11764},[411,2045],[390,11766],{"className":11767,"style":800},[717],[390,11769,805],{"className":11770},[804],[390,11772],{"className":11773,"style":800},[717],[390,11775,11777,11780],{"className":11776},[402],[390,11778],{"className":11779,"style":2748},[406],[390,11781,11783,11786],{"className":11782},[411],[390,11784,413],{"className":11785},[411,412],[390,11787,11789],{"className":11788},[2021],[390,11790,11792],{"className":11791},[845],[390,11793,11795],{"className":11794},[850],[390,11796,11798],{"className":11797,"style":2748},[854],[390,11799,11800,11803],{"style":2751},[390,11801],{"className":11802,"style":2038},[862],[390,11804,11806],{"className":11805},[2042,2043,2044,2045],[390,11807,713],{"className":11808},[411,2045],". The combine work ",[390,11811,11813],{"className":11812},[393],[390,11814,11816,11843,11861],{"className":11815,"ariaHidden":398},[397],[390,11817,11819,11822,11825,11828,11831,11834,11837,11840],{"className":11818},[402],[390,11820],{"className":11821,"style":600},[406],[390,11823,7577],{"className":11824,"style":7576},[411,412],[390,11826,608],{"className":11827},[607],[390,11829,413],{"className":11830},[411,412],[390,11832,616],{"className":11833},[615],[390,11835],{"className":11836,"style":800},[717],[390,11838,805],{"className":11839},[804],[390,11841],{"className":11842,"style":800},[717],[390,11844,11846,11849,11852,11855,11858],{"className":11845},[402],[390,11847],{"className":11848,"style":407},[406],[390,11850,413],{"className":11851},[411,412],[390,11853],{"className":11854,"style":800},[717],[390,11856,805],{"className":11857},[804],[390,11859],{"className":11860,"style":800},[717],[390,11862,11864,11867,11870,11873,11911],{"className":11863},[402],[390,11865],{"className":11866,"style":2723},[406],[390,11868,1139],{"className":11869,"style":1138},[411,412],[390,11871,608],{"className":11872},[607],[390,11874,11876,11879],{"className":11875},[411],[390,11877,413],{"className":11878},[411,412],[390,11880,11882],{"className":11881},[2021],[390,11883,11885],{"className":11884},[845],[390,11886,11888],{"className":11887},[850],[390,11889,11891],{"className":11890,"style":2748},[854],[390,11892,11893,11896],{"style":2751},[390,11894],{"className":11895,"style":2038},[862],[390,11897,11899],{"className":11898},[2042,2043,2044,2045],[390,11900,11902,11905,11908],{"className":11901},[411,2045],[390,11903,713],{"className":11904},[411,2045],[390,11906,3954],{"className":11907},[916,2045],[390,11909,8675],{"className":11910},[411,412,2045],[390,11912,616],{"className":11913},[615]," (take\n",[390,11916,11918],{"className":11917},[393],[390,11919,11921,11939],{"className":11920,"ariaHidden":398},[397],[390,11922,11924,11927,11930,11933,11936],{"className":11923},[402],[390,11925],{"className":11926,"style":407},[406],[390,11928,8675],{"className":11929},[411,412],[390,11931],{"className":11932,"style":800},[717],[390,11934,805],{"className":11935},[804],[390,11937],{"className":11938,"style":800},[717],[390,11940,11942,11945],{"className":11941},[402],[390,11943],{"className":11944,"style":1464},[406],[390,11946,668],{"className":11947},[411],") is polynomially ",[385,11950,11951],{},"smaller"," than the watershed, which is Case 1, so",[390,11954,11956],{"className":11955},[772],[390,11957,11959],{"className":11958},[393],[390,11960,11962,11989],{"className":11961,"ariaHidden":398},[397],[390,11963,11965,11968,11971,11974,11977,11980,11983,11986],{"className":11964},[402],[390,11966],{"className":11967,"style":600},[406],[390,11969,723],{"className":11970,"style":722},[411,412],[390,11972,608],{"className":11973},[607],[390,11975,413],{"className":11976},[411,412],[390,11978,616],{"className":11979},[615],[390,11981],{"className":11982,"style":800},[717],[390,11984,805],{"className":11985},[804],[390,11987],{"className":11988,"style":800},[717],[390,11990,11992,11996,11999,12002,12033,12036],{"className":11991},[402],[390,11993],{"className":11994,"style":11995},[406],"height:1.1141em;vertical-align:-0.25em;",[390,11997,441],{"className":11998},[411],[390,12000,608],{"className":12001},[607],[390,12003,12005,12008],{"className":12004},[411],[390,12006,413],{"className":12007},[411,412],[390,12009,12011],{"className":12010},[2021],[390,12012,12014],{"className":12013},[845],[390,12015,12017],{"className":12016},[850],[390,12018,12021],{"className":12019,"style":12020},[854],"height:0.8641em;",[390,12022,12024,12027],{"style":12023},"top:-3.113em;margin-right:0.05em;",[390,12025],{"className":12026,"style":2038},[862],[390,12028,12030],{"className":12029},[2042,2043,2044,2045],[390,12031,713],{"className":12032},[411,2045],[390,12034,616],{"className":12035},[615],[390,12037,617],{"className":12038},[411],[381,12040,12041,12042,12083],{},"The recursion has so many leaves (",[390,12043,12045],{"className":12044},[393],[390,12046,12048],{"className":12047,"ariaHidden":398},[397],[390,12049,12051,12054],{"className":12050},[402],[390,12052],{"className":12053,"style":2748},[406],[390,12055,12057,12060],{"className":12056},[411],[390,12058,413],{"className":12059},[411,412],[390,12061,12063],{"className":12062},[2021],[390,12064,12066],{"className":12065},[845],[390,12067,12069],{"className":12068},[850],[390,12070,12072],{"className":12071,"style":2748},[854],[390,12073,12074,12077],{"style":2751},[390,12075],{"className":12076,"style":2038},[862],[390,12078,12080],{"className":12079},[2042,2043,2044,2045],[390,12081,713],{"className":12082},[411,2045]," of them) that they dominate the modest\nlinear work per level.",[381,12085,12086,9520,12089,10449,12193,8624,12226,12259,12260,11809,12368,12498,12499,12502,12503,3036,12838,12954],{},[416,12087,12088],{},"Example 4, root-dominated.",[390,12090,12092],{"className":12091},[393],[390,12093,12095,12122,12158],{"className":12094,"ariaHidden":398},[397],[390,12096,12098,12101,12104,12107,12110,12113,12116,12119],{"className":12097},[402],[390,12099],{"className":12100,"style":600},[406],[390,12102,723],{"className":12103,"style":722},[411,412],[390,12105,608],{"className":12106},[607],[390,12108,413],{"className":12109},[411,412],[390,12111,616],{"className":12112},[615],[390,12114],{"className":12115,"style":800},[717],[390,12117,805],{"className":12118},[804],[390,12120],{"className":12121,"style":800},[717],[390,12123,12125,12128,12131,12134,12137,12140,12143,12146,12149,12152,12155],{"className":12124},[402],[390,12126],{"className":12127,"style":600},[406],[390,12129,713],{"className":12130},[411],[390,12132],{"className":12133,"style":718},[717],[390,12135,723],{"className":12136,"style":722},[411,412],[390,12138,608],{"className":12139},[607],[390,12141,413],{"className":12142},[411,412],[390,12144,696],{"className":12145},[411],[390,12147,616],{"className":12148},[615],[390,12150],{"className":12151,"style":912},[717],[390,12153,917],{"className":12154},[916],[390,12156],{"className":12157,"style":912},[717],[390,12159,12161,12164],{"className":12160},[402],[390,12162],{"className":12163,"style":2748},[406],[390,12165,12167,12170],{"className":12166},[411],[390,12168,413],{"className":12169},[411,412],[390,12171,12173],{"className":12172},[2021],[390,12174,12176],{"className":12175},[845],[390,12177,12179],{"className":12178},[850],[390,12180,12182],{"className":12181,"style":2748},[854],[390,12183,12184,12187],{"style":2751},[390,12185],{"className":12186,"style":2038},[862],[390,12188,12190],{"className":12189},[2042,2043,2044,2045],[390,12191,713],{"className":12192},[411,2045],[390,12194,12196],{"className":12195},[393],[390,12197,12199,12217],{"className":12198,"ariaHidden":398},[397],[390,12200,12202,12205,12208,12211,12214],{"className":12201},[402],[390,12203],{"className":12204,"style":407},[406],[390,12206,421],{"className":12207},[411,412],[390,12209],{"className":12210,"style":800},[717],[390,12212,805],{"className":12213},[804],[390,12215],{"className":12216,"style":800},[717],[390,12218,12220,12223],{"className":12219},[402],[390,12221],{"className":12222,"style":1464},[406],[390,12224,713],{"className":12225},[411],[390,12227,12229],{"className":12228},[393],[390,12230,12232,12250],{"className":12231,"ariaHidden":398},[397],[390,12233,12235,12238,12241,12244,12247],{"className":12234},[402],[390,12236],{"className":12237,"style":5075},[406],[390,12239,6498],{"className":12240},[411,412],[390,12242],{"className":12243,"style":800},[717],[390,12245,805],{"className":12246},[804],[390,12248],{"className":12249,"style":800},[717],[390,12251,12253,12256],{"className":12252},[402],[390,12254],{"className":12255,"style":1464},[406],[390,12257,713],{"className":12258},[411],",\nwatershed ",[390,12261,12263],{"className":12262},[393],[390,12264,12266,12359],{"className":12265,"ariaHidden":398},[397],[390,12267,12269,12272,12350,12353,12356],{"className":12268},[402],[390,12270],{"className":12271,"style":7791},[406],[390,12273,12275,12278],{"className":12274},[411],[390,12276,413],{"className":12277},[411,412],[390,12279,12281],{"className":12280},[2021],[390,12282,12284],{"className":12283},[845],[390,12285,12287],{"className":12286},[850],[390,12288,12290],{"className":12289,"style":7791},[854],[390,12291,12292,12295],{"style":2751},[390,12293],{"className":12294,"style":2038},[862],[390,12296,12298],{"className":12297},[2042,2043,2044,2045],[390,12299,12301,12344,12347],{"className":12300},[411,2045],[390,12302,12304,12310],{"className":12303},[1213,2045],[390,12305,12307],{"className":12306},[1213,2045],[390,12308,1219],{"className":12309,"style":1218},[411,1217,2045],[390,12311,12313],{"className":12312},[2021],[390,12314,12316,12336],{"className":12315},[845,846],[390,12317,12319,12333],{"className":12318},[850],[390,12320,12322],{"className":12321,"style":10673},[854],[390,12323,12324,12327],{"style":7845},[390,12325],{"className":12326,"style":7849},[862],[390,12328,12330],{"className":12329},[2042,7853,3651,2045],[390,12331,713],{"className":12332},[411,2045],[390,12334,937],{"className":12335},[936],[390,12337,12339],{"className":12338},[850],[390,12340,12342],{"className":12341,"style":7866},[854],[390,12343],{},[390,12345],{"className":12346,"style":7872},[717,2045],[390,12348,713],{"className":12349},[411,2045],[390,12351],{"className":12352,"style":800},[717],[390,12354,805],{"className":12355},[804],[390,12357],{"className":12358,"style":800},[717],[390,12360,12362,12365],{"className":12361},[402],[390,12363],{"className":12364,"style":407},[406],[390,12366,413],{"className":12367},[411,412],[390,12369,12371],{"className":12370},[393],[390,12372,12374,12401,12445],{"className":12373,"ariaHidden":398},[397],[390,12375,12377,12380,12383,12386,12389,12392,12395,12398],{"className":12376},[402],[390,12378],{"className":12379,"style":600},[406],[390,12381,7577],{"className":12382,"style":7576},[411,412],[390,12384,608],{"className":12385},[607],[390,12387,413],{"className":12388},[411,412],[390,12390,616],{"className":12391},[615],[390,12393],{"className":12394,"style":800},[717],[390,12396,805],{"className":12397},[804],[390,12399],{"className":12400,"style":800},[717],[390,12402,12404,12407,12436,12439,12442],{"className":12403},[402],[390,12405],{"className":12406,"style":2748},[406],[390,12408,12410,12413],{"className":12409},[411],[390,12411,413],{"className":12412},[411,412],[390,12414,12416],{"className":12415},[2021],[390,12417,12419],{"className":12418},[845],[390,12420,12422],{"className":12421},[850],[390,12423,12425],{"className":12424,"style":2748},[854],[390,12426,12427,12430],{"style":2751},[390,12428],{"className":12429,"style":2038},[862],[390,12431,12433],{"className":12432},[2042,2043,2044,2045],[390,12434,713],{"className":12435},[411,2045],[390,12437],{"className":12438,"style":800},[717],[390,12440,805],{"className":12441},[804],[390,12443],{"className":12444,"style":800},[717],[390,12446,12448,12451,12454,12457,12495],{"className":12447},[402],[390,12449],{"className":12450,"style":2723},[406],[390,12452,4922],{"className":12453},[411],[390,12455,608],{"className":12456},[607],[390,12458,12460,12463],{"className":12459},[411],[390,12461,413],{"className":12462},[411,412],[390,12464,12466],{"className":12465},[2021],[390,12467,12469],{"className":12468},[845],[390,12470,12472],{"className":12471},[850],[390,12473,12475],{"className":12474,"style":2748},[854],[390,12476,12477,12480],{"style":2751},[390,12478],{"className":12479,"style":2038},[862],[390,12481,12483],{"className":12482},[2042,2043,2044,2045],[390,12484,12486,12489,12492],{"className":12485},[411,2045],[390,12487,668],{"className":12488},[411,2045],[390,12490,917],{"className":12491},[916,2045],[390,12493,8675],{"className":12494},[411,412,2045],[390,12496,616],{"className":12497},[615],"\nis polynomially ",[385,12500,12501],{},"larger",", a Case 3 candidate. Check regularity:\n",[390,12504,12506],{"className":12505},[393],[390,12507,12509,12548,12607,12719,12814],{"className":12508,"ariaHidden":398},[397],[390,12510,12512,12515,12518,12521,12524,12527,12530,12533,12536,12539,12542,12545],{"className":12511},[402],[390,12513],{"className":12514,"style":600},[406],[390,12516,421],{"className":12517},[411,412],[390,12519],{"className":12520,"style":718},[717],[390,12522,7577],{"className":12523,"style":7576},[411,412],[390,12525,608],{"className":12526},[607],[390,12528,413],{"className":12529},[411,412],[390,12531,7551],{"className":12532},[411],[390,12534,6498],{"className":12535},[411,412],[390,12537,616],{"className":12538},[615],[390,12540],{"className":12541,"style":800},[717],[390,12543,805],{"className":12544},[804],[390,12546],{"className":12547,"style":800},[717],[390,12549,12551,12554,12557,12560,12563,12566,12569,12598,12601,12604],{"className":12550},[402],[390,12552],{"className":12553,"style":2723},[406],[390,12555,713],{"className":12556},[411],[390,12558],{"className":12559,"style":718},[717],[390,12561,608],{"className":12562},[607],[390,12564,413],{"className":12565},[411,412],[390,12567,696],{"className":12568},[411],[390,12570,12572,12575],{"className":12571},[615],[390,12573,616],{"className":12574},[615],[390,12576,12578],{"className":12577},[2021],[390,12579,12581],{"className":12580},[845],[390,12582,12584],{"className":12583},[850],[390,12585,12587],{"className":12586,"style":2748},[854],[390,12588,12589,12592],{"style":2751},[390,12590],{"className":12591,"style":2038},[862],[390,12593,12595],{"className":12594},[2042,2043,2044,2045],[390,12596,713],{"className":12597},[411,2045],[390,12599],{"className":12600,"style":800},[717],[390,12602,805],{"className":12603},[804],[390,12605],{"className":12606,"style":800},[717],[390,12608,12610,12613,12681,12710,12713,12716],{"className":12609},[402],[390,12611],{"className":12612,"style":6661},[406],[390,12614,12616,12619,12678],{"className":12615},[411],[390,12617],{"className":12618},[607,1044],[390,12620,12622],{"className":12621},[3253],[390,12623,12625,12670],{"className":12624},[845,846],[390,12626,12628,12667],{"className":12627},[850],[390,12629,12631,12645,12653],{"className":12630,"style":6680},[854],[390,12632,12633,12636],{"style":3266},[390,12634],{"className":12635,"style":2212},[862],[390,12637,12639],{"className":12638},[2042,2043,2044,2045],[390,12640,12642],{"className":12641},[411,2045],[390,12643,713],{"className":12644},[411,2045],[390,12646,12647,12650],{"style":3281},[390,12648],{"className":12649,"style":2212},[862],[390,12651],{"className":12652,"style":3289},[3288],[390,12654,12655,12658],{"style":3292},[390,12656],{"className":12657,"style":2212},[862],[390,12659,12661],{"className":12660},[2042,2043,2044,2045],[390,12662,12664],{"className":12663},[411,2045],[390,12665,668],{"className":12666},[411,2045],[390,12668,937],{"className":12669},[936],[390,12671,12673],{"className":12672},[850],[390,12674,12676],{"className":12675,"style":3314},[854],[390,12677],{},[390,12679],{"className":12680},[615,1044],[390,12682,12684,12687],{"className":12683},[411],[390,12685,413],{"className":12686},[411,412],[390,12688,12690],{"className":12689},[2021],[390,12691,12693],{"className":12692},[845],[390,12694,12696],{"className":12695},[850],[390,12697,12699],{"className":12698,"style":2748},[854],[390,12700,12701,12704],{"style":2751},[390,12702],{"className":12703,"style":2038},[862],[390,12705,12707],{"className":12706},[2042,2043,2044,2045],[390,12708,713],{"className":12709},[411,2045],[390,12711],{"className":12712,"style":800},[717],[390,12714,805],{"className":12715},[804],[390,12717],{"className":12718,"style":800},[717],[390,12720,12722,12725,12793,12796,12799,12802,12805,12808,12811],{"className":12721},[402],[390,12723],{"className":12724,"style":6661},[406],[390,12726,12728,12731,12790],{"className":12727},[411],[390,12729],{"className":12730},[607,1044],[390,12732,12734],{"className":12733},[3253],[390,12735,12737,12782],{"className":12736},[845,846],[390,12738,12740,12779],{"className":12739},[850],[390,12741,12743,12757,12765],{"className":12742,"style":6680},[854],[390,12744,12745,12748],{"style":3266