[{"data":1,"prerenderedAt":13129},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":374,"course-wordcounts":12998,"ref-card-index":13054},[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":45,"blurb":376,"body":377,"description":12960,"extension":12961,"meta":12962,"module":29,"navigation":12964,"path":46,"practice":12965,"rawbody":12980,"readingTime":12981,"seo":12986,"sources":12987,"status":12994,"stem":12995,"summary":49,"topics":12996,"__hash__":12997},"course\u002F01.algorithms\u002F02.divide-and-conquer\u002F03.selection.md","",{"type":378,"value":379,"toc":12941},"minimark",[380,507,512,721,899,903,1044,1124,1199,1544,1614,1619,1691,1786,1793,2286,2351,2778,2785,2943,2950,2954,2997,3083,3087,3169,3208,3462,3786,3918,4258,4613,4617,4699,4775,5038,5318,5827,5965,6614,6645,7103,8176,8852,8856,8919,8923,8934,9610,9772,9776,9838,10144,10164,10261,10543,10737,10761,11011,11281,11285,11364,11394,11479,11524,11528,11579,11819,11905,12124,12828,12937],[381,382,383,384,388,389,414,415,466,467,485,486,489,490,501,502,506],"p",{},"How do you find the ",[385,386,387],"strong",{},"median"," of ",[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"," numbers? The obvious answer, sorting them\nand reading off the middle, costs ",[390,416,418],{"className":417},[393],[390,419,421],{"className":420,"ariaHidden":398},[397],[390,422,424,428,432,437,440,445,455,458,461],{"className":423},[402],[390,425],{"className":426,"style":427},[406],"height:1em;vertical-align:-0.25em;",[390,429,431],{"className":430},[411],"Θ",[390,433,436],{"className":434},[435],"mopen","(",[390,438,413],{"className":439},[411,412],[390,441],{"className":442,"style":444},[443],"mspace","margin-right:0.1667em;",[390,446,449],{"className":447},[448],"mop",[390,450,454],{"className":451,"style":453},[411,452],"mathrm","margin-right:0.0139em;","log",[390,456],{"className":457,"style":444},[443],[390,459,413],{"className":460},[411,412],[390,462,465],{"className":463},[464],"mclose",")",". But the median, and more\ngenerally the ",[390,468,470],{"className":469},[393],[390,471,473],{"className":472,"ariaHidden":398},[397],[390,474,476,480],{"className":475},[402],[390,477],{"className":478,"style":479},[406],"height:0.6944em;",[390,481,484],{"className":482,"style":483},[411,412],"margin-right:0.0315em;","k","-th smallest element, can be found in ",[385,487,488],{},"linear"," time.",[491,492,493],"sup",{},[494,495,500],"a",{"href":496,"ariaDescribedBy":497,"dataFootnoteRef":376,"id":499},"#user-content-fn-clrs-select",[498],"footnote-label","user-content-fnref-clrs-select","1"," The\ninsight is that selection asks for ",[503,504,505],"em",{},"less"," than sorting: we want one element's\nvalue, not the full ordering, and we can stop as soon as we have it.",[508,509,511],"h2",{"id":510},"the-selection-problem","The selection problem",[513,514,516],"callout",{"type":515},"problem",[381,517,518,521,522,388,553,568,569,584,585,641,642,645,646,662,663,701,702,720],{},[385,519,520],{},"Input:"," an array ",[390,523,525],{"className":524},[393],[390,526,528],{"className":527,"ariaHidden":398},[397],[390,529,531,534,538,542,546,549],{"className":530},[402],[390,532],{"className":533,"style":427},[406],[390,535,537],{"className":536},[411,412],"A",[390,539,541],{"className":540},[435],"[",[390,543,545],{"className":544},[411],"1..",[390,547,413],{"className":548},[411,412],[390,550,552],{"className":551},[464],"]",[390,554,556],{"className":555},[393],[390,557,559],{"className":558,"ariaHidden":398},[397],[390,560,562,565],{"className":561},[402],[390,563],{"className":564,"style":407},[406],[390,566,413],{"className":567},[411,412]," distinct numbers and an integer ",[390,570,572],{"className":571},[393],[390,573,575],{"className":574,"ariaHidden":398},[397],[390,576,578,581],{"className":577},[402],[390,579],{"className":580,"style":479},[406],[390,582,484],{"className":583,"style":483},[411,412]," with\n",[390,586,588],{"className":587},[393],[390,589,591,613,632],{"className":590,"ariaHidden":398},[397],[390,592,594,598,601,605,610],{"className":593},[402],[390,595],{"className":596,"style":597},[406],"height:0.7804em;vertical-align:-0.136em;",[390,599,500],{"className":600},[411],[390,602],{"className":603,"style":604},[443],"margin-right:0.2778em;",[390,606,609],{"className":607},[608],"mrel","≤",[390,611],{"className":612,"style":604},[443],[390,614,616,620,623,626,629],{"className":615},[402],[390,617],{"className":618,"style":619},[406],"height:0.8304em;vertical-align:-0.136em;",[390,621,484],{"className":622,"style":483},[411,412],[390,624],{"className":625,"style":604},[443],[390,627,609],{"className":628},[608],[390,630],{"className":631,"style":604},[443],[390,633,635,638],{"className":634},[402],[390,636],{"className":637,"style":407},[406],[390,639,413],{"className":640},[411,412],".\n",[385,643,644],{},"Output:"," the element of ",[390,647,649],{"className":648},[393],[390,650,652],{"className":651,"ariaHidden":398},[397],[390,653,655,659],{"className":654},[402],[390,656],{"className":657,"style":658},[406],"height:0.6833em;",[390,660,537],{"className":661},[411,412]," that is larger than exactly ",[390,664,666],{"className":665},[393],[390,667,669,691],{"className":668,"ariaHidden":398},[397],[390,670,672,676,679,683,688],{"className":671},[402],[390,673],{"className":674,"style":675},[406],"height:0.7778em;vertical-align:-0.0833em;",[390,677,484],{"className":678,"style":483},[411,412],[390,680],{"className":681,"style":682},[443],"margin-right:0.2222em;",[390,684,687],{"className":685},[686],"mbin","−",[390,689],{"className":690,"style":682},[443],[390,692,694,698],{"className":693},[402],[390,695],{"className":696,"style":697},[406],"height:0.6444em;",[390,699,500],{"className":700},[411]," of the\nothers — the ",[385,703,704,719],{},[390,705,707],{"className":706},[393],[390,708,710],{"className":709,"ariaHidden":398},[397],[390,711,713,716],{"className":712},[402],[390,714],{"className":715,"style":479},[406],[390,717,484],{"className":718,"style":483},[411,412],"-th order statistic",".",[381,722,723,724,758,759,792,793,863,864,898],{},"Special cases: ",[390,725,727],{"className":726},[393],[390,728,730,749],{"className":729,"ariaHidden":398},[397],[390,731,733,736,739,742,746],{"className":732},[402],[390,734],{"className":735,"style":479},[406],[390,737,484],{"className":738,"style":483},[411,412],[390,740],{"className":741,"style":604},[443],[390,743,745],{"className":744},[608],"=",[390,747],{"className":748,"style":604},[443],[390,750,752,755],{"className":751},[402],[390,753],{"className":754,"style":697},[406],[390,756,500],{"className":757},[411]," is the minimum, ",[390,760,762],{"className":761},[393],[390,763,765,783],{"className":764,"ariaHidden":398},[397],[390,766,768,771,774,777,780],{"className":767},[402],[390,769],{"className":770,"style":479},[406],[390,772,484],{"className":773,"style":483},[411,412],[390,775],{"className":776,"style":604},[443],[390,778,745],{"className":779},[608],[390,781],{"className":782,"style":604},[443],[390,784,786,789],{"className":785},[402],[390,787],{"className":788,"style":407},[406],[390,790,413],{"className":791},[411,412]," the maximum, and\n",[390,794,796],{"className":795},[393],[390,797,799,817],{"className":798,"ariaHidden":398},[397],[390,800,802,805,808,811,814],{"className":801},[402],[390,803],{"className":804,"style":479},[406],[390,806,484],{"className":807,"style":483},[411,412],[390,809],{"className":810,"style":604},[443],[390,812,745],{"className":813},[608],[390,815],{"className":816,"style":604},[443],[390,818,820,823],{"className":819},[402],[390,821],{"className":822,"style":427},[406],[390,824,827,833,836,839,842,846,849,852,855,859],{"className":825},[826],"minner",[390,828,832],{"className":829,"style":831},[435,830],"delimcenter","top:0em;","⌊",[390,834,436],{"className":835},[435],[390,837,413],{"className":838},[411,412],[390,840],{"className":841,"style":682},[443],[390,843,845],{"className":844},[686],"+",[390,847],{"className":848,"style":682},[443],[390,850,500],{"className":851},[411],[390,853,465],{"className":854},[464],[390,856,858],{"className":857},[411],"\u002F2",[390,860,862],{"className":861,"style":831},[464,830],"⌋"," the median. The minimum and maximum are easy in ",[390,865,867],{"className":866},[393],[390,868,870,889],{"className":869,"ariaHidden":398},[397],[390,871,873,877,880,883,886],{"className":872},[402],[390,874],{"className":875,"style":876},[406],"height:0.6667em;vertical-align:-0.0833em;",[390,878,413],{"className":879},[411,412],[390,881],{"className":882,"style":682},[443],[390,884,687],{"className":885},[686],[390,887],{"className":888,"style":682},[443],[390,890,892,895],{"className":891},[402],[390,893],{"className":894,"style":697},[406],[390,896,500],{"className":897},[411],"\ncomparisons. The median is the interesting case, and the algorithms below solve\nit as a byproduct of solving general selection.",[508,900,902],{"id":901},"quickselect-partition-then-recurse-on-one-side","Quickselect: partition, then recurse on one side",[381,904,905,907,908,911,912,915,916,949,950,968,969,1027,1028,1043],{},[494,906,39],{"href":40}," partitions around a pivot\nand recurses on ",[385,909,910],{},"both"," halves. But for\nselection we only care about ",[503,913,914],{},"one"," of them. After partitioning ",[390,917,919],{"className":918},[393],[390,920,922],{"className":921,"ariaHidden":398},[397],[390,923,925,928,931,934,937,941,946],{"className":924},[402],[390,926],{"className":927,"style":427},[406],[390,929,537],{"className":930},[411,412],[390,932,541],{"className":933},[435],[390,935,381],{"className":936},[411,412],[390,938,940],{"className":939},[411],"..",[390,942,945],{"className":943,"style":944},[411,412],"margin-right:0.0278em;","r",[390,947,552],{"className":948},[464]," around\na pivot that lands at index ",[390,951,953],{"className":952},[393],[390,954,956],{"className":955,"ariaHidden":398},[397],[390,957,959,963],{"className":958},[402],[390,960],{"className":961,"style":962},[406],"height:0.625em;vertical-align:-0.1944em;",[390,964,967],{"className":965,"style":966},[411,412],"margin-right:0.0359em;","q",", the pivot is the ",[390,970,972],{"className":971},[393],[390,973,975,996,1015],{"className":974,"ariaHidden":398},[397],[390,976,978,981,984,987,990,993],{"className":977},[402],[390,979],{"className":980,"style":427},[406],[390,982,436],{"className":983},[435],[390,985,967],{"className":986,"style":966},[411,412],[390,988],{"className":989,"style":682},[443],[390,991,687],{"className":992},[686],[390,994],{"className":995,"style":682},[443],[390,997,999,1003,1006,1009,1012],{"className":998},[402],[390,1000],{"className":1001,"style":1002},[406],"height:0.7778em;vertical-align:-0.1944em;",[390,1004,381],{"className":1005},[411,412],[390,1007],{"className":1008,"style":682},[443],[390,1010,845],{"className":1011},[686],[390,1013],{"className":1014,"style":682},[443],[390,1016,1018,1021,1024],{"className":1017},[402],[390,1019],{"className":1020,"style":427},[406],[390,1022,500],{"className":1023},[411],[390,1025,465],{"className":1026},[464],"-th smallest\nelement of the subarray. Compare that rank to ",[390,1029,1031],{"className":1030},[393],[390,1032,1034],{"className":1033,"ariaHidden":398},[397],[390,1035,1037,1040],{"className":1036},[402],[390,1038],{"className":1039,"style":479},[406],[390,1041,484],{"className":1042,"style":483},[411,412],":",[1045,1046,1047,1071,1090],"ul",{},[1048,1049,1050,1051,1066,1067,1070],"li",{},"if it equals ",[390,1052,1054],{"className":1053},[393],[390,1055,1057],{"className":1056,"ariaHidden":398},[397],[390,1058,1060,1063],{"className":1059},[402],[390,1061],{"className":1062,"style":479},[406],[390,1064,484],{"className":1065,"style":483},[411,412],", the pivot ",[503,1068,1069],{},"is"," the answer;",[1048,1072,1073,1074,1089],{},"if ",[390,1075,1077],{"className":1076},[393],[390,1078,1080],{"className":1079,"ariaHidden":398},[397],[390,1081,1083,1086],{"className":1082},[402],[390,1084],{"className":1085,"style":479},[406],[390,1087,484],{"className":1088,"style":483},[411,412]," is smaller, the answer lies in the left part, so recurse there;",[1048,1091,1073,1092,1107,1108,1123],{},[390,1093,1095],{"className":1094},[393],[390,1096,1098],{"className":1097,"ariaHidden":398},[397],[390,1099,1101,1104],{"className":1100},[402],[390,1102],{"className":1103,"style":479},[406],[390,1105,484],{"className":1106,"style":483},[411,412]," is larger, the answer lies in the right part; recurse there, adjusting\n",[390,1109,1111],{"className":1110},[393],[390,1112,1114],{"className":1113,"ariaHidden":398},[397],[390,1115,1117,1120],{"className":1116},[402],[390,1118],{"className":1119,"style":479},[406],[390,1121,484],{"className":1122,"style":483},[411,412]," to skip the elements we discarded.",[381,1125,1126,1127,1166,1167,720,1191],{},"Throwing away the side that cannot contain the answer is what 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partitions once around ",[390,1474,1476],{"className":1475},[393],[390,1477,1479],{"className":1478,"ariaHidden":398},[397],[390,1480,1482,1485],{"className":1481},[402],[390,1483],{"className":1484,"style":407},[406],[390,1486,1488],{"className":1487},[411,412],"x",", then recurses into only the side holding rank ",[390,1491,1493],{"className":1492},[393],[390,1494,1496],{"className":1495,"ariaHidden":398},[397],[390,1497,1499,1502],{"className":1498},[402],[390,1500],{"className":1501,"style":479},[406],[390,1503,484],{"className":1504,"style":483},[411,412]," (here ",[390,1507,1509],{"className":1508},[393],[390,1510,1512,1532],{"className":1511,"ariaHidden":398},[397],[390,1513,1515,1519,1522,1525,1529],{"className":1514},[402],[390,1516],{"className":1517,"style":1518},[406],"height:0.7335em;vertical-align:-0.0391em;",[390,1520,484],{"className":1521,"style":483},[411,412],[390,1523],{"className":1524,"style":604},[443],[390,1526,1528],{"className":1527},[608],"\u003C",[390,1530],{"className":1531,"style":604},[443],[390,1533,1535,1539],{"className":1534},[402],[390,1536],{"className":1537,"style":1538},[406],"height:0.6595em;",[390,1540,1542],{"className":1541},[411,412],"i",", the left part) and discards the other.",[1545,1546,1550],"pre",{"className":1547,"code":1548,"language":1549,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Quickselect}(A, p, r, k)$ — return the $k$-th smallest of $A[p..r]$\nnumber: 1\nif $p = r$ then\n  return $A[p]$ \u002F\u002F only element: the answer\n$q \\gets$ call $\\textsc{Randomized-Partition}(A, p, r)$\n$i \\gets q - p + 1$ \u002F\u002F pivot rank in A[p..r]\nif $k = i$ then\n  return $A[q]$ \u002F\u002F pivot is the k-th smallest\nelse if $k \u003C i$ then\n  return call $\\textsc{Quickselect}(A, p, q - 1, k)$ \u002F\u002F recurse left\nelse\n  return call $\\textsc{Quickselect}(A, q + 1, r, k - i)$ \u002F\u002F recurse right, shift k\n","algorithm",[1551,1552,1553,1559,1564,1569,1574,1579,1584,1589,1594,1599,1604,1609],"code",{"__ignoreMap":376},[390,1554,1556],{"class":1555,"line":6},"line",[390,1557,1558],{},"caption: $\\textsc{Quickselect}(A, p, r, k)$ — return the $k$-th smallest of $A[p..r]$\n",[390,1560,1561],{"class":1555,"line":18},[390,1562,1563],{},"number: 1\n",[390,1565,1566],{"class":1555,"line":24},[390,1567,1568],{},"if $p = r$ then\n",[390,1570,1571],{"class":1555,"line":73},[390,1572,1573],{},"  return $A[p]$ \u002F\u002F only element: the answer\n",[390,1575,1576],{"class":1555,"line":102},[390,1577,1578],{},"$q \\gets$ call $\\textsc{Randomized-Partition}(A, p, r)$\n",[390,1580,1581],{"class":1555,"line":108},[390,1582,1583],{},"$i \\gets q - p + 1$ \u002F\u002F pivot rank in A[p..r]\n",[390,1585,1586],{"class":1555,"line":116},[390,1587,1588],{},"if $k = i$ then\n",[390,1590,1591],{"class":1555,"line":196},[390,1592,1593],{},"  return $A[q]$ \u002F\u002F pivot is the k-th smallest\n",[390,1595,1596],{"class":1555,"line":202},[390,1597,1598],{},"else if $k \u003C i$ then\n",[390,1600,1601],{"class":1555,"line":283},[390,1602,1603],{},"  return call $\\textsc{Quickselect}(A, p, q - 1, k)$ \u002F\u002F recurse left\n",[390,1605,1606],{"class":1555,"line":333},[390,1607,1608],{},"else\n",[390,1610,1611],{"class":1555,"line":354},[390,1612,1613],{},"  return call $\\textsc{Quickselect}(A, q + 1, r, k - i)$ \u002F\u002F recurse right, shift k\n",[1615,1616,1618],"h3",{"id":1617},"expected-linear-time","Expected linear time",[381,1620,1621,1622,1646,1647,1649,1650,1666,1667,1690],{},"Partition costs ",[390,1623,1625],{"className":1624},[393],[390,1626,1628],{"className":1627,"ariaHidden":398},[397],[390,1629,1631,1634,1637,1640,1643],{"className":1630},[402],[390,1632],{"className":1633,"style":427},[406],[390,1635,431],{"className":1636},[411],[390,1638,436],{"className":1639},[435],[390,1641,413],{"className":1642},[411,412],[390,1644,465],{"className":1645},[464],". The win over quicksort is that we recurse on only\n",[385,1648,914],{}," side. With a randomized pivot, the partition splits the array at a\nuniformly random rank, so on average we discard a constant fraction each time.\nIntuitively, a random pivot lands in the middle half of the array with\nprobability ",[390,1651,1653],{"className":1652},[393],[390,1654,1656],{"className":1655,"ariaHidden":398},[397],[390,1657,1659,1662],{"className":1658},[402],[390,1660],{"className":1661,"style":427},[406],[390,1663,1665],{"className":1664},[411],"1\u002F2",", in which case the surviving side has at most ",[390,1668,1670],{"className":1669},[393],[390,1671,1673],{"className":1672,"ariaHidden":398},[397],[390,1674,1676,1679,1683,1686],{"className":1675},[402],[390,1677],{"className":1678,"style":427},[406],[390,1680,1682],{"className":1681},[411],"3",[390,1684,413],{"className":1685},[411,412],[390,1687,1689],{"className":1688},[411],"\u002F4","\nelements. The expected work satisfies, roughly,",[390,1692,1695],{"className":1693},[1694],"katex-display",[390,1696,1698],{"className":1697},[393],[390,1699,1701,1730,1763],{"className":1700,"ariaHidden":398},[397],[390,1702,1704,1707,1712,1715,1718,1721,1724,1727],{"className":1703},[402],[390,1705],{"className":1706,"style":427},[406],[390,1708,1711],{"className":1709,"style":1710},[411,412],"margin-right:0.1389em;","T",[390,1713,436],{"className":1714},[435],[390,1716,413],{"className":1717},[411,412],[390,1719,465],{"className":1720},[464],[390,1722],{"className":1723,"style":604},[443],[390,1725,609],{"className":1726},[608],[390,1728],{"className":1729,"style":604},[443],[390,1731,1733,1736,1739,1742,1745,1748,1751,1754,1757,1760],{"className":1732},[402],[390,1734],{"className":1735,"style":427},[406],[390,1737,1711],{"className":1738,"style":1710},[411,412],[390,1740,436],{"className":1741},[435],[390,1743,1682],{"className":1744},[411],[390,1746,413],{"className":1747},[411,412],[390,1749,1689],{"className":1750},[411],[390,1752,465],{"className":1753},[464],[390,1755],{"className":1756,"style":682},[443],[390,1758,845],{"className":1759},[686],[390,1761],{"className":1762,"style":682},[443],[390,1764,1766,1769,1772,1775,1778,1781],{"className":1765},[402],[390,1767],{"className":1768,"style":427},[406],[390,1770,431],{"className":1771},[411],[390,1773,436],{"className":1774},[435],[390,1776,413],{"className":1777},[411,412],[390,1779,465],{"className":1780},[464],[390,1782,1785],{"className":1783},[1784],"mpunct",",",[381,1787,1788,1789,1792],{},"a ",[503,1790,1791],{},"geometric"," (not branching) recurrence. Unrolling it,",[390,1794,1796],{"className":1795},[1694],[390,1797,1799],{"className":1798},[393],[390,1800,1802,1829,2123,2146,2241,2265],{"className":1801,"ariaHidden":398},[397],[390,1803,1805,1808,1811,1814,1817,1820,1823,1826],{"className":1804},[402],[390,1806],{"className":1807,"style":427},[406],[390,1809,1711],{"className":1810,"style":1710},[411,412],[390,1812,436],{"className":1813},[435],[390,1815,413],{"className":1816},[411,412],[390,1818,465],{"className":1819},[464],[390,1821],{"className":1822,"style":604},[443],[390,1824,609],{"className":1825},[608],[390,1827],{"className":1828,"style":604},[443],[390,1830,1832,1836,1840,1844,1847,2114,2117,2120],{"className":1831},[402],[390,1833],{"className":1834,"style":1835},[406],"height:1.8em;vertical-align:-0.65em;",[390,1837,1839],{"className":1838},[411,412],"c",[390,1841],{"className":1842,"style":1843},[443],"margin-right:-0.1667em;",[390,1845],{"className":1846,"style":444},[443],[390,1848,1850,1858,1861,1864,1867,1870,1960,1963,1966,1969,1972,2086,2089,2092,2095,2098,2101,2105,2108],{"className":1849},[826],[390,1851,1853],{"className":1852,"style":831},[435,830],[390,1854,436],{"className":1855},[1856,1857],"delimsizing","size2",[390,1859,413],{"className":1860},[411,412],[390,1862],{"className":1863,"style":682},[443],[390,1865,845],{"className":1866},[686],[390,1868],{"className":1869,"style":682},[443],[390,1871,1873,1877,1957],{"className":1872},[411],[390,1874],{"className":1875},[435,1876],"nulldelimiter",[390,1878,1881],{"className":1879},[1880],"mfrac",[390,1882,1886,1948],{"className":1883},[1884,1885],"vlist-t","vlist-t2",[390,1887,1890,1943],{"className":1888},[1889],"vlist-r",[390,1891,1895,1917,1928],{"className":1892,"style":1894},[1893],"vlist","height:0.8451em;",[390,1896,1898,1903],{"style":1897},"top:-2.655em;",[390,1899],{"className":1900,"style":1902},[1901],"pstrut","height:3em;",[390,1904,1910],{"className":1905},[1906,1907,1908,1909],"sizing","reset-size6","size3","mtight",[390,1911,1913],{"className":1912},[411,1909],[390,1914,1916],{"className":1915},[411,1909],"4",[390,1918,1920,1923],{"style":1919},"top:-3.23em;",[390,1921],{"className":1922,"style":1902},[1901],[390,1924],{"className":1925,"style":1927},[1926],"frac-line","border-bottom-width:0.04em;",[390,1929,1931,1934],{"style":1930},"top:-3.394em;",[390,1932],{"className":1933,"style":1902},[1901],[390,1935,1937],{"className":1936},[1906,1907,1908,1909],[390,1938,1940],{"className":1939},[411,1909],[390,1941,1682],{"className":1942},[411,1909],[390,1944,1947],{"className":1945},[1946],"vlist-s","​",[390,1949,1951],{"className":1950},[1889],[390,1952,1955],{"className":1953,"style":1954},[1893],"height:0.345em;",[390,1956],{},[390,1958],{"className":1959},[464,1876],[390,1961,413],{"className":1962},[411,412],[390,1964],{"className":1965,"style":682},[443],[390,1967,845],{"className":1968},[686],[390,1970],{"className":1971,"style":682},[443],[390,1973,1975,2059],{"className":1974},[826],[390,1976,1978,1985,2053],{"className":1977},[826],[390,1979,1981],{"className":1980,"style":831},[435,830],[390,1982,436],{"className":1983},[1856,1984],"size1",[390,1986,1988,1991,2050],{"className":1987},[411],[390,1989],{"className":1990},[435,1876],[390,1992,1994],{"className":1993},[1880],[390,1995,1997,2042],{"className":1996},[1884,1885],[390,1998,2000,2039],{"className":1999},[1889],[390,2001,2003,2017,2025],{"className":2002,"style":1894},[1893],[390,2004,2005,2008],{"style":1897},[390,2006],{"className":2007,"style":1902},[1901],[390,2009,2011],{"className":2010},[1906,1907,1908,1909],[390,2012,2014],{"className":2013},[411,1909],[390,2015,1916],{"className":2016},[411,1909],[390,2018,2019,2022],{"style":1919},[390,2020],{"className":2021,"style":1902},[1901],[390,2023],{"className":2024,"style":1927},[1926],[390,2026,2027,2030],{"style":1930},[390,2028],{"className":2029,"style":1902},[1901],[390,2031,2033],{"className":2032},[1906,1907,1908,1909],[390,2034,2036],{"className":2035},[411,1909],[390,2037,1682],{"className":2038},[411,1909],[390,2040,1947],{"className":2041},[1946],[390,2043,2045],{"className":2044},[1889],[390,2046,2048],{"className":2047,"style":1954},[1893],[390,2049],{},[390,2051],{"className":2052},[464,1876],[390,2054,2056],{"className":2055,"style":831},[464,830],[390,2057,465],{"className":2058},[1856,1984],[390,2060,2063],{"className":2061},[2062],"msupsub",[390,2064,2066],{"className":2065},[1884],[390,2067,2069],{"className":2068},[1889],[390,2070,2073],{"className":2071,"style":2072},[1893],"height:1.054em;",[390,2074,2076,2080],{"style":2075},"top:-3.3029em;margin-right:0.05em;",[390,2077],{"className":2078,"style":2079},[1901],"height:2.7em;",[390,2081,2083],{"className":2082},[1906,1907,1908,1909],[390,2084,1198],{"className":2085},[411,1909],[390,2087],{"className":2088,"style":444},[443],[390,2090,413],{"className":2091},[411,412],[390,2093],{"className":2094,"style":682},[443],[390,2096,845],{"className":2097},[686],[390,2099],{"className":210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single-side recurrence shrinks the subproblem by a constant factor each level, so the bars 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sum to a geometric ",[390,2755,2757],{"className":2756},[393],[390,2758,2760],{"className":2759,"ariaHidden":398},[397],[390,2761,2763,2766,2769,2772,2775],{"className":2762},[402],[390,2764],{"className":2765,"style":427},[406],[390,2767,431],{"className":2768},[411],[390,2770,436],{"className":2771},[435],[390,2773,413],{"className":2774},[411,412],[390,2776,465],{"className":2777},[464],[381,2779,2780,2781,2784],{},"The catch is the same as quicksort's: the ",[385,2782,2783],{},"worst\ncase",", with every pivot maximally unbalanced, is",[390,2786,2788],{"className":2787},[1694],[390,2789,2791],{"className":2790},[393],[390,2792,2794,2821,2845,2866,2893],{"className":2793,"ariaHidden":398},[397],[390,2795,2797,2800,2803,2806,2809,2812,2815,2818],{"className":2796},[402],[390,2798],{"className":2799,"style":427},[406],[390,2801,1711],{"className":2802,"style":1710},[411,412],[390,2804,436],{"className":2805},[435],[390,2807,413],{"className":2808},[411,412],[390,2810,465],{"className":2811},[464],[390,2813],{"className":2814,"style":604},[443],[390,2816,745],{"className":2817},[608],[390,2819],{"className":2820,"style":604},[443],[390,2822,2824,2827,2830,2833,2836,2839,2842],{"className":2823},[402],[390,2825],{"className":2826,"style":427},[406],[390,2828,1711],{"className":2829,"style":1710},[411,412],[390,2831,436],{"className":2832},[435],[390,2834,413],{"className":2835},[411,412],[390,2837],{"className":2838,"style":682},[443],[390,2840,687],{"className":2841},[686],[390,2843],{"className":2844,"style":682},[443],[390,2846,2848,2851,2854,2857,2860,2863],{"className":2847},[402],[390,2849],{"className":2850,"style":427},[406],[390,2852,500],{"className":2853},[411],[390,2855,465],{"className":2856},[464],[390,2858],{"className":2859,"style":682},[443],[390,2861,845],{"className":2862},[686],[390,2864],{"className":2865,"style":682},[443],[390,2867,2869,2872,2875,2878,2881,2884,2887,2890],{"className":2868},[402],[390,2870],{"className":2871,"style":427},[406],[390,2873,431],{"className":2874},[411],[390,2876,436],{"className":2877},[435],[390,2879,413],{"className":2880},[411,412],[390,2882,465],{"className":2883},[464],[390,2885],{"className":2886,"style":604},[443],[390,2888,745],{"className":2889},[608],[390,2891],{"className":2892,"style":604},[443],[390,2894,2896,2900,2903,2906,2937,2940],{"className":2895},[402],[390,2897],{"className":2898,"style":2899},[406],"height:1.1141em;vertical-align:-0.25em;",[390,2901,431],{"className":2902},[411],[390,2904,436],{"className":2905},[435],[390,2907,2909,2912],{"className":2908},[411],[390,2910,413],{"className":2911},[411,412],[390,2913,2915],{"className":2914},[2062],[390,2916,2918],{"className":2917},[1884],[390,2919,2921],{"className":2920},[1889],[390,2922,2925],{"className":2923,"style":2924},[1893],"height:0.8641em;",[390,2926,2928,2931],{"style":2927},"top:-3.113em;margin-right:0.05em;",[390,2929],{"className":2930,"style":2079},[1901],[390,2932,2934],{"className":2933},[1906,1907,1908,1909],[390,2935,1198],{"className":2936},[411,1909],[390,2938,465],{"className":2939},[464],[390,2941,720],{"className":2942},[411],[381,2944,2945,2946,2949],{},"Randomization makes that astronomically unlikely, but it is still possible. Can\nwe ",[503,2947,2948],{},"guarantee"," linear time?",[508,2951,2953],{"id":2952},"median-of-medians-a-guaranteed-good-pivot","Median of medians: a guaranteed-good pivot",[381,2955,2956,2957,2981,2982,2985,2986,2993,2994,720],{},"The deterministic algorithm of Blum, Floyd, Pratt, Rivest, and Tarjan (1973)\nachieves worst-case ",[390,2958,2960],{"className":2959},[393],[390,2961,2963],{"className":2962,"ariaHidden":398},[397],[390,2964,2966,2969,2972,2975,2978],{"className":2965},[402],[390,2967],{"className":2968,"style":427},[406],[390,2970,2340],{"className":2971,"style":944},[411,412],[390,2973,436],{"className":2974},[435],[390,2976,413],{"className":2977},[411,412],[390,2979,465],{"className":2980},[464]," by spending a little effort to choose a pivot that\nis ",[503,2983,2984],{},"provably"," not too extreme.",[491,2987,2988],{},[494,2989,1682],{"href":2990,"ariaDescribedBy":2991,"dataFootnoteRef":376,"id":2992},"#user-content-fn-clrs-mom",[498],"user-content-fnref-clrs-mom"," The trick is to pick the pivot as a ",[385,2995,2996],{},"median of\nmedians",[1545,2998,3000],{"className":1547,"code":2999,"language":1549,"meta":376,"style":376},"caption: $\\textsc{Select}(A, k)$ — deterministic $k$-th smallest, worst-case $O(n)$\nnumber: 2\nif $A$ has at most $5$ elements then\n  return the $k$-th smallest of $A$ by direct sorting\ndivide $A$ into $\\ceil{n\u002F5}$ groups of $5$ elements (last group may be smaller)\nforeach group do\n  find that group's median by sorting its $\\le 5$ elements\nlet $M$ be the array of the $\\ceil{n\u002F5}$ group medians\n$x \\gets$ call $\\textsc{Select}(M, \\ceil{|M|\u002F2})$ \u002F\u002F median of medians\n$q \\gets$ partition $A$ around the pivot $x$, returning its rank $i$\nif $k = i$ then\n  return $x$\nelse if $k \u003C i$ then\n  return call $\\textsc{Select}(A[\\,\\text{left part}\\,], k)$\nelse\n  return call $\\textsc{Select}(A[\\,\\text{right part}\\,], k - i)$\n",[1551,3001,3002,3007,3012,3017,3022,3027,3032,3037,3042,3047,3052,3056,3061,3066,3072,3077],{"__ignoreMap":376},[390,3003,3004],{"class":1555,"line":6},[390,3005,3006],{},"caption: $\\textsc{Select}(A, k)$ — deterministic $k$-th smallest, worst-case $O(n)$\n",[390,3008,3009],{"class":1555,"line":18},[390,3010,3011],{},"number: 2\n",[390,3013,3014],{"class":1555,"line":24},[390,3015,3016],{},"if $A$ has at most $5$ elements then\n",[390,3018,3019],{"class":1555,"line":73},[390,3020,3021],{},"  return the $k$-th smallest of $A$ by direct sorting\n",[390,3023,3024],{"class":1555,"line":102},[390,3025,3026],{},"divide $A$ into $\\ceil{n\u002F5}$ groups of $5$ elements (last group may be smaller)\n",[390,3028,3029],{"class":1555,"line":108},[390,3030,3031],{},"foreach group do\n",[390,3033,3034],{"class":1555,"line":116},[390,3035,3036],{},"  find that group's median by sorting its $\\le 5$ elements\n",[390,3038,3039],{"class":1555,"line":196},[390,3040,3041],{},"let $M$ be the array of the $\\ceil{n\u002F5}$ group medians\n",[390,3043,3044],{"class":1555,"line":202},[390,3045,3046],{},"$x \\gets$ call $\\textsc{Select}(M, \\ceil{|M|\u002F2})$ \u002F\u002F median of medians\n",[390,3048,3049],{"class":1555,"line":283},[390,3050,3051],{},"$q \\gets$ partition $A$ around the pivot $x$, returning its rank $i$\n",[390,3053,3054],{"class":1555,"line":333},[390,3055,1588],{},[390,3057,3058],{"class":1555,"line":354},[390,3059,3060],{},"  return $x$\n",[390,3062,3064],{"class":1555,"line":3063},13,[390,3065,1598],{},[390,3067,3069],{"class":1555,"line":3068},14,[390,3070,3071],{},"  return call $\\textsc{Select}(A[\\,\\text{left part}\\,], k)$\n",[390,3073,3075],{"class":1555,"line":3074},15,[390,3076,1608],{},[390,3078,3080],{"class":1555,"line":3079},16,[390,3081,3082],{},"  return call $\\textsc{Select}(A[\\,\\text{right part}\\,], k - i)$\n",[1615,3084,3086],{"id":3085},"why-groups-of-five-give-a-good-pivot","Why groups of five give a good pivot",[381,3088,3089,3090,3120,3121,3124,3125,3168],{},"Here is the core of it. Picture the ",[390,3091,3093],{"className":3092},[393],[390,3094,3096],{"className":3095,"ariaHidden":398},[397],[390,3097,3099,3102],{"className":3098},[402],[390,3100],{"className":3101,"style":427},[406],[390,3103,3105,3109,3112,3116],{"className":3104},[826],[390,3106,3108],{"className":3107,"style":831},[435,830],"⌈",[390,3110,413],{"className":3111},[411,412],[390,3113,3115],{"className":3114},[411],"\u002F5",[390,3117,3119],{"className":3118,"style":831},[464,830],"⌉"," groups as ",[385,3122,3123],{},"columns",", each\nsorted top (large) to bottom (small), and now imagine the columns reordered\nleft to right by their medians. The pivot ",[390,3126,3128],{"className":3127},[393],[390,3129,3131,3149],{"className":3130,"ariaHidden":398},[397],[390,3132,3134,3137,3140,3143,3146],{"className":3133},[402],[390,3135],{"className":3136,"style":407},[406],[390,3138,1488],{"className":3139},[411,412],[390,3141],{"className":3142,"style":604},[443],[390,3144,745],{"className":3145},[608],[390,3147],{"className":3148,"style":604},[443],[390,3150,3152,3155],{"className":3151},[402],[390,3153],{"className":3154,"style":658},[406],[390,3156,3160],{"className":3157},[3158,3159],"enclosing","textsc",[390,3161,3164],{"className":3162},[411,3163],"text",[390,3165,3167],{"className":3166},[411],"MoM"," is the median of\nthat middle row, so it sits dead-center in the grid below. (We assume distinct\nelements for simplicity.)",[381,3170,3171,3172,3199,3200,3203,3204,3207],{},"The pivot is built in stages: chop the array into groups of five, sort each\ngroup to expose its median, collect those ",[390,3173,3175],{"className":3174},[393],[390,3176,3178],{"className":3177,"ariaHidden":398},[397],[390,3179,3181,3184],{"className":3180},[402],[390,3182],{"className":3183,"style":427},[406],[390,3185,3187,3190,3193,3196],{"className":3186},[826],[390,3188,3108],{"className":3189,"style":831},[435,830],[390,3191,413],{"className":3192},[411,412],[390,3194,3115],{"className":3195},[411],[390,3197,3119],{"className":3198,"style":831},[464,830]," medians, and ",[385,3201,3202],{},"recurse","\nto find ",[503,3205,3206],{},"their"," median — the median of medians.",[1200,3209,3211,3398],{"className":3210},[1203,1204],[1206,3212,3216],{"xmlns":1208,"width":3213,"height":3214,"viewBox":3215},"412.404","133.844","-75 -75 309.303 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9.141-29.437",[1243],[1468,3752,3754,3755,3770,3771,720],{"className":3753},[1471],"Grid of groups of five with the median of medians ",[390,3756,3758],{"className":3757},[393],[390,3759,3761],{"className":3760,"ariaHidden":398},[397],[390,3762,3764,3767],{"className":3763},[402],[390,3765],{"className":3766,"style":407},[406],[390,3768,1488],{"className":3769},[411,412],", showing the regions guaranteed at least or at most ",[390,3772,3774],{"className":3773},[393],[390,3775,3777],{"className":3776,"ariaHidden":398},[397],[390,3778,3780,3783],{"className":3779},[402],[390,3781],{"className":3782,"style":407},[406],[390,3784,1488],{"className":3785},[411,412],[381,3787,3788,3789,3818,3819,3834,3835,3862,3863,3866,3867,3894,3895,3898,3899,3917],{},"Consider the columns whose median is ",[390,3790,3792],{"className":3791},[393],[390,3793,3795,3809],{"className":3794,"ariaHidden":398},[397],[390,3796,3798,3802,3806],{"className":3797},[402],[390,3799],{"className":3800,"style":3801},[406],"height:0.7719em;vertical-align:-0.136em;",[390,3803,3805],{"className":3804},[608],"≥",[390,3807],{"className":3808,"style":604},[443],[390,3810,3812,3815],{"className":3811},[402],[390,3813],{"className":3814,"style":407},[406],[390,3816,1488],{"className":3817},[411,412],": that is, ",[390,3820,3822],{"className":3821},[393],[390,3823,3825],{"className":3824,"ariaHidden":398},[397],[390,3826,3828,3831],{"className":3827},[402],[390,3829],{"className":3830,"style":407},[406],[390,3832,1488],{"className":3833},[411,412],"'s own column and\nthe roughly half of the ",[390,3836,3838],{"className":3837},[393],[390,3839,3841],{"className":3840,"ariaHidden":398},[397],[390,3842,3844,3847],{"className":3843},[402],[390,3845],{"className":3846,"style":427},[406],[390,3848,3850,3853,3856,3859],{"className":3849},[826],[390,3851,3108],{"className":3852,"style":831},[435,830],[390,3854,413],{"className":3855},[411,412],[390,3857,3115],{"className":3858},[411],[390,3860,3119],{"className":3861,"style":831},[464,830]," columns to its right. In each such column,\nthe median ",[503,3864,3865],{},"and"," the two elements above it (the two larger ones) are all\n",[390,3868,3870],{"className":3869},[393],[390,3871,3873,3885],{"className":3872,"ariaHidden":398},[397],[390,3874,3876,3879,3882],{"className":3875},[402],[390,3877],{"className":3878,"style":3801},[406],[390,3880,3805],{"className":3881},[608],[390,3883],{"className":3884,"style":604},[443],[390,3886,3888,3891],{"className":3887},[402],[390,3889],{"className":3890,"style":407},[406],[390,3892,1488],{"className":3893},[411,412],". That is ",[385,3896,3897],{},"3 elements per column",", across about half of the ",[390,3900,3902],{"className":3901},[393],[390,3903,3905],{"className":3904,"ariaHidden":398},[397],[390,3906,3908,3911,3914],{"className":3907},[402],[390,3909],{"className":3910,"style":427},[406],[390,3912,413],{"className":3913},[411,412],[390,3915,3115],{"className":3916},[411]," columns:",[390,3919,3921],{"className":3920},[1694],[390,3922,3924],{"className":3923},[393],[390,3925,3927,3991,4009,4182],{"className":3926,"ariaHidden":398},[397],[390,3928,3930,3933,3937,3940,3976,3979,3982,3985,3988],{"className":3929},[402],[390,3931],{"className":3932,"style":427},[406],[390,3934,3936],{"className":3935},[411],"#",[390,3938],{"className":3939,"style":444},[443],[390,3941,3943,3947,3950,3957,3960,3963,3966,3969,3972],{"className":3942},[826],[390,3944,3946],{"className":3945,"style":831},[435,830],"{",[390,3948],{"className":3949,"style":444},[443],[390,3951,3953],{"className":3952},[411,3163],[390,3954,3956],{"className":3955},[411],"elements",[390,3958],{"className":3959,"style":604},[443],[390,3961,3805],{"className":3962},[608],[390,3964],{"className":3965,"style":604},[443],[390,3967,1488],{"className":3968},[411,412],[390,3970],{"className":3971,"style":444},[443],[390,3973,3975],{"className":3974,"style":831},[464,830],"}",[390,3977],{"className":3978,"style":604},[443],[390,3980],{"className":3981,"style":604},[443],[390,3983,3805],{"className":3984},[608],[390,3986],{"className":3987,"style":604},[443],[390,3989],{"className":3990,"style":604},[443],[390,3992,3994,3997,4000,4003,4006],{"className":3993},[402],[390,3995],{"className":3996,"style":697},[406],[390,3998,1682],{"className":3999},[411],[390,4001],{"className":4002,"style":682},[443],[390,4004,2142],{"className":4005},[686],[390,4007],{"className":4008,"style":682},[443],[390,4010,4012,4016,4166,4169,4172,4176,4179],{"className":4011},[402],[390,4013],{"className":4014,"style":4015},[406],"height:2.4em;vertical-align:-0.95em;",[390,4017,4019,4025,4088,4091,4094,4097,4160],{"className":4018},[826],[390,4020,4022],{"className":4021,"style":831},[435,830],[390,4023,436],{"className":4024},[1856,1908],[390,4026,4028,4031,4085],{"className":4027},[411],[390,4029],{"className":4030},[435,1876],[390,4032,4034],{"className":4033},[1880],[390,4035,4037,4076],{"className":4036},[1884,1885],[390,4038,4040,4073],{"className":4039},[1889],[390,4041,4043,4054,4062],{"className":4042,"style":2171},[1893],[390,4044,4045,4048],{"style":2174},[390,4046],{"className":4047,"style":1902},[1901],[390,4049,4051],{"className":4050},[411],[390,4052,1198],{"className":4053},[411],[390,4055,4056,4059],{"style":1919},[390,4057],{"className":4058,"style":1902},[1901],[390,4060],{"className":4061,"style":1927},[1926],[390,4063,4064,4067],{"style":2207},[390,4065],{"className":4066,"style":1902},[1901],[390,4068,4070],{"className":4069},[411],[390,4071,500],{"className":4072},[411],[390,4074,1947],{"className":4075},[1946],[390,4077,4079],{"className":4078},[1889],[390,4080,4083],{"className":4081,"style":4082},[1893],"height:0.686em;",[390,4084],{},[390,4086],{"className":4087},[464,1876],[390,4089],{"className":4090,"style":682},[443],[390,4092,2142],{"className":4093},[686],[390,4095],{"className":4096,"style":682},[443],[390,4098,4100,4103,4157],{"className":4099},[411],[390,4101],{"className":4102},[435,1876],[390,4104,4106],{"className":4105},[1880],[390,4107,4109,4149],{"className":4108},[1884,1885],[390,4110,4112,4146],{"className":4111},[1889],[390,4113,4116,4127,4135],{"className":4114,"style":4115},[1893],"height:1.1076em;",[390,4117,4118,4121],{"style":2174},[390,4119],{"className":4120,"style":1902},[1901],[390,4122,4124],{"className":4123},[411],[390,4125,3417],{"className":4126},[411],[390,4128,4129,4132],{"style":1919},[390,4130],{"className":4131,"style":1902},[1901],[390,4133],{"className":4134,"style":1927},[1926],[390,4136,4137,4140],{"style":2207},[390,4138],{"className":4139,"style":1902},[1901],[390,4141,4143],{"className":4142},[411],[390,4144,413],{"className":4145},[411,412],[390,4147,1947],{"className":4148},[1946],[390,4150,4152],{"className":4151},[1889],[390,4153,4155],{"className":4154,"style":4082},[1893],[390,4156],{},[390,4158],{"className":4159},[464,1876],[390,4161,4163],{"className":4162,"style":831},[464,830],[390,4164,465],{"className":4165},[1856,1908],[390,4167],{"className":4168,"style":604},[443],[390,4170],{"className":4171,"style":604},[443],[390,4173,4175],{"className":4174},[608],"≈",[390,4177],{"className":4178,"style":604},[443],[390,4180],{"className":4181,"style":604},[443],[390,4183,4185,4189,4255],{"className":4184},[402],[390,4186],{"className":4187,"style":4188},[406],"height:2.0074em;vertical-align:-0.686em;",[390,4190,4192,4195,4252],{"className":4191},[411],[390,4193],{"className":4194},[435,1876],[390,4196,4198],{"className":4197},[1880],[390,4199,4201,4244],{"className":4200},[1884,1885],[390,4202,4204,4241],{"className":4203},[1889],[390,4205,4207,4219,4227],{"className":4206,"style":2171},[1893],[390,4208,4209,4212],{"style":2174},[390,4210],{"className":4211,"style":1902},[1901],[390,4213,4215],{"className":4214},[411],[390,4216,4218],{"className":4217},[411],"10",[390,4220,4221,4224],{"style":1919},[390,4222],{"className":4223,"style":1902},[1901],[390,4225],{"className":4226,"style":1927},[1926],[390,4228,4229,4232],{"style":2207},[390,4230],{"className":4231,"style":1902},[1901],[390,4233,4235,4238],{"className":4234},[411],[390,4236,1682],{"className":4237},[411],[390,4239,413],{"className":4240},[411,412],[390,4242,1947],{"className":4243},[1946],[390,4245,4247],{"className":4246},[1889],[390,4248,4250],{"className":4249,"style":4082},[1893],[390,4251],{},[390,4253],{"className":4254},[464,1876],[390,4256,720],{"className":4257},[411],[381,4259,4260,4261,4296,4297,4324,4325,4328,4391,4392,4420,4421,4454,4455,4482,4483,4454,4516,641,4544,4581,4582,4597,4598,2350],{},"So at least ",[390,4262,4264],{"className":4263},[393],[390,4265,4267,4280],{"className":4266,"ariaHidden":398},[397],[390,4268,4270,4274,4277],{"className":4269},[402],[390,4271],{"className":4272,"style":4273},[406],"height:0.4831em;",[390,4275,4175],{"className":4276},[608],[390,4278],{"className":4279,"style":604},[443],[390,4281,4283,4286,4289,4292],{"className":4282},[402],[390,4284],{"className":4285,"style":427},[406],[390,4287,1682],{"className":4288},[411],[390,4290,413],{"className":4291},[411,412],[390,4293,4295],{"className":4294},[411],"\u002F10"," elements are ",[390,4298,4300],{"className":4299},[393],[390,4301,4303,4315],{"className":4302,"ariaHidden":398},[397],[390,4304,4306,4309,4312],{"className":4305},[402],[390,4307],{"className":4308,"style":3801},[406],[390,4310,3805],{"className":4311},[608],[390,4313],{"className":4314,"style":604},[443],[390,4316,4318,4321],{"className":4317},[402],[390,4319],{"className":4320,"style":407},[406],[390,4322,1488],{"className":4323},[411,412],", which forces ",[385,4326,4327],{},"at most",[390,4329,4331],{"className":4330},[393],[390,4332,4334,4352,4376],{"className":4333,"ariaHidden":398},[397],[390,4335,4337,4340,4343,4346,4349],{"className":4336},[402],[390,4338],{"className":4339,"style":876},[406],[390,4341,413],{"className":4342},[411,412],[390,4344],{"className":4345,"style":682},[443],[390,4347,687],{"className":4348},[686],[390,4350],{"className":4351,"style":682},[443],[390,4353,4355,4358,4361,4364,4367,4370,4373],{"className":4354},[402],[390,4356],{"className":4357,"style":427},[406],[390,4359,1682],{"className":4360},[411],[390,4362,413],{"className":4363},[411,412],[390,4365,4295],{"className":4366},[411],[390,4368],{"className":4369,"style":604},[443],[390,4371,745],{"className":4372},[608],[390,4374],{"className":4375,"style":604},[443],[390,4377,4379,4382,4385,4388],{"className":4378},[402],[390,4380],{"className":4381,"style":427},[406],[390,4383,1366],{"className":4384},[411],[390,4386,413],{"className":4387},[411,412],[390,4389,4295],{"className":4390},[411]," to be ",[390,4393,4395],{"className":4394},[393],[390,4396,4398,4411],{"className":4397,"ariaHidden":398},[397],[390,4399,4401,4405,4408],{"className":4400},[402],[390,4402],{"className":4403,"style":4404},[406],"height:0.5782em;vertical-align:-0.0391em;",[390,4406,1528],{"className":4407},[608],[390,4409],{"className":4410,"style":604},[443],[390,4412,4414,4417],{"className":4413},[402],[390,4415],{"className":4416,"style":407},[406],[390,4418,1488],{"className":4419},[411,412],". By the symmetric argument (the dashed block)\nat least ",[390,4422,4424],{"className":4423},[393],[390,4425,4427,4439],{"className":4426,"ariaHidden":398},[397],[390,4428,4430,4433,4436],{"className":4429},[402],[390,4431],{"className":4432,"style":4273},[406],[390,4434,4175],{"className":4435},[608],[390,4437],{"className":4438,"style":604},[443],[390,4440,4442,4445,4448,4451],{"className":4441},[402],[390,4443],{"className":4444,"style":427},[406],[390,4446,1682],{"className":4447},[411],[390,4449,413],{"className":4450},[411,412],[390,4452,4295],{"className":4453},[411]," are ",[390,4456,4458],{"className":4457},[393],[390,4459,4461,4473],{"className":4460,"ariaHidden":398},[397],[390,4462,4464,4467,4470],{"className":4463},[402],[390,4465],{"className":4466,"style":3801},[406],[390,4468,609],{"className":4469},[608],[390,4471],{"className":4472,"style":604},[443],[390,4474,4476,4479],{"className":4475},[402],[390,4477],{"className":4478,"style":407},[406],[390,4480,1488],{"className":4481},[411,412],", so at most ",[390,4484,4486],{"className":4485},[393],[390,4487,4489,4501],{"className":4488,"ariaHidden":398},[397],[390,4490,4492,4495,4498],{"className":4491},[402],[390,4493],{"className":4494,"style":4273},[406],[390,4496,4175],{"className":4497},[608],[390,4499],{"className":4500,"style":604},[443],[390,4502,4504,4507,4510,4513],{"className":4503},[402],[390,4505],{"className":4506,"style":427},[406],[390,4508,1366],{"className":4509},[411],[390,4511,413],{"className":4512},[411,412],[390,4514,4295],{"className":4515},[411],[390,4517,4519],{"className":4518},[393],[390,4520,4522,4535],{"className":4521,"ariaHidden":398},[397],[390,4523,4525,4528,4532],{"className":4524},[402],[390,4526],{"className":4527,"style":4404},[406],[390,4529,4531],{"className":4530},[608],">",[390,4533],{"className":4534,"style":604},[443],[390,4536,4538,4541],{"className":4537},[402],[390,4539],{"className":4540,"style":407},[406],[390,4542,1488],{"className":4543},[411,412],[385,4545,4546,4547,4580],{},"Whichever side we recurse into has at most ",[390,4548,4550],{"className":4549},[393],[390,4551,4553,4565],{"className":4552,"ariaHidden":398},[397],[390,4554,4556,4559,4562],{"className":4555},[402],[390,4557],{"className":4558,"style":4273},[406],[390,4560,4175],{"className":4561},[608],[390,4563],{"className":4564,"style":604},[443],[390,4566,4568,4571,4574,4577],{"className":4567},[402],[390,4569],{"className":4570,"style":427},[406],[390,4572,1366],{"className":4573},[411],[390,4575,413],{"className":4576},[411,412],[390,4578,4295],{"className":4579},[411]," elements",", so the\npivot is provably never too lopsided. (Five is the smallest odd group size that\nmakes the recurrence below close; groups of ",[390,4583,4585],{"className":4584},[393],[390,4586,4588],{"className":4587,"ariaHidden":398},[397],[390,4589,4591,4594],{"className":4590},[402],[390,4592],{"className":4593,"style":697},[406],[390,4595,1682],{"className":4596},[411]," fail because their fractions\nsum to exactly ",[390,4599,4601],{"className":4600},[393],[390,4602,4604],{"className":4603,"ariaHidden":398},[397],[390,4605,4607,4610],{"className":4606},[402],[390,4608],{"className":4609,"style":697},[406],[390,4611,500],{"className":4612},[411],[1615,4614,4616],{"id":4615},"solving-the-recurrence","Solving the recurrence",[381,4618,4619,4620,4644,4645,4669,4670,4694,4695,4698],{},"Tallying the work: splitting into groups and finding their medians is ",[390,4621,4623],{"className":4622},[393],[390,4624,4626],{"className":4625,"ariaHidden":398},[397],[390,4627,4629,4632,4635,4638,4641],{"className":4628},[402],[390,4630],{"className":4631,"style":427},[406],[390,4633,2340],{"className":4634,"style":944},[411,412],[390,4636,436],{"className":4637},[435],[390,4639,413],{"className":4640},[411,412],[390,4642,465],{"className":4643},[464],"\n(each group is sorted in ",[390,4646,4648],{"className":4647},[393],[390,4649,4651],{"className":4650,"ariaHidden":398},[397],[390,4652,4654,4657,4660,4663,4666],{"className":4653},[402],[390,4655],{"className":4656,"style":427},[406],[390,4658,2340],{"className":4659,"style":944},[411,412],[390,4661,436],{"className":4662},[435],[390,4664,500],{"className":4665},[411],[390,4667,465],{"className":4668},[464],"). Partitioning is ",[390,4671,4673],{"className":4672},[393],[390,4674,4676],{"className":4675,"ariaHidden":398},[397],[390,4677,4679,4682,4685,4688,4691],{"className":4678},[402],[390,4680],{"className":4681,"style":427},[406],[390,4683,2340],{"className":4684,"style":944},[411,412],[390,4686,436],{"className":4687},[435],[390,4689,413],{"className":4690},[411,412],[390,4692,465],{"className":4693},[464],". There are ",[385,4696,4697],{},"two","\nrecursive calls:",[1045,4700,4701,4751],{},[1048,4702,4703,4704,4731,4732,4750],{},"finding the median of the ",[390,4705,4707],{"className":4706},[393],[390,4708,4710],{"className":4709,"ariaHidden":398},[397],[390,4711,4713,4716],{"className":4712},[402],[390,4714],{"className":4715,"style":427},[406],[390,4717,4719,4722,4725,4728],{"className":4718},[826],[390,4720,3108],{"className":4721,"style":831},[435,830],[390,4723,413],{"className":4724},[411,412],[390,4726,3115],{"className":4727},[411],[390,4729,3119],{"className":4730,"style":831},[464,830]," medians, a subproblem of size ",[390,4733,4735],{"className":4734},[393],[390,4736,4738],{"className":4737,"ariaHidden":398},[397],[390,4739,4741,4744,4747],{"className":4740},[402],[390,4742],{"className":4743,"style":427},[406],[390,4745,413],{"className":4746},[411,412],[390,4748,3115],{"className":4749},[411],";",[1048,4752,4753,4754,720],{},"recursing into the surviving side, a subproblem of size at most ",[390,4755,4757],{"className":4756},[393],[390,4758,4760],{"className":4759,"ariaHidden":398},[397],[390,4761,4763,4766,4769,4772],{"className":4762},[402],[390,4764],{"className":4765,"style":427},[406],[390,4767,1366],{"className":4768},[411],[390,4770,413],{"className":4771},[411,412],[390,4773,4295],{"className":4774},[411],[390,4776,4778],{"className":4777},[1694],[390,4779,4781],{"className":4780},[393],[390,4782,4784,4811,4913,5017],{"className":4783,"ariaHidden":398},[397],[390,4785,4787,4790,4793,4796,4799,4802,4805,4808],{"className":4786},[402],[390,4788],{"className":4789,"style":427},[406],[390,4791,1711],{"className":4792,"style":1710},[411,412],[390,4794,436],{"className":4795},[435],[390,4797,413],{"className":4798},[411,412],[390,4800,465],{"className":4801},[464],[390,4803],{"className":4804,"style":604},[443],[390,4806,609],{"className":4807},[608],[390,4809],{"className":4810,"style":604},[443],[390,4812,4814,4818,4821,4824,4827,4904,4907,4910],{"className":4813},[402],[390,4815],{"className":4816,"style":4817},[406],"height:1.836em;vertical-align:-0.686em;",[390,4819,1711],{"className":4820,"style":1710},[411,412],[390,4822],{"className":4823,"style":1843},[443],[390,4825],{"className":4826,"style":444},[443],[390,4828,4830,4836,4898],{"className":4829},[826],[390,4831,4833],{"className":4832,"style":831},[435,830],[390,4834,436],{"className":4835},[1856,1857],[390,4837,4839,4842,4895],{"className":4838},[411],[390,4840],{"className":4841},[435,1876],[390,4843,4845],{"className":4844},[1880],[390,4846,4848,4887],{"className":4847},[1884,1885],[390,4849,4851,4884],{"className":4850},[1889],[390,4852,4854,4865,4873],{"className":4853,"style":4115},[1893],[390,4855,4856,4859],{"style":2174},[390,4857],{"className":4858,"style":1902},[1901],[390,4860,4862],{"className":4861},[411],[390,4863,3417],{"className":4864},[411],[390,4866,4867,4870],{"style":1919},[390,4868],{"className":4869,"style":1902},[1901],[390,4871],{"className":4872,"style":1927},[1926],[390,4874,4875,4878],{"style":2207},[390,4876],{"className":4877,"style":1902},[1901],[390,4879,4881],{"className":4880},[411],[390,4882,413],{"className":4883},[411,412],[390,4885,1947],{"className":4886},[1946],[390,4888,4890],{"className":4889},[1889],[390,4891,4893],{"className":4892,"style":4082},[1893],[390,4894],{},[390,4896],{"className":4897},[464,1876],[390,4899,4901],{"className":4900,"style":831},[464,830],[390,4902,465],{"className":4903},[1856,1857],[390,4905],{"className":4906,"style":682},[443],[390,4908,845],{"className":4909},[686],[390,4911],{"className":4912,"style":682},[443],[390,4914,4916,4919,4922,4925,4928,5008,5011,5014],{"className":4915},[402],[390,4917],{"className":4918,"style":4015},[406],[390,4920,1711],{"className":4921,"style":1710},[411,412],[390,4923],{"className":4924,"style":1843},[443],[390,4926],{"className":4927,"style":444},[443],[390,4929,4931,4937,5002],{"className":4930},[826],[390,4932,4934],{"className":4933,"style":831},[435,830],[390,4935,436],{"className":4936},[1856,1908],[390,4938,4940,4943,4999],{"className":4939},[411],[390,4941],{"className":4942},[435,1876],[390,4944,4946],{"className":4945},[1880],[390,4947,4949,4991],{"className":4948},[1884,1885],[390,4950,4952,4988],{"className":4951},[1889],[390,4953,4955,4966,4974],{"className":4954,"style":2171},[1893],[390,4956,4957,4960],{"style":2174},[390,4958],{"className":4959,"style":1902},[1901],[390,4961,4963],{"className":4962},[411],[390,4964,4218],{"className":4965},[411],[390,4967,4968,4971],{"style":1919},[390,4969],{"className":4970,"style":1902},[1901],[390,4972],{"className":4973,"style":1927},[1926],[390,4975,4976,4979],{"style":2207},[390,4977],{"className":4978,"style":1902},[1901],[390,4980,4982,4985],{"className":4981},[411],[390,4983,1366],{"className":4984},[411],[390,4986,413],{"className":4987},[411,412],[390,4989,1947],{"className":4990},[1946],[390,4992,4994],{"className":4993},[1889],[390,4995,4997],{"className":4996,"style":4082},[1893],[390,4998],{},[390,5000],{"className":5001},[464,1876],[390,5003,5005],{"className":5004,"style":831},[464,830],[390,5006,465],{"className":5007},[1856,1908],[390,5009],{"className":5010,"style":682},[443],[390,5012,845],{"className":5013},[686],[390,5015],{"className":5016,"style":682},[443],[390,5018,5020,5023,5026,5029,5032,5035],{"className":5019},[402],[390,5021],{"className":5022,"style":427},[406],[390,5024,2340],{"className":5025,"style":944},[411,412],[390,5027,436],{"className":5028},[435],[390,5030,413],{"className":5031},[411,412],[390,5033,465],{"className":5034},[464],[390,5036,720],{"className":5037},[411],[381,5039,5040,5041,5044,5045,5309,5310,5313,5314,5317],{},"The two fractions are what make this work. ",[385,5042,5043],{},"Observe"," that\n",[390,5046,5048],{"className":5047},[393],[390,5049,5051,5134,5217,5300],{"className":5050,"ariaHidden":398},[397],[390,5052,5054,5057,5125,5128,5131],{"className":5053},[402],[390,5055],{"className":5056,"style":2551},[406],[390,5058,5060,5063,5122],{"className":5059},[411],[390,5061],{"className":5062},[435,1876],[390,5064,5066],{"className":5065},[1880],[390,5067,5069,5114],{"className":5068},[1884,1885],[390,5070,5072,5111],{"className":5071},[1889],[390,5073,5075,5089,5097],{"className":5074,"style":1894},[1893],[390,5076,5077,5080],{"style":1897},[390,5078],{"className":5079,"style":1902},[1901],[390,5081,5083],{"className":5082},[1906,1907,1908,1909],[390,5084,5086],{"className":5085},[411,1909],[390,5087,3417],{"className":5088},[411,1909],[390,5090,5091,5094],{"style":1919},[390,5092],{"className":5093,"style":1902},[1901],[390,5095],{"className":5096,"style":1927},[1926],[390,5098,5099,5102],{"style":1930},[390,5100],{"className":5101,"style":1902},[1901],[390,5103,5105],{"className":5104},[1906,1907,1908,1909],[390,5106,5108],{"className":5107},[411,1909],[390,5109,500],{"className":5110},[411,1909],[390,5112,1947],{"className":5113},[1946],[390,5115,5117],{"className":5116},[1889],[390,5118,5120],{"className":5119,"style":1954},[1893],[390,5121],{},[390,5123],{"className":5124},[464,1876],[390,5126],{"className":5127,"style":682},[443],[390,5129,845],{"className":5130},[686],[390,5132],{"className":5133,"style":682},[443],[390,5135,5137,5140,5208,5211,5214],{"className":5136},[402],[390,5138],{"className":5139,"style":2551},[406],[390,5141,5143,5146,5205],{"className":5142},[411],[390,5144],{"className":5145},[435,1876],[390,5147,5149],{"className":5148},[1880],[390,5150,5152,5197],{"className":5151},[1884,1885],[390,5153,5155,5194],{"className":5154},[1889],[390,5156,5158,5172,5180],{"className":5157,"style":1894},[1893],[390,5159,5160,5163],{"style":1897},[390,5161],{"className":5162,"style":1902},[1901],[390,5164,5166],{"className":5165},[1906,1907,1908,1909],[390,5167,5169],{"className":5168},[411,1909],[390,5170,4218],{"className":5171},[411,1909],[390,5173,5174,5177],{"style":1919},[390,5175],{"className":5176,"style":1902},[1901],[390,5178],{"className":5179,"style":1927},[1926],[390,5181,5182,5185],{"style":1930},[390,5183],{"className":5184,"style":1902},[1901],[390,5186,5188],{"className":5187},[1906,1907,1908,1909],[390,5189,5191],{"className":5190},[411,1909],[390,5192,1366],{"className":5193},[411,1909],[390,5195,1947],{"className":5196},[1946],[390,5198,5200],{"className":5199},[1889],[390,5201,5203],{"className":5202,"style":1954},[1893],[390,5204],{},[390,5206],{"className":5207},[464,1876],[390,5209],{"className":5210,"style":604},[443],[390,5212,745],{"className":5213},[608],[390,5215],{"className":5216,"style":604},[443],[390,5218,5220,5223,5291,5294,5297],{"className":5219},[402],[390,5221],{"className":5222,"style":2551},[406],[390,5224,5226,5229,5288],{"className":5225},[411],[390,5227],{"className":5228},[435,1876],[390,5230,5232],{"className":5231},[1880],[390,5233,5235,5280],{"className":5234},[1884,1885],[390,5236,5238,5277],{"className":5237},[1889],[390,5239,5241,5255,5263],{"className":5240,"style":1894},[1893],[390,5242,5243,5246],{"style":1897},[390,5244],{"className":5245,"style":1902},[1901],[390,5247,5249],{"className":5248},[1906,1907,1908,1909],[390,5250,5252],{"className":5251},[411,1909],[390,5253,4218],{"className":5254},[411,1909],[390,5256,5257,5260],{"style":1919},[390,5258],{"className":5259,"style":1902},[1901],[390,5261],{"className":5262,"style":1927},[1926],[390,5264,5265,5268],{"style":1930},[390,5266],{"className":5267,"style":1902},[1901],[390,5269,5271],{"className":5270},[1906,1907,1908,1909],[390,5272,5274],{"className":5273},[411,1909],[390,5275,1235],{"className":5276},[411,1909],[390,5278,1947],{"className":5279},[1946],[390,5281,5283],{"className":5282},[1889],[390,5284,5286],{"className":5285,"style":1954},[1893],[390,5287],{},[390,5289],{"className":5290},[464,1876],[390,5292],{"className":5293,"style":604},[443],[390,5295,1528],{"className":5296},[608],[390,5298],{"className":5299,"style":604},[443],[390,5301,5303,5306],{"className":5302},[402],[390,5304],{"className":5305,"style":697},[406],[390,5307,500],{"className":5308},[411],": the two subproblems together\nare ",[503,5311,5312],{},"strictly smaller"," than the input. That is ",[385,5315,5316],{},"shrinkage"," — the total work\ncontracts by a constant factor at every level.",[1200,5319,5321,5453],{"className":5320},[1203,1204],[1206,5322,5326],{"xmlns":1208,"width":5323,"height":5324,"viewBox":5325},"396.792","64.979","-75 -75 297.594 48.734",[1213,5327,5328,5331,5334,5352,5355,5379,5383,5390,5426,5429,5432,5447,5450],{"stroke":1215,"style":1216},[1218,5329],{"fill":1233,"d":5330},"M-65.403-52.153V-72.07h284.527v19.917ZM219.124-72.07",[1218,5332],{"fill":1220,"d":5333},"M-65.403-52.153V-72.07h56.905v19.917ZM-8.498-72.07",[1213,5335,5336,5343,5346],{"stroke":1233},[1213,5337,5339],{"transform":5338},"translate(25.05 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",[390,5458,5460],{"className":5459},[393],[390,5461,5463,5546,5629,5712],{"className":5462,"ariaHidden":398},[397],[390,5464,5466,5469,5537,5540,5543],{"className":5465},[402],[390,5467],{"className":5468,"style":2551},[406],[390,5470,5472,5475,5534],{"className":5471},[411],[390,5473],{"className":5474},[435,1876],[390,5476,5478],{"className":5477},[1880],[390,5479,5481,5526],{"className":5480},[1884,1885],[390,5482,5484,5523],{"className":5483},[1889],[390,5485,5487,5501,5509],{"className":5486,"style":1894},[1893],[390,5488,5489,5492],{"style":1897},[390,5490],{"className":5491,"style":1902},[1901],[390,5493,5495],{"className":5494},[1906,1907,1908,1909],[390,5496,5498],{"className":5497},[411,1909],[390,5499,3417],{"className":5500},[411,1909],[390,5502,5503,5506],{"style":1919},[390,5504],{"className":5505,"style":1902},[1901],[390,5507],{"className":5508,"style":1927},[1926],[390,5510,5511,5514],{"style":1930},[390,5512],{"className":5513,"style":1902},[1901],[390,5515,5517],{"className":5516},[1906,1907,1908,1909],[390,5518,5520],{"className":5519},[411,1909],[390,5521,500],{"className":5522},[411,1909],[390,5524,1947],{"className":5525},[1946],[390,5527,5529],{"className":5528},[1889],[390,5530,5532],{"className":5531,"style":1954},[1893],[390,5533],{},[390,5535],{"className":5536},[464,1876],[390,5538],{"className":5539,"style":682},[443],[390,5541,845],{"className":5542},[686],[390,5544],{"className":5545,"style":682},[443],[390,5547,5549,5552,5620,5623,5626],{"className":5548},[402],[390,5550],{"className":5551,"style":2551},[406],[390,5553,5555,5558,5617],{"className":5554},[411],[390,5556],{"className":5557},[435,1876],[390,5559,5561],{"className":5560},[1880],[390,5562,5564,5609],{"className":5563},[1884,1885],[390,5565,5567,5606],{"className":5566},[1889],[390,5568,5570,5584,5592],{"className":5569,"style":1894},[1893],[390,5571,5572,5575],{"style":1897},[390,5573],{"className":5574,"style":1902},[1901],[390,5576,5578],{"className":5577},[1906,1907,1908,1909],[390,5579,5581],{"className":5580},[411,1909],[390,5582,4218],{"className":5583},[411,1909],[390,5585,5586,5589],{"style":1919},[390,5587],{"className":5588,"style":1902},[1901],[390,5590],{"className":5591,"style":1927},[1926],[390,5593,5594,5597],{"style":1930},[390,5595],{"className":5596,"style":1902},[1901],[390,5598,5600],{"className":5599},[1906,1907,1908,1909],[390,5601,5603],{"className":5602},[411,1909],[390,5604,1366],{"className":5605},[411,1909],[390,5607,1947],{"className":5608},[1946],[390,5610,5612],{"className":5611},[1889],[390,5613,5615],{"className":5614,"style":1954},[1893],[390,5616],{},[390,5618],{"className":5619},[464,1876],[390,5621],{"className":5622,"style":604},[443],[390,5624,745],{"className":5625},[608],[390,5627],{"className":5628,"style":604},[443],[390,5630,5632,5635,5703,5706,5709],{"className":5631},[402],[390,5633],{"className":5634,"style":2551},[406],[390,5636,5638,5641,5700],{"className":5637},[411],[390,5639],{"className":5640},[435,1876],[390,5642,5644],{"className":5643},[1880],[390,5645,5647,5692],{"className":5646},[1884,1885],[390,5648,5650,5689],{"className":5649},[1889],[390,5651,5653,5667,5675],{"className":5652,"style":1894},[1893],[390,5654,5655,5658],{"style":1897},[390,5656],{"className":5657,"style":1902},[1901],[390,5659,5661],{"className":5660},[1906,1907,1908,1909],[390,5662,5664],{"className":5663},[411,1909],[390,5665,4218],{"className":5666},[411,1909],[390,5668,5669,5672],{"style":1919},[390,5670],{"className":5671,"style":1902},[1901],[390,5673],{"className":5674,"style":1927},[1926],[390,5676,5677,5680],{"style":1930},[390,5678],{"className":5679,"style":1902},[1901],[390,5681,5683],{"className":5682},[1906,1907,1908,1909],[390,5684,5686],{"className":5685},[411,1909],[390,5687,1235],{"className":5688},[411,1909],[390,5690,1947],{"className":5691},[1946],[390,5693,5695],{"className":5694},[1889],[390,5696,5698],{"className":5697,"style":1954},[1893],[390,5699],{},[390,5701],{"className":5702},[464,1876],[390,5704],{"className":5705,"style":604},[443],[390,5707,1528],{"className":5708},[608],[390,5710],{"className":5711,"style":604},[443],[390,5713,5715,5718],{"className":5714},[402],[390,5716],{"className":5717,"style":697},[406],[390,5719,500],{"className":5720},[411]," of the input; the leftover ",[390,5723,5725],{"className":5724},[393],[390,5726,5728],{"className":5727,"ariaHidden":398},[397],[390,5729,5731,5734],{"className":5730},[402],[390,5732],{"className":5733,"style":2551},[406],[390,5735,5737,5740,5799],{"className":5736},[411],[390,5738],{"className":5739},[435,1876],[390,5741,5743],{"className":5742},[1880],[390,5744,5746,5791],{"className":5745},[1884,1885],[390,5747,5749,5788],{"className":5748},[1889],[390,5750,5752,5766,5774],{"className":5751,"style":1894},[1893],[390,5753,5754,5757],{"style":1897},[390,5755],{"className":5756,"style":1902},[1901],[390,5758,5760],{"className":5759},[1906,1907,1908,1909],[390,5761,5763],{"className":5762},[411,1909],[390,5764,4218],{"className":5765},[411,1909],[390,5767,5768,5771],{"style":1919},[390,5769],{"className":5770,"style":1902},[1901],[390,5772],{"className":5773,"style":1927},[1926],[390,5775,5776,5779],{"style":1930},[390,5777],{"className":5778,"style":1902},[1901],[390,5780,5782],{"className":5781},[1906,1907,1908,1909],[390,5783,5785],{"className":5784},[411,1909],[390,5786,500],{"className":5787},[411,1909],[390,5789,1947],{"className":5790},[1946],[390,5792,5794],{"className":5793},[1889],[390,5795,5797],{"className":5796,"style":1954},[1893],[390,5798],{},[390,5800],{"className":5801},[464,1876]," gap is the shrinkage that forces ",[390,5804,5806],{"className":5805},[393],[390,5807,5809],{"className":5808,"ariaHidden":398},[397],[390,5810,5812,5815,5818,5821,5824],{"className":5811},[402],[390,5813],{"className":5814,"style":427},[406],[390,5816,2340],{"className":5817,"style":944},[411,412],[390,5819,436],{"className":5820},[435],[390,5822,413],{"className":5823},[411,412],[390,5825,465],{"className":5826},[464],[381,5828,5829,5832,5833,5908,5909,5912,5913,5964],{},[385,5830,5831],{},"For inspiration",", first consider the single-call cousin\n",[390,5834,5836],{"className":5835},[393],[390,5837,5839,5868,5898],{"className":5838,"ariaHidden":398},[397],[390,5840,5842,5845,5850,5853,5856,5859,5862,5865],{"className":5841},[402],[390,5843],{"className":5844,"style":427},[406],[390,5846,5849],{"className":5847,"style":5848},[411,412],"margin-right:0.1076em;","f",[390,5851,436],{"className":5852},[435],[390,5854,413],{"className":5855},[411,412],[390,5857,465],{"className":5858},[464],[390,5860],{"className":5861,"style":604},[443],[390,5863,609],{"className":5864},[608],[390,5866],{"className":5867,"style":604},[443],[390,5869,5871,5874,5877,5880,5883,5886,5889,5892,5895],{"className":5870},[402],[390,5872],{"className":5873,"style":427},[406],[390,5875,5849],{"className":5876,"style":5848},[411,412],[390,5878,436],{"className":5879},[435],[390,5881,413],{"className":5882},[411,412],[390,5884,858],{"className":5885},[411],[390,5887,465],{"className":5888},[464],[390,5890],{"className":5891,"style":682},[443],[390,5893,845],{"className":5894},[686],[390,5896],{"className":5897,"style":682},[443],[390,5899,5901,5904],{"className":5900},[402],[390,5902],{"className":5903,"style":479},[406],[390,5905,5907],{"className":5906},[411,412],"bn",". The\n",[494,5910,5911],{"href":23},"master theorem"," gives ",[390,5914,5916],{"className":5915},[393],[390,5917,5919,5946],{"className":5918,"ariaHidden":398},[397],[390,5920,5922,5925,5928,5931,5934,5937,5940,5943],{"className":5921},[402],[390,5923],{"className":5924,"style":427},[406],[390,5926,5849],{"className":5927,"style":5848},[411,412],[390,5929,436],{"className":5930},[435],[390,5932,413],{"className":5933},[411,412],[390,5935,465],{"className":5936},[464],[390,5938],{"className":5939,"style":604},[443],[390,5941,745],{"className":5942},[608],[390,5944],{"className":5945,"style":604},[443],[390,5947,5949,5952,5955,5958,5961],{"className":5948},[402],[390,5950],{"className":5951,"style":427},[406],[390,5953,2340],{"className":5954,"style":944},[411,412],[390,5956,436],{"className":5957},[435],[390,5959,413],{"className":5960},[411,412],[390,5962,465],{"className":5963},[464],";\nunrolling shows why directly,",[390,5966,5968],{"className":5967},[1694],[390,5969,5971],{"className":5970},[393],[390,5972,5974,6001,6110,6128,6235,6323,6341,6368,6572,6599],{"className":5973,"ariaHidden":398},[397],[390,5975,5977,5980,5983,5986,5989,5992,5995,5998],{"className":5976},[402],[390,5978],{"className":5979,"style":427},[406],[390,5981,5849],{"className":5982,"style":5848},[411,412],[390,5984,436],{"className":5985},[435],[390,5987,413],{"className":5988},[411,412],[390,5990,465],{"className":5991},[464],[390,5993],{"className":5994,"style":604},[443],[390,5996,609],{"className":5997},[608],[390,5999],{"className":6000,"style":604},[443],[390,6002,6004,6008,6011,6014,6017,6101,6104,6107],{"className":6003},[402],[390,6005],{"className":6006,"style":6007},[406],"height:1.2em;vertical-align:-0.35em;",[390,6009,5849],{"className":6010,"style":5848},[411,412],[390,6012],{"className":6013,"style":1843},[443],[390,6015],{"className":6016,"style":444},[443],[390,6018,6020,6026,6095],{"className":6019},[826],[390,6021,6023],{"className":6022,"style":831},[435,830],[390,6024,436],{"className":6025},[1856,1984],[390,6027,6029,6032,6092],{"className":6028},[411],[390,6030],{"className":6031},[435,1876],[390,6033,6035],{"className":6034},[1880],[390,6036,6038,6084],{"className":6037},[1884,1885],[390,6039,6041,6081],{"className":6040},[1889],[390,6042,6045,6059,6067],{"className":6043,"style":6044},[1893],"height:0.6954em;",[390,6046,6047,6050],{"style":1897},[390,6048],{"className":6049,"style":1902},[1901],[390,6051,6053],{"className":6052},[1906,1907,1908,1909],[390,6054,6056],{"className":6055},[411,1909],[390,6057,1198],{"className":6058},[411,1909],[390,6060,6061,6064],{"style":1919},[390,6062],{"className":6063,"style":1902},[1901],[390,6065],{"className":6066,"style":1927},[1926],[390,6068,6069,6072],{"style":1930},[390,6070],{"className":6071,"style":1902},[1901],[390,6073,6075],{"className":6074},[1906,1907,1908,1909],[390,6076,6078],{"className":6077},[411,1909],[390,6079,413],{"className":6080},[411,412,1909],[390,6082,1947],{"className":6083},[1946],[390,6085,6087],{"className":6086},[1889],[390,6088,6090],{"className":6089,"style":1954},[1893],[390,6091],{},[390,6093],{"className":6094},[464,1876],[390,6096,6098],{"className":6097,"style":831},[464,830],[390,6099,465],{"className":6100},[1856,1984],[390,6102],{"className":6103,"style":682},[443],[390,6105,845],{"className":6106},[686],[390,6108],{"className":6109,"style":682},[443],[390,6111,6113,6116,6119,6122,6125],{"className":6112},[402],[390,6114],{"className":6115,"style":619},[406],[390,6117,5907],{"className":6118},[411,412],[390,6120],{"className":6121,"style":604},[443],[390,6123,609],{"className":6124},[608],[390,6126],{"className":6127,"style":604},[443],[390,6129,6131,6134,6137,6140,6143,6226,6229,6232],{"className":6130},[402],[390,6132],{"className":6133,"style":6007},[406],[390,6135,5849],{"className":6136,"style":5848},[411,412],[390,6138],{"className":6139,"style":1843},[443],[390,6141],{"className":6142,"style":444},[443],[390,6144,6146,6152,6220],{"className":6145},[826],[390,6147,6149],{"className":6148,"style":831},[435,830],[390,6150,436],{"className":6151},[1856,1984],[390,6153,6155,6158,6217],{"className":6154},[411],[390,6156],{"className":6157},[435,1876],[390,6159,6161],{"className":6160},[1880],[390,6162,6164,6209],{"className":6163},[1884,1885],[390,6165,6167,6206],{"className":6166},[1889],[390,6168,6170,6184,6192],{"className":6169,"style":6044},[1893],[390,6171,6172,6175],{"style":1897},[390,6173],{"className":6174,"style":1902},[1901],[390,6176,6178],{"className":6177},[1906,1907,1908,1909],[390,6179,6181],{"className":6180},[411,1909],[390,6182,1916],{"className":6183},[411,1909],[390,6185,6186,6189],{"style":1919},[390,6187],{"className":6188,"style":1902},[1901],[390,6190],{"className":6191,"style":1927},[1926],[390,6193,6194,6197],{"style":1930},[390,6195],{"className":6196,"style":1902},[1901],[390,6198,6200],{"className":6199},[1906,1907,1908,1909],[390,6201,6203],{"className":6202},[411,1909],[390,6204,413],{"className":6205},[411,412,1909],[390,6207,1947],{"className":6208},[1946],[390,6210,6212],{"className":6211},[1889],[390,6213,6215],{"className":6214,"style":1954},[1893],[390,6216],{},[390,6218],{"className":6219},[464,1876],[390,6221,6223],{"className":6222,"style":831},[464,830],[390,6224,465],{"className":6225},[1856,1984],[390,6227],{"className":6228,"style":682},[443],[390,6230,845],{"className":6231},[686],[390,6233],{"className":6234,"style":682},[443],[390,6236,6238,6242,6246,6314,6317,6320],{"className":6237},[402],[390,6239],{"className":6240,"style":6241},[406],"height:1.0404em;vertical-align:-0.345em;",[390,6243,6245],{"className":6244},[411,412],"b",[390,6247,6249,6252,6311],{"className":6248},[411],[390,6250],{"className":6251},[435,1876],[390,6253,6255],{"className":6254},[1880],[390,6256,6258,6303],{"className":6257},[1884,1885],[390,6259,6261,6300],{"className":6260},[1889],[390,6262,6264,6278,6286],{"className":6263,"style":6044},[1893],[390,6265,6266,6269],{"style":1897},[390,6267],{"className":6268,"style":1902},[1901],[390,6270,6272],{"className":6271},[1906,1907,1908,1909],[390,6273,6275],{"className":6274},[411,1909],[390,6276,1198],{"className":6277},[411,1909],[390,6279,6280,6283],{"style":1919},[390,6281],{"className":6282,"style":1902},[1901],[390,6284],{"className":6285,"style":1927},[1926],[390,6287,6288,6291],{"style":1930},[390,6289],{"className":6290,"style":1902},[1901],[390,6292,6294],{"className":6293},[1906,1907,1908,1909],[390,6295,6297],{"className":6296},[411,1909],[390,6298,413],{"className":6299},[411,412,1909],[390,6301,1947],{"className":6302},[1946],[390,6304,6306],{"className":6305},[1889],[390,6307,6309],{"className":6308,"style":1954},[1893],[390,6310],{},[390,6312],{"className":6313},[464,1876],[390,6315],{"className":6316,"style":682},[443],[390,6318,845],{"className":6319},[686],[390,6321],{"className":6322,"style":682},[443],[390,6324,6326,6329,6332,6335,6338],{"className":6325},[402],[390,6327],{"className":6328,"style":619},[406],[390,6330,5907],{"className":6331},[411,412],[390,6333],{"className":6334,"style":604},[443],[390,6336,609],{"className":6337},[608],[390,6339],{"className":6340,"style":604},[443],[390,6342,6344,6347,6350,6353,6356,6359,6362,6365],{"className":6343},[402],[390,6345],{"className":6346,"style":427},[406],[390,6348,5849],{"className":6349,"style":5848},[411,412],[390,6351,436],{"className":6352},[435],[390,6354,500],{"className":6355},[411],[390,6357,465],{"className":6358},[464],[390,6360],{"className":6361,"style":682},[443],[390,6363,845],{"className":6364},[686],[390,6366],{"className":6367,"style":682},[443],[390,6369,6371,6375,6563,6566,6569],{"className":6370},[402],[390,6372],{"className":6373,"style":6374},[406],"height:1.2301em;vertical-align:-0.35em;",[390,6376,6378,6384,6387,6390,6393,6396,6465,6468,6471,6474,6542,6545,6548,6551,6554,6557],{"className":6377},[826],[390,6379,6381],{"className":6380,"style":831},[435,830],[390,6382,436],{"className":6383},[1856,1984],[390,6385,5907],{"className":6386},[411,412],[390,6388],{"className":6389,"style":682},[443],[390,6391,845],{"className":6392},[686],[390,6394],{"className":6395,"style":682},[443],[390,6397,6399,6402,6462],{"className":6398},[411],[390,6400],{"className":6401},[435,1876],[390,6403,6405],{"className":6404},[1880],[390,6406,6408,6454],{"className":6407},[1884,1885],[390,6409,6411,6451],{"className":6410},[1889],[390,6412,6415,6429,6437],{"className":6413,"style":6414},[1893],"height:0.8801em;",[390,6416,6417,6420],{"style":1897},[390,6418],{"className":6419,"style":1902},[1901],[390,6421,6423],{"className":6422},[1906,1907,1908,1909],[390,6424,6426],{"className":6425},[411,1909],[390,6427,1198],{"className":6428},[411,1909],[390,6430,6431,6434],{"style":1919},[390,6432],{"className":6433,"style":1902},[1901],[390,6435],{"className":6436,"style":1927},[1926],[390,6438,6439,6442],{"style":1930},[390,6440],{"className":6441,"style":1902},[1901],[390,6443,6445],{"className":6444},[1906,1907,1908,1909],[390,6446,6448],{"className":6447},[411,1909],[390,6449,5907],{"className":6450},[411,412,1909],[390,6452,1947],{"className":6453},[1946],[390,6455,6457],{"className":6456},[1889],[390,6458,6460],{"className":6459,"style":1954},[1893],[390,6461],{},[390,6463],{"className":6464},[464,1876],[390,6466],{"className":6467,"style":682},[443],[390,6469,845],{"className":6470},[686],[390,6472],{"className":6473,"style":682},[443],[390,6475,6477,6480,6539],{"className":6476},[411],[390,6478],{"className":6479},[435,1876],[390,6481,6483],{"className":6482},[1880],[390,6484,6486,6531],{"className":6485},[1884,1885],[390,6487,6489,6528],{"className":6488},[1889],[390,6490,6492,6506,6514],{"className":6491,"style":6414},[1893],[390,6493,6494,6497],{"style":1897},[390,6495],{"className":6496,"style":1902},[1901],[390,6498,6500],{"className":6499},[1906,1907,1908,1909],[390,6501,6503],{"className":6502},[411,1909],[390,6504,1916],{"className":6505},[411,1909],[390,6507,6508,6511],{"style":1919},[390,6509],{"className":6510,"style":1902},[1901],[390,6512],{"className":6513,"style":1927},[1926],[390,6515,6516,6519],{"style":1930},[390,6517],{"className":6518,"style":1902},[1901],[390,6520,6522],{"className":6521},[1906,1907,1908,1909],[390,6523,6525],{"className":6524},[411,1909],[390,6526,5907],{"className":6527},[411,412,1909],[390,6529,1947],{"className":6530},[1946],[390,6532,6534],{"className":6533},[1889],[390,6535,6537],{"className":6536,"style":1954},[1893],[390,6538],{},[390,6540],{"className":6541},[464,1876],[390,6543],{"className":6544,"style":682},[443],[390,6546,845],{"className":6547},[686],[390,6549],{"className":6550,"style":682},[443],[390,6552,2104],{"className":6553},[826],[390,6555],{"className":6556,"style":444},[443],[390,6558,6560],{"className":6559,"style":831},[464,830],[390,6561,465],{"className":6562},[1856,1984],[390,6564],{"className":6565,"style":604},[443],[390,6567,609],{"className":6568},[608],[390,6570],{"className":6571,"style":604},[443],[390,6573,6575,6578,6581,6584,6587,6590,6593,6596],{"className":6574},[402],[390,6576],{"className":6577,"style":427},[406],[390,6579,5849],{"className":6580,"style":5848},[411,412],[390,6582,436],{"className":6583},[435],[390,6585,500],{"className":6586},[411],[390,6588,465],{"className":6589},[464],[390,6591],{"className":6592,"style":682},[443],[390,6594,845],{"className":6595},[686],[390,6597],{"className":6598,"style":682},[443],[390,6600,6602,6605,6608,6611],{"className":6601},[402],[390,6603],{"className":6604,"style":479},[406],[390,6606,1198],{"className":6607},[411],[390,6609,5907],{"className":6610},[411,412],[390,6612,720],{"className":6613},[411],[381,6615,6616,6617,6641,6642,6644],{},"The