},[390,12746],{"className":12747,"style":2212},[862],[390,12749,12751],{"className":12750},[2042,2043,2044,2045],[390,12752,12754],{"className":12753},[411,2045],[390,12755,713],{"className":12756},[411,2045],[390,12758,12759,12762],{"style":3281},[390,12760],{"className":12761,"style":2212},[862],[390,12763],{"className":12764,"style":3289},[3288],[390,12766,12767,12770],{"style":3292},[390,12768],{"className":12769,"style":2212},[862],[390,12771,12773],{"className":12772},[2042,2043,2044,2045],[390,12774,12776],{"className":12775},[411,2045],[390,12777,668],{"className":12778},[411,2045],[390,12780,937],{"className":12781},[936],[390,12783,12785],{"className":12784},[850],[390,12786,12788],{"className":12787,"style":3314},[854],[390,12789],{},[390,12791],{"className":12792},[615,1044],[390,12794,7577],{"className":12795,"style":7576},[411,412],[390,12797,608],{"className":12798},[607],[390,12800,413],{"className":12801},[411,412],[390,12803,616],{"className":12804},[615],[390,12806],{"className":12807,"style":800},[717],[390,12809,1089],{"className":12810},[804],[390,12812],{"className":12813,"style":800},[717],[390,12815,12817,12820,12823,12826,12829,12832,12835],{"className":12816},[402],[390,12818],{"className":12819,"style":600},[406],[390,12821,9576],{"className":12822,"style":9575},[411,412],[390,12824],{"className":12825,"style":718},[717],[390,12827,7577],{"className":12828,"style":7576},[411,412],[390,12830,608],{"className":12831},[607],[390,12833,413],{"className":12834},[411,412],[390,12836,616],{"className":12837},[615],[390,12839,12841],{"className":12840},[393],[390,12842,12844,12862,12945],{"className":12843,"ariaHidden":398},[397],[390,12845,12847,12850,12853,12856,12859],{"className":12846},[402],[390,12848],{"className":12849,"style":5075},[406],[390,12851,9576],{"className":12852,"style":9575},[411,412],[390,12854],{"className":12855,"style":800},[717],[390,12857,805],{"className":12858},[804],[390,12860],{"className":12861,"style":800},[717],[390,12863,12865,12868,12936,12939,12942],{"className":12864},[402],[390,12866],{"className":12867,"style":6661},[406],[390,12869,12871,12874,12933],{"className":12870},[411],[390,12872],{"className":12873},[607,1044],[390,12875,12877],{"className":12876},[3253],[390,12878,12880,12925],{"className":12879},[845,846],[390,12881,12883,12922],{"className":12882},[850],[390,12884,12886,12900,12908],{"className":12885,"style":6680},[854],[390,12887,12888,12891],{"style":3266},[390,12889],{"className":12890,"style":2212},[862],[390,12892,12894],{"className":12893},[2042,2043,2044,2045],[390,12895,12897],{"className":12896},[411,2045],[390,12898,713],{"className":12899},[411,2045],[390,12901,12902,12905],{"style":3281},[390,12903],{"className":12904,"style":2212},[862],[390,12906],{"className":12907,"style":3289},[3288],[390,12909,12910,12913],{"style":3292},[390,12911],{"className":12912,"style":2212},[862],[390,12914,12916],{"className":12915},[2042,2043,2044,2045],[390,12917,12919],{"className":12918},[411,2045],[390,12920,668],{"className":12921},[411,2045],[390,12923,937],{"className":12924},[936],[390,12926,12928],{"className":12927},[850],[390,12929,12931],{"className":12930,"style":3314},[854],[390,12932],{},[390,12934],{"className":12935},[615,1044],[390,12937],{"className":12938,"style":800},[717],[390,12940,5408],{"className":12941},[804],[390,12943],{"className":12944,"style":800},[717],[390,12946,12948,12951],{"className":12947},[402],[390,12949],{"className":12950,"style":1464},[406],[390,12952,668],{"className":12953},[411],". Regularity holds, so",[390,12956,12958],{"className":12957},[772],[390,12959,12961],{"className":12960},[393],[390,12962,12964,12991],{"className":12963,"ariaHidden":398},[397],[390,12965,12967,12970,12973,12976,12979,12982,12985,12988],{"className":12966},[402],[390,12968],{"className":12969,"style":600},[406],[390,12971,723],{"className":12972,"style":722},[411,412],[390,12974,608],{"className":12975},[607],[390,12977,413],{"className":12978},[411,412],[390,12980,616],{"className":12981},[615],[390,12983],{"className":12984,"style":800},[717],[390,12986,805],{"className":12987},[804],[390,12989],{"className":12990,"style":800},[717],[390,12992,12994,12997,13000,13003,13032,13035],{"className":12993},[402],[390,12995],{"className":12996,"style":11995},[406],[390,12998,441],{"className":12999},[411],[390,13001,608],{"className":13002},[607],[390,13004,13006,13009],{"className":13005},[411],[390,13007,413],{"className":13008},[411,412],[390,13010,13012],{"className":13011},[2021],[390,13013,13015],{"className":13014},[845],[390,13016,13018],{"className":13017},[850],[390,13019,13021],{"className":13020,"style":12020},[854],[390,13022,13023,13026],{"style":12023},[390,13024],{"className":13025,"style":2038},[862],[390,13027,13029],{"className":13028},[2042,2043,2044,2045],[390,13030,713],{"className":13031},[411,2045],[390,13033,616],{"className":13034},[615],[390,13036,617],{"className":13037},[411],[381,13039,13040],{},"The root's quadratic work swamps the tree beneath it.",[444,13042,13044],{"id":13043},"choosing-a-method","Choosing a method",[381,13046,13047,13048],{},"The three methods are complementary, and Erickson in particular urges fluency\nwith all of them:",[1315,13049,13050],{},[421,13051,1943],{"href":13052,"ariaDescribedBy":13053,"dataFootnoteRef":376,"id":13054},"#user-content-fn-erickson-methods",[1321],"user-content-fnref-erickson-methods",[638,13056,13057,13070,13143],{},[641,13058,13059,13062,13063,1283,13066,13069],{},[416,13060,13061],{},"Recursion tree:"," fastest for ",[385,13064,13065],{},"building intuition",[385,13067,13068],{},"guessing"," the answer;\nshows where the work concentrates.",[641,13071,13072,13062,13075,13078,13079,13142],{},[416,13073,13074],{},"Master Theorem:",[385,13076,13077],{},"getting the answer"," when the recurrence fits\nthe ",[390,13080,13082],{"className":13081},[393],[390,13083,13085,13124],{"className":13084,"ariaHidden":398},[397],[390,13086,13088,13091,13094,13097,13100,13103,13106,13109,13112,13115,13118,13121],{"className":13087},[402],[390,13089],{"className":13090,"style":600},[406],[390,13092,421],{"className":13093},[411,412],[390,13095],{"className":13096,"style":718},[717],[390,13098,723],{"className":13099,"style":722},[411,412],[390,13101,608],{"className":13102},[607],[390,13104,413],{"className":13105},[411,412],[390,13107,7551],{"className":13108},[411],[390,13110,6498],{"className":13111},[411,412],[390,13113,616],{"className":13114},[615],[390,13116],{"className":13117,"style":912},[717],[390,13119,917],{"className":13120},[916],[390,13122],{"className":13123,"style":912},[717],[390,13125,13127,13130,13133,13136,13139],{"className":13126},[402],[390,13128],{"className":13129,"style":600},[406],[390,13131,7577],{"className":13132,"style":7576},[411,412],[390,13134,608],{"className":13135},[607],[390,13137,413],{"className":13138},[411,412],[390,13140,616],{"className":13141},[615]," template; no derivation needed, but it has gaps.",[641,13144,13145,13148,13149,13152,13153,13259],{},[416,13146,13147],{},"Substitution:"," the ",[385,13150,13151],{},"rigorous"," method that always works and produces a proof;\nuse it to certify a guess, or when the other two do not apply (uneven splits,\nrecurrences like ",[390,13154,13156],{"className":13155},[393],[390,13157,13159,13186,13217,13250],{"className":13158,"ariaHidden":398},[397],[390,13160,13162,13165,13168,13171,13174,13177,13180,13183],{"className":13161},[402],[390,13163],{"className":13164,"style":600},[406],[390,13166,723],{"className":13167,"style":722},[411,412],[390,13169,608],{"className":13170},[607],[390,13172,413],{"className":13173},[411,412],[390,13175,616],{"className":13176},[615],[390,13178],{"className":13179,"style":800},[717],[390,13181,805],{"className":13182},[804],[390,13184],{"className":13185,"style":800},[717],[390,13187,13189,13192,13195,13198,13201,13205,13208,13211,13214],{"className":13188},[402],[390,13190],{"className":13191,"style":600},[406],[390,13193,723],{"className":13194,"style":722},[411,412],[390,13196,608],{"className":13197},[607],[390,13199,413],{"className":13200},[411,412],[390,13202,13204],{"className":13203},[411],"\u002F3",[390,13206,616],{"className":13207},[615],[390,13209],{"className":13210,"style":912},[717],[390,13212,917],{"className":13213},[916],[390,13215],{"className":13216,"style":912},[717],[390,13218,13220,13223,13226,13229,13232,13235,13238,13241,13244,13247],{"className":13219},[402],[390,13221],{"className":13222,"style":600},[406],[390,13224,723],{"className":13225,"style":722},[411,412],[390,13227,608],{"className":13228},[607],[390,13230,713],{"className":13231},[411],[390,13233,413],{"className":13234},[411,412],[390,13236,13204],{"className":13237},[411],[390,13239,616],{"className":13240},[615],[390,13242],{"className":13243,"style":912},[717],[390,13245,917],{"className":13246},[916],[390,13248],{"className":13249,"style":912},[717],[390,13251,13253,13256],{"className":13252},[402],[390,13254],{"className":13255,"style":407},[406],[390,13257,413],{"className":13258},[411,412],").",[381,13261,13262],{},"In