constant-factor-shrinkage plus linear cleanup work sums to ",[390,6618,6620],{"className":6619},[393],[390,6621,6623],{"className":6622,"ariaHidden":398},[397],[390,6624,6626,6629,6632,6635,6638],{"className":6625},[402],[390,6627],{"className":6628,"style":427},[406],[390,6630,2340],{"className":6631,"style":944},[411,412],[390,6633,436],{"className":6634},[435],[390,6636,413],{"className":6637},[411,412],[390,6639,465],{"className":6640},[464],". The\nmedian-of-medians recurrence has ",[503,6643,4697],{}," calls instead of one, but the same\nshrinkage idea carries it through. State it as a clean general\nlemma:",[513,6646,6648],{"type":6647},"lemma",[381,6649,6650,6653,6654,6696,6697,6772,6773,6696,6874,6944,6945,6998,6999,7051,7052,720],{},[385,6651,6652],{},"Lemma (Shrinkage)."," Suppose ",[390,6655,6657],{"className":6656},[393],[390,6658,6660,6687],{"className":6659,"ariaHidden":398},[397],[390,6661,6663,6666,6669,6672,6675,6678,6681,6684],{"className":6662},[402],[390,6664],{"className":6665,"style":427},[406],[390,6667,1711],{"className":6668,"style":1710},[411,412],[390,6670,436],{"className":6671},[435],[390,6673,413],{"className":6674},[411,412],[390,6676,465],{"className":6677},[464],[390,6679],{"className":6680,"style":604},[443],[390,6682,609],{"className":6683},[608],[390,6685],{"className":6686,"style":604},[443],[390,6688,6690,6693],{"className":6689},[402],[390,6691],{"className":6692,"style":407},[406],[390,6694,1839],{"className":6695},[411,412]," for ",[390,6698,6700],{"className":6699},[393],[390,6701,6703,6721],{"className":6702,"ariaHidden":398},[397],[390,6704,6706,6709,6712,6715,6718],{"className":6705},[402],[390,6707],{"className":6708,"style":3801},[406],[390,6710,413],{"className":6711},[411,412],[390,6713],{"className":6714,"style":604},[443],[390,6716,609],{"className":6717},[608],[390,6719],{"className":6720,"style":604},[443],[390,6722,6724,6728],{"className":6723},[402],[390,6725],{"className":6726,"style":6727},[406],"height:0.5806em;vertical-align:-0.15em;",[390,6729,6731,6734],{"className":6730},[411],[390,6732,413],{"className":6733},[411,412],[390,6735,6737],{"className":6736},[2062],[390,6738,6740,6763],{"className":6739},[1884,1885],[390,6741,6743,6760],{"className":6742},[1889],[390,6744,6747],{"className":6745,"style":6746},[1893],"height:0.3011em;",[390,6748,6750,6753],{"style":6749},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[390,6751],{"className":6752,"style":2079},[1901],[390,6754,6756],{"className":6755},[1906,1907,1908,1909],[390,6757,6759],{"className":6758},[411,1909],"0",[390,6761,1947],{"className":6762},[1946],[390,6764,6766],{"className":6765},[1889],[390,6767,6770],{"className":6768,"style":6769},[1893],"height:0.15em;",[390,6771],{},", and\n",[390,6774,6776],{"className":6775},[393],[390,6777,6779,6806,6834,6865],{"className":6778,"ariaHidden":398},[397],[390,6780,6782,6785,6788,6791,6794,6797,6800,6803],{"className":6781},[402],[390,6783],{"className":6784,"style":427},[406],[390,6786,1711],{"className":6787,"style":1710},[411,412],[390,6789,436],{"className":6790},[435],[390,6792,413],{"className":6793},[411,412],[390,6795,465],{"className":6796},[464],[390,6798],{"className":6799,"style":604},[443],[390,6801,609],{"className":6802},[608],[390,6804],{"className":6805,"style":604},[443],[390,6807,6809,6812,6815,6818,6822,6825,6828,6831],{"className":6808},[402],[390,6810],{"className":6811,"style":427},[406],[390,6813,1711],{"className":6814,"style":1710},[411,412],[390,6816,436],{"className":6817},[435],[390,6819,6821],{"className":6820},[411,412],"λn",[390,6823,465],{"className":6824},[464],[390,6826],{"className":6827,"style":682},[443],[390,6829,845],{"className":6830},[686],[390,6832],{"className":6833,"style":682},[443],[390,6835,6837,6840,6843,6846,6850,6853,6856,6859,6862],{"className":6836},[402],[390,6838],{"className":6839,"style":427},[406],[390,6841,1711],{"className":6842,"style":1710},[411,412],[390,6844,436],{"className":6845},[435],[390,6847,6849],{"className":6848},[411,412],"μ",[390,6851,413],{"className":6852},[411,412],[390,6854,465],{"className":6855},[464],[390,6857],{"className":6858,"style":682},[443],[390,6860,845],{"className":6861},[686],[390,6863],{"className":6864,"style":682},[443],[390,6866,6868,6871],{"className":6867},[402],[390,6869],{"className":6870,"style":479},[406],[390,6872,5907],{"className":6873},[411,412],[390,6875,6877],{"className":6876},[393],[390,6878,6880,6898],{"className":6879,"ariaHidden":398},[397],[390,6881,6883,6886,6889,6892,6895],{"className":6882},[402],[390,6884],{"className":6885,"style":4404},[406],[390,6887,413],{"className":6888},[411,412],[390,6890],{"className":6891,"style":604},[443],[390,6893,4531],{"className":6894},[608],[390,6896],{"className":6897,"style":604},[443],[390,6899,6901,6904],{"className":6900},[402],[390,6902],{"className":6903,"style":6727},[406],[390,6905,6907,6910],{"className":6906},[411],[390,6908,413],{"className":6909},[411,412],[390,6911,6913],{"className":6912},[2062],[390,6914,6916,6936],{"className":6915},[1884,1885],[390,6917,6919,6933],{"className":6918},[1889],[390,6920,6922],{"className":6921,"style":6746},[1893],[390,6923,6924,6927],{"style":6749},[390,6925],{"className":6926,"style":2079},[1901],[390,6928,6930],{"className":6929},[1906,1907,1908,1909],[390,6931,6759],{"className":6932},[411,1909],[390,6934,1947],{"className":6935},[1946],[390,6937,6939],{"className":6938},[1889],[390,6940,6942],{"className":6941,"style":6769},[1893],[390,6943],{},", where\n",[390,6946,6948],{"className":6947},[393],[390,6949,6951,6989],{"className":6950,"ariaHidden":398},[397],[390,6952,6954,6958,6962,6965,6968,6971,6974,6977,6980,6983,6986],{"className":6953},[402],[390,6955],{"className":6956,"style":6957},[406],"height:0.8889em;vertical-align:-0.1944em;",[390,6959,6961],{"className":6960},[411,412],"λ",[390,6963,1785],{"className":6964},[1784],[390,6966],{"className":6967,"style":444},[443],[390,6969,6849],{"className":6970},[411,412],[390,6972,1785],{"className":6973},[1784],[390,6975],{"className":6976,"style":444},[443],[390,6978,6245],{"className":6979},[411,412],[390,6981],{"className":6982,"style":604},[443],[390,6984,4531],{"className":6985},[608],[390,6987],{"className":6988,"style":604},[443],[390,6990,6992,6995],{"className":6991},[402],[390,6993],{"className":6994,"style":697},[406],[390,6996,6759],{"className":6997},[411]," are constants with ",[390,7000,7002],{"className":7001},[393],[390,7003,7005,7023,7042],{"className":7004,"ariaHidden":398},[397],[390,7006,7008,7011,7014,7017,7020],{"className":7007},[402],[390,7009],{"className":7010,"style":675},[406],[390,7012,6961],{"className":7013},[411,412],[390,7015],{"className":7016,"style":682},[443],[390,7018,845],{"className":7019},[686],[390,7021],{"className":7022,"style":682},[443],[390,7024,7026,7030,7033,7036,7039],{"className":7025},[402],[390,7027],{"className":7028,"style":7029},[406],"height:0.7335em;vertical-align:-0.1944em;",[390,7031,6849],{"className":7032},[411,412],[390,7034],{"className":7035,"style":604},[443],[390,7037,1528],{"className":7038},[608],[390,7040],{"className":7041,"style":604},[443],[390,7043,7045,7048],{"className":7044},[402],[390,7046],{"className":7047,"style":697},[406],[390,7049,500],{"className":7050},[411],". Then\n",[390,7053,7055],{"className":7054},[393],[390,7056,7058,7085],{"className":7057,"ariaHidden":398},[397],[390,7059,7061,7064,7067,7070,7073,7076,7079,7082],{"className":7060},[402],[390,7062],{"className":7063,"style":427},[406],[390,7065,1711],{"className":7066,"style":1710},[411,412],[390,7068,436],{"className":7069},[435],[390,7071,413],{"className":7072},[411,412],[390,7074,465],{"className":7075},[464],[390,7077],{"className":7078,"style":604},[443],[390,7080,745],{"className":7081},[608],[390,7083],{"className":7084,"style":604},[443],[390,7086,7088,7091,7094,7097,7100],{"className":7087},[402],[390,7089],{"className":7090,"style":427},[406],[390,7092,2340],{"className":7093,"style":944},[411,412],[390,7095,436],{"className":7096},[435],[390,7098,413],{"className":7099},[411,412],[390,7101,465],{"className":7102},[464],[513,7104,7106,7206,7377,7781],{"type":7105},"proof",[381,7107,7108,7111,7112,7127,7128,7171,7172,7205],{},[385,7109,7110],{},"Proof."," By induction on ",[390,7113,7115],{"className":7114},[393],[390,7116,7118],{"className":7117,"ariaHidden":398},[397],[390,7119,7121,7124],{"className":7120},[402],[390,7122],{"className":7123,"style":407},[406],[390,7125,413],{"className":7126},[411,412]," we show ",[390,7129,7131],{"className":7130},[393],[390,7132,7134,7161],{"className":7133,"ariaHidden":398},[397],[390,7135,7137,7140,7143,7146,7149,7152,7155,7158],{"className":7136},[402],[390,7138],{"className":7139,"style":427},[406],[390,7141,1711],{"className":7142,"style":1710},[411,412],[390,7144,436],{"className":7145},[435],[390,7147,413],{"className":7148},[411,412],[390,7150,465],{"className":7151},[464],[390,7153],{"className":7154,"style":604},[443],[390,7156,609],{"className":7157},[608],[390,7159],{"className":7160,"style":604},[443],[390,7162,7164,7167],{"className":7163},[402],[390,7165],{"className":7166,"style":407},[406],[390,7168,7170],{"className":7169},[411,412],"an"," for all ",[390,7173,7175],{"className":7174},[393],[390,7176,7178,7196],{"className":7177,"ariaHidden":398},[397],[390,7179,7181,7184,7187,7190,7193],{"className":7180},[402],[390,7182],{"className":7183,"style":3801},[406],[390,7185,413],{"className":7186},[411,412],[390,7188],{"className":7189,"style":604},[443],[390,7191,3805],{"className":7192},[608],[390,7194],{"className":7195,"style":604},[443],[390,7197,7199,7202],{"className":7198},[402],[390,7200],{"className":7201,"style":697},[406],[390,7203,500],{"className":7204},[411],", where",[390,7207,7209],{"className":7208},[1694],[390,7210,7212],{"className":7211},[393],[390,7213,7215,7233],{"className":7214,"ariaHidden":398},[397],[390,7216,7218,7221,7224,7227,7230],{"className":7217},[402],[390,7219],{"className":7220,"style":407},[406],[390,7222,494],{"className":7223},[411,412],[390,7225],{"className":7226,"style":604},[443],[390,7228,745],{"className":7229},[608],[390,7231],{"className":7232,"style":604},[443],[390,7234,7236,7239,7246,7249,7371,7374],{"className":7235},[402],[390,7237],{"className":7238,"style":4015},[406],[390,7240,7242],{"className":7241},[448],[390,7243,7245],{"className":7244},[411,452],"max",[390,7247],{"className":7248,"style":444},[443],[390,7250,7252,7258,7261,7264,7267,7271,7274,7362,7365],{"className":7251},[826],[390,7253,7255],{"className":7254,"style":831},[435,830],[390,7256,3946],{"className":7257},[1856,1908],[390,7259],{"className":7260,"style":444},[443],[390,7262,1839],{"className":7263},[411,412],[390,7265,1785],{"className":7266},[1784],[390,7268,7270],{"className":7269},[443]," ",[390,7272],{"className":7273,"style":444},[443],[390,7275,7277,7280,7359],{"className":7276},[411],[390,7278],{"className":7279},[435,1876],[390,7281,7283],{"className":7282},[1880],[390,7284,7286,7350],{"className":7285},[1884,1885],[390,7287,7289,7347],{"className":7288},[1889],[390,7290,7293,7328,7336],{"className":7291,"style":7292},[1893],"height:1.3714em;",[390,7294,7295,7298],{"style":2174},[390,7296],{"className":7297,"style":1902},[1901],[390,7299,7301,7304,7307,7310,7313,7316,7319,7322,7325],{"className":7300},[411],[390,7302,500],{"className":7303},[411],[390,7305],{"className":7306,"style":682},[443],[390,7308,687],{"className":7309},[686],[390,7311],{"className":7312,"style":682},[443],[390,7314,6961],{"className":7315},[411,412],[390,7317],{"className":7318,"style":682},[443],[390,7320,687],{"className":7321},[686],[390,7323],{"className":7324,"style":682},[443],[390,7326,6849],{"className":7327},[411,412],[390,7329,7330,7333],{"style":1919},[390,7331],{"className":7332,"style":1902},[1901],[390,7334],{"className":7335,"style":1927},[1926],[390,7337,7338,7341],{"style":2207},[390,7339],{"className":7340,"style":1902},[1901],[390,7342,7344],{"className":7343},[411],[390,7345,6245],{"className":7346},[411,412],[390,7348,1947],{"className":7349},[1946],[390,7351,7353],{"className":7352},[1889],[390,7354,7357],{"className":7355,"style":7356},[1893],"height:0.8804em;",[390,7358],{},[390,7360],{"className":7361},[464,1876],[390,7363],{"className":7364,"style":444},[443],[390,7366,7368],{"className":7367,"style":831},[464,830],[390,7369,3975],{"className":7370},[1856,1908],[390,7372],{"className":7373,"style":444},[443],[390,7375,720],{"className":7376},[411],[381,7378,7379,7380,7413,7414,7465,7466,7536,7537,7615,7616,7686,7687,5912,7702,7780],{},"Note ",[390,7381,7383],{"className":7382},[393],[390,7384,7386,7404],{"className":7385,"ariaHidden":398},[397],[390,7387,7389,7392,7395,7398,7401],{"className":7388},[402],[390,7390],{"className":7391,"style":4404},[406],[390,7393,494],{"className":7394},[411,412],[390,7396],{"className":7397,"style":604},[443],[390,7399,4531],{"className":7400},[608],[390,7402],{"className":7403,"style":604},[443],[390,7405,7407,7410],{"className":7406},[402],[390,7408],{"className":7409,"style":697},[406],[390,7411,6759],{"className":7412},[411]," is well-defined precisely because ",[390,7415,7417],{"className":7416},[393],[390,7418,7420,7438,7456],{"className":7419,"ariaHidden":398},[397],[390,7421,7423,7426,7429,7432,7435],{"className":7422},[402],[390,7424],{"className":7425,"style":675},[406],[390,7427,6961],{"className":7428},[411,412],[390,7430],{"className":7431,"style":682},[443],[390,7433,845],{"className":7434},[686],[390,7436],{"className":7437,"style":682},[443],[390,7439,7441,7444,7447,7450,7453],{"className":7440},[402],[390,7442],{"className":7443,"style":7029},[406],[390,7445,6849],{"className":7446},[411,412],[390,7448],{"className":7449,"style":604},[443],[390,7451,1528],{"className":7452},[608],[390,7454],{"className":7455,"style":604},[443],[390,7457,7459,7462],{"className":7458},[402],[390,7460],{"className":7461,"style":697},[406],[390,7463,500],{"className":7464},[411],". The base\ncases ",[390,7467,7469],{"className":7468},[393],[390,7470,7472,7490],{"className":7471,"ariaHidden":398},[397],[390,7473,7475,7478,7481,7484,7487],{"className":7474},[402],[390,7476],{"className":7477,"style":3801},[406],[390,7479,413],{"className":7480},[411,412],[390,7482],{"className":7483,"style":604},[443],[390,7485,609],{"className":7486},[608],[390,7488],{"className":7489,"style":604},[443],[390,7491,7493,7496],{"className":7492},[402],[390,7494],{"className":7495,"style":6727},[406],[390,7497,7499,7502],{"className":7498},[411],[390,7500,413],{"className":7501},[411,412],[390,7503,7505],{"className":7504},[2062],[390,7506,7508,7528],{"className":7507},[1884,1885],[390,7509,7511,7525],{"className":7510},[1889],[390,7512,7514],{"className":7513,"style":6746},[1893],[390,7515,7516,7519],{"style":6749},[390,7517],{"className":7518,"style":2079},[1901],[390,7520,7522],{"className":7521},[1906,1907,1908,1909],[390,7523,6759],{"className":7524},[411,1909],[390,7526,1947],{"className":7527},[1946],[390,7529,7531],{"className":7530},[1889],[390,7532,7534],{"className":7533,"style":6769},[1893],[390,7535],{}," hold since ",[390,7538,7540],{"className":7539},[393],[390,7541,7543,7570,7588,7606],{"className":7542,"ariaHidden":398},[397],[390,7544,7546,7549,7552,7555,7558,7561,7564,7567],{"className":7545},[402],[390,7547],{"className":7548,"style":427},[406],[390,7550,1711],{"className":7551,"style":1710},[411,412],[390,7553,436],{"className":7554},[435],[390,7556,413],{"className":7557},[411,412],[390,7559,465],{"className":7560},[464],[390,7562],{"className":7563,"style":604},[443],[390,7565,609],{"className":7566},[608],[390,7568],{"className":7569,"style":604},[443],[390,7571,7573,7576,7579,7582,7585],{"className":7572},[402],[390,7574],{"className":7575,"style":3801},[406],[390,7577,1839],{"className":7578},[411,412],[390,7580],{"className":7581,"style":604},[443],[390,7583,609],{"className":7584},[608],[390,7586],{"className":7587,"style":604},[443],[390,7589,7591,7594,7597,7600,7603],{"className":7590},[402],[390,7592],{"className":7593,"style":3801},[406],[390,7595,494],{"className":7596},[411,412],[390,7598],{"className":7599,"style":604},[443],[390,7601,609],{"className":7602},[608],[390,7604],{"className":7605,"style":604},[443],[390,7607,7609,7612],{"className":7608},[402],[390,7610],{"className":7611,"style":407},[406],[390,7613,7170],{"className":7614},[411,412],". For the inductive step\n",[390,7617,7619],{"className":7618},[393],[390,7620,7622,7640],{"className":7621,"ariaHidden":398},[397],[390,7623,7625,7628,7631,7634,7637],{"className":7624},[402],[390,7626],{"className":7627,"style":4404},[406],[390,7629,413],{"className":7630},[411,412],[390,7632],{"className":7633,"style":604},[443],[390,7635,4531],{"className":7636},[608],[390,7638],{"className":7639,"style":604},[443],[390,7641,7643,7646],{"className":7642},[402],[390,7644],{"className":7645,"style":6727},[406],[390,7647,7649,7652],{"className":7648},[411],[390,7650,413],{"className":7651},[411,412],[390,7653,7655],{"className":7654},[2062],[390,7656,7658,7678],{"className":7657},[1884,1885],[390,7659,7661,7675],{"className":7660},[1889],[390,7662,7664],{"className":7663,"style":6746},[1893],[390,7665,7666,7669],{"style":6749},[390,7667],{"className":7668,"style":2079},[1901],[390,7670,7672],{"className":7671},[1906,1907,1908,1909],[390,7673,6759],{"className":7674},[411,1909],[390,7676,1947],{"className":7677},[1946],[390,7679,7681],{"className":7680},[1889],[390,7682,7684],{"className":7683,"style":6769},[1893],[390,7685],{},", the choice of 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so",[390,7782,7784],{"className":7783},[1694],[390,7785,7787],{"className":7786},[393],[390,7788,7790,7817,7844,7874,7892,7911,7935,7953,7983,8004,8031,8061,8082,8106,8124,8154],{"className":7789,"ariaHidden":398},[397],[390,7791,7793,7796,7799,7802,7805,7808,7811,7814],{"className":7792},[402],[390,7794],{"className":7795,"style":427},[406],[390,7797,1711],{"className":7798,"style":1710},[411,412],[390,7800,436],{"className":7801},[435],[390,7803,413],{"className":7804},[411,412],[390,7806,465],{"className":7807},[464],[390,7809],{"className":7810,"style":604},[443],[390,7812,609],{"className":7813},[608],[390,7815],{"className":7816,"style":604},[443],[390,7818,7820,7823,7826,7829,7832,7835,7838,7841],{"className":7819},[402],[390,7821],{"className":7822,"style":427},[406],[390,7824,1711],{"className":7825,"style":1710},[411,412],[390,7827,436],{"className":7828},[435],[390,7830,6821],{"className":7831},[411,412],[390,7833,465],{"className":7834},[464],[390,7836],{"className":7837,"style":682},[443],[390,7839,845],{"className":7840},[686],[390,7842],{"className":7843,"style":682},[443],[390,7845,7847,7850,7853,7856,7859,7862,7865,7868,7871],{"className":7846},[402],[390,7848],{"className":7849,"style":427},[406],[390,7851,1711],{"className":7852,"style":1710},[411,412],[390,7854,436],{"className":7855},[435],[390,7857,6849],{"className":7858},[411,412],[390,7860,413],{"className":7861},[411,412],[390,7863,465],{"className":7864},[464],[390,7866],{"className":7867,"style":682},[443],[390,7869,845],{"className":7870},[686],[390,7872],{"className":7873,"style":682},[443],[390,7875,7877,7880,7883,7886,7889],{"className":7876},[402],[390,7878],{"className":7879,"style":619},[406],[390,7881,5907],{"className":7882},[411,412],[390,7884],{"className":7885,"style":604},[443],[390,7887,609],{"className":7888},[608],[390,7890],{"className":7891,"style":604},[443],[390,7893,7895,7898,7902,7905,7908],{"className":7894},[402],[390,7896],{"className":7897,"style":675},[406],[390,7899,7901],{"className":7900},[411,412],"aλn",[390,7903],{"className":7904,"style":682},[443],[390,7906,845],{"className":7907},[686],[390,7909],{"className":7910,"style":682},[443],[390,7912,7914,7917,7920,7923,7926,7929,7932],{"className":7913},[402],[390,7915],{"className":7916,"style":1002},[406],[390,7918,494],{"className":7919},[411,412],[390,7921,6849],{"className":7922},[411,412],[390,7924,413],{"className":7925},[411,412],[390,7927],{"className":7928,"style":682},[443],[390,7930,845],{"className":7931},[686],[390,7933],{"className":7934,"style":682},[443],[390,7936,7938,7941,7944,7947,7950],{"className":7937},[402],[390,7939],{"className":7940,"style":479},[406],[390,7942,5907],{"className":7943},[411,412],[390,7945],{"className":7946,"style":604},[443],[390,7948,745],{"className":7949},[608],[390,7951],{"className":7952,"style":604},[443],[390,7954,7956,7959,7965,7968,7971,7974,7977,7980],{"className":7955},[402],[390,7957],{"className":7958,"style":6007},[406],[390,7960,7962],{"className":7961},[435],[390,7963,436],{"className":7964},[1856,1984],[390,7966,494],{"className":7967},[411,412],[390,7969,436],{"className":7970},[435],[390,7972,6961],{"className":7973},[411,412],[390,7975],{"className":7976,"style":682},[443],[390,7978,845],{"className":7979},[686],[390,7981],{"className":7982,"style":682},[443],[390,7984,7986,7989,7992,7995,7998,8001],{"className":7985},[402],[390,7987],{"className":7988,"style":427},[406],[390,7990,6849],{"className":7991},[411,412],[390,7993,465],{"className":7994},[464],[390,7996],{"className":7997,"style":682},[443],[390,7999,845],{"className":8000},[686],[390,8002],{"className":8003,"style":682},[443],[390,8005,8007,8010,8013,8019,8022,8025,8028],{"className":8006},[402],[390,8008],{"className":8009,"style":6007},[406],[390,8011,6245],{"className":8012},[411,412],[390,8014,8016],{"className":8015},[464],[390,8017,465],{"className":8018},[1856,1984],[390,8020,413],{"className":8021},[411,412],[390,8023],{"className":8024,"style":604},[443],[390,8026,609],{"className":8027},[608],[390,8029],{"className":8030,"style":604},[443],[390,8032,8034,8037,8043,8046,8049,8052,8055,8058],{"className":8033},[402],[390,8035],{"className":8036,"style":6007},[406],[390,8038,8040],{"className":8039},[435],[390,8041,436],{"className":8042},[1856,1984],[390,8044,494],{"className":8045},[411,412],[390,8047,436],{"className":8048},[435],[390,8050,6961],{"className":8051},[411,412],[390,8053],{"className":8054,"style":682},[443],[390,8056,845],{"className":8057},[686],[390,8059],{"className":8060,"style":682},[443],[390,8062,8064,8067,8070,8073,8076,8079],{"className":8063},[402],[390,8065],{"className":8066,"style":427},[406],[390,8068,6849],{"className":8069},[411,412],[390,8071,465],{"className":8072},[464],[390,8074],{"className":8075,"style":682},[443],[390,8077,845],{"className":8078},[686],[390,8080],{"className":8081,"style":682},[443],[390,8083,8085,8088,8091,8094,8097,8100,8103],{"className":8084},[402],[390,8086],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gives ",[390,8495,8497],{"className":8496},[393],[390,8498,8500,8527],{"className":8499,"ariaHidden":398},[397],[390,8501,8503,8506,8509,8512,8515,8518,8521,8524],{"className":8502},[402],[390,8504],{"className":8505,"style":427},[406],[390,8507,1711],{"className":8508,"style":1710},[411,412],[390,8510,436],{"className":8511},[435],[390,8513,413],{"className":8514},[411,412],[390,8516,465],{"className":8517},[464],[390,8519],{"className":8520,"style":604},[443],[390,8522,745],{"className":8523},[608],[390,8525],{"className":8526,"style":604},[443],[390,8528,8530,8533,8536,8539,8542],{"className":8529},[402],[390,8531],{"className":8532,"style":427},[406],[390,8534,2340],{"className":8535,"style":944},[411,412],[390,8537,436],{"className":8538},[435],[390,8540,413],{"className":8541},[411,412],[390,8543,465],{"className":8544},[464],", that is, ",[385,8547,8548],{},"worst-case linear\ntime",". Had the fractions summed to ",[390,8551,8553],{"className":8552},[393],[390,8554,8556],{"className":8555,"ariaHidden":398},[397],[390,8557,8559,8562],{"className":8558},[402],[390,8560],{"className":8561,"style":697},[406],[390,8563,500],{"className":8564},[411]," or more (as for groups of 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the lemma's hypothesis fails: the per-level savings\nvanish, the recursion tree carries ",[390,8764,8766],{"className":8765},[393],[390,8767,8769],{"className":8768,"ariaHidden":398},[397],[390,8770,8772,8775,8781,8784],{"className":8771},[402],[390,8773],{"className":8774,"style":6957},[406],[390,8776,8778],{"className":8777},[448],[390,8779,454],{"className":8780,"style":453},[411,452],[390,8782],{"className":8783,"style":444},[443],[390,8785,413],{"className":8786},[411,412]," levels of ",[390,8789,8791],{"className":8790},[393],[390,8792,8794],{"className":8793,"ariaHidden":398},[397],[390,8795,8797,8800,8803,8806,8809],{"className":8796},[402],[390,8798],{"className":8799,"style":427},[406],[390,8801,431],{"className":8802},[411],[390,8804,436],{"className":8805},[435],[390,8807,413],{"className":8808},[411,412],[390,8810,465],{"className":8811},[464]," work, and the\nbound degrades to ",[390,8814,8816],{"className":8815},[393],[390,8817,8819],{"className":8818,"ariaHidden":398},[397],[390,8820,8822,8825,8828,8831,8834,8837,8843,8846,8849],{"className":8821},[402],[390,8823],{"className":8824,"style":427},[406],[390,8826,431],{"className":8827},[411],[390,8829,436],{"className":8830},[435],[390,8832,413],{"className":8833},[411,412],[390,8835],{"className":8836,"style":444},[443],[390,8838,8840],{"className":8839},[448],[390,8841,454],{"className":8842,"style":453},[411,452],[390,8844],{"className":8845,"style":444},[443],[390,8847,413],{"className":8848},[411,412],[390,8850,465],{"className":8851},[464],[508,8853,8855],{"id":8854},"which-to-use","Which to use",[381,8857,8858,8859,8862,8863,8866,8867,8874,8875,8878,8879,8918],{},"The median-of-medians algorithm is a landmark: it proves selection is\n",[503,8860,8861],{},"possible"," in worst-case linear time. But its constant factor is large (all\nthat grouping and recursive median-finding), so in practice ",[385,8864,8865],{},"randomized\nquickselect is faster"," and is the algorithm of choice.",[491,8868,8869],{},[494,8870,1916],{"href":8871,"ariaDescribedBy":8872,"dataFootnoteRef":376,"id":8873},"#user-content-fn-skiena-select",[498],"user-content-fnref-skiena-select"," The deterministic\nversion shines when a worst-case guarantee is mandatory, or, most importantly,\nas a ",[385,8876,8877],{},"pivot-selection subroutine"," that can be bolted onto quicksort to give\nit a worst-case ",[390,8880,8882],{"className":8881},[393],[390,8883,8885],{"className":8884,"ariaHidden":398},[397],[390,8886,8888,8891,8894,8897,8900,8903,8909,8912,8915],{"className":8887},[402],[390,8889],{"className":8890,"style":427},[406],[390,8892,431],{"className":8893},[411],[390,8895,436],{"className":8896},[435],[390,8898,413],{"className":8899},[411,412],[390,8901],{"className":8902,"style":444},[443],[390,8904,8906],{"className":8905},[448],[390,8907,454],{"className":8908,"style":453},[411,452],[390,8910],{"className":8911,"style":444},[443],[390,8913,413],{"className":8914},[411,412],[390,8916,465],{"className":8917},[464]," bound.",[508,8920,8922],{"id":8921},"bonus-closest-pair-of-points","Bonus: Closest Pair of Points",[381,8924,8925,8926,8929,8930,8933],{},"This is often called ",[967,8927,8928],{},"the most beautiful divide-and-conquer algorithm,"," and\nalso the scariest to get right. It is worth seeing because, like\nmedian-of-medians, its speed hinges on a geometric ",[503,8931,8932],{},"counting"," argument that\ncaps how much work the combine step can possibly do.",[513,8935,8936],{"type":515},[381,8937,8938,8940,8941,8956,8957,8985,8986,641,9112,9114,9115,9191,9192,6944,9245,720],{},[385,8939,520],{}," ",[390,8942,8944],{"className":8943},[393],[390,8945,8947],{"className":8946,"ariaHidden":398},[397],[390,8948,8950,8953],{"className":8949},[402],[390,8951],{"className":8952,"style":407},[406],[390,8954,413],{"className":8955},[411,412]," points ",[390,8958,8960],{"className":8959},[393],[390,8961,8963],{"className":8962,"ariaHidden":398},[397],[390,8964,8966,8969,8973,8976,8979,8982],{"className":8965},[402],[390,8967],{"className":8968,"style":427},[406],[390,8970,8972],{"className":8971,"style":1710},[411,412],"P",[390,8974,541],{"className":8975},[435],[390,8977,545],{"className":8978},[411],[390,8980,413],{"className":8981},[411,412],[390,8983,552],{"className":8984},[464]," in the plane, each a pair ",[390,8987,8989],{"className":8988},[393],[390,8990,8992,9010],{"className":8991,"ariaHidden":398},[397],[390,8993,8995,8998,9001,9004,9007],{"className":8994},[402],[390,8996],{"className":8997,"style":962},[406],[390,8999,381],{"className":9000},[411,412],[390,9002],{"className":9003,"style":604},[443],[390,9005,745],{"className":9006},[608],[390,9008],{"className":9009,"style":604},[443],[390,9011,9013,9017,9020,9061,9064,9067,9109],{"className":9012},[402],[390,9014],{"className":9015,"style":9016},[406],"height:1.0361em;vertical-align:-0.2861em;",[390,9018,436],{"className":9019},[435],[390,9021,9023,9026],{"className":9022},[411],[390,9024,381],{"className":9025},[411,412],[390,9027,9029],{"className":9028},[2062],[390,9030,9032,9053],{"className":9031},[1884,1885],[390,9033,9035,9050],{"className":9034},[1889],[390,9036,9039],{"className":9037,"style":9038},[1893],"height:0.1514em;",[390,9040,9041,9044],{"style":6749},[390,9042],{"className":9043,"style":2079},[1901],[390,9045,9047],{"className":9046},[1906,1907,1908,1909],[390,9048,1488],{"className":9049},[411,412,1909],[390,9051,1947],{"className":9052},[1946],[390,9054,9056],{"className":9055},[1889],[390,9057,9059],{"className":9058,"style":6769},[1893],[390,9060],{},[390,9062,1785],{"className":9063},[1784],[390,9065],{"className":9066,"style":444},[443],[390,9068,9070,9073],{"className":9069},[411],[390,9071,381],{"className":9072},[411,412],[390,9074,9076],{"className":9075},[2062],[390,9077,9079,9100],{"className":9078},[1884,1885],[390,9080,9082,9097],{"className":9081},[1889],[390,9083,9085],{"className":9084,"style":9038},[1893],[390,9086,9087,9090],{"style":6749},[390,9088],{"className":9089,"style":2079},[1901],[390,9091,9093],{"className":9092},[1906,1907,1908,1909],[390,9094,9096],{"className":9095,"style":966},[411,412,1909],"y",[390,9098,1947],{"className":9099},[1946],[390,9101,9103],{"className":9102},[1889],[390,9104,9107],{"className":9105,"style":9106},[1893],"height:0.2861em;",[390,9108],{},[390,9110,465],{"className":9111},[464],[385,9113,644],{}," indices ",[390,9116,9118],{"className":9117},[393],[390,9119,9121,9179],{"className":9120,"ariaHidden":398},[397],[390,9122,9124,9127,9130,9133,9176],{"className":9123},[402],[390,9125],{"className":9126,"style":6957},[406],[390,9128,1542],{"className":9129},[411,412],[390,9131],{"className":9132,"style":604},[443],[390,9134,9136,9169,9173],{"className":9135},[608],[390,9137,9139],{"className":9138},[608],[390,9140,9143],{"className":9141},[411,9142],"vbox",[390,9144,9147],{"className":9145},[9146],"thinbox",[390,9148,9151,9154,9165],{"className":9149},[9150],"rlap",[390,9152],{"className":9153,"style":6957},[406],[390,9155,9158],{"className":9156},[9157],"inner",[390,9159,9161],{"className":9160},[411],[390,9162,9164],{"className":9163},[608],"",[390,9166],{"className":9167},[9168],"fix",[390,9170],{"className":9171},[443,9172],"nobreak",[390,9174,745],{"className":9175},[608],[390,9177],{"className":9178,"style":604},[443],[390,9180,9182,9186],{"className":9181},[402],[390,9183],{"className":9184,"style":9185},[406],"height:0.854em;vertical-align:-0.1944em;",[390,9187,9190],{"className":9188,"style":9189},[411,412],"margin-right:0.0572em;","j"," minimizing the Euclidean distance\n",[390,9193,9195],{"className":9194},[393],[390,9196,9198],{"className":9197,"ariaHidden":398},[397],[390,9199,9201,9204,9211,9214,9217,9220,9223,9226,9229,9232,9235,9238,9241],{"className":9200},[402],[390,9202],{"className":9203,"style":427},[406],[390,9205,9207],{"className":9206},[448],[390,9208,9210],{"className":9209},[411,452],"dist",[390,9212,436],{"className":9213},[435],[390,9215,8972],{"className":9216,"style":1710},[411,412],[390,9218,541],{"className":9219},[435],[390,9221,1542],{"className":9222},[411,412],[390,9224,552],{"className":9225},[464],[390,9227,1785],{"className":9228},[1784],[390,9230],{"className":9231,"style":444},[443],[390,9233,8972],{"className":9234,"style":1710},[411,412],[390,9236,541],{"className":9237},[435],[390,9239,9190],{"className":9240,"style":9189},[411,412],[390,9242,9244],{"className":9243},[464],"])",[390,9246,9248],{"className":9247},[393],[390,9249,9251,9290],{"className":9250,"ariaHidden":398},[397],[390,9252,9254,9257,9263,9266,9269,9272,9275,9278,9281,9284,9287],{"className":9253},[402],[390,9255],{"className":9256,"style":427},[406],[390,9258,9260],{"className":9259},[448],[390,9261,9210],{"className":9262},[411,452],[390,9264,436],{"className":9265},[435],[390,9267,381],{"className":9268},[411,412],[390,9270,1785],{"className":9271},[1784],[390,9273],{"className":9274,"style":444},[443],[390,9276,967],{"className":9277,"style":966},[411,412],[390,9279,465],{"className":9280},[464],[390,9282],{"className":9283,"style":604},[443],[390,9285,745],{"className":9286},[608],[390,9288],{"className":9289,"style":604},[443],[390,9291,9293,9297],{"className":9292},[402],[390,9294],{"className":9295,"style":9296},[406],"height:1.24em;vertical-align:-0.3231em;",[390,9298,9301],{"className":9299},[411,9300],"sqrt",[390,9302,9304,9601],{"className":9303},[1884,1885],[390,9305,9307,9598],{"className":9306},[1889],[390,9308,9311,9578],{"className":9309,"style":9310},[1893],"height:0.9169em;",[390,9312,9316,9320],{"className":9313,"style":9315},[9314],"svg-align","top:-3.2em;",[390,9317],{"className":9318,"style":9319},[1901],"height:3.2em;",[390,9321,9324,9327,9367,9370,9373,9376,9417,9448,9451,9454,9457,9460,9500,9503,9506,9509,9549],{"className":9322,"style":9323},[411],"padding-left:1em;",[390,9325,436],{"className":9326},[435],[390,9328,9330,9333],{"className":9329},[411],[390,9331,381],{"className":9332},[411,412],[390,9334,9336],{"className":9335},[2062],[390,9337,9339,9359],{"className":9338},[1884,1885],[390,9340,9342,9356],{"className":9341},[1889],[390,9343,9345],{"className":9344,"style":9038},[1893],[390,9346,9347,9350],{"style":6749},[390,9348],{"className":9349,"style":2079},[1901],[390,9351,9353],{"className":9352},[1906,1907,1908,1909],[390,9354,1488],{"className":9355},[411,412,1909],[390,9357,1947],{"className":9358},[1946],[390,9360,9362],{"className":9361},[1889],[390,9363,9365],{"className":9364,"style":6769},[1893],[390,9366],{},[390,9368],{"className":9369,"style":682},[443],[390,9371,687],{"className":9372},[686],[390,9374],{"className":9375,"style":682},[443],[390,9377,9379,9382],{"className":9378},[411],[390,9380,967],{"className":9381,"style":966},[411,412],[390,9383,9385],{"className":9384},[2062],[390,9386,9388,9409],{"className":9387},[1884,1885],[390,9389,9391,9406],{"className":9390},[1889],[390,9392,9394],{"className":9393,"style":9038},[1893],[390,9395,9397,9400],{"style":9396},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[390,9398],{"className":9399,"style":2079},[1901],[390,9401,9403],{"className":9402},[1906,1907,1908,1909],[390,9404,1488],{"className":9405},[411,412,1909],[390,9407,1947],{"className":9408},[1946],[390,9410,9412],{"className":9411},[1889],[390,9413,9415],{"className":9414,"style":6769},[1893],[390,9416],{},[390,9418,9420,9423],{"className":9419},[464],[390,9421,465],{"className":9422},[464],[390,9424,9426],{"className":9425},[2062],[390,9427,9429],{"className":9428},[1884],[390,9430,9432],{"className":9431},[1889],[390,9433,9436],{"className":9434,"style":9435},[1893],"height:0.7401em;",[390,9437,9439,9442],{"style":9438},"top:-2.989em;margin-right:0.05em;",[390,9440],{"className":9441,"style":2079},[1901],[390,9443,9445],{"className":9444},[1906,1907,1908,1909],[390,9446,1198],{"className":9447},[411,1909],[390,9449],{"className":9450,"style":682},[443],[390,9452,845],{"className":9453},[686],[390,9455],{"className":9456,"style":682},[443],[390,9458,436],{"className":9459},[435],[390,9461,9463,9466],{"className":9462},[411],[390,9464,381],{"className":9465},[411,412],[390,9467,9469],{"className":9468},[2062],[390,9470,9472,9492],{"className":9471},[1884,1885],[390,9473,9475,9489],{"className":9474},[1889],[390,9476,9478],{"className":9477,"style":9038},[1893],[390,9479,9480,9483],{"style":6749},[390,9481],{"className":9482,"style":2079},[1901],[390,9484,9486],{"className":9485},[1906,1907,1908,1909],[390,9487,9096],{"className":9488,"style":966},[411,412,1909],[390,9490,1947],{"className":9491},[1946],[390,9493,9495],{"className":9494},[1889],[390,9496,9498],{"className":9497,"style":9106},[1893],[390,9499],{},[390,9501],{"className":9502,"style":682},[443],[390,9504,687],{"className":9505},[686],[390,9507],{"className":9508,"style":682},[443],[390,9510,9512,9515],{"className":9511},[411],[390,9513,967],{"className":9514,"style":966},[411,412],[390,9516,9518],{"className":9517},[2062],[390,9519,9521,9541],{"className":9520},[1884,1885],[390,9522,9524,9538],{"className":9523},[1889],[390,9525,9527],{"className":9526,"style":9038},[1893],[390,9528,9529,9532],{"style":9396},[390,9530],{"className":9531,"style":2079},[1901],[390,9533,9535],{"className":9534},[1906,1907,1908,1909],[390,9536,9096],{"className":9537,"style":966},[411,412,1909],[390,9539,1947],{"className":9540},[1946],[390,9542,9544],{"className":9543},[1889],[390,9545,9547],{"className":9546,"style":9106},[1893],[390,9548],{},[390,9550,9552,9555],{"className":9551},[464],[390,9553,465],{"className":9554},[464],[390,9556,9558],{"className":9557},[2062],[390,9559,9561],{"className":9560},[1884],[390,9562,9564],{"className":9563},[1889],[390,9565,9567],{"className":9566,"style":9435},[1893],[390,9568,9569,9572],{"style":9438},[390,9570],{"className":9571,"style":2079},[1901],[390,9573,9575],{"className":9574},[1906,1907,1908,1909],[390,9576,1198],{"className":9577},[411,1909],[390,9579,9581,9584],{"style":9580},"top:-2.8769em;",[390,9582],{"className":9583,"style":9319},[1901],[390,9585,9589],{"className":9586,"style":9588},[9587],"hide-tail","min-width:1.02em;height:1.28em;",[1206,9590,9595],{"xmlns":1208,"width":9591,"height":9592,"viewBox":9593,"preserveAspectRatio":9594},"400em","1.28em","0 0 400000 1296","xMinYMin slice",[1218,9596],{"d":9597},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[390,9599,1947],{"className":9600},[1946],[390,9602,9604],{"className":9603},[1889],[390,9605,9608],{"className":9606,"style":9607},[1893],"height:0.3231em;",[390,9609],{},[381,9611,9612,9613,9637,9638,9719,9720,9771],{},"We treat one distance computation and one distance comparison as ",[390,9614,9616],{"className":9615},[393],[390,9617,9619],{"className":9618,"ariaHidden":398},[397],[390,9620,9622,9625,9628,9631,9634],{"className":9621},[402],[390,9623],{"className":9624,"style":427},[406],[390,9626,2340],{"className":9627,"style":944},[411,412],[390,9629,436],{"className":9630},[435],[390,9632,500],{"className":9633},[411],[390,9635,465],{"className":9636},[464],". The\nnaïve algorithm, two nested loops over all ",[390,9639,9641],{"className":9640},[393],[390,9642,9644],{"className":9643,"ariaHidden":398},[397],[390,9645,9647,9650],{"className":9646},[402],[390,9648],{"className":9649,"style":6007},[406],[390,9651,9653,9659,9713],{"className":9652},[411],[390,9654,9656],{"className":9655,"style":831},[435,830],[390,9657,436],{"className":9658},[1856,1984],[390,9660,9662],{"className":9661},[1880],[390,9663,9665,9705],{"className":9664},[1884,1885],[390,9666,9668,9702],{"className":9667},[1889],[390,9669,9672,9687],{"className":9670,"style":9671},[1893],"height:0.7454em;",[390,9673,9675,9678],{"style":9674},"top:-2.355em;",[390,9676],{"className":9677,"style":2079},[1901],[390,9679,9681],{"className":9680},[1906,1907,1908,1909],[390,9682,9684],{"className":9683},[411,1909],[390,9685,1198],{"className":9686},[411,1909],[390,9688,9690,9693],{"style":9689},"top:-3.144em;",[390,9691],{"className":9692,"style":2079},[1901],[390,9694,9696],{"className":9695},[1906,1907,1908,1909],[390,9697,9699],{"className":9698},[411,1909],[390,9700,413],{"className":9701},[411,412,1909],[390,9703,1947],{"className":9704},[1946],[390,9706,9708],{"className":9707},[1889],[390,9709,9711],{"className":9710,"style":1954},[1893],[390,9712],{},[390,9714,9716],{"className":9715,"style":831},[464,830],[390,9717,465],{"className":9718},[1856,1984]," pairs, runs in\n",[390,9721,9723],{"className":9722},[393],[390,9724,9726],{"className":9725,"ariaHidden":398},[397],[390,9727,9729,9733,9736,9739,9768],{"className":9728},[402],[390,9730],{"className":9731,"style":9732},[406],"height:1.0641em;vertical-align:-0.25em;",[390,9734,431],{"className":9735},[411],[390,9737,436],{"className":9738},[435],[390,9740,9742,9745],{"className":9741},[411],[390,9743,413],{"className":9744},[411,412],[390,9746,9748],{"className":9747},[2062],[390,9749,9751],{"className":9750},[1884],[390,9752,9754],{"className":9753},[1889],[390,9755,9757],{"className":9756,"style":2727},[1893],[390,9758,9759,9762],{"style":2730},[390,9760],{"className":9761,"style":2079},[1901],[390,9763,9765],{"className":9764},[1906,1907,1908,1909],[390,9766,1198],{"className":9767},[411,1909],[390,9769,465],{"className":9770},[464],". Can divide-and-conquer beat it?",[1615,9773,9775],{"id":9774},"the-idea-and-the-trap","The idea, and the trap",[381,9777,9778,8940,9781,9796,9797,9812,9813,9815,9816,9833,9834,9837],{},[385,9779,9780],{},"Split",[390,9782,9784],{"className":9783},[393],[390,9785,9787],{"className":9786,"ariaHidden":398},[397],[390,9788,9790,9793],{"className":9789},[402],[390,9791],{"className":9792,"style":658},[406],[390,9794,8972],{"className":9795,"style":1710},[411,412]," by the median ",[390,9798,9800],{"className":9799},[393],[390,9801,9803],{"className":9802,"ariaHidden":398},[397],[390,9804,9806,9809],{"className":9805},[402],[390,9807],{"className":9808,"style":407},[406],[390,9810,1488],{"className":9811},[411,412],"-coordinate into a left half and a right half,\n",[385,9814,3202],{}," on each to get the closest pair within each side, and let ",[390,9817,9819],{"className":9818},[393],[390,9820,9822],{"className":9821,"ariaHidden":398},[397],[390,9823,9825,9828],{"className":9824},[402],[390,9826],{"className":9827,"style":479},[406],[390,9829,9832],{"className":9830,"style":9831},[411,412],"margin-right:0.0379em;","δ","\nbe the smaller of the two returned distances. The danger is the ",[385,9835,9836],{},"combine","\nstep: the true closest pair might straddle the dividing line, so we still have\nto look across it.",[381,9839,9840,9841,9895,9896,9950,9951,10067,10068,10118,10119,10143],{},"The trap is doing this naïvely. If ",[390,9842,9844],{"className":9843},[393],[390,9845,9847],{"className":9846,"ariaHidden":398},[397],[390,9848,9850,9853],{"className":9849},[402],[390,9851],{"className":9852,"style":6727},[406],[390,9854,9856,9859],{"className":9855},[411],[390,9857,413],{"className":9858},[411,412],[390,9860,9862],{"className":9861},[2062],[390,9863,9865,9887],{"className":9864},[1884,1885],[390,9866,9868,9884],{"className":9867},[1889],[390,9869,9872],{"className":9870,"style":9871},[1893],"height:0.3283em;",[390,9873,9874,9877],{"style":6749},[390,9875],{"className":9876,"style":2079},[1901],[390,9878,9880],{"className":9879},[1906,1907,1908,1909],[390,9881,9883],{"className":9882},[411,412,1909],"L",[390,9885,1947],{"className":9886},[1946],[390,9888,9890],{"className":9889},[1889],[390,9891,9893],{"className":9892,"style":6769},[1893],[390,9894],{}," left points and ",[390,9897,9899],{"className":9898},[393],[390,9900,9902],{"className":9901,"ariaHidden":398},[397],[390,9903,9905,9908],{"className":9904},[402],[390,9906],{"className":9907,"style":6727},[406],[390,9909,9911,9914],{"className":9910},[411],[390,9912,413],{"className":9913},[411,412],[390,9915,9917],{"className":9916},[2062],[390,9918,9920,9942],{"className":9919},[1884,1885],[390,9921,9923,9939],{"className":9922},[1889],[390,9924,9926],{"className":9925,"style":9871},[1893],[390,9927,9928,9931],{"style":6749},[390,9929],{"className":9930,"style":2079},[1901],[390,9932,9934],{"className":9933},[1906,1907,1908,1909],[390,9935,9938],{"className":9936,"style":9937},[411,412,1909],"margin-right:0.0077em;","R",[390,9940,1947],{"className":9941},[1946],[390,9943,9945],{"className":9944},[1889],[390,9946,9948],{"className":9947,"style":6769},[1893],[390,9949],{}," right points lie\nnear the line, checking every cross pair costs ",[390,9952,9954],{"className":9953},[393],[390,9955,9957,10018],{"className":9956,"ariaHidden":398},[397],[390,9958,9960,9963,9966,9969,10009,10012,10015],{"className":9959},[402],[390,9961],{"className":9962,"style":427},[406],[390,9964,431],{"className":9965},[411],[390,9967,436],{"className":9968},[435],[390,9970,9972,9975],{"className":9971},[411],[390,9973,413],{"className":9974},[411,412],[390,9976,9978],{"className":9977},[2062],[390,9979,9981,10001],{"className":9980},[1884,1885],[390,9982,9984,9998],{"className":9983},[1889],[390,9985,9987],{"className":9986,"style":9871},[1893],[390,9988,9989,9992],{"style":6749},[390,9990],{"className":9991,"style":2079},[1901],[390,9993,9995],{"className":9994},[1906,1907,1908,1909],[390,9996,9883],{"className":9997},[411,412,1909],[390,9999,1947],{"className":10000},[1946],[390,10002,10004],{"className":10003},[1889],[390,10005,10007],{"className":10006,"style":6769},[1893],[390,10008],{},[390,10010],{"className":10011,"style":682},[443],[390,10013,2142],{"className":10014},[686],[390,10016],{"className":10017,"style":682},[443],[390,10019,10021,10024,10064],{"className":10020},[402],[390,10022],{"className":10023,"style":427},[406],[390,10025,10027,10030],{"className":10026},[411],[390,10028,413],{"className":10029},[411,412],[390,10031,10033],{"className":10032},[2062],[390,10034,10036,10056],{"className":10035},[1884,1885],[390,10037,10039,10053],{"className":10038},[1889],[390,10040,10042],{"className":10041,"style":9871},[1893],[390,10043,10044,10047],{"style":6749},[390,10045],{"className":10046,"style":2079},[1901],[390,10048,10050],{"className":10049},[1906,1907,1908,1909],[390,10051,9938],{"className":10052,"style":9937},[411,412,1909],[390,10054,1947],{"className":10055},[1946],[390,10057,10059],{"className":10058},[1889],[390,10060,10062],{"className":10061,"style":6769},[1893],[390,10063],{},[390,10065,465],{"className":10066},[464],", which can\nbe ",[390,10069,10071],{"className":10070},[393],[390,10072,10074],{"className":10073,"ariaHidden":398},[397],[390,10075,10077,10080,10083,10086,10115],{"className":10076},[402],[390,10078],{"className":10079,"style":9732},[406],[390,10081,431],{"className":10082},[411],[390,10084,436],{"className":10085},[435],[390,10087,10089,10092],{"className":10088},[411],[390,10090,413],{"className":10091},[411,412],[390,10093,10095],{"className":10094},[2062],[390,10096,10098],{"className":10097},[1884],[390,10099,10101],{"className":10100},[1889],[390,10102,10104],{"className":10103,"style":2727},[1893],[390,10105,10106,10109],{"style":2730},[390,10107],{"className":10108,"style":2079},[1901],[390,10110,10112],{"className":10111},[1906,1907,1908,1909],[390,10113,1198],{"className":10114},[411,1909],[390,10116,465],{"className":10117},[464],", no better than brute force. The whole game is to show that\nonly ",[390,10120,10122],{"className":10121},[393],[390,10123,10125],{"className":10124,"ariaHidden":398},[397],[390,10126,10128,10131,10134,10137,10140],{"className":10127},[402],[390,10129],{"className":10130,"style":427},[406],[390,10132,2340],{"className":10133,"style":944},[411,412],[390,10135,436],{"className":10136},[435],[390,10138,413],{"className":10139},[411,412],[390,10141,465],{"className":10142},[464]," cross-comparisons are ever needed.",[1615,10145,10147,10148,10163],{"id":10146},"the-δ-strip-and-the-7-neighbor-bound","The ",[390,10149,10151],{"className":10150},[393],[390,10152,10154],{"className":10153,"ariaHidden":398},[397],[390,10155,10157,10160],{"className":10156},[402],[390,10158],{"className":10159,"style":479},[406],[390,10161,9832],{"className":10162,"style":9831},[411,412],"-strip and the 7-neighbor bound",[381,10165,10166,10167,10182,10183,10185,10186,10201,10202,10205,10206,10224,10225,10240,10241,10244,10245,10260],{},"Any cross pair closer than ",[390,10168,10170],{"className":10169},[393],[390,10171,10173],{"className":10172,"ariaHidden":398},[397],[390,10174,10176,10179],{"className":10175},[402],[390,10177],{"className":10178,"style":479},[406],[390,10180,9832],{"className":10181,"style":9831},[411,412]," must have ",[385,10184,910],{}," points within horizontal\ndistance ",[390,10187,10189],{"className":10188},[393],[390,10190,10192],{"className":10191,"ariaHidden":398},[397],[390,10193,10195,10198],{"className":10194},[402],[390,10196],{"className":10197,"style":479},[406],[390,10199,9832],{"className":10200,"style":9831},[411,412]," of the dividing line, i.e. inside a vertical ",[385,10203,10204],{},"strip"," of\nwidth ",[390,10207,10209],{"className":10208},[393],[390,10210,10212],{"className":10211,"ariaHidden":398},[397],[390,10213,10215,10218,10221],{"className":10214},[402],[390,10216],{"className":10217,"style":479},[406],[390,10219,1198],{"className":10220},[411],[390,10222,9832],{"className":10223,"style":9831},[411,412],". So discard everything outside the strip. Then sort the strip's\npoints by ",[390,10226,10228],{"className":10227},[393],[390,10229,10231],{"className":10230,"ariaHidden":398},[397],[390,10232,10234,10237],{"className":10233},[402],[390,10235],{"className":10236,"style":962},[406],[390,10238,9096],{"className":10239,"style":966},[411,412],"-coordinate