practice: sketch the tree to guess, apply the Master Theorem if it fits, and\nreach for substitution whenever you need a guarantee rather than a hunch.",[444,13264,13266],{"id":13265},"takeaways","Takeaways",[638,13268,13269,13385,13489,13495,13706,13944],{},[641,13270,13271,13272,13275,13276,13300,13301,617],{},"A ",[416,13273,13274],{},"recursive algorithm induces a recurrence",": ",[390,13277,13279],{"className":13278},[393],[390,13280,13282],{"className":13281,"ariaHidden":398},[397],[390,13283,13285,13288,13291,13294,13297],{"className":13284},[402],[390,13286],{"className":13287,"style":600},[406],[390,13289,723],{"className":13290,"style":722},[411,412],[390,13292,608],{"className":13293},[607],[390,13295,413],{"className":13296},[411,412],[390,13298,616],{"className":13299},[615]," = local work + cost of\nrecursive calls on smaller inputs. Merge sort gives ",[390,13302,13304],{"className":13303},[393],[390,13305,13307,13334,13367],{"className":13306,"ariaHidden":398},[397],[390,13308,13310,13313,13316,13319,13322,13325,13328,13331],{"className":13309},[402],[390,13311],{"className":13312,"style":600},[406],[390,13314,723],{"className":13315,"style":722},[411,412],[390,13317,608],{"className":13318},[607],[390,13320,413],{"className":13321},[411,412],[390,13323,616],{"className":13324},[615],[390,13326],{"className":13327,"style":800},[717],[390,13329,805],{"className":13330},[804],[390,13332],{"className":13333,"style":800},[717],[390,13335,13337,13340,13343,13346,13349,13352,13355,13358,13361,13364],{"className":13336},[402],[390,13338],{"className":13339,"style":600},[406],[390,13341,713],{"className":13342},[411],[390,13344,723],{"className":13345,"style":722},[411,412],[390,13347,608],{"className":13348},[607],[390,13350,413],{"className":13351},[411,412],[390,13353,696],{"className":13354},[411],[390,13356,616],{"className":13357},[615],[390,13359],{"className":13360,"style":912},[717],[390,13362,917],{"className":13363},[916],[390,13365],{"className":13366,"style":912},[717],[390,13368,13370,13373,13376,13379,13382],{"className":13369},[402],[390,13371],{"className":13372,"style":600},[406],[390,13374,441],{"className":13375},[411],[390,13377,608],{"className":13378},[607],[390,13380,413],{"className":13381},[411,412],[390,13383,616],{"className":13384},[615],[641,13386,547,13387,13390,13391,13415,13416,13449,13450,617],{},[416,13388,13389],{},"recursion tree"," sums the per-node work; for merge sort every level costs\n",[390,13392,13394],{"className":13393},[393],[390,13395,13397],{"className":13396,"ariaHidden":398},[397],[390,13398,13400,13403,13406,13409,13412],{"className":13399},[402],[390,13401],{"className":13402,"style":600},[406],[390,13404,441],{"className":13405},[411],[390,13407,608],{"className":13408},[607],[390,13410,413],{"className":13411},[411,412],[390,13413,616],{"className":13414},[615]," across ",[390,13417,13419],{"className":13418},[393],[390,13420,13422],{"className":13421,"ariaHidden":398},[397],[390,13423,13425,13428,13431,13434,13440,13443,13446],{"className":13424},[402],[390,13426],{"className":13427,"style":600},[406],[390,13429,441],{"className":13430},[411],[390,13432,608],{"className":13433},[607],[390,13435,13437],{"className":13436},[1213],[390,13438,1219],{"className":13439,"style":1218},[411,1217],[390,13441],{"className":13442,"style":718},[717],[390,13444,413],{"className":13445},[411,412],[390,13447,616],{"className":13448},[615]," levels, giving ",[390,13451,13453],{"className":13452},[393],[390,13454,13456],{"className":13455,"ariaHidden":398},[397],[390,13457,13459,13462,13465,13468,13471,13474,13480,13483,13486],{"className":13458},[402],[390,13460],{"className":13461,"style":600},[406],[390,13463,441],{"className":13464},[411],[390,13466,608],{"className":13467},[607],[390,13469,413],{"className":13470},[411,412],[390,13472],{"className":13473,"style":718},[717],[390,13475,13477],{"className":13476},[1213],[390,13478,1219],{"className":13479,"style":1218},[411,1217],[390,13481],{"className":13482,"style":718},[717],[390,13484,413],{"className":13485},[411,412],[390,13487,616],{"className":13488},[615],[641,13490,13491,13494],{},[416,13492,13493],{},"Substitution"," guesses the form and proves it by induction; it is the only\nalways-applicable, fully rigorous method. Strengthen the hypothesis if a\nresidual term blocks the step.",[641,13496,547,13497,13500,13501,9520,13504,13564,13565,13615,13616,13666,13667,617],{},[416,13498,13499],{},"combine cost drives the answer."," Counting inversions has the merge-sort\n",[385,13502,13503],{},"shape",[390,13505,13507],{"className":13506},[393],[390,13508,13510,13546],{"className":13509,"ariaHidden":398},[397],[390,13511,13513,13516,13519,13522,13525,13528,13531,13534,13537,13540,13543],{"className":13512},[402],[390,13514],{"className":13515,"style":600},[406],[390,13517,713],{"className":13518},[411],[390,13520],{"className":13521,"style":718},[717],[390,13523,723],{"className":13524,"style":722},[411,412],[390,13526,608],{"className":13527},[607],[390,13529,413],{"className":13530},[411,412],[390,13532,696],{"className":13533},[411],[390,13535,616],{"className":13536},[615],[390,13538],{"className":13539,"style":912},[717],[390,13541,917],{"className":13542},[916],[390,13544],{"className":13545,"style":912},[717],[390,13547,13549,13552,13555,13561],{"className":13548},[402],[390,13550],{"className":13551,"style":600},[406],[390,13553,608],{"className":13554},[607],[390,13556,13558],{"className":13557},[411,471],[390,13559,487],{"className":13560},[411],[390,13562,616],{"className":13563},[615],", but a naive ",[390,13566,13568],{"className":13567},[393],[390,13569,13571],{"className":13570,"ariaHidden":398},[397],[390,13572,13574,13577,13580,13583,13612],{"className":13573},[402],[390,13575],{"className":13576,"style":2723},[406],[390,13578,441],{"className":13579},[411],[390,13581,608],{"className":13582},[607],[390,13584,13586,13589],{"className":13585},[411],[390,13587,413],{"className":13588},[411,412],[390,13590,13592],{"className":13591},[2021],[390,13593,13595],{"className":13594},[845],[390,13596,13598],{"className":13597},[850],[390,13599,13601],{"className":13600,"style":2748},[854],[390,13602,13603,13606],{"style":2751},[390,13604],{"className":13605,"style":2038},[862],[390,13607,13609],{"className":13608},[2042,2043,2044,2045],[390,13610,713],{"className":13611},[411,2045],[390,13613,616],{"className":13614},[615]," combine gives\n",[390,13617,13619],{"className":13618},[393],[390,13620,13622],{"className":13621,"ariaHidden":398},[397],[390,13623,13625,13628,13631,13634,13663],{"className":13624},[402],[390,13626],{"className":13627,"style":2723},[406],[390,13629,441],{"className":13630},[411],[390,13632,608],{"className":13633},[607],[390,13635,13637,13640],{"className":13636},[411],[390,13638,413],{"className":13639},[411,412],[390,13641,13643],{"className":13642},[2021],[390,13644,13646],{"className":13645},[845],[390,13647,13649],{"className":13648},[850],[390,13650,13652],{"className":13651,"style":2748},[854],[390,13653,13654,13657],{"style":2751},[390,13655],{"className":13656,"style":2038},[862],[390,13658,13660],{"className":13659},[2042,2043,2044,2045],[390,13661,713],{"className":13662},[411,2045],[390,13664,616],{"className":13665},[615]," overall, for no gain. Only a linear combine recovers ",[390,13668,13670],{"className":13669},[393],[390,13671,13673],{"className":13672,"ariaHidden":398},[397],[390,13674,13676,13679,13682,13685,13688,13691,13697,13700,13703],{"className":13675},[402],[390,13677],{"className":13678,"style":600},[406],[390,13680,441],{"className":13681},[411],[390,13683,608],{"className":13684},[607],[390,13686,413],{"className":13687},[411,412],[390,13689],{"className":13690,"style":718},[717],[390,13692,13694],{"className":13693},[1213],[390,13695,1219],{"className":13696,"style":1218},[411,1217],[390,13698],{"className":13699,"style":718},[717],[390,13701,413],{"className":13702},[411,412],[390,13704,616],{"className":13705},[615],[641,13707,547,13708,13711,13712,13802,13803,13827,13828,13918,13919,13943],{},[416,13709,13710],{},"Master Theorem"," solves ",[390,13713,13715],{"className":13714},[393],[390,13716,13718,13745,13784],{"className":13717,"ariaHidden":398},[397],[390,13719,13721,13724,13727,13730,13733,13736,13739,13742],{"className":13720},[402],[390,13722],{"className":13723,"style":600},[406],[390,13725,723],{"className":13726,"style":722},[411,412],[390,13728,608],{"className":13729},[607],[390,13731,413],{"className":13732},[411,412],[390,13734,616],{"className":13735},[615],[390,13737],{"className":13738,"style":800},[717],[390,13740,805],{"className":13741},[804],[390,13743],{"className":13744,"style":800},[717],[390,13746,13748,13751,13754,13757,13760,13763,13766,13769,13772,13775,13778,13781],{"className":13747},[402],[390,13749],{"className":13750,"style":600},[406],[390,13752,421],{"className":13753},[411,412],[390,13755],{"className":13756,"style":718},[717],[390,13758,723],{"className":13759,"style":722},[411,412],[390,13761,608],{"className":13762},[607],[390,13764,413],{"className":13765},[411,412],[390,13767,7551],{"className":13768},[411],[390,13770,6498],{"className":13771},[411,412],[390,13773,616],{"className":13774},[615],[390,13776],{"className":13777,"style":912},[717],[390,13779,917],{"className":13780},[916],[390,13782],{"className":13783,"style":912},[717],[390,13785,13787,13790,13793,13796,13799],{"className":13786},[402],[390,13788],{"className":13789,"style":600},[406],[390,13791,7577],{"className":13792,"style":7576},[411,412],[390,13794,608],{"className":13795},[607],[390,13797,413],{"className":13798},[411,412],[390,13800,616],{"className":13801},[615]," by comparing ",[390,13804,13806],{"className":13805},[393],[390,13807,13809],{"className":13808,"ariaHidden":398},[397],[390,13810,13812,13815,13818,13821,13824],{"className":13811},[402],[390,13813],{"className":13814,"style":600},[406],[390,13816,7577],{"className":13817,"style":7576},[411,412],[390,13819,608],{"className":13820},[607],[390,13822,413],{"className":13823},[411,412],[390,13825,616],{"className":13826},[615]," to\nthe watershed ",[390,13829,13831],{"className":13830},[393],[390,13832,13834],{"className":13833,"ariaHidden":398},[397],[390,13835,13837,13840],{"className":13836},[402],[390,13838],{"className":13839,"style":7791},[406],[390,13841,13843,13846],{"className":13842},[411],[390,13844,413],{"className":13845},[411,412],[390,13847,13849],{"className":13848},[2021],[390,13850,13852],{"className":13851},[845],[390,13853,13855],{"className":13854},[850],[390,13856,13858],{"className":13857,"style":7791},[854],[390,13859,13860,13863],{"style":2751},[390,13861],{"className":13862,"style":2038},[862],[390,13864,13866],{"className":13865},[2042,2043,2044,2045],[390,13867,13869,13912,13915],{"className":13868},[411,2045],[390,13870,13872,13878],{"className":13871},[1213,2045],[390,13873,13875],{"className":13874},[1213,2045],[390,13876,1219],{"className":13877,"style":1218},[411,1217,2045],[390,13879,13881],{"className":13880},[2021],[390,13882,13884,13904],{"className":13883},[845,846],[390,13885,13887,13901],{"className":13886},[850],[390,13888,13890],{"className":13889,"style":7842},[854],[390,13891,13892,13895],{"style":7845},[390,13893],{"className":13894,"style":7849},[862],[390,13896,13898],{"className":13897},[2042,7853,3651,2045],[390,13899,6498],{"className":13900},[411,412,2045],[390,13902,937],{"className":13903},[936],[390,13905,13907],{"className":13906},[850],[390,13908,13910],{"className":13909,"style":7866},[854],[390,13911],{},[390,13913],{"className":13914,"style":7872},[717,2045],[390,13916,421],{"className":13917},[411,412,2045],": leaves win (Case 1), they tie (Case 2, extra\n",[390,13920,13922],{"className":13921},[393],[390,13923,13925],{"className":13924,"ariaHidden":398},[397],[390,13926,13928,13931,13937,13940],{"className":13927},[402],[390,13929],{"className":13930,"style":462},[406],[390,13932,13934],{"className":13933},[1213],[390,13935,1219],{"className":13936,"style":1218},[411,1217],[390,13938],{"className":13939,"style":718},[717],[390,13941,413],{"className":13942},[411,412],"), or the root wins (Case 3, needs regularity).",[641,13945,13946,13947,13949,13950,13965],{},"The cases have ",[416,13948,10196],{},"; when ",[390,13951,13953],{"className":13952},[393],[390,13954,13956],{"className":13955,"ariaHidden":398},[397],[390,13957,13959,13962],{"className":13958},[402],[390,13960],{"className":13961,"style":462},[406],[390,13963,7577],{"className":13964,"style":7576},[411,412]," is only non-polynomially separated from the\nwatershed, fall back to the tree or substitution.",[13967,13968,13971,13976],"section",{"className":13969,"dataFootnotes":376},[13970],"footnotes",[444,13972,13975],{"className":13973,"id":1321},[13974],"sr-only","Footnotes",[13977,13978,13979,13993,14045,14147],"ol",{},[641,13980,13982,13985,13986],{"id":13981},"user-content-fn-skiena-floors",[416,13983,13984],{},"Skiena",", §2.7–2.10 — Logarithms, Recurrences, Divide-and-Conquer: justification for dropping floors and ceilings in recurrences since they perturb the answer by lower-order amounts. ",[421,13987,13992],{"href":13988,"ariaLabel":13989,"className":13990,"dataFootnoteBackref":376},"#user-content-fnref-skiena-floors","Back to reference 1",[13991],"data-footnote-backref","↩",[641,13994,13996,13999,14000,14039,14040],{"id":13995},"user-content-fn-clrs-tree",[416,13997,13998],{},"CLRS",", Ch. 4 — Divide-and-Conquer: the recursion-tree derivation that merge sort runs in ",[390,14001,14003],{"className":14002},[393],[390,14004,14006],{"className":14005,"ariaHidden":398},[397],[390,14007,14009,14012,14015,14018,14021,14024,14030,14033,14036],{"className":14008},[402],[390,14010],{"className":14011,"style":600},[406],[390,14013,441],{"className":14014},[411],[390,14016,608],{"className":14017},[607],[390,14019,413],{"className":14020},[411,412],[390,14022],{"className":14023,"style":718},[717],[390,14025,14027],{"className":14026},[1213],[390,14028,1219],{"className":14029,"style":1218},[411,1217],[390,14031],{"className":14032,"style":718},[717],[390,14034,413],{"className":14035},[411,412],[390,14037,616],{"className":14038},[615]," time. ",[421,14041,13992],{"href":14042,"ariaLabel":14043,"className":14044,"dataFootnoteBackref":376},"#user-content-fnref-clrs-tree","Back to reference 2",[13991],[641,14046,14048,14050,14051,14141,14142],{"id":14047},"user-content-fn-clrs-master",[416,14049,13998],{},", Ch. 4 — Divide-and-Conquer: the Master Theorem for ",[390,14052,14054],{"className":14053},[393],[390,14055,14057,14084,14123],{"className":14056,"ariaHidden":398},[397],[390,14058,14060,14063,14066,14069,14072,14075,14078,14081],{"className":14059},[402],[390,14061],{"className":14062,"style":600},[406],[390,14064,723],{"className":14065,"style":722},[411,412],[390,14067,608],{"className":14068},[607],[390,14070,413],{"className":14071},[411,412],[390,14073,616],{"className":14074},[615],[390,14076],{"className":14077,"style":800},[717],[390,14079,805],{"className":14080},[804],[390,14082],{"className":14083,"style":800},[717],[390,14085,14087,14090,14093,14096,14099,14102,14105,14108,14111,14114,14117,14120],{"className":14086},[402],[390,14088],{"className":14089,"style":600},[406],[390,14091,421],{"className":14092},[411,412],[390,14094],{"className":14095,"style":718},[717],[390,14097,723],{"className":14098,"style":722},[411,412],[390,14100,608],{"className":14101},[607],[390,14103,413],{"className":14104},[411,412],[390,14106,7551],{"className":14107},[411],[390,14109,6498],{"className":14110},[411,412],[390,14112,616],{"className":14113},[615],[390,14115],{"className":14116,"style":912},[717],[390,14118,917],{"className":14119},[916],[390,14121],{"className":14122,"style":912},[717],[390,14124,14126,14129,14132,14135,14138],{"className":14125},[402],[390,14127],{"className":14128,"style":600},[406],[390,14130,7577],{"className":14131,"style":7576},[411,412],[390,14133,608],{"className":14134},[607],[390,14136,413],{"className":14137},[411,412],[390,14139,616],{"className":14140},[615]," and the gaps where it does not apply. ",[421,14143,13992],{"href":14144,"ariaLabel":14145,"className":14146,"dataFootnoteBackref":376},"#user-content-fnref-clrs-master","Back to reference 