and walk up it; the magic is that for each point you\nonly need to compare against the ",[385,10242,10243],{},"next 7"," points in ",[390,10246,10248],{"className":10247},[393],[390,10249,10251],{"className":10250,"ariaHidden":398},[397],[390,10252,10254,10257],{"className":10253},[402],[390,10255],{"className":10256,"style":962},[406],[390,10258,9096],{"className":10259,"style":966},[411,412],"-order.",[1200,10262,10264,10475],{"className":10263},[1203,1204],[1206,10265,10269],{"xmlns":1208,"width":10266,"height":10267,"viewBox":10268},"247.416","213.403","-75 -75 185.562 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51.850",[1243],[1213,10468,10469,10472],{"fill":1408,"stroke":1408},[1218,10470],{"fill":1233,"d":10471},"M39.507-34.935H15.58",[1218,10473],{"fill":1233,"d":10474,"style":3741},"M17.46-32.536c-.38-1.44-1.227-2.12-2.08-2.4.853-.28 1.7-.96 2.08-2.4",[1468,10476,10478,10479,8940,10494,10509,10510,10527,10528,720],{"className":10477},[1471],"The width-",[390,10480,10482],{"className":10481},[393],[390,10483,10485],{"className":10484,"ariaHidden":398},[397],[390,10486,10488,10491],{"className":10487},[402],[390,10489],{"className":10490,"style":697},[406],[390,10492,1198],{"className":10493},[411],[390,10495,10497],{"className":10496},[393],[390,10498,10500],{"className":10499,"ariaHidden":398},[397],[390,10501,10503,10506],{"className":10502},[402],[390,10504],{"className":10505,"style":479},[406],[390,10507,9832],{"className":10508,"style":9831},[411,412]," strip around the mid line with the 8 cells bounding the 7 neighbors above point ",[390,10511,10513],{"className":10512},[393],[390,10514,10516],{"className":10515,"ariaHidden":398},[397],[390,10517,10519,10522],{"className":10518},[402],[390,10520],{"className":10521,"style":658},[406],[390,10523,10526],{"className":10524,"style":10525},[411,412],"margin-right:0.0502em;","B"," at ",[390,10529,10531],{"className":10530},[393],[390,10532,10534],{"className":10533,"ariaHidden":398},[397],[390,10535,10537,10540],{"className":10536},[402],[390,10538],{"className":10539,"style":1538},[406],[390,10541,1542],{"className":10542},[411,412],[381,10544,10545,10548,10549,10573,10574,10598,10599,10614,10615,10630,10631,10668,10669,10693,10694,10733,10734,10736],{},[385,10546,10547],{},"Why 7?"," Fix a strip point ",[390,10550,10552],{"className":10551},[393],[390,10553,10555],{"className":10554,"ariaHidden":398},[397],[390,10556,10558,10561,10564,10567,10570],{"className":10557},[402],[390,10559],{"className":10560,"style":427},[406],[390,10562,10526],{"className":10563,"style":10525},[411,412],[390,10565,541],{"className":10566},[435],[390,10568,1542],{"className":10569},[411,412],[390,10571,552],{"className":10572},[464]," and look at the points ",[390,10575,10577],{"className":10576},[393],[390,10578,10580],{"className":10579,"ariaHidden":398},[397],[390,10581,10583,10586,10589,10592,10595],{"className":10582},[402],[390,10584],{"className":10585,"style":427},[406],[390,10587,10526],{"className":10588,"style":10525},[411,412],[390,10590,541],{"className":10591},[435],[390,10593,9190],{"className":10594,"style":9189},[411,412],[390,10596,552],{"className":10597},[464]," above it\n(larger ",[390,10600,10602],{"className":10601},[393],[390,10603,10605],{"className":10604,"ariaHidden":398},[397],[390,10606,10608,10611],{"className":10607},[402],[390,10609],{"className":10610,"style":962},[406],[390,10612,9096],{"className":10613,"style":966},[411,412],") that could beat ",[390,10616,10618],{"className":10617},[393],[390,10619,10621],{"className":10620,"ariaHidden":398},[397],[390,10622,10624,10627],{"className":10623},[402],[390,10625],{"className":10626,"style":479},[406],[390,10628,9832],{"className":10629,"style":9831},[411,412],". Any such point lies in the\n",[390,10632,10634],{"className":10633},[393],[390,10635,10637,10659],{"className":10636,"ariaHidden":398},[397],[390,10638,10640,10643,10646,10649,10652,10656],{"className":10639},[402],[390,10641],{"className":10642,"style":675},[406],[390,10644,1198],{"className":10645},[411],[390,10647,9832],{"className":10648,"style":9831},[411,412],[390,10650],{"className":10651,"style":682},[443],[390,10653,10655],{"className":10654},[686],"×",[390,10657],{"className":10658,"style":682},[443],[390,10660,10662,10665],{"className":10661},[402],[390,10663],{"className":10664,"style":479},[406],[390,10666,9832],{"className":10667,"style":9831},[411,412]," box sitting just above ",[390,10670,10672],{"className":10671},[393],[390,10673,10675],{"className":10674,"ariaHidden":398},[397],[390,10676,10678,10681,10684,10687,10690],{"className":10677},[402],[390,10679],{"className":10680,"style":427},[406],[390,10682,10526],{"className":10683,"style":10525},[411,412],[390,10685,541],{"className":10686},[435],[390,10688,1542],{"className":10689},[411,412],[390,10691,552],{"className":10692},[464]," inside the strip. Tile\nthat box with a grid of ",[390,10695,10697],{"className":10696},[393],[390,10698,10700,10721],{"className":10699,"ariaHidden":398},[397],[390,10701,10703,10706,10709,10712,10715,10718],{"className":10702},[402],[390,10704],{"className":10705,"style":427},[406],[390,10707,9832],{"className":10708,"style":9831},[411,412],[390,10710,858],{"className":10711},[411],[390,10713],{"className":10714,"style":682},[443],[390,10716,10655],{"className":10717},[686],[390,10719],{"className":10720,"style":682},[443],[390,10722,10724,10727,10730],{"className":10723},[402],[390,10725],{"className":10726,"style":427},[406],[390,10728,9832],{"className":10729,"style":9831},[411,412],[390,10731,858],{"className":10732},[411]," cells — there are exactly\n",[385,10735,1289],{}," of them.",[513,10738,10739],{"type":6647},[381,10740,10741,10744,10745,10760],{},[385,10742,10743],{},"Claim."," No such cell can contain ",[390,10746,10748],{"className":10747},[393],[390,10749,10751],{"className":10750,"ariaHidden":398},[397],[390,10752,10754,10757],{"className":10753},[402],[390,10755],{"className":10756,"style":697},[406],[390,10758,1198],{"className":10759},[411]," points of the strip.",[513,10762,10763],{"type":7105},[381,10764,10765,10767,10768,10792,10793,10820,10821,10839,10840,10995,10996],{},[385,10766,7110],{}," If two points ",[390,10769,10771],{"className":10770},[393],[390,10772,10774],{"className":10773,"ariaHidden":398},[397],[390,10775,10777,10780,10783,10786,10789],{"className":10776},[402],[390,10778],{"className":10779,"style":962},[406],[390,10781,967],{"className":10782,"style":966},[411,412],[390,10784,1785],{"className":10785},[1784],[390,10787],{"className":10788,"style":444},[443],[390,10790,945],{"className":10791,"style":944},[411,412]," shared a cell, they would lie on the same side of\nthe dividing line, so their distance is already ",[390,10794,10796],{"className":10795},[393],[390,10797,10799,10811],{"className":10798,"ariaHidden":398},[397],[390,10800,10802,10805,10808],{"className":10801},[402],[390,10803],{"className":10804,"style":3801},[406],[390,10806,3805],{"className":10807},[608],[390,10809],{"className":10810,"style":604},[443],[390,10812,10814,10817],{"className":10813},[402],[390,10815],{"className":10816,"style":479},[406],[390,10818,9832],{"className":10819,"style":9831},[411,412]," (the recursive\nguarantee). But two points in a ",[390,10822,10824],{"className":10823},[393],[390,10825,10827],{"className":10826,"ariaHidden":398},[397],[390,10828,10830,10833,10836],{"className":10829},[402],[390,10831],{"className":10832,"style":427},[406],[390,10834,9832],{"className":10835,"style":9831},[411,412],[390,10837,858],{"className":10838},[411],"-square are at most\n",[390,10841,10843],{"className":10842},[393],[390,10844,10846,10986],{"className":10845,"ariaHidden":398},[397],[390,10847,10849,10853,10921,10977,10980,10983],{"className":10848},[402],[390,10850],{"className":10851,"style":10852},[406],"height:1.2522em;vertical-align:-0.345em;",[390,10854,10856,10859,10918],{"className":10855},[411],[390,10857],{"className":10858},[435,1876],[390,10860,10862],{"className":10861},[1880],[390,10863,10865,10910],{"className":10864},[1884,1885],[390,10866,10868,10907],{"className":10867},[1889],[390,10869,10871,10885,10893],{"className":10870,"style":6414},[1893],[390,10872,10873,10876],{"style":1897},[390,10874],{"className":10875,"style":1902},[1901],[390,10877,10879],{"className":10878},[1906,1907,1908,1909],[390,10880,10882],{"className":10881},[411,1909],[390,10883,1198],{"className":10884},[411,1909],[390,10886,10887,10890],{"style":1919},[390,10888],{"className":10889,"style":1902},[1901],[390,10891],{"className":10892,"style":1927},[1926],[390,10894,10895,10898],{"style":1930},[390,10896],{"className":10897,"style":1902},[1901],[390,10899,10901],{"className":10900},[1906,1907,1908,1909],[390,10902,10904],{"className":10903},[411,1909],[390,10905,9832],{"className":10906,"style":9831},[411,412,1909],[390,10908,1947],{"className":10909},[1946],[390,10911,10913],{"className":10912},[1889],[390,10914,10916],{"className":10915,"style":1954},[1893],[390,10917],{},[390,10919],{"className":10920},[464,1876],[390,10922,10924],{"className":10923},[411,9300],[390,10925,10927,10968],{"className":10926},[1884,1885],[390,10928,10930,10965],{"className":10929},[1889],[390,10931,10934,10948],{"className":10932,"style":10933},[1893],"height:0.9072em;",[390,10935,10938,10941],{"className":10936,"style":10937},[9314],"top:-3em;",[390,10939],{"className":10940,"style":1902},[1901],[390,10942,10945],{"className":10943,"style":10944},[411],"padding-left:0.833em;",[390,10946,1198],{"className":10947},[411],[390,10949,10951,10954],{"style":10950},"top:-2.8672em;",[390,10952],{"className":10953,"style":1902},[1901],[390,10955,10958],{"className":10956,"style":10957},[9587],"min-width:0.853em;height:1.08em;",[1206,10959,10962],{"xmlns":1208,"width":9591,"height":10960,"viewBox":10961,"preserveAspectRatio":9594},"1.08em","0 0 400000 1080",[1218,10963],{"d":10964},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[390,10966,1947],{"className":10967},[1946],[390,10969,10971],{"className":10970},[1889],[390,10972,10975],{"className":10973,"style":10974},[1893],"height:0.1328em;",[390,10976],{},[390,10978],{"className":10979,"style":604},[443],[390,10981,1528],{"className":10982},[608],[390,10984],{"className":10985,"style":604},[443],[390,10987,10989,10992],{"className":10988},[402],[390,10990],{"className":10991,"style":479},[406],[390,10993,9832],{"className":10994,"style":9831},[411,412]," apart — contradiction. ",[390,10997,10999],{"className":10998},[393],[390,11000,11002],{"className":11001,"ariaHidden":398},[397],[390,11003,11005,11008],{"className":11004},[402],[390,11006],{"className":11007,"style":8160},[406],[390,11009,8175],{"className":11010},[411,8174],[381,11012,11013,11014,11029,11030,11054,11055,11070,11071,11086,11087,584,11111,11201,11202,11205,11206,11230,11231,720],{},"So the box holds at most ",[390,11015,11017],{"className":11016},[393],[390,11018,11020],{"className":11019,"ariaHidden":398},[397],[390,11021,11023,11026],{"className":11022},[402],[390,11024],{"className":11025,"style":697},[406],[390,11027,1289],{"className":11028},[411]," points, one of which is ",[390,11031,11033],{"className":11032},[393],[390,11034,11036],{"className":11035,"ariaHidden":398},[397],[390,11037,11039,11042,11045,11048,11051],{"className":11038},[402],[390,11040],{"className":11041,"style":427},[406],[390,11043,10526],{"className":11044,"style":10525},[411,412],[390,11046,541],{"className":11047},[435],[390,11049,1542],{"className":11050},[411,412],[390,11052,552],{"className":11053},[464]," itself: at most ",[390,11056,11058],{"className":11057},[393],[390,11059,11061],{"className":11060,"ariaHidden":398},[397],[390,11062,11064,11067],{"className":11063},[402],[390,11065],{"className":11066,"style":697},[406],[390,11068,1366],{"className":11069},[411],"\ncandidates. Hence in ",[390,11072,11074],{"className":11073},[393],[390,11075,11077],{"className":11076,"ariaHidden":398},[397],[390,11078,11080,11083],{"className":11079},[402],[390,11081],{"className":11082,"style":962},[406],[390,11084,9096],{"className":11085,"style":966},[411,412],"-sorted order it suffices to compare ",[390,11088,11090],{"className":11089},[393],[390,11091,11093],{"className":11092,"ariaHidden":398},[397],[390,11094,11096,11099,11102,11105,11108],{"className":11095},[402],[390,11097],{"className":11098,"style":427},[406],[390,11100,10526],{"className":11101,"style":10525},[411,412],[390,11103,541],{"className":11104},[435],[390,11106,1542],{"className":11107},[411,412],[390,11109,552],{"className":11110},[464],[390,11112,11114],{"className":11113},[393],[390,11115,11117,11141,11189],{"className":11116,"ariaHidden":398},[397],[390,11118,11120,11123,11126,11129,11132,11135,11138],{"className":11119},[402],[390,11121],{"className":11122,"style":427},[406],[390,11124,10526],{"className":11125,"style":10525},[411,412],[390,11127,541],{"className":11128},[435],[390,11130,1542],{"className":11131},[411,412],[390,11133],{"className":11134,"style":682},[443],[390,11136,845],{"className":11137},[686],[390,11139],{"className":11140,"style":682},[443],[390,11142,11144,11147,11150,11153,11156,11159,11162,11165,11168,11171,11174,11177,11180,11183,11186],{"className":11143},[402],[390,11145],{"className":11146,"style":427},[406],[390,11148,500],{"className":11149},[411],[390,11151,552],{"className":11152},[464],[390,11154,1785],{"className":11155},[1784],[390,11157],{"className":11158,"style":444},[443],[390,11160,2752],{"className":11161},[826],[390,11163],{"className":11164,"style":444},[443],[390,11166,1785],{"className":11167},[1784],[390,11169],{"className":11170,"style":444},[443],[390,11172,10526],{"className":11173,"style":10525},[411,412],[390,11175,541],{"className":11176},[435],[390,11178,1542],{"className":11179},[411,412],[390,11181],{"className":11182,"style":682},[443],[390,11184,845],{"className":11185},[686],[390,11187],{"className":11188,"style":682},[443],[390,11190,11192,11195,11198],{"className":11191},[402],[390,11193],{"className":11194,"style":427},[406],[390,11196,1366],{"className":11197},[411],[390,11199,552],{"className":11200},[464]," — a ",[385,11203,11204],{},"constant"," number of checks per point. The combine\nstep is ",[390,11207,11209],{"className":11208},[393],[390,11210,11212],{"className":11211,"ariaHidden":398},[397],[390,11213,11215,11218,11221,11224,11227],{"className":11214},[402],[390,11216],{"className":11217,"style":427},[406],[390,11219,2340],{"className":11220,"style":944},[411,412],[390,11222,436],{"className":11223},[435],[390,11225,413],{"className":11226},[411,412],[390,11228,465],{"className":11229},[464],", not ",[390,11232,11234],{"className":11233},[393],[390,11235,11237],{"className":11236,"ariaHidden":398},[397],[390,11238,11240,11243,11246,11249,11278],{"className":11239},[402],[390,11241],{"className":11242,"style":9732},[406],[390,11244,431],{"className":11245},[411],[390,11247,436],{"className":11248},[435],[390,11250,11252,11255],{"className":11251},[411],[390,11253,413],{"className":11254},[411,412],[390,11256,11258],{"className":11257},[2062],[390,11259,11261],{"className":11260},[1884],[390,11262,11264],{"className":11263},[1889],[390,11265,11267],{"className":11266,"style":2727},[1893],[390,11268,11269,11272],{"style":2730},[390,11270],{"className":11271,"style":2079},[1901],[390,11273,11275],{"className":11274},[1906,1907,1908,1909],[390,11276,1198],{"className":11277},[411,1909],[390,11279,465],{"className":11280},[464],[1615,11282,11284],{"id":11283},"the-algorithm","The algorithm",[381,11286,11287,11288,11290,11291,11306,11307,11327,11328,11343,11344,11363],{},"The whole thing rests on having the points pre-sorted ",[385,11289,910],{}," ways. Sort once\nby ",[390,11292,11294],{"className":11293},[393],[390,11295,11297],{"className":11296,"ariaHidden":398},[397],[390,11298,11300,11303],{"className":11299},[402],[390,11301],{"className":11302,"style":407},[406],[390,11304,1488],{"className":11305},[411,412]," (call it ",[390,11308,11310],{"className":11309},[393],[390,11311,11313],{"className":11312,"ariaHidden":398},[397],[390,11314,11316,11319,11322],{"className":11315},[402],[390,11317],{"className":11318,"style":658},[406],[390,11320,537],{"className":11321},[411,412],[390,11323,11326],{"className":11324,"style":11325},[411,412],"margin-right:0.0785em;","X",") and once by ",[390,11329,11331],{"className":11330},[393],[390,11332,11334],{"className":11333,"ariaHidden":398},[397],[390,11335,11337,11340],{"className":11336},[402],[390,11338],{"className":11339,"style":962},[406],[390,11341,9096],{"className":11342,"style":966},[411,412]," (",[390,11345,11347],{"className":11346},[393],[390,11348,11350],{"className":11349,"ariaHidden":398},[397],[390,11351,11353,11356,11359],{"className":11352},[402],[390,11354],{"className":11355,"style":658},[406],[390,11357,537],{"className":11358},[411,412],[390,11360,11362],{"className":11361,"style":682},[411,412],"Y",") up front; the recursion then\nthreads these sorted orders down without re-sorting.",[1545,11365,11367],{"className":1547,"code":11366,"language":1549,"meta":376,"style":376},"caption: $\\textsc{Closest-Pair}(P)$ — closest pair in $O(n\\log n)$\nnumber: 3\n$AX \\gets P$ sorted by $x$-coordinate\n$AY \\gets P$ sorted by $y$-coordinate\nreturn call $\\textsc{Closest-Pair-Rec}(AX, AY)$\n",[1551,11368,11369,11374,11379,11384,11389],{"__ignoreMap":376},[390,11370,11371],{"class":1555,"line":6},[390,11372,11373],{},"caption: $\\textsc{Closest-Pair}(P)$ — closest pair in $O(n\\log n)$\n",[390,11375,11376],{"class":1555,"line":18},[390,11377,11378],{},"number: 3\n",[390,11380,11381],{"class":1555,"line":24},[390,11382,11383],{},"$AX \\gets P$ sorted by $x$-coordinate\n",[390,11385,11386],{"class":1555,"line":73},[390,11387,11388],{},"$AY \\gets P$ sorted by $y$-coordinate\n",[390,11390,11391],{"class":1555,"line":102},[390,11392,11393],{},"return call $\\textsc{Closest-Pair-Rec}(AX, AY)$\n",[1545,11395,11397],{"className":1547,"code":11396,"language":1549,"meta":376,"style":376},"caption: $\\textsc{Closest-Pair-Rec}(AX, AY)$ — $AX$ sorted by $x$, $AY$ by $y$\nnumber: 4\n$n \\gets |AX|$\nif $n \u003C 2$ then\n  return $\\bot$ \u002F\u002F no pair exists\nif $n = 2$ then\n  return $(AX[1], AX[2])$\n$\\textit{mid} \\gets AX[\\ceil{n\u002F2}]$ \u002F\u002F line through median x\n$LX \\gets AX[1 \\,..\\, \\ceil{n\u002F2}]$; $\\quad RX \\gets AX[\\ceil{n\u002F2}+1 \\,..\\, n]$ \u002F\u002F split by index (ties safe)\n$LY, RY \\gets$ the same split applied to $AY$ \u002F\u002F keeps halves y-sorted, O(n)\n$(uL, vL) \\gets$ call $\\textsc{Closest-Pair-Rec}(LX, LY)$ \u002F\u002F best pair on left\n$(uR, vR) \\gets$ call $\\textsc{Closest-Pair-Rec}(RX, RY)$ \u002F\u002F best pair on right\n$\\delta \\gets \\min\\set{\\operatorname{dist}(uL, vL),\\ \\operatorname{dist}(uR, vR)}$\n$B \\gets [\\,p \\in AY : \\textit{mid}_x - \\delta \\le p_x \\le \\textit{mid}_x + \\delta\\,]$ \u002F\u002F strip, still y-sorted\n$(uM, vM) \\gets$ call $\\textsc{Best-Pair}(B, \\delta)$\nreturn the closest of $(uL, vL)$, $(uR, vR)$, $(uM, vM)$\n",[1551,11398,11399,11404,11409,11414,11419,11424,11429,11434,11439,11444,11449,11454,11459,11464,11469,11474],{"__ignoreMap":376},[390,11400,11401],{"class":1555,"line":6},[390,11402,11403],{},"caption: $\\textsc{Closest-Pair-Rec}(AX, AY)$ — $AX$ sorted by $x$, $AY$ by $y$\n",[390,11405,11406],{"class":1555,"line":18},[390,11407,11408],{},"number: 4\n",[390,11410,11411],{"class":1555,"line":24},[390,11412,11413],{},"$n \\gets |AX|$\n",[390,11415,11416],{"class":1555,"line":73},[390,11417,11418],{},"if $n \u003C 2$ then\n",[390,11420,11421],{"class":1555,"line":102},[390,11422,11423],{},"  return $\\bot$ \u002F\u002F no pair exists\n",[390,11425,11426],{"class":1555,"line":108},[390,11427,11428],{},"if $n = 2$ then\n",[390,11430,11431],{"class":1555,"line":116},[390,11432,11433],{},"  return $(AX[1], AX[2])$\n",[390,11435,11436],{"class":1555,"line":196},[390,11437,11438],{},"$\\textit{mid} \\gets AX[\\ceil{n\u002F2}]$ \u002F\u002F line through median x\n",[390,11440,11441],{"class":1555,"line":202},[390,11442,11443],{},"$LX \\gets AX[1 \\,..\\, \\ceil{n\u002F2}]$; $\\quad RX \\gets AX[\\ceil{n\u002F2}+1 \\,..\\, n]$ \u002F\u002F split by index (ties safe)\n",[390,11445,11446],{"class":1555,"line":283},[390,11447,11448],{},"$LY, RY \\gets$ the same split applied to $AY$ \u002F\u002F keeps halves y-sorted, O(n)\n",[390,11450,11451],{"class":1555,"line":333},[390,11452,11453],{},"$(uL, vL) \\gets$ call $\\textsc{Closest-Pair-Rec}(LX, LY)$ \u002F\u002F best pair on left\n",[390,11455,11456],{"class":1555,"line":354},[390,11457,11458],{},"$(uR, vR) \\gets$ call $\\textsc{Closest-Pair-Rec}(RX, RY)$ \u002F\u002F best pair on right\n",[390,11460,11461],{"class":1555,"line":3063},[390,11462,11463],{},"$\\delta \\gets \\min\\set{\\operatorname{dist}(uL, vL),\\ \\operatorname{dist}(uR, vR)}$\n",[390,11465,11466],{"class":1555,"line":3068},[390,11467,11468],{},"$B \\gets [\\,p \\in AY : \\textit{mid}_x - \\delta \\le p_x \\le \\textit{mid}_x + \\delta\\,]$ \u002F\u002F strip, still y-sorted\n",[390,11470,11471],{"class":1555,"line":3074},[390,11472,11473],{},"$(uM, vM) \\gets$ call $\\textsc{Best-Pair}(B, \\delta)$\n",[390,11475,11476],{"class":1555,"line":3079},[390,11477,11478],{},"return the closest of $(uL, vL)$, $(uR, vR)$, $(uM, vM)$\n",[1545,11480,11482],{"className":1547,"code":11481,"language":1549,"meta":376,"style":376},"caption: $\\textsc{Best-Pair}(B, \\delta)$ — best pair inside the strip; $B$ sorted by $y$\nnumber: 5\n$(u_m, v_m) \\gets (\\bot, \\bot)$; $\\quad d_m \\gets \\delta$ \u002F\u002F best pair so far + its dist\nfor $i \\gets 1$ to $|B|$ do\n  for $j \\gets i + 1$ to $\\min\\set{i + 7,\\ |B|}$ do \u002F\u002F only 7 neighbors each!\n    if $\\operatorname{dist}(B[i], B[j]) \u003C d_m$ then\n      $(u_m, v_m) \\gets (B[i], B[j])$; $\\quad d_m \\gets \\operatorname{dist}(B[i], B[j])$\nreturn $(u_m, v_m)$\n",[1551,11483,11484,11489,11494,11499,11504,11509,11514,11519],{"__ignoreMap":376},[390,11485,11486],{"class":1555,"line":6},[390,11487,11488],{},"caption: $\\textsc{Best-Pair}(B, \\delta)$ — best pair inside the strip; $B$ sorted by $y$\n",[390,11490,11491],{"class":1555,"line":18},[390,11492,11493],{},"number: 5\n",[390,11495,11496],{"class":1555,"line":24},[390,11497,11498],{},"$(u_m, v_m) \\gets (\\bot, \\bot)$; $\\quad d_m \\gets \\delta$ \u002F\u002F best pair so far + its dist\n",[390,11500,11501],{"class":1555,"line":73},[390,11502,11503],{},"for $i \\gets 1$ to $|B|$ do\n",[390,11505,11506],{"class":1555,"line":102},[390,11507,11508],{},"  for $j \\gets i + 1$ to $\\min\\set{i + 7,\\ |B|}$ do \u002F\u002F only 7 neighbors each!