3",[13991],[641,14148,14150,8624,14153,14156,14157],{"id":14149},"user-content-fn-erickson-methods",[416,14151,14152],{},"Erickson",[385,14154,14155],{},"Algorithms",", Ch. 1–2 — Recursion; Backtracking & Divide-and-Conquer: the case for fluency with recursion trees, substitution, and the Master Theorem as complementary methods. ",[421,14158,13992],{"href":14159,"ariaLabel":14160,"className":14161,"dataFootnoteBackref":376},"#user-content-fnref-erickson-methods","Back to reference 4",[13991],[14163,14164,14165],"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":14167},[14168,14169,14170,14171,14172,14175,14176,14177],{"id":446,"depth":18,"text":447},{"id":1326,"depth":18,"text":1327},{"id":2820,"depth":18,"text":2821},{"id":5215,"depth":18,"text":5216},{"id":7482,"depth":18,"text":7483,"children":14173},[14174],{"id":10355,"depth":24,"text":10356},{"id":13043,"depth":18,"text":13044},{"id":13265,"depth":18,"text":13266},{"id":1321,"depth":18,"text":13975},"When an algorithm solves a problem by calling smaller copies of itself, its\nrunning time obeys an equation that refers to itself: the cost on an input of\nsize n is some local work plus the cost of the recursive calls on smaller\ninputs. Such an equation is a recurrence. Counting loops, as in the previous\nlesson, no longer suffices; we need\ntechniques to turn a recurrence into a closed Θ-bound. This lesson develops\nthree, in increasing order of power and precision.","md",{"moduleNumber":6,"lessonNumber":24,"order":14181},103,true,[14184,14188,14192,14195],{"title":14185,"slug":14186,"difficulty":14187},"Sqrt(x)","sqrtx","Easy",{"title":14189,"slug":14190,"difficulty":14191},"Pow(x, n)","powx-n","Medium",{"title":14193,"slug":14194,"difficulty":14191},"Search in Rotated Sorted Array","search-in-rotated-sorted-array",{"title":14196,"slug":14197,"difficulty":14191},"Different Ways to Add Parentheses","different-ways-to-add-parentheses","---\ntitle: Recurrences and the Master Theorem\nmodule: Foundations\nmoduleNumber: 1\nlessonNumber: 3\norder: 103\nsummary: >\n  Recursive and divide-and-conquer algorithms describe their own running time\n  with a recurrence: $T(n)$ in terms of $T$ on smaller inputs. We solve\n  recurrences three ways — drawing the recursion tree, guessing-and-verifying by\n  induction, and applying the Master Theorem — using merge sort as the running\n  example.\ntopics: [Recurrences]\nsources:\n  - book: CLRS\n    ref: \"Ch. 4 — Divide-and-Conquer\"\n  - book: Skiena\n    ref: \"§2.7–2.10 — Logarithms, Recurrences, Divide-and-Conquer\"\n  - book: Erickson\n    ref: \"Ch. 1–2 — Recursion; Backtracking & Divide-and-Conquer\"\npractice:\n  - title: 'Sqrt(x)'\n    slug: sqrtx\n    difficulty: Easy\n  - title: 'Pow(x, n)'\n    slug: powx-n\n    difficulty: Medium\n  - title: 'Search in Rotated Sorted Array'\n    slug: search-in-rotated-sorted-array\n    difficulty: Medium\n  - title: 'Different Ways to Add Parentheses'\n    slug: different-ways-to-add-parentheses\n    difficulty: Medium\n---\n\nWhen an algorithm solves a problem by calling smaller copies of itself, its\nrunning time obeys an equation that refers to _itself_: the cost on an input of\nsize $n$ is some local work plus the cost of the recursive calls on smaller\ninputs. Such an equation is a **recurrence**. Counting loops, as in [the previous\nlesson](\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis), no longer suffices; we need\ntechniques to turn a recurrence into a closed $\\Theta$-bound. This lesson develops\nthree, in increasing order of power and precision.\n\n## From a recursive algorithm to a recurrence\n\n$\\textsc{Divide-and-conquer}$ is the paradigm all three books use to introduce\nrecurrences. It has three steps: **divide** the instance into subproblems,\n**conquer** them by recursion, and **combine** their solutions. [**Merge sort**](\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort)\nsplits the array in half, sorts each half recursively, and merges the two sorted\nhalves.\n\n```algorithm\ncaption: $\\textsc{Merge-Sort}(A, p, r)$ — sort $A[p..r]$\nnumber: 1\nif $p \u003C r$ then\n  $q \\gets \\floor{(p + r) \u002F 2}$ \u002F\u002F midpoint\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\nreturn $A$\n```\n\nThe $\\textsc{Merge}$ subroutine walks the two sorted halves with two pointers, repeatedly\ncopying the smaller front element into the output. It touches each of the $n$\nelements a constant number of times, so it costs $\\Theta(n)$.\n\nNow read the cost off the structure. On an array of size $n$:\n\n- **Divide** is computing the midpoint, $\\Theta(1)$.\n- **Conquer** is two recursive calls, each on $n\u002F2$ elements, costing $2\\,T(n\u002F2)$.\n- **Combine** is the merge, $\\Theta(n)$.\n\nAdding these (and noting that a one-element array is sorted at no cost) gives the\nrecurrence\n\n$$\nT(n) =\n\\begin{cases}\n\\Theta(1) & \\text{if } n = 1, \\\\[2pt]\n2\\,T(n\u002F2) + \\Theta(n) & \\text{if } n > 1.\n\\end{cases}\n$$\n\nWe can write this compactly, and as an _inequality_ (since the\ncombine step costs _at most_ linear), as\n$$\nT(n) \\le 2\\,T(n\u002F2) + O(n) \\;\\Longrightarrow\\; T(n) = O(n\\log n),\n$$\nand the bulk of the work is justifying that implication. This is the equation we\nmust solve. (We freely write $n\u002F2$ rather than\n$\\floor{n\u002F2}$ and $\\ceil{n\u002F2}$; the floors and ceilings change the answer by\nlower-order amounts that the asymptotics absorb. Skiena and CLRS both justify\ndropping them.[^skiena-floors]) Throughout we also assume a constant base case, which lets us\nignore the boundary condition when finding the asymptotic order.\n\n## Method 1: the recursion tree\n\nThe most intuitive method _draws_ the recurrence. Each node is a subproblem\nlabeled with the **non-recursive** work it does; its children are the subproblems\nit spawns. Summing all node labels gives $T(n)$.\n\nFor merge sort, the root does $cn$ work and has two children of size $n\u002F2$. Each\nof those does $c(n\u002F2)$ work and has two children of size $n\u002F4$, and so on, until\nthe leaves are size-$1$ subproblems.\n\n$$\n% caption: Recursion tree for merge sort, with each level summing to $cn$.\n\\begin{tikzpicture}[\n  every node\u002F.style={draw, rounded corners, minimum size=6mm, inner sep=2pt},\n  edge\u002F.style={draw, -}, node distance=8mm and 4mm]\n  % level 0\n  \\node (r) {$cn$};\n  % level 1\n  \\node (a) [below left=12mm and 16mm of r] {$c\\tfrac{n}{2}$};\n  \\node (b) [below right=12mm and 16mm of r] {$c\\tfrac{n}{2}$};\n  % level 2\n  \\node (a1) [below left=12mm and 4mm of a] {$c\\tfrac{n}{4}$};\n  \\node (a2) [below right=12mm and 4mm of a] {$c\\tfrac{n}{4}$};\n  \\node (b1) [below left=12mm and 4mm of b] {$c\\tfrac{n}{4}$};\n  \\node (b2) [below right=12mm and 4mm of b] {$c\\tfrac{n}{4}$};\n  \\draw (r) -- (a); \\draw (r) -- (b);\n  \\draw (a) -- (a1); \\draw (a) -- (a2);\n  \\draw (b) -- (b1); \\draw (b) -- (b2);\n  % per-level sums on the right\n  \\node[draw=none] at ($(r) + (5.4,0)$) {$\\rightarrow\\; cn$};\n  \\node[draw=none] at ($(b) + (3.0,0)$) {$\\rightarrow\\; 2\\cdot c\\tfrac{n}{2} = cn$};\n  \\node[draw=none] at ($(b2) + (2.0,0)$) {$\\rightarrow\\; 4\\cdot c\\tfrac{n}{4} = cn$};\n  \\node[draw=none] at ($(b2) + (-3.0,-1.0)$) {$\\vdots$};\n\\end{tikzpicture}\n$$\n\nThe key observation: **every level sums to $cn$.** The root level is $cn$; the\nnext level is $2 \\cdot c(n\u002F2) = cn$; the level below is $4 \\cdot c(n\u002F4) = cn$. The\nsubproblem sizes shrink by half each level, so the tree has\n$\\log_2 n + 1$ levels (from size $n$ down to size $1$), and the bottom level holds\nthe $n$ size-$1$ leaves. Therefore\n\n$$\nT(n) = \\underbrace{cn}_{\\text{per level}} \\times \\underbrace{(\\log_2 n + 1)}_{\\text{levels}}\n= cn \\log_2 n + cn = \\Theta(n \\log n).\n$$\n\nThis is the result: **merge sort runs in $\\Theta(n \\log n)$ time**,\nstrictly better than insertion sort's $\\Theta(n^2)$.[^clrs-tree] The recursion tree also\nexposes _why_: the per-level work stays flat at $cn$ while the depth is only\nlogarithmic.\n\nThe tree is a derivation, not yet a proof — it asks us to trust the level sums\nand the level count. When a recurrence is irregular (unequal splits, work that\nisn't a clean power of $n$), the tree still gives a reliable _guess_, which we\nthen certify with the next method.\n\n## Method 2: substitution (guess and verify)\n\nThe **substitution method** is the rigorous one: guess the form of the\nanswer, then prove it by **induction** on $n$. It is the only method that always\nworks, and the only one that produces a complete proof.\n\nWe verify the guess $T(n) = O(n \\log n)$ for the merge-sort recurrence\n$T(n) = 2\\,T(n\u002F2) + cn$. We claim there is a constant $d > 0$ with\n$T(n) \\le d\\,n \\log_2 n$ for all $n \\ge 2$.\n\n**Inductive hypothesis.** Assume the bound holds for all sizes smaller than $n$;\nin particular $T(n\u002F2) \\le d\\,\\frac{n}{2}\\log_2\\frac{n}{2}$.\n\n**Inductive step.** Substitute into the recurrence:\n$$\n\\begin{aligned}\nT(n) &= 2\\,T(n\u002F2) + cn \\\\\n&\\le 2\\left(d\\,\\tfrac{n}{2}\\log_2\\tfrac{n}{2}\\right) + cn \\\\\n&= d\\,n\\,(\\log_2 n - 1) + cn \\\\\n&= d\\,n\\log_2 n - dn + cn \\\\\n&= d\\,n\\log_2 n - (d - c)\\,n.\n\\end{aligned}\n$$\nThis is $\\le d\\,n\\log_2 n$ exactly when $d \\ge c$. So choosing any $d \\ge c$\n(and picking the base-case constant large enough to cover $n = 2$) completes the\ninduction. Hence $T(n) = O(n\\log n)$.\n\n$$\n% caption: The substitution method: assume the bound on smaller inputs, substitute into\n%          the recurrence, and check that the leftover residual term lets the same bound\n%          re-emerge for $n$.\n\\begin{tikzpicture}[\n  box\u002F.style={draw, rounded corners, align=center, inner sep=4pt, minimum height=10mm},\n  node distance=6mm,\n  >={Stealth[length=2.5mm]}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[box, fill=acc!15] (h) {assume\\\\$T(n\u002F2)\\le d\\,\\tfrac{n}{2}\\log\\tfrac{n}{2}$};\n  \\node[box, right=14mm of h] (s) {substitute into\\\\$T(n)=2T(n\u002F2)+cn$};\n  \\node[box, fill=acc!15, right=14mm of s] (c) {conclude\\\\$T(n)\\le d\\,n\\log n$};\n  \\draw[->] (h) -- (s);\n  \\draw[->] (s) -- (c);\n  \\node[box, fill=red!16, below=10mm of s, align=center] (r) {residual $-(d-c)\\,n\\le 0$\\\\holds iff $d\\ge c$};\n  \\draw[->] (s) -- (r);\n  \\draw[->] (r) -- (c);\n\\end{tikzpicture}\n$$\n\nA symmetric argument with the inequality reversed gives $T(n) = \\Omega(n\\log n)$,\nand together they yield $T(n) = \\Theta(n\\log n)$, confirming the tree.\n\nTwo warnings the books repeat:\n\n- **Guess the right form.** Substitution verifies a guess; it cannot invent one.\n  Use the recursion tree (or the Master Theorem below) to _find_ the candidate.\n- **Beware the off-by-a-term trap.** A guess like $T(n) \\le d\\,n$ for a\n  $\\Theta(n\\log n)$ recurrence will _seem_ to almost work but leave an\n  unkillable residual term. Strengthening the hypothesis (subtracting a\n  lower-order term, as in $T(n) \\le d\\,n\\log n - bn$) is the standard fix.\n\n## A second example: counting inversions\n\nA second divide-and-conquer problem makes the point sharply. Its\nrecurrence has the _same shape_ as merge sort but a different combine cost, and\nthe combine cost is exactly the thing you must get right. An **inversion**\nof a list $\\langle a_1,\\dots,a_n\\rangle$ is a pair $(i,j)$ with $i \u003C j$ but\n$a_i > a_j$; the number of inversions measures how far from sorted the list is\n(a sorted list has $0$, a reversed list has $\\binom{n}{2}$). The task: given\n$A[1..n]$, return $\\operatorname{ninv}(A)$.\n\nThe brute-force algorithm compares every pair and runs in $\\Theta(n^2)$. To beat\nit, **mimic merge sort**: split $A$ in half, recursively count inversions inside\neach half, then count the **cross inversions**, the pairs with one element in the\nleft half and one in the right. That gives a recurrence of the merge-sort form,\n\n$$\nT(n) \\le 2\\,T(n\u002F2) + (\\text{cost of counting cross inversions}).\n$$\n\nThe three kinds of inversion partition cleanly along the split. On\n$A = \\langle 2,4,1,3\\rangle$, the inversions _within_ each half are counted by\nrecursion; the **cross** pairs, a left element greater than a right element, are\nexactly what the combine step must tally.\n\n$$\n% caption: Cross inversions on $\\langle 2,4,1,3\\rangle$ split into halves\n%          $\\langle 2,4\\rangle$ and $\\langle 1,3\\rangle$. Each red arc is a cross\n%          inversion (a left element bigger than a right one): $(2,1),(4,1),(4,3)$.\n%          Within-half inversions are handled by recursion; the combine step counts only\n%          these crossing pairs.\n\\begin{tikzpicture}[\n    cell\u002F.style={draw, minimum size=8mm, font=\\small},\n    lbl\u002F.style={font=\\footnotesize},\n    >={Stealth[length=2.2mm]}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-0.6,-2.1) rectangle (5.9,1.3);\n  % left half\n  \\node[cell, fill=acc!15] (L1) at (0,0) {2};\n  \\node[cell, fill=acc!15, right=0mm of L1] (L2) {4};\n  % gap, then right half\n  \\node[cell, right=8mm of L2] (R1) {1};\n  \\node[cell, right=0mm of R1] (R2) {3};\n  \\node[lbl, above=1mm of L1.north west, anchor=south west] {left half};\n  \\node[lbl, above=1mm of R2.north east, anchor=south east] {right half};\n  \\draw[dashed, black!45] ($(L2.north east)+(0.4,0.55)$) -- ($(L2.south east)+(0.4,-0.95)$);\n  % cross inversions (red arcs), routed below\n  \\draw[->, red!75!black] (L1.south) .. controls +(0,-0.7) and +(0,-0.7) .. (R1.south);\n  \\draw[->, red!75!black] (L2.south) .. controls +(0,-1.1) and +(0,-1.1) .. (R1.south);\n  \\draw[->, red!75!black] (L2.south) .. controls +(0,-0.5) and +(0,-0.5) .. (R2.south);\n  \\node[lbl, red!75!black, align=center] at (2.65,-1.85) {$3$ cross inversions: $(2{>}1),\\,(4{>}1),\\,(4{>}3)$};\n\\end{tikzpicture}\n$$\n\nCounting cross inversions _naively_, with a double loop over the two halves, costs\n$\\Theta(n^2)$, so the recurrence becomes $T(n) \\le 2\\,T(n\u002F2) + bn^2$. Feed that to\nthe recursion tree: the per-level work is now $b\\,n^2,\\ 2\\cdot b(n\u002F2)^2 =\n\\tfrac12 bn^2,\\ \\dots$, which _shrinks geometrically_, so the root dominates and the\ntree sums to $\\Theta(n^2)$. That is no improvement. The split bought us nothing because\nthe combine step is as expensive as the brute force.\n\n$$\n% caption: Naive counting-inversions tree: unlike merge sort's flat tree, per-level work\n%          shrinks geometrically, so the root dominates and the sum is $\\Theta(n^2)$.\n\\begin{tikzpicture}[\n  every node\u002F.style={draw, rounded corners, minimum size=6mm, inner sep=2pt},\n  node distance=8mm and 4mm]\n  % level 0\n  \\node (r) {$bn^2$};\n  % level 1\n  \\node (a) [below left=12mm and 16mm of r] {$b(\\tfrac{n}{2})^2$};\n  \\node (b) [below right=12mm and 16mm of r] {$b(\\tfrac{n}{2})^2$};\n  % level 2\n  \\node (a1) [below left=12mm and 4mm of a] {$b(\\tfrac{n}{4})^2$};\n  \\node (a2) [below right=12mm and 4mm of a] {$b(\\tfrac{n}{4})^2$};\n  \\node (b1) [below left=12mm and 4mm of b] {$b(\\tfrac{n}{4})^2$};\n  \\node (b2) [below right=12mm and 4mm of b] {$b(\\tfrac{n}{4})^2$};\n  \\draw (r) -- (a); \\draw (r) -- (b);\n  \\draw (a) -- (a1); \\draw (a) -- (a2);\n  \\draw (b) -- (b1); \\draw (b) -- (b2);\n  % per-level sums on the right\n  \\node[draw=none] at ($(r) + (6.0,0)$) {$\\rightarrow\\; bn^2$};\n  \\node[draw=none] at ($(b) + (3.7,0)$) {$\\rightarrow\\; \\tfrac12 bn^2$};\n  \\node[draw=none] at ($(b2) + (2.6,0)$) {$\\rightarrow\\; \\tfrac14 bn^2$};\n  \\node[draw=none] at ($(b2) + (-3.0,-1.0)$) {$\\vdots$};\n\\end{tikzpicture}\n$$\n\nThe lesson the recurrence teaches is therefore a _demand_: the combine step must\nrun in $O(n)$, not $O(n^2)$. If we can count cross inversions in linear time,\nwhich one can, by counting them _while merging_ the two sorted halves, the\nrecurrence collapses to $T(n) \\le 2\\,T(n\u002F2) + O(n)$, the merge-sort recurrence,\nand we get $\\Theta(n\\log n)$. The recurrence both predicts\nthe running time and tells you precisely how fast the combine step has to be for\ndivide-and-conquer to pay off.\n\n## Method 3: the Master Theorem\n\nMerge sort's recurrence is one instance of a common pattern. The **Master\nTheorem** solves every recurrence of the form\n\n$$\nT(n) = a\\,T(n\u002Fb) + f(n),\n$$\n\nwhere $a \\ge 1$ and $b > 1$ are constants and $f(n)$ is the divide-and-combine\nwork. Here $a$ is the number of subproblems, $n\u002Fb$ is each subproblem's size, and\n$f(n)$ is the work done outside the recursion.\n\nThe theorem compares $f(n)$ against the **watershed function**\n$n^{\\log_b a}$, the total cost of the leaves, which equals the number of leaves\n$a^{\\log_b n} = n^{\\log_b a}$ times the constant base-case cost. Which of the two\ndominates determines the answer.\n\n$$\n% caption: The Master Theorem weighs the root's combine work $f(n)$ against the leaves'\n%          total cost, the watershed $n^{\\log_b a}$.