\n",[390,11510,11511],{"class":1555,"line":108},[390,11512,11513],{},"    if $\\operatorname{dist}(B[i], B[j]) \u003C d_m$ then\n",[390,11515,11516],{"class":1555,"line":116},[390,11517,11518],{},"      $(u_m, v_m) \\gets (B[i], B[j])$; $\\quad d_m \\gets \\operatorname{dist}(B[i], B[j])$\n",[390,11520,11521],{"class":1555,"line":196},[390,11522,11523],{},"return $(u_m, v_m)$\n",[1615,11525,11527],{"id":11526},"running-time-and-correctness","Running time and correctness",[381,11529,11530,11531,11553,11554,11578],{},"The split, the strip extraction, and ",[390,11532,11534],{"className":11533},[393],[390,11535,11537],{"className":11536,"ariaHidden":398},[397],[390,11538,11540,11543],{"className":11539},[402],[390,11541],{"className":11542,"style":658},[406],[390,11544,11546],{"className":11545},[3158,3159],[390,11547,11549],{"className":11548},[411,3163],[390,11550,11552],{"className":11551},[411],"Best-Pair"," are each ",[390,11555,11557],{"className":11556},[393],[390,11558,11560],{"className":11559,"ariaHidden":398},[397],[390,11561,11563,11566,11569,11572,11575],{"className":11562},[402],[390,11564],{"className":11565,"style":427},[406],[390,11567,2340],{"className":11568,"style":944},[411,412],[390,11570,436],{"className":11571},[435],[390,11573,413],{"className":11574},[411,412],[390,11576,465],{"className":11577},[464],", and\nthere are two half-size recursive calls:",[390,11580,11582],{"className":11581},[1694],[390,11583,11585],{"className":11584},[393],[390,11586,11588,11615,11722,11756,11783],{"className":11587,"ariaHidden":398},[397],[390,11589,11591,11594,11597,11600,11603,11606,11609,11612],{"className":11590},[402],[390,11592],{"className":11593,"style":427},[406],[390,11595,1711],{"className":11596,"style":1710},[411,412],[390,11598,436],{"className":11599},[435],[390,11601,413],{"className":11602},[411,412],[390,11604,465],{"className":11605},[464],[390,11607],{"className":11608,"style":604},[443],[390,11610,609],{"className":11611},[608],[390,11613],{"className":11614,"style":604},[443],[390,11616,11618,11621,11624,11627,11630,11633,11636,11713,11716,11719],{"className":11617},[402],[390,11619],{"className":11620,"style":4817},[406],[390,11622,1198],{"className":11623},[411],[390,11625],{"className":11626,"style":444},[443],[390,11628,1711],{"className":11629,"style":1710},[411,412],[390,11631],{"className":11632,"style":1843},[443],[390,11634],{"className":11635,"style":444},[443],[390,11637,11639,11645,11707],{"className":11638},[826],[390,11640,11642],{"className":11641,"style":831},[435,830],[390,11643,436],{"className":11644},[1856,1857],[390,11646,11648,11651,11704],{"className":11647},[411],[390,11649],{"className":11650},[435,1876],[390,11652,11654],{"className":11653},[1880],[390,11655,11657,11696],{"className":11656},[1884,1885],[390,11658,11660,11693],{"className":11659},[1889],[390,11661,11663,11674,11682],{"className":11662,"style":4115},[1893],[390,11664,11665,11668],{"style":2174},[390,11666],{"className":11667,"style":1902},[1901],[390,11669,11671],{"className":11670},[411],[390,11672,1198],{"className":11673},[411],[390,11675,11676,11679],{"style":1919},[390,11677],{"className":11678,"style":1902},[1901],[390,11680],{"className":11681,"style":1927},[1926],[390,11683,11684,11687],{"style":2207},[390,11685],{"className":11686,"style":1902},[1901],[390,11688,11690],{"className":11689},[411],[390,11691,413],{"className":11692},[411,412],[390,11694,1947],{"className":11695},[1946],[390,11697,11699],{"className":11698},[1889],[390,11700,11702],{"className":11701,"style":4082},[1893],[390,11703],{},[390,11705],{"className":11706},[464,1876],[390,11708,11710],{"className":11709,"style":831},[464,830],[390,11711,465],{"className":11712},[1856,1857],[390,11714],{"className":11715,"style":682},[443],[390,11717,845],{"className":11718},[686],[390,11720],{"className":11721,"style":682},[443],[390,11723,11725,11728,11731,11734,11737,11740,11743,11746,11750,11753],{"className":11724},[402],[390,11726],{"className":11727,"style":427},[406],[390,11729,2340],{"className":11730,"style":944},[411,412],[390,11732,436],{"className":11733},[435],[390,11735,413],{"className":11736},[411,412],[390,11738,465],{"className":11739},[464],[390,11741],{"className":11742,"style":604},[443],[390,11744],{"className":11745,"style":604},[443],[390,11747,11749],{"className":11748},[608],"⟹",[390,11751],{"className":11752,"style":604},[443],[390,11754],{"className":11755,"style":604},[443],[390,11757,11759,11762,11765,11768,11771,11774,11777,11780],{"className":11758},[402],[390,11760],{"className":11761,"style":427},[406],[390,11763,1711],{"className":11764,"style":1710},[411,412],[390,11766,436],{"className":11767},[435],[390,11769,413],{"className":11770},[411,412],[390,11772,465],{"className":11773},[464],[390,11775],{"className":11776,"style":604},[443],[390,11778,745],{"className":11779},[608],[390,11781],{"className":11782,"style":604},[443],[390,11784,11786,11789,11792,11795,11798,11801,11807,11810,11813,11816],{"className":11785},[402],[390,11787],{"className":11788,"style":427},[406],[390,11790,2340],{"className":11791,"style":944},[411,412],[390,11793,436],{"className":11794},[435],[390,11796,413],{"className":11797},[411,412],[390,11799],{"className":11800,"style":444},[443],[390,11802,11804],{"className":11803},[448],[390,11805,454],{"className":11806,"style":453},[411,452],[390,11808],{"className":11809,"style":444},[443],[390,11811,413],{"className":11812},[411,412],[390,11814,465],{"className":11815},[464],[390,11817,1785],{"className":11818},[1784],[381,11820,11821,11822,11825,11826,11865,11866,720],{},"by the master theorem, the same shape as\n",[494,11823,11824],{"href":34},"mergesort",". The initial sorts add a\none-time ",[390,11827,11829],{"className":11828},[393],[390,11830,11832],{"className":11831,"ariaHidden":398},[397],[390,11833,11835,11838,11841,11844,11847,11850,11856,11859,11862],{"className":11834},[402],[390,11836],{"className":11837,"style":427},[406],[390,11839,2340],{"className":11840,"style":944},[411,412],[390,11842,436],{"className":11843},[435],[390,11845,413],{"className":11846},[411,412],[390,11848],{"className":11849,"style":444},[443],[390,11851,11853],{"className":11852},[448],[390,11854,454],{"className":11855,"style":453},[411,452],[390,11857],{"className":11858,"style":444},[443],[390,11860,413],{"className":11861},[411,412],[390,11863,465],{"className":11864},[464],", so the total stays ",[390,11867,11869],{"className":11868},[393],[390,11870,11872],{"className":11871,"ariaHidden":398},[397],[390,11873,11875,11878,11881,11884,11887,11890,11896,11899,11902],{"className":11874},[402],[390,11876],{"className":11877,"style":427},[406],[390,11879,2340],{"className":11880,"style":944},[411,412],[390,11882,436],{"className":11883},[435],[390,11885,413],{"className":11886},[411,412],[390,11888],{"className":11889,"style":444},[443],[390,11891,11893],{"className":11892},[448],[390,11894,454],{"className":11895,"style":453},[411,452],[390,11897],{"className":11898,"style":444},[443],[390,11900,413],{"className":11901},[411,412],[390,11903,465],{"className":11904},[464],[381,11906,11907,11910,11911,8277,11926,11929,11930,11951,11952,11976,11977,11992,11993,12008,12009,12033,12034,12058,12059,12123],{},[385,11908,11909],{},"Correctness"," is an easy induction on ",[390,11912,11914],{"className":11913},[393],[390,11915,11917],{"className":11916,"ariaHidden":398},[397],[390,11918,11920,11923],{"className":11919},[402],[390,11921],{"className":11922,"style":407},[406],[390,11924,413],{"className":11925},[411,412],[503,11927,11928],{},"given"," that ",[390,11931,11933],{"className":11932},[393],[390,11934,11936],{"className":11935,"ariaHidden":398},[397],[390,11937,11939,11942],{"className":11938},[402],[390,11940],{"className":11941,"style":658},[406],[390,11943,11945],{"className":11944},[3158,3159],[390,11946,11948],{"className":11947},[411,3163],[390,11949,11552],{"className":11950},[411],"\nfinds any closest cross-strip pair. That assumption is exactly the 7-neighbor\nclaim above: for each ",[390,11953,11955],{"className":11954},[393],[390,11956,11958],{"className":11957,"ariaHidden":398},[397],[390,11959,11961,11964,11967,11970,11973],{"className":11960},[402],[390,11962],{"className":11963,"style":427},[406],[390,11965,10526],{"className":11966,"style":10525},[411,412],[390,11968,541],{"className":11969},[435],[390,11971,1542],{"className":11972},[411,412],[390,11974,552],{"className":11975},[464],", every later strip point that could improve on\n",[390,11978,11980],{"className":11979},[393],[390,11981,11983],{"className":11982,"ariaHidden":398},[397],[390,11984,11986,11989],{"className":11985},[402],[390,11987],{"className":11988,"style":479},[406],[390,11990,9832],{"className":11991,"style":9831},[411,412]," lies in one of the ",[390,11994,11996],{"className":11995},[393],[390,11997,11999],{"className":11998,"ariaHidden":398},[397],[390,12000,12002,12005],{"className":12001},[402],[390,12003],{"className":12004,"style":697},[406],[390,12006,1366],{"className":12007},[411]," cells of ",[390,12010,12012],{"className":12011},[393],[390,12013,12015],{"className":12014,"ariaHidden":398},[397],[390,12016,12018,12021,12024,12027,12030],{"className":12017},[402],[390,12019],{"className":12020,"style":427},[406],[390,12022,10526],{"className":12023,"style":10525},[411,412],[390,12025,541],{"className":12026},[435],[390,12028,1542],{"className":12029},[411,412],[390,12031,552],{"className":12032},[464],"'s grid (the cell holding ",[390,12035,12037],{"className":12036},[393],[390,12038,12040],{"className":12039,"ariaHidden":398},[397],[390,12041,12043,12046,12049,12052,12055],{"className":12042},[402],[390,12044],{"className":12045,"style":427},[406],[390,12047,10526],{"className":12048,"style":10525},[411,412],[390,12050,541],{"className":12051},[435],[390,12053,1542],{"className":12054},[411,412],[390,12056,552],{"className":12057},[464],"\naside), and each cell holds at most one point — so scanning ",[390,12060,12062],{"className":12061},[393],[390,12063,12065,12089,12111],{"className":12064,"ariaHidden":398},[397],[390,12066,12068,12071,12074,12077,12080,12083,12086],{"className":12067},[402],[390,12069],{"className":12070,"style":427},[406],[390,12072,10526],{"className":12073,"style":10525},[411,412],[390,12075,541],{"className":12076},[435],[390,12078,1542],{"className":12079},[411,412],[390,12081],{"className":12082,"style":682},[443],[390,12084,845],{"className":12085},[686],[390,12087],{"className":12088,"style":682},[443],[390,12090,12092,12096,12099,12102,12105,12108],{"className":12091},[402],[390,12093],{"className":12094,"style":12095},[406],"height:0.7429em;vertical-align:-0.0833em;",[390,12097,545],{"className":12098},[411],[390,12100,1542],{"className":12101},[411,412],[390,12103],{"className":12104,"style":682},[443],[390,12106,845],{"className":12107},[686],[390,12109],{"className":12110,"style":682},[443],[390,12112,12114,12117,12120],{"className":12113},[402],[390,12115],{"className":12116,"style":427},[406],[390,12118,1366],{"className":12119},[411],[390,12121,552],{"className":12122},[464]," cannot\nmiss it.",[1045,12125,12126,12175,12275,12315,12642,12645],{},[1048,12127,12128,12131,12132,12147,12148,12150,12151,720],{},[385,12129,12130],{},"Selection"," finds the ",[390,12133,12135],{"className":12134},[393],[390,12136,12138],{"className":12137,"ariaHidden":398},[397],[390,12139,12141,12144],{"className":12140},[402],[390,12142],{"className":12143,"style":479},[406],[390,12145,484],{"className":12146,"style":483},[411,412],"-th smallest element; it needs ",[503,12149,505],{}," than\nsorting, so it can run in ",[390,12152,12154],{"className":12153},[393],[390,12155,12157],{"className":12156,"ariaHidden":398},[397],[390,12158,12160,12163,12166,12169,12172],{"className":12159},[402],[390,12161],{"className":12162,"style":427},[406],[390,12164,2340],{"className":12165,"style":944},[411,412],[390,12167,436],{"className":12168},[435],[390,12170,413],{"className":12171},[411,412],[390,12173,465],{"className":12174},[464],[1048,12176,12177,12199,12200,12224,12225,720],{},[390,12178,12180],{"className":12179},[393],[390,12181,12183],{"className":12182,"ariaHidden":398},[397],[390,12184,12186,12189],{"className":12185},[402],[390,12187],{"className":12188,"style":6957},[406],[390,12190,12192],{"className":12191},[3158,3159],[390,12193,12195],{"className":12194},[411,3163],[390,12196,12198],{"className":12197},[411],"Quickselect"," = quicksort's partition, but recurse into only the side that\nholds the answer; expected ",[390,12201,12203],{"className":12202},[393],[390,12204,12206],{"className":12205,"ariaHidden":398},[397],[390,12207,12209,12212,12215,12218,12221],{"className":12208},[402],[390,12210],{"className":12211,"style":427},[406],[390,12213,2340],{"className":12214,"style":944},[411,412],[390,12216,436],{"className":12217},[435],[390,12219,413],{"className":12220},[411,412],[390,12222,465],{"className":12223},[464]," via a geometric (non-branching) recurrence,\nworst case ",[390,12226,12228],{"className":12227},[393],[390,12229,12231],{"className":12230,"ariaHidden":398},[397],[390,12232,12234,12237,12240,12243,12272],{"className":12233},[402],[390,12235],{"className":12236,"style":9732},[406],[390,12238,431],{"className":12239},[411],[390,12241,436],{"className":12242},[435],[390,12244,12246,12249],{"className":12245},[411],[390,12247,413],{"className":12248},[411,412],[390,12250,12252],{"className":12251},[2062],[390,12253,12255],{"className":12254},[1884],[390,12256,12258],{"className":12257},[1889],[390,12259,12261],{"className":12260,"style":2727},[1893],[390,12262,12263,12266],{"style":2730},[390,12264],{"className":12265,"style":2079},[1901],[390,12267,12269],{"className":12268},[1906,1907,1908,1909],[390,12270,1198],{"className":12271},[411,1909],[390,12273,465],{"className":12274},[464],[1048,12276,12277,12280,12281,12314],{},[385,12278,12279],{},"Median of medians"," chooses a provably balanced pivot using groups of five,\nguaranteeing the recursion drops at least ",[390,12282,12284],{"className":12283},[393],[390,12285,12287,12299],{"className":12286,"ariaHidden":398},[397],[390,12288,12290,12293,12296],{"className":12289},[402],[390,12291],{"className":12292,"style":4273},[406],[390,12294,4175],{"className":12295},[608],[390,12297],{"className":12298,"style":604},[443],[390,12300,12302,12305,12308,12311],{"className":12301},[402],[390,12303],{"className":12304,"style":427},[406],[390,12306,1682],{"className":12307},[411],[390,12309,413],{"className":12310},[411,412],[390,12312,4295],{"className":12313},[411]," elements per side.",[1048,12316,12317,12318,12432,12433,12614,12615,720],{},"That balance yields ",[390,12319,12321],{"className":12320},[393],[390,12322,12324,12351,12381,12414],{"className":12323,"ariaHidden":398},[397],[390,12325,12327,12330,12333,12336,12339,12342,12345,12348],{"className":12326},[402],[390,12328],{"className":12329,"style":427},[406],[390,12331,1711],{"className":12332,"style":1710},[411,412],[390,12334,436],{"className":12335},[435],[390,12337,413],{"className":12338},[411,412],[390,12340,465],{"className":12341},[464],[390,12343],{"className":12344,"style":604},[443],[390,12346,609],{"className":12347},[608],[390,12349],{"className":12350,"style":604},[443],[390,12352,12354,12357,12360,12363,12366,12369,12372,12375,12378],{"className":12353},[402],[390,12355],{"className":12356,"style":427},[406],[390,12358,1711],{"className":12359,"style":1710},[411,412],[390,12361,436],{"className":12362},[435],[390,12364,413],{"className":12365},[411,412],[390,12367,3115],{"className":12368},[411],[390,12370,465],{"className":12371},[464],[390,12373],{"className":12374,"style":682},[443],[390,12376,845],{"className":12377},[686],[390,12379],{"className":12380,"style":682},[443],[390,12382,12384,12387,12390,12393,12396,12399,12402,12405,12408,12411],{"className":12383},[402],[390,12385],{"className":12386,"style":427},[406],[390,12388,1711],{"className":12389,"style":1710},[411,412],[390,12391,436],{"className":12392},[435],[390,12394,1366],{"className":12395},[411],[390,12397,413],{"className":12398},[411,412],[390,12400,4295],{"className":12401},[411],[390,12403,465],{"className":12404},[464],[390,12406],{"className":12407,"style":682},[443],[390,12409,845],{"className":12410},[686],[390,12412],{"className":12413,"style":682},[443],[390,12415,12417,12420,12423,12426,12429],{"className":12416},[402],[390,12418],{"className":12419,"style":427},[406],[390,12421,2340],{"className":12422,"style":944},[411,412],[390,12424,436],{"className":12425},[435],[390,12427,413],{"className":12428},[411,412],[390,12430,465],{"className":12431},[464],", and because\n",[390,12434,12436],{"className":12435},[393],[390,12437,12439,12522,12605],{"className":12438,"ariaHidden":398},[397],[390,12440,12442,12445,12513,12516,12519],{"className":12441},[402],[390,12443],{"className":12444,"style":2551},[406],[390,12446,12448,12451,12510],{"className":12447},[411],[390,12449],{"className":12450},[435,1876],[390,12452,12454],{"className":12453},[1880],[390,12455,12457,12502],{"className":12456},[1884,1885],[390,12458,12460,12499],{"className":12459},[1889],[390,12461,12463,12477,12485],{"className":12462,"style":1894},[1893],[390,12464,12465,12468],{"style":1897},[390,12466],{"className":12467,"style":1902},[1901],[390,12469,12471],{"className":12470},[1906,1907,1908,1909],[390,12472,12474],{"className":12473},[411,1909],[390,12475,3417],{"className":12476},[411,1909],[390,12478,12479,12482],{"style":1919},[390,12480],{"className":12481,"style":1902},[1901],[390,12483],{"className":12484,"style":1927},[1926],[390,12486,12487,12490],{"style":1930},[390,12488],{"className":12489,"style":1902},[1901],[390,12491,12493],{"className":12492},[1906,1907,1908,1909],[390,12494,12496],{"className":12495},[411,1909],[390,12497,500],{"className":12498},[411,1909],[390,12500,1947],{"className":12501},[1946],[390,12503,12505],{"className":12504},[1889],[390,12506,12508],{"className":12507,"style":1954},[1893],[390,12509],{},[390,12511],{"className":12512},[464,1876],[390,12514],{"className":12515,"style":682},[443],[390,12517,845],{"className":12518},[686],[390,12520],{"className":12521,"style":682},[443],[390,12523,12525,12528,12596,12599,12602],{"className":12524},[402],[390,12526],{"className":12527,"style":2551},[406],[390,12529,12531,12534,12593],{"className":12530},[411],[390,12532],{"className":12533},[435,1876],[390,12535,12537],{"className":12536},[1880],[390,12538,12540,12585],{"className":12539},[1884,1885],[390,12541,12543,12582],{"className":12542},[1889],[390,12544,12546,12560,12568],{"className":12545,"style":1894},[1893],[390,12547,12548,12551],{"style":1897},[390,12549],{"className":12550,"style":1902},[1901],[390,12552,12554],{"className":12553},[1906,1907,1908,1909],[390,12555,12557],{"className":12556},[411,1909],[390,12558,4218],{"className":12559},[411,1909],[390,12561,12562,12565],{"style":1919},[390,12563],{"className":12564,"style":1902},[1901],[390,12566],{"className":12567,"style":1927},[1926],[390,12569,12570,12573],{"style":1930},[390,12571],{"className":12572,"style":1902},[1901],[390,12574,12576],{"className":12575},[1906,1907,1908,1909],[390,12577,12579],{"className":12578},[411,1909],[390,12580,1366],{"className":12581},[411,1909],[390,12583,1947],{"className":12584},[1946],[390,12586,12588],{"className":12587},[1889],[390,12589,12591],{"className":12590,"style":1954},[1893],[390,12592],{},[390,12594],{"className":12595},[464,1876],[390,12597],{"className":12598,"style":604},[443],[390,12600,1528],{"className":12601},[608],[390,12603],{"className":12604,"style":604},[443],[390,12606,12608,12611],{"className":12607},[402],[390,12609],{"className":12610,"style":697},[406],[390,12612,500],{"className":12613},[411],", the recurrence solves to ",[385,12616,12617,12618],{},"worst-case\n",[390,12619,12621],{"className":12620},[393],[390,12622,12624],{"className":12623,"ariaHidden":398},[397],[390,12625,12627,12630,12633,12636,12639],{"className":12626},[402],[390,12628],{"className":12629,"style":427},[406],[390,12631,2340],{"className":12632,"style":944},[411,412],[390,12634,436],{"className":12635},[435],[390,12637,413],{"className":12638},[411,412],[390,12640,465],{"className":12641},[464],[1048,12643,12644],{},"In practice randomized quickselect wins on constants; median-of-medians\nmatters for guarantees and as a quicksort pivot rule.",[1048,12646,12647,12650,12651,12666,12667,12685,12686,12701,12702,720],{},[385,12648,12649],{},"Closest pair of points"," is divide-and-conquer with the same flavor: split\nby median ",[390,12652,12654],{"className":12653},[393],[390,12655,12657],{"className":12656,"ariaHidden":398},[397],[390,12658,12660,12663],{"className":12659},[402],[390,12661],{"className":12662,"style":407},[406],[390,12664,1488],{"className":12665},[411,412],", recurse on each half, then combine over a width-",[390,12668,12670],{"className":12669},[393],[390,12671,12673],{"className":12672,"ariaHidden":398},[397],[390,12674,12676,12679,12682],{"className":12675},[402],[390,12677],{"className":12678,"style":479},[406],[390,12680,1198],{"className":12681},[411],[390,12683,9832],{"className":12684,"style":9831},[411,412],"\nstrip. A geometric argument caps the strip work at ",[390,12687,12689],{"className":12688},[393],[390,12690,12692],{"className":12691,"ariaHidden":398},[397],[390,12693,12695,12698],{"className":12694},[402],[390,12696],{"className":12697,"style":697},[406],[390,12699,1366],{"className":12700},[411]," comparisons per point,\ngiving ",[390,12703,12705],{"className":12704},[393],[390,12706,12708,12735,12768,12795],{"className":12707,"ariaHidden":398},[397],[390,12709,12711,12714,12717,12720,12723,12726,12729,12732],{"className":12710},[402],[390,12712],{"className":12713,"style":427},[406],[390,12715,1711],{"className":12716,"style":1710},[411,412],[390,12718,436],{"className":12719},[435],[390,12721,413],{"className":12722},[411,412],[390,12724,465],{"className":12725},[464],[390,12727],{"className":12728,"style":604},[443],[390,12730,609],{"className":12731},[608],[390,12733],{"className":12734,"style":604},[443],[390,12736,12738,12741,12744,12747,12750,12753,12756,12759,12762,12765],{"className":12737},[402],[390,12739],{"className":12740,"style":427},[406],[390,12742,1198],{"className":12743},[411],[390,12745,1711],{"className":12746,"style":1710},[411,412],[390,12748,436],{"className":12749},[435],[390,12751,413],{"className":12752},[411,412],[390,12754,858],{"className":12755},[411],[390,12757,465],{"className":12758},[464],[390,12760],{"className":12761,"style":682},[443],[390,12763,845],{"className":12764},[686],[390,12766],{"className":12767,"style":682},[443],[390,12769,12771,12774,12777,12780,12783,12786,12789,12792],{"className":12770},[402],[390,12772],{"className":12773,"style":427},[406],[390,12775,2340],{"className":12776,"style":944},[411,412],[390,12778,436],{"className":12779},[435],[390,12781,413],{"className":12782},[411,412],[390,12784,465],{"className":12785},[464],[390,12787],{"className":12788,"style":604},[443],[390,12790,745],{"className":12791},[608],[390,12793],{"className":12794,"style":604},[443],[390,12796,12798,12801,12804,12807,12810,12813,12819,12822,12825],{"className":12797},[402],[390,12799],{"className":12800,"style":427},[406],[390,12802,2340],{"className":12803,"style":944},[411,412],[390,12805,436],{"className":12806},[435],[390,12808,413],{"className":12809},[411,412],[390,12811],{"className":12812,"style":444},[443],[390,12814,12816],{"className":12815},[448],[390,12817,454],{"className":12818,"style":453},[411,452],[390,12820],{"className":12821,"style":444},[443],[390,12823,413],{"className":12824},[411,412],[390,12826,465],{"className":12827},[464],[12829,12830,12833,12838],"section",{"className":12831,"dataFootnotes":376},[12832],"footnotes",[508,12834,12837],{"className":12835,"id":498},[12836],"sr-only","Footnotes",[12839,12840,12841,12871,12886,12922],"ol",{},[1048,12842,12844,12847,12848,12863,12864],{"id":12843},"user-content-fn-clrs-select",[385,12845,12846],{},"CLRS",", Ch. 9 — Medians and Order Statistics: selecting the ",[390,12849,12851],{"className":12850},[393],[390,12852,12854],{"className":12853,"ariaHidden":398},[397],[390,12855,12857,12860],{"className":12856},[402],[390,12858],{"className":12859,"style":479},[406],[390,12861,484],{"className":12862,"style":483},[411,412],"-th order statistic in linear time without fully sorting. ",[494,12865,12870],{"href":12866,"ariaLabel":12867,"className":12868,"dataFootnoteBackref":376},"#user-content-fnref-clrs-select","Back to reference 1",[12869],"data-footnote-backref","↩",[1048,12872,12874,8277,12877,12880,12881],{"id":12873},"user-content-fn-erickson-qsel",[385,12875,12876],{},"Erickson",[503,12878,12879],{},"Algorithms",", Ch. 1 — Recursion: quickselect adapting quicksort's partition to recurse into only the side that holds the answer. ",[494,12882,12870],{"href":12883,"ariaLabel":12884,"className":12885,"dataFootnoteBackref":376},"#user-content-fnref-erickson-qsel","Back to reference 2",[12869],[1048,12887,12889,12891,12892,12916,12917],{"id":12888},"user-content-fn-clrs-mom",[385,12890,12846],{},", Ch. 9 — Medians and Order Statistics: the Blum–Floyd–Pratt–Rivest–Tarjan median-of-medians algorithm achieving worst-case ",[390,12893,12895],{"className":12894},[393],[390,12896,12898],{"className":12897,"ariaHidden":398},[397],[390,12899,12901,12904,12907,12910,12913],{"className":12900},[402],[390,12902],{"className":12903,"style":427},[406],[390,12905,2340],{"className":12906,"style":944},[411,412],[390,12908,436],{"className":12909},[435],[390,12911,413],{"className":12912},[411,412],[390,12914,465],{"className":12915},[464]," via groups of five. ",[494,12918,12870],{"href":12919,"ariaLabel":12920,"className":12921,"dataFootnoteBackref":376},"#user-content-fnref-clrs-mom","Back to reference 3",[12869],[1048,12923,12925,8277,12928,12931,12932],{"id":12924},"user-content-fn-skiena-select",[385,12926,12927],{},"Skiena",[503,12929,12930],{},"The Algorithm Design Manual",", §4 — Sorting and Searching: randomized quickselect as the practical choice over deterministic median-of-medians. ",[494,12933,12870],{"href":12934,"ariaLabel":12935,"className":12936,"dataFootnoteBackref":376},"#user-content-fnref-skiena-select","Back to reference 4",[12869],[12938,12939,12940],"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":12942},[12943,12944,12947,12951,12952,12959],{"id":510,"depth":18,"text":511},{"id":901,"depth":18,"text":902,"children":12945},[12946],{"id":1617,"depth":24,"text":1618},{"id":2952,"depth":18,"text":2953,"children":12948},[12949,12950],{"id":3085,"depth":24,"text":3086},{"id":4615,"depth":24,"text":4616},{"id":8854,"depth":18,"text":8855},{"id":8921,"depth":18,"text":8922,"children":12953},[12954,12955,12957,12958],{"id":9774,"depth":24,"text":9775},{"id":10146,"depth":24,"text":12956},"The δ-strip and the 7-neighbor bound",{"id":11283,"depth":24,"text":11284},{"id":11526,"depth":24,"text":11527},{"id":498,"depth":18,"text":12837},"How do you find the median of n numbers? The obvious answer, sorting them\nand reading off the middle, costs Θ(nlogn). But the median, and more\ngenerally the k-th smallest element, can be found in linear time.1 The\ninsight is that selection asks for less than sorting: we want one element's\nvalue, not the full ordering, and we can stop as soon as we have it.","md",{"moduleNumber":18,"lessonNumber":24,"order":12963},203,true,[12966,12970,12973,12976],{"title":12967,"slug":12968,"difficulty":12969},"Kth Largest Element in an Array","kth-largest-element-in-an-array","Medium",{"title":12971,"slug":12972,"difficulty":12969},"K Closest Points to Origin","k-closest-points-to-origin",{"title":12974,"slug":12975,"difficulty":12969},"Wiggle Sort II","wiggle-sort-ii",{"title":12977,"slug":12978,"difficulty":12979},"Median of Two Sorted Arrays","median-of-two-sorted-arrays","Hard","---\ntitle: Linear-Time Selection\nmodule: Divide & Conquer\nmoduleNumber: 2\nlessonNumber: 3\norder: 203\nsummary: >-\n  Finding the $k$-th smallest element looks like it should require sorting, but\n  it does not. Quickselect adapts quicksort's partition to recurse on just one\n  side, achieving expected $O(n)$. The median-of-medians algorithm guarantees a\n  good pivot with the groups-of-five trick, pushing the worst case down to a\n  provable $O(n)$.\ntopics: [Order Statistics, Divide & Conquer]\nsources:\n  - book: CLRS\n    ref: \"Ch. 9 — Medians and Order Statistics\"\n  - book: Skiena\n    ref: \"§4 — Sorting and Searching\"\n  - book: Erickson\n    ref: \"Ch. 1 — Recursion\"\npractice:\n  - title: 'Kth Largest Element in an Array'\n    slug: kth-largest-element-in-an-array\n    difficulty: Medium\n  - title: 'K Closest Points to Origin'\n    slug: k-closest-points-to-origin\n    difficulty: Medium\n  - title: 'Wiggle Sort II'\n    slug: wiggle-sort-ii\n    difficulty: Medium\n  - title: 'Median of Two Sorted Arrays'\n    slug: median-of-two-sorted-arrays\n    difficulty: Hard\n---\n\nHow do you find the **median** of $n$ numbers? The obvious answer, sorting them\nand reading off the middle, costs $\\Theta(n\\log n)$. But the median, and more\ngenerally the $k$-th smallest element, can be found in **linear** time.[^clrs-select] The\ninsight is that selection asks for _less_ than sorting: we want one element's\nvalue, not the full ordering, and we can stop as soon as we have it.\n\n## The selection problem\n\n> **Input:** an array $A[1..n]$ of $n$ distinct numbers and an integer $k$ with\n> $1 \\le k \\le n$.\n> **Output:** the element of $A$ that is larger than exactly $k - 1$ of the\n> others — the **$k$-th order statistic**.\n\nSpecial cases: $k = 1$ is the minimum, $k = n$ the maximum, and\n$k = \\floor{(n+1)\u002F2}$ the median. The minimum and maximum are easy in $n - 1$\ncomparisons. The median is the interesting case, and the algorithms below solve\nit as a byproduct of solving general selection.\n\n## Quickselect: partition, then recurse on one side\n\n[Quicksort](\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort) partitions around a pivot\nand recurses on **both** halves. But for\nselection we only care about _one_ of them. After partitioning $A[p..r]$ around\na pivot that lands at index $q$, the pivot is the $(q - p + 1)$-th smallest\nelement of the subarray. Compare that rank to $k$:\n\n- if it equals $k$, the pivot _is_ the answer;\n- if $k$ is smaller, the answer lies in the left part, so recurse there;\n- if $k$ is larger, the answer lies in the right part; recurse there, adjusting\n  $k$ to skip the elements we discarded.\n\nThrowing away the side that cannot contain the answer is what turns\n$\\Theta(n\\log n)$ into $\\Theta(n)$.[^erickson-qsel]\n\n$$\n% caption: Quickselect partitions once around $x$, then recurses into only the side\n%          holding rank $k$ (here $k\u003Ci$, the left part) and discards the other.\n\\begin{tikzpicture}[>=Stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  % three regions; the pivot lands at its final rank i (= its sorted index)\n  \\draw[fill=acc!8]   (0,0)   rectangle (3.4,0.8);\n  \\draw[fill=acc!25, draw=acc, very thick] (3.4,0) rectangle (4.4,0.8);\n  \\draw[fill=red!8]    (4.4,0) rectangle (8.0,0.8);\n  \\node at (1.7,0.4) {$\u003C\\,x$};\n  \\node at (3.9,0.4) {$x$};\n  \\node at (6.2,0.4) {$>\\,x$};\n  % endpoints + pivot label\n  \\node[above] at (0,0.8)   {\\footnotesize $p$};\n  \\node[above] at (8.0,0.8) {\\footnotesize $r$};\n  \\node[above] at (3.9,0.85) {\\footnotesize pivot at rank $i$};\n  % left = the side holding rank k: recurse (accent)\n  \\draw[\u003C->, acc] (0,-0.25) -- (3.4,-0.25) node[midway,below] {\\footnotesize $A[p..i-1]$};\n  \\node[acc, font=\\scriptsize] at (1.7,-0.92) {recurse (holds $k$)};\n  % pivot's final index\n  \\draw[->] (3.9,-0.25) -- (3.9,-0.55) node[below] {\\footnotesize $i$};\n  % right = discarded side (red)\n  \\draw[\u003C->, red!75!black] (4.4,-0.25) -- (8.0,-0.25) node[midway,below] {\\footnotesize $A[i+1..r]$};\n  \\node[red!75!black, font=\\scriptsize] at (6.2,-0.92) {discarded};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{Quickselect}(A, p, r, k)$ — return the $k$-th smallest of $A[p..r]$\nnumber: 1\nif $p = r$ then\n  return $A[p]$ \u002F\u002F only element: the answer\n$q \\gets$ call $\\textsc{Randomized-Partition}(A, p, r)$\n$i \\gets q - p + 1$ \u002F\u002F pivot rank in A[p..r]\nif $k = i$ then\n  return $A[q]$ \u002F\u002F pivot is the k-th smallest\nelse if $k \u003C i$ then\n  return call $\\textsc{Quickselect}(A, p, q - 1, k)$ \u002F\u002F recurse left\nelse\n  return call $\\textsc{Quickselect}(A, q + 1, r, k - i)$ \u002F\u002F recurse right, shift k\n```\n\n### Expected linear time\n\nPartition costs $\\Theta(n)$. The win over quicksort is that we recurse on only\n**one** side. With a randomized pivot, the partition splits the array at a\nuniformly random rank, so on average we discard a constant fraction each time.\nIntuitively, a random pivot lands in the middle half of the array with\nprobability $1\u002F2$, in which case the surviving side has at most $3n\u002F4$\nelements. The expected work satisfies, roughly,\n\n$$\nT(n) \\le T(3n\u002F4) + \\Theta(n),\n$$\n\na _geometric_ (not branching) recurrence. Unrolling it,\n\n$$\nT(n) \\le c\\!\\left(n + \\tfrac{3}{4}n + \\left(\\tfrac{3}{4}\\right)^2 n + \\cdots\\right)\n= cn \\cdot \\frac{1}{1 - 3\u002F4} = 4cn = \\Theta(n).\n$$\n\nThe geometric series converges to a constant, so the total is linear. (A\ncareful CLRS-style analysis with indicator variables gives the same\n$\\mathbb{E}[T(n)] = O(n)$.)\n\n$$\n% caption: A single-side recurrence shrinks the subproblem by a constant factor each\n%          level, so the bars $n, \\tfrac34 n, (\\tfrac34)^2 n, \\dots$ sum to a geometric\n%          $\\Theta(n)$.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]}, y=1cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[fill=acc!30] (0,0) rectangle (6.0,0.5); \\node[anchor=west, font=\\scriptsize] at (6.15,0.25) {$n$};\n  \\draw[fill=acc!22] (0,-0.7) rectangle (4.5,-0.2); \\node[anchor=west, font=\\scriptsize] at (6.15,-0.45) {$\\tfrac34 n$};\n  \\draw[fill=acc!16] (0,-1.4) rectangle (3.375,-0.9); \\node[anchor=west, font=\\scriptsize] at (6.15,-1.15) {$(\\tfrac34)^2 n$};\n  \\draw[fill=acc!10] (0,-2.1) rectangle (2.53,-1.6); \\node[anchor=west, font=\\scriptsize] at (6.15,-1.85) {$(\\tfrac34)^3 n$};\n  \\node[font=\\scriptsize] at (1.0,-2.5) {$\\vdots$};\n  \\node[anchor=west, acc, font=\\scriptsize] at (6.15,-2.55) {sum $=4n$};\n\\end{tikzpicture}\n$$\n\nThe catch is the same as quicksort's: the **worst\ncase**, with every pivot maximally unbalanced, is\n\n$$\nT(n) = T(n - 1) + \\Theta(n) = \\Theta(n^2).\n$$\n\nRandomization makes that astronomically unlikely, but it is still possible. Can\nwe _guarantee_ linear time?\n\n## Median of medians: a guaranteed-good pivot\n\nThe deterministic algorithm of Blum, Floyd, Pratt, Rivest, and Tarjan (1973)\nachieves worst-case $O(n)$ by spending a little effort to choose a pivot that\nis _provably_ not too extreme.[^clrs-mom] The trick is to pick the pivot as a **median of\nmedians**.\n\n```algorithm\ncaption: $\\textsc{Select}(A, k)$ — deterministic $k$-th smallest, worst-case $O(n)$\nnumber: 2\nif $A$ has at most $5$ elements then\n  return the $k$-th smallest of $A$ by direct sorting\ndivide $A$ into $\\ceil{n\u002F5}$ groups of $5$ elements (last group may be smaller)\nforeach group do\n  find that group's median by sorting its $\\le 5$ elements\nlet $M$ be the array of the $\\ceil{n\u002F5}$ group medians\n$x \\gets$ call $\\textsc{Select}(M, \\ceil{|M|\u002F2})$ \u002F\u002F median of medians\n$q \\gets$ partition $A$ around the pivot $x$, returning its rank $i$\nif $k = i$ then\n  return $x$\nelse if $k \u003C i$ then\n  return call $\\textsc{Select}(A[\\,\\text{left part}\\,], k)$\nelse\n  return call $\\textsc{Select}(A[\\,\\text{right part}\\,], k - i)$\n```\n\n### Why groups of five give a good pivot\n\nHere is the core of it. Picture the $\\ceil{n\u002F5}$ groups as **columns**, each\nsorted top (large) to bottom (small), and now imagine the columns reordered\nleft to right by their medians. The pivot $x = \\textsc{MoM}$ is the median of\nthat middle row, so it sits dead-center in the grid below. (We assume distinct\nelements for simplicity.)\n\nThe pivot is built in stages: chop the array into groups of five, sort each\ngroup to expose its median, collect those $\\ceil{n\u002F5}$ medians, and **recurse**\nto find _their_ median — the median of medians.\n\n$$\n% caption: Building the pivot: split into groups of $5$, sort each to surface its median\n%          (accent cell), gather the $\\ceil{n\u002F5}$ medians, then recurse on that smaller\n%          array to get the median of medians $x$.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]}, x=1cm, y=1cm,\n  cell\u002F.style={draw, minimum width=5mm, minimum height=5mm, font=\\scriptsize, inner sep=0}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % three sorted groups of five (columns): plain cells in rows 0,1,3,4 ...\n  \\foreach \\gx in {0,1,2} {\n    \\foreach \\r in {0,1,3,4} \\node[cell] at (\\gx, -\\r*0.55) {};\n  }\n  % ... and the centre-row median highlighted per column\n  \\foreach \\gx in {0,1,2} \\node[cell, fill=acc!18, draw=acc, very thick] at (\\gx,-1.1) {};\n  \\node[font=\\scriptsize] at (1, 0.62) {$\\lceil n\u002F5\\rceil$ sorted groups};\n  \\node[font=\\scriptsize, anchor=east, acc] at (-0.45,-1.1) {medians};\n  % arrow: gather medians into a row\n  \\draw[->, red!75!black, thick] (2.55,-1.1) -- (3.45,-1.1);\n  \\node[font=\\scriptsize, red!75!black] at (3.0,-0.7) {gather};\n  % the gathered median array\n  \\foreach \\x in {0,1,2} \\node[cell, fill=acc!18, draw=acc, very thick] at (4+\\x*0.55,-1.1) {};\n  \\node[font=\\scriptsize] at (4.55,-0.5) {$\\lceil n\u002F5\\rceil$ medians};\n  % arrow: recurse to find median of medians\n  \\draw[->, red!75!black, thick] (5.5,-1.1) -- (6.4,-1.1);\n  \\node[font=\\scriptsize, red!75!black, align=center] at (5.95,-1.55) {recurse:\\\\Select};\n  % the median of medians x\n  \\node[cell, fill=acc!30, draw=acc, very thick] at (6.95,-1.1) {};\n  \\node[font=\\scriptsize, anchor=west] at (7.3,-1.1) {$x=$ MoM};\n\\end{tikzpicture}\n$$\n\n$$\n% caption: Grid of groups of five with the median of medians $x$, showing the regions\n%          guaranteed at least or at most $x$.\n\\begin{tikzpicture}[x=8mm, y=8mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\fill[acc!15] (-0.45,-2.45) rectangle (4.45,-1.55);\n  % grid of dots\n  \\foreach \\c in {0,...,4} {\n    \\foreach \\r in {0,...,4} {\n      \\node[circle, fill=acc, inner sep=1.1pt] (n\\c\\r) at (\\c,-\\r) {};\n    }\n  }\n  % the guaranteed >= x region (upper-right block): pivot + medians to its right + the two above each\n  % (the median row runs through the pivot; leave a gap around it so the rule clears the $x$ glyph)\n  \\draw[thick, red] (1.5,0.5) -- (1.5,-2) -- (1.8,-2);\n  \\draw[thick, red] (2.2,-2) -- (4.5,-2);\n  % the guaranteed \u003C= x region (lower-left block)\n  \\draw[thick, red!60!black, dashed] (2.5,-4.5) -- (2.5,-2) -- (2.2,-2);\n  \\draw[thick, red!60!black, dashed] (1.8,-2) -- (-0.5,-2);\n  % the median-of-medians: an empty circle marks its position; the $x$ label sits\n  % fully OUTSIDE the grid (far right) with a long leader threaded between the dot\n  % rows (y between -1 and -2) so it crosses no dots\n  \\node[draw=acc, circle, very thick, fill=acc!30, minimum size=4mm, inner sep=0] at (2,-2) {};\n  \\node[font=\\scriptsize] at (5.5,-1.5) {$x$};\n  \\draw[->, thin] (5.25,-1.55) -- (2.2,-1.83);\n  % region labels moved outside the grid, each with a leader arrow into its block\n  \\node[font=\\scriptsize, anchor=west] at (5.0,-0.5) {each $\\ge x$};\n  \\draw[->, thin] (4.95,-0.5) -- (3.6,-0.6);\n  \\node[font=\\scriptsize, anchor=east] at (-1.0,-3.6) {each $\\le x$};\n  \\draw[->, thin] (-0.95,-3.6) -- (0.4,-3.2);\n  \\node[font=\\scriptsize, gray] at (2.0,1.6) {middle row (medians, incr.\\ order)};\n  \\draw[->, gray] (-0.7,-4.9) -- (-0.7,0.9);\n  \\node[font=\\scriptsize, gray, anchor=east] at (-0.9,-2) {incr.};\n\\end{tikzpicture}\n$$\n\nConsider the columns whose median is $\\ge x$: that is, $x$'s own column and\nthe roughly half of the $\\ceil{n\u002F5}$ columns to its right. In each such column,\nthe median _and_ the two elements above it (the two larger ones) are all\n$\\ge x$. That is **3 elements per column**, across about half of the $n\u002F5$ columns:\n\n$$\n\\#\\set{\\,\\text{elements} \\ge x\\,} \\;\\ge\\; 3 \\cdot \\left(\\frac{1}{2}\\cdot\\frac{n}{5} \\right) \\;\\approx\\; \\frac{3n}{10}.\n$$\n\nSo at least $\\approx 3n\u002F10$ elements are $\\ge x$, which forces **at most**\n$n - 3n\u002F10 = 7n\u002F10$ to be $\u003C x$. By the symmetric argument (the dashed block)\nat least $\\approx 3n\u002F10$ are $\\le x$, so at most $\\approx 7n\u002F10$ are $> x$.\n**Whichever side we recurse into has at most $\\approx 7n\u002F10$ elements**, so the\npivot is provably never too lopsided. (Five is the smallest odd group size that\nmakes the recurrence below close; groups of $3$ fail because their fractions\nsum to exactly $1$.)\n\n### Solving the recurrence\n\nTallying the work: splitting into groups and finding their medians is $O(n)$\n(each group is sorted in $O(1)$). Partitioning is $O(n)$. There are **two**\nrecursive calls:\n\n- finding the median of the $\\ceil{n\u002F5}$ medians, a subproblem of size $n\u002F5$;\n- recursing into the surviving side, a subproblem of size at most $7n\u002F10$.\n\n$$\nT(n) \\le T\\!\\left(\\frac{n}{5}\\right) + T\\!\\left(\\frac{7n}{10}\\right) + O(n).\n$$\n\nThe two fractions are what make this work. **Observe** that\n$\\tfrac{1}{5} + \\tfrac{7}{10} = \\tfrac{9}{10} \u003C 1$: the two subproblems together\nare _strictly smaller_ than the input. That is **shrinkage** — the total work\ncontracts by a constant factor at every level.\n\n$$\n% caption: The two recursive calls span $\\tfrac15 + \\tfrac{7}{10} = \\tfrac{9}{10} \u003C 1$ of\n%          the input; the leftover $\\tfrac{1}{10}$ gap is the shrinkage that forces\n%          $O(n)$.\n\\begin{tikzpicture}[font=\\scriptsize, >={Stealth[round]}, x=1cm, y=1cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw (0,0) rectangle (10,0.7);\n  \\draw[fill=acc!18] (0,0) rectangle (2,0.7);   \\node at (1,0.35) {$\\tfrac n5$};\n  \\draw[fill=acc!32] (2,0) rectangle (9,0.7);    \\node at (5.5,0.35) {$\\tfrac{7n}{10}$};\n  \\draw[fill=black!8] (9,0) rectangle (10,0.7);  \\node[font=\\scriptsize] at (9.5,0.35) {gap};\n  \\node[anchor=west, font=\\scriptsize] at (0,-0.55) {median of $\\tfrac n5$ medians};\n  \\draw[->] (1,-0.35) -- (1,-0.05);\n  \\node[anchor=east, font=\\scriptsize] at (9,-0.55) {surviving side};\n  \\draw[->] (5.5,-0.35) -- (5.5,-0.05);\n\\end{tikzpicture}\n$$\n\n**For inspiration**, first consider the single-call cousin\n$f(n) \\le f(n\u002F2) + bn$. The\n[master theorem](\u002Falgorithms\u002Ffoundations\u002Frecurrences) gives $f(n) = O(n)$;\nunrolling shows why directly,\n\n$$\nf(n) \\le f\\!\\left(\\tfrac{n}{2}\\right) + bn\n\\le f\\!\\left(\\tfrac{n}{4}\\right) + b\\tfrac{n}{2} + bn\n\\le f(1) + \\left(bn + \\tfrac{bn}{2} + \\tfrac{bn}{4} + \\cdots\\right)\n\\le f(1) + 2bn.\n$$\n\nThe constant-factor-shrinkage plus linear cleanup work sums to $O(n)$. The\nmedian-of-medians recurrence has _two_ calls instead of one, but the same\nshrinkage idea carries it through. State it as a clean general\nlemma:\n\n> **Lemma (Shrinkage).** Suppose $T(n) \\le c$ for $n \\le n_0$, and\n> $T(n) \\le T(\\lambda n) + T(\\mu n) + bn$ for $n > n_0$, where\n> $\\lambda, \\mu, b > 0$ are constants with $\\lambda + \\mu \u003C 1$. Then\n> $T(n) = O(n)$.\n\n> **Proof.** By induction on $n$ we show $T(n) \\le an$ for all $n \\ge 1$, where\n>\n> $$\n> a = \\max\\set{\\,c,\\ \\frac{b}{1 - \\lambda - \\mu}\\,}.\n> $$\n>\n> Note $a > 0$ is well-defined precisely because $\\lambda + \\mu \u003C 1$. The base\n> cases $n \\le n_0$ hold since $T(n) \\le c \\le a \\le an$. For the inductive step\n> $n > n_0$, the choice of $a$ gives $b \\le a(1 - \\lambda - \\mu)$, so\n>\n> $$\n> T(n) \\le T(\\lambda n) + T(\\mu n) + bn\n> \\le a\\lambda n + a\\mu n + bn\n> = \\bigl(a(\\lambda + \\mu) + b\\bigr)n\n> \\le \\bigl(a(\\lambda + \\mu) + a(1 - \\lambda - \\mu)\\bigr)n = an. \\qquad\\blacksquare\n> $$\n\nPlugging in $\\lambda = \\tfrac15$, $\\mu = \\tfrac{7}{10}$ (so\n$\\lambda + \\mu = \\tfrac{9}{10}$) gives $T(n) = O(n)$, that is, **worst-case linear\ntime**. Had the fractions summed to $1$ or more (as for groups of $3$, where\n$\\tfrac13 + \\tfrac23 = 1$), the lemma's hypothesis fails: the per-level savings\nvanish, the recursion tree carries $\\log n$ levels of $\\Theta(n)$ work, and the\nbound degrades to $\\Theta(n\\log n)$.\n\n## Which to use\n\nThe median-of-medians algorithm is a landmark: it proves selection is\n_possible_ in worst-case linear time. But its constant factor is large (all\nthat grouping and recursive median-finding), so in practice **randomized\nquickselect is faster** and is the algorithm of choice.[^skiena-select] The deterministic\nversion shines when a worst-case guarantee is mandatory, or, most importantly,\nas a **pivot-selection subroutine** that can be bolted onto quicksort to give\nit a worst-case $\\Theta(n\\log n)$ bound.\n\n## Bonus: Closest Pair of Points\n\nThis is often called \"the most beautiful divide-and-conquer algorithm,\" and\nalso the scariest to get right. It is worth seeing because, like\nmedian-of-medians, its speed hinges on a geometric _counting_ argument that\ncaps how much work the combine step can possibly do.\n\n> **Input:** $n$ points $P[1..n]$ in the plane, each a pair $p = (p_x, p_y)$.\n> **Output:** indices $i \\ne j$ minimizing the Euclidean distance\n> $\\operatorname{dist}(P[i], P[j])$, where\n> $\\operatorname{dist}(p, q) = \\sqrt{(p_x - q_x)^2 + (p_y - q_y)^2}$.\n\nWe treat one distance computation and one distance comparison as $O(1)$. The\nnaïve algorithm, two nested loops over all $\\binom{n}{2}$ pairs, runs in\n$\\Theta(n^2)$. Can divide-and-conquer beat it?\n\n### The idea, and the trap\n\n**Split** $P$ by the median $x$-coordinate into a left half and a right half,\n**recurse** on each to get the closest pair within each side, and let $\\delta$\nbe the smaller of the two returned distances. The danger is the **combine**\nstep: the true closest pair might straddle the dividing line, so we still have\nto look across it.\n\nThe trap is doing this naïvely. If $n_L$ left points and $n_R$ right points lie\nnear the line, checking every cross pair costs $\\Theta(n_L \\cdot n_R)$, which can\nbe $\\Theta(n^2)$, no better than brute force. The whole game is to show that\nonly $O(n)$ cross-comparisons are ever needed.\n\n### The $\\delta$-strip and the 7-neighbor bound\n\nAny cross pair closer than $\\delta$ must have **both** points within horizontal\ndistance $\\delta$ of the dividing line, i.e. inside a vertical **strip** of\nwidth $2\\delta$. So discard everything outside the strip. Then sort the strip's\npoints by $y$-coordinate and walk up it; the magic is that for each point you\nonly need to compare against the **next 7** points in $y$-order.\n\n$$\n% caption: The width-$2$ $\\delta$ strip around the mid line with the 8 cells bounding the\n%          7 neighbors above point $B$ at $i$.\n\\begin{tikzpicture}[x=9mm, y=9mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  % dividing line\n  \\draw[dashed] (0,-0.4) -- (0,4.4) node[above, font=\\scriptsize] {mid line};\n  % strip of width 2*delta\n  \\fill[acc!12] (-1,-0.4) rectangle (1,4.4);\n  \\draw[gray] (-1,-0.4) -- (-1,4.4);\n  \\draw[gray] (1,-0.4) -- (1,4.4);\n  \\draw[\u003C->] (-1,-0.7) -- (0,-0.7) node[midway, below, font=\\scriptsize] {$\\delta$};\n  \\draw[\u003C->] (0,-0.7) -- (1,-0.7) node[midway, below, font=\\scriptsize] {$\\delta$};\n  % the point B[i] and the box of candidates above it (label moved outside the strip)\n  \\fill[acc] (-0.4,0.6) circle (1.6pt);\n  \\node[font=\\scriptsize, anchor=east] at (-1.7,0.1) {$B[i]$};\n  \\draw[->, thin] (-1.6,0.15) -- (-0.45,0.55);\n  % d\u002F2 x d\u002F2 grid: 2 columns each side x 2 rows = 8 cells of side delta\u002F2 above B[i]\n  \\draw[acc, very thick] (-1,0.6) rectangle (1,1.6);\n  \\foreach \\gx in {-1,-0.5,0,0.5,1} { \\draw[acc!45] (\\gx,0.6) -- (\\gx,1.6); }\n  \\draw[acc!45] (-1,1.1) -- (1,1.1);\n  \\node[font=\\scriptsize, acc, anchor=west] at (1.85,1.1) {$8$ cells, side $\\delta\u002F2$};\n  \\draw[->, acc] (1.7,1.1) -- (1.05,1.1);\n  % a far-away point B[i+8] — too far, so flag it red with a leader\n  \\fill[red!75!black] (0.5,3.4) circle (1.6pt);\n  \\node[font=\\scriptsize, red!75!black, anchor=west] at (1.7,3.4) {$B[i{+}8]$: too far};\n  \\draw[->, thin, red!75!black] (1.6,3.4) -- (0.65,3.4);\n\\end{tikzpicture}\n$$\n\n**Why 7?** Fix a strip point $B[i]$ and look at the points $B[j]$ above it\n(larger $y$) that could beat $\\delta$. Any such point lies in the\n$2\\delta \\times \\delta$ box sitting just above $B[i]$ inside the strip. Tile\nthat box with a grid of $\\delta\u002F2 \\times \\delta\u002F2$ cells — there are exactly\n**8** of them.\n\n> **Claim.** No such cell can contain $2$ points of the strip.\n\n> **Proof.** If two points $q, r$ shared a cell, they would lie on the same side of\n> the dividing line, so their distance is already $\\ge \\delta$ (the recursive\n> guarantee). But two points in a $\\delta\u002F2$-square are at most\n> $\\frac{\\delta}{2}\\sqrt{2} \u003C \\delta$ apart — contradiction. $\\blacksquare$\n\nSo the box holds at most $8$ points, one of which is $B[i]$ itself: at most $7$\ncandidates. Hence in $y$-sorted order it suffices to compare $B[i]$ with\n$B[i+1], \\dots, B[i+7]$ — a **constant** number of checks per point. The combine\nstep is $O(n)$, not $\\Theta(n^2)$.\n\n### The algorithm\n\nThe whole thing rests on having the points pre-sorted **both** ways. Sort once\nby $x$ (call it $AX$) and once by $y$ ($AY$) up front; the recursion then\nthreads these sorted orders down without re-sorting.\n\n```algorithm\ncaption: $\\textsc{Closest-Pair}(P)$ — closest pair in $O(n\\log n)$\nnumber: 3\n$AX \\gets P$ sorted by $x$-coordinate\n$AY \\gets P$ sorted by $y$-coordinate\nreturn call $\\textsc{Closest-Pair-Rec}(AX, AY)$\n```\n\n```algorithm\ncaption: $\\textsc{Closest-Pair-Rec}(AX, AY)$ — $AX$ sorted by $x$, $AY$ by $y$\nnumber: 4\n$n \\gets |AX|$\nif $n \u003C 2$ then\n  return $\\bot$ \u002F\u002F no pair exists\nif $n = 2$ then\n  return $(AX[1], AX[2])$\n$\\textit{mid} \\gets AX[\\ceil{n\u002F2}]$ \u002F\u002F line through median x\n$LX \\gets AX[1 \\,..\\, \\ceil{n\u002F2}]$; $\\quad RX \\gets AX[\\ceil{n\u002F2}+1 \\,..\\, n]$ \u002F\u002F split by index (ties safe)\n$LY, RY \\gets$ the same split applied to $AY$ \u002F\u002F keeps halves y-sorted, O(n)\n$(uL, vL) \\gets$ call $\\textsc{Closest-Pair-Rec}(LX, LY)$ \u002F\u002F best pair on left\n$(uR, vR) \\gets$ call $\\textsc{Closest-Pair-Rec}(RX, RY)$ \u002F\u002F best pair on right\n$\\delta \\gets \\min\\set{\\operatorname{dist}(uL, vL),\\ \\operatorname{dist}(uR, vR)}$\n$B \\gets [\\,p \\in AY : \\textit{mid}_x - \\delta \\le p_x \\le \\textit{mid}_x + \\delta\\,]$ \u002F\u002F strip, still y-sorted\n$(uM, vM) \\gets$ call $\\textsc{Best-Pair}(B, \\delta)$\nreturn the closest of $(uL, vL)$, $(uR, vR)$, $(uM, vM)$\n```\n\n```algorithm\ncaption: $\\textsc{Best-Pair}(B, \\delta)$ — best pair inside the strip; $B$ sorted by $y$\nnumber: 5\n$(u_m, v_m) \\gets (\\bot, \\bot)$; $\\quad d_m \\gets \\delta$ \u002F\u002F best pair so far + its dist\nfor $i \\gets 1$ to $|B|$ do\n  for $j \\gets i + 1$ to $\\min\\set{i + 7,\\ |B|}$ do \u002F\u002F only 7 neighbors each!\n    if $\\operatorname{dist}(B[i], B[j]) \u003C d_m$ then\n      $(u_m, v_m) \\gets (B[i], B[j])$; $\\quad d_m \\gets \\operatorname{dist}(B[i], B[j])$\nreturn $(u_m, v_m)$\n```\n\n### Running time and correctness\n\nThe split, the strip extraction, and $\\textsc{Best-Pair}$ are each $O(n)$, and\nthere are two half-size recursive calls:\n\n$$\nT(n) \\le 2\\,T\\!\\left(\\frac{n}{2}\\right) + O(n) \\;\\Longrightarrow\\; T(n) = O(n\\log n),\n$$\n\nby the master theorem, the same shape as\n[mergesort](\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort). The initial sorts add a\none-time $O(n\\log n)$, so the total stays $O(n\\log n)$.\n\n**Correctness** is an easy induction on $n$, _given_ that $\\textsc{Best-Pair}$\nfinds any closest cross-strip pair. That assumption is exactly the 7-neighbor\nclaim above: for each $B[i]$, every later strip point that could improve on\n$\\delta$ lies in one of the $7$ cells of $B[i]$'s grid (the cell holding $B[i]$\naside), and each cell holds at most one point — so scanning $B[i+1..i+7]$ cannot\nmiss it.\n\n- **Selection** finds the $k$-th smallest element; it needs _less_ than\n  sorting, so it can run in $O(n)$.\n- $\\textsc{Quickselect}$ = quicksort's partition, but recurse into only the side that\n  holds the answer; expected $O(n)$ via a geometric (non-branching) recurrence,\n  worst case $\\Theta(n^2)$.\n- **Median of medians** chooses a provably balanced pivot using groups of five,\n  guaranteeing the recursion drops at least $\\approx 3n\u002F10$ elements per side.\n- That balance yields $T(n) \\le T(n\u002F5) + T(7n\u002F10) + O(n)$, and because\n  $\\tfrac{1}{5} + \\tfrac{7}{10} \u003C 1$, the recurrence solves to **worst-case\n  $O(n)$**.\n- In practice randomized quickselect wins on constants; median-of-medians\n  matters for guarantees and as a quicksort pivot rule.\n- **Closest pair of points** is divide-and-conquer with the same flavor: split\n  by median $x$, recurse on each half, then combine over a width-$2\\delta$\n  strip. A geometric argument caps the strip work at $7$ comparisons per point,\n  giving $T(n) \\le 2T(n\u002F2) + O(n) = O(n\\log n)$.\n\n[^clrs-select]: **CLRS**, Ch. 9 — Medians and Order Statistics: selecting the $k$-th order statistic in linear time without fully sorting.\n[^erickson-qsel]: **Erickson**, _Algorithms_, Ch. 1 — Recursion: quickselect adapting quicksort's partition to recurse into only the side that holds the answer.\n[^clrs-mom]: **CLRS**, Ch. 9 — Medians and Order Statistics: the Blum–Floyd–Pratt–Rivest–Tarjan median-of-medians algorithm achieving worst-case $O(n)$ via groups of five.\n[^skiena-select]: **Skiena**, _The Algorithm Design Manual_, §4 — Sorting and Searching: randomized quickselect as the practical choice over deterministic median-of-medians.\n",{"text":12982,"minutes":12983,"time":12984,"words":12985},"11 min read",10.24,614400,2048,{"title":45,"description":12960},[12988,12990,12992],{"book":12846,"ref":12989},"Ch. 9 — Medians and Order 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