\n\\begin{tikzpicture}[\n  every node\u002F.style={draw, rounded corners, minimum size=6mm, inner sep=2pt},\n  node distance=8mm and 4mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  % root\n  \\node[fill=acc!15] (r) {$f(n)$};\n  % level 1\n  \\node (a) [below left=11mm and 14mm of r] {$f(n\u002Fb)$};\n  \\node (b) [below right=11mm and 14mm of r] {$f(n\u002Fb)$};\n  \\node[draw=none] (dots) [below=2mm of r, yshift=-9mm] {$\\cdots$};\n  % leaves\n  \\node[fill=black!8] (l1) [below=20mm of a] {$\\Theta(1)$};\n  \\node[fill=black!8] (l2) [below=20mm of b] {$\\Theta(1)$};\n  \\node[draw=none] (ldots) at ($(l1)!0.5!(l2)$) {$\\cdots$};\n  \\draw (r) -- (a); \\draw (r) -- (b);\n  \\draw[dashed] (a) -- (l1); \\draw[dashed] (b) -- (l2);\n  % annotations\n  \\node[draw=none, right=3mm of r, align=left] {root work $= f(n)$};\n  \\node[draw=none, below=4mm of ldots, align=center] {$n^{\\log_b a}$ leaves $\\;\\Rightarrow\\;$ watershed $n^{\\log_b a}$};\n\\end{tikzpicture}\n$$\n\n> **Theorem (Master).** Let $T(n) = a\\,T(n\u002Fb) + f(n)$ with $a \\ge 1$, $b > 1$.\n> Let $\\epsilon > 0$ be a constant. Then:\n>\n> **Case 1 (leaves dominate).** If $f(n) = O\\!\\left(n^{\\log_b a - \\epsilon}\\right)$\n> (that is, $f$ is _polynomially smaller_ than the watershed), then\n> $$T(n) = \\Theta\\!\\left(n^{\\log_b a}\\right).$$\n>\n> **Case 2 (balanced).** If $f(n) = \\Theta\\!\\left(n^{\\log_b a}\\right)$ ($f$\n> matches the watershed), then\n> $$T(n) = \\Theta\\!\\left(n^{\\log_b a}\\,\\log n\\right).$$\n>\n> **Case 3 (root dominates).** If $f(n) = \\Omega\\!\\left(n^{\\log_b a + \\epsilon}\\right)$\n> ($f$ is _polynomially larger_ than the watershed) and the **regularity\n> condition** $a\\,f(n\u002Fb) \\le k\\,f(n)$ holds for some constant $k \u003C 1$ and all\n> large $n$, then\n> $$T(n) = \\Theta\\!\\bigl(f(n)\\bigr).$$\n\nThe intuition matches the recursion tree. Compare the work at the root, $f(n)$,\nto the work at the leaves, $n^{\\log_b a}$. In Case 1 the tree is leaf-heavy and\nthe answer is the leaf count. In Case 3 the root work dwarfs everything below it\nand the answer is $f(n)$. In Case 2 the work is spread evenly across all\n$\\Theta(\\log n)$ levels, as we saw for merge sort, giving the extra $\\log n$\nfactor.\n\n$$\n% caption: Where the work concentrates in the Master Theorem's three cases.\n\\begin{tikzpicture}[scale=1.0]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\def\\bar#1#2#3{\\filldraw[acc!75,draw=acc] ({#1-#3\u002F2},{#2}) rectangle ({#1+#3\u002F2},{#2+0.4});}\n  \\bar{0}{0}{0.5} \\bar{0}{-0.52}{0.9} \\bar{0}{-1.04}{1.5} \\bar{0}{-1.56}{2.4}\n  \\bar{4.3}{0}{1.5} \\bar{4.3}{-0.52}{1.5} \\bar{4.3}{-1.04}{1.5} \\bar{4.3}{-1.56}{1.5}\n  \\bar{8.6}{0}{2.4} \\bar{8.6}{-0.52}{1.5} \\bar{8.6}{-1.04}{0.9} \\bar{8.6}{-1.56}{0.5}\n  \\node[font=\\small\\bfseries] at (0,0.75) {Case 1};\n  \\node[font=\\small\\bfseries] at (4.3,0.75) {Case 2};\n  \\node[font=\\small\\bfseries] at (8.6,0.75) {Case 3};\n  \\node[font=\\footnotesize] at (0,-2.5) {leaves dominate};\n  \\node[font=\\footnotesize] at (4.3,-2.5) {every level equal};\n  \\node[font=\\footnotesize] at (8.6,-2.5) {root dominates};\n  \\node[font=\\footnotesize] at (0,-3.05) {$\\Theta(n^{\\log_b a})$};\n  \\node[font=\\footnotesize] at (4.3,-3.05) {$\\Theta(n^{\\log_b a}\\log n)$};\n  \\node[font=\\footnotesize] at (8.6,-3.05) {$\\Theta(f(n))$};\n\\end{tikzpicture}\n$$\n\nEach panel stacks the per-level work from root (top) to leaves (bottom); the bar\nwidth is the work at that level. The case is decided by which end is heavier.\n\nA caution from CLRS: the cases do not cover every recurrence.[^clrs-master] There are **gaps**.\nWhen $f$ is larger than the watershed but not _polynomially_ larger (e.g.\n$f(n) = n^{\\log_b a}\\log n$), or when the regularity condition fails,\nthe Master Theorem simply does not apply. Fall back to the recursion tree or\nsubstitution.\n\n### Worked examples\n\n**Example 1, merge sort.** $T(n) = 2\\,T(n\u002F2) + \\Theta(n)$. Here $a = 2$,\n$b = 2$, so $n^{\\log_b a} = n^{\\log_2 2} = n^1 = n$. And $f(n) = \\Theta(n) =\n\\Theta(n^{\\log_b a})$, which is Case 2. Therefore\n$$\nT(n) = \\Theta(n \\log n),\n$$\nrecovering exactly what the tree and substitution gave.\n\n**Example 2, [binary search](\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer).**\n$T(n) = T(n\u002F2) + \\Theta(1)$: one subproblem of\nhalf size, constant work to pick the side. Here $a = 1$, $b = 2$, so\n$n^{\\log_2 1} = n^0 = 1$. Then $f(n) = \\Theta(1) = \\Theta(n^{\\log_b a})$, Case 2\nagain, and\n$$\nT(n) = \\Theta(\\log n).\n$$\n\n**Example 3, leaf-dominated.** $T(n) = 4\\,T(n\u002F2) + n$. Now $a = 4$, $b = 2$, so\n$n^{\\log_2 4} = n^2$. The combine work $f(n) = n = O(n^{2 - \\epsilon})$ (take\n$\\epsilon = 1$) is polynomially _smaller_ than the watershed, which is Case 1, so\n$$\nT(n) = \\Theta(n^2).\n$$\nThe recursion has so many leaves ($n^2$ of them) that they dominate the modest\nlinear work per level.\n\n**Example 4, root-dominated.** $T(n) = 2\\,T(n\u002F2) + n^2$. Here $a = 2$, $b = 2$,\nwatershed $n^{\\log_2 2} = n$. The combine work $f(n) = n^2 = \\Omega(n^{1+\\epsilon})$\nis polynomially _larger_, a Case 3 candidate. Check regularity:\n$a\\,f(n\u002Fb) = 2\\,(n\u002F2)^2 = \\tfrac12 n^2 = \\tfrac12 f(n) \\le k\\,f(n)$ with\n$k = \\tfrac12 \u003C 1$. Regularity holds, so\n$$\nT(n) = \\Theta(n^2).\n$$\nThe root's quadratic work swamps the tree beneath it.\n\n## Choosing a method\n\nThe three methods are complementary, and Erickson in particular urges fluency\nwith all of them:[^erickson-methods]\n\n- **Recursion tree:** fastest for _building intuition_ and _guessing_ the answer;\n  shows where the work concentrates.\n- **Master Theorem:** fastest for _getting the answer_ when the recurrence fits\n  the $a\\,T(n\u002Fb) + f(n)$ template; no derivation needed, but it has gaps.\n- **Substitution:** the _rigorous_ method that always works and produces a proof;\n  use it to certify a guess, or when the other two do not apply (uneven splits,\n  recurrences like $T(n) = T(n\u002F3) + T(2n\u002F3) + n$).\n\nIn practice: sketch the tree to guess, apply the Master Theorem if it fits, and\nreach for substitution whenever you need a guarantee rather than a hunch.\n\n## Takeaways\n\n- A **recursive algorithm induces a recurrence**: $T(n)$ = local work + cost of\n  recursive calls on smaller inputs. Merge sort gives $T(n) = 2T(n\u002F2) + \\Theta(n)$.\n- The **recursion tree** sums the per-node work; for merge sort every level costs\n  $\\Theta(n)$ across $\\Theta(\\log n)$ levels, giving $\\Theta(n\\log n)$.\n- **Substitution** guesses the form and proves it by induction; it is the only\n  always-applicable, fully rigorous method. Strengthen the hypothesis if a\n  residual term blocks the step.\n- The **combine cost drives the answer.** Counting inversions has the merge-sort\n  _shape_ $2\\,T(n\u002F2) + (\\text{combine})$, but a naive $\\Theta(n^2)$ combine gives\n  $\\Theta(n^2)$ overall, for no gain. Only a linear combine recovers $\\Theta(n\\log n)$.\n- The **Master Theorem** solves $T(n) = a\\,T(n\u002Fb) + f(n)$ by comparing $f(n)$ to\n  the watershed $n^{\\log_b a}$: leaves win (Case 1), they tie (Case 2, extra\n  $\\log n$), or the root wins (Case 3, needs regularity).\n- The cases have **gaps**; when $f$ is only non-polynomially separated from the\n  watershed, fall back to the tree or substitution.\n\n[^skiena-floors]: **Skiena**, §2.7–2.10 — Logarithms, Recurrences, Divide-and-Conquer: justification for dropping floors and ceilings in recurrences since they perturb the answer by lower-order amounts.\n[^clrs-tree]: **CLRS**, Ch. 4 — Divide-and-Conquer: the recursion-tree derivation that merge sort runs in $\\Theta(n\\log n)$ time.\n[^clrs-master]: **CLRS**, Ch. 4 — Divide-and-Conquer: the Master Theorem for $T(n) = a\\,T(n\u002Fb) + f(n)$ and the gaps where it does not apply.\n[^erickson-methods]: **Erickson**, _Algorithms_, Ch. 1–2 — Recursion; Backtracking & Divide-and-Conquer: the case for fluency with recursion trees, substitution, and the Master Theorem as complementary methods.\n",{"text":14200,"minutes":14201,"time":14202,"words":14203},"9 min read",8.69,521400,1738,{"title":22,"description":14178},[14206,14208,14210],{"book":13998,"ref":14207},"Ch. 4 — Divide-and-Conquer",{"book":13984,"ref":14209},"§2.7–2.10 — Logarithms, Recurrences, Divide-and-Conquer",{"book":14152,"ref":14211},"Ch. 1–2 — Recursion; Backtracking & 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