[{"data":1,"prerenderedAt":20263},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":374,"course-wordcounts":20132,"ref-card-index":20188},[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":16,"blurb":376,"body":377,"description":20091,"extension":20092,"meta":20093,"module":5,"navigation":20095,"path":17,"practice":20096,"rawbody":20114,"readingTime":20115,"seo":20120,"sources":20121,"status":20128,"stem":20129,"summary":20,"topics":20130,"__hash__":20131},"course\u002F01.algorithms\u002F01.foundations\u002F02.asymptotic-analysis.md","",{"type":378,"value":379,"toc":20069},"minimark",[380,389,394,402,571,598,629,1159,1260,1529,1644,1648,1778,1868,2249,2260,2264,2267,2389,2392,2681,2854,2858,2946,2950,2953,3349,3494,3694,4052,4881,4885,4888,5286,5390,5394,5870,5940,6129,6399,6403,6406,6916,7067,7635,7865,7972,8572,8936,8955,9058,9460,9642,10059,10980,11261,11906,11910,11982,12566,13347,13714,13718,13761,14349,14847,14878,15224,15528,15532,15535,15932,16097,16152,16404,16408,16411,16658,16698,16728,16781,17176,17385,17563,18264,18682,19304,19314,19318,19920,20065],[381,382,383,384,388],"p",{},"In the previous lesson we saw the same algorithm, insertion sort, cost\nquadratically many comparisons on one input and linearly many on another, all on\nthe same machine. To compare algorithms ",[385,386,387],"em",{},"as algorithms",", independent of the\nhardware and the particular input, we need two things: a model of computation\nabstract enough to ignore the machine, and a notation coarse enough to ignore\nconstants. This lesson supplies both.",[390,391,393],"h2",{"id":392},"the-ram-model-of-computation","The RAM model of computation",[381,395,396,397,401],{},"We analyze algorithms against an idealized machine, the ",[398,399,400],"strong",{},"random-access machine\n(RAM)",". It has the properties all three books assume, usually tacitly:",[403,404,405,409,491,498],"ul",{},[406,407,408],"li",{},"Instructions execute one at a time, no concurrency.",[406,410,411,412,436,437,436,453,436,469,486,487,490],{},"The basic operations, namely arithmetic (",[413,414,417],"span",{"className":415},[416],"katex",[413,418,422],{"className":419,"ariaHidden":421},[420],"katex-html","true",[413,423,426,431],{"className":424},[425],"base",[413,427],{"className":428,"style":430},[429],"strut","height:0.6667em;vertical-align:-0.0833em;",[413,432,435],{"className":433},[434],"mord","+",", ",[413,438,440],{"className":439},[416],[413,441,443],{"className":442,"ariaHidden":421},[420],[413,444,446,449],{"className":445},[425],[413,447],{"className":448,"style":430},[429],[413,450,452],{"className":451},[434],"−",[413,454,456],{"className":455},[416],[413,457,459],{"className":458,"ariaHidden":421},[420],[413,460,462,465],{"className":461},[425],[413,463],{"className":464,"style":430},[429],[413,466,468],{"className":467},[434],"×",[413,470,472],{"className":471},[416],[413,473,475],{"className":474,"ariaHidden":421},[420],[413,476,478,482],{"className":477},[425],[413,479],{"className":480,"style":481},[429],"height:1em;vertical-align:-0.25em;",[413,483,485],{"className":484},[434],"\u002F","),\ncomparisons, data movement (load, store, copy), and control flow, each take a\n",[398,488,489],{},"constant"," amount of time.",[406,492,493,494,497],{},"Memory is an unbounded array of cells, and accessing any cell by its index\ncosts the same constant (this is what ",[385,495,496],{},"random access"," means).",[406,499,500,501,505,506,553,554,570],{},"Each cell holds an integer or float of ",[502,503,504],"q",{},"reasonable"," size, roughly ",[413,507,509],{"className":508},[416],[413,510,512],{"className":511,"ariaHidden":421},[420],[413,513,515,518,524,529,539,544,548],{"className":514},[425],[413,516],{"className":517,"style":481},[429],[413,519,523],{"className":520,"style":522},[434,521],"mathnormal","margin-right:0.0278em;","O",[413,525,528],{"className":526},[527],"mopen","(",[413,530,533],{"className":531},[532],"mop",[413,534,538],{"className":535,"style":537},[434,536],"mathrm","margin-right:0.0139em;","log",[413,540],{"className":541,"style":543},[542],"mspace","margin-right:0.1667em;",[413,545,547],{"className":546},[434,521],"n",[413,549,552],{"className":550},[551],"mclose",")","\nbits for an input of size ",[413,555,557],{"className":556},[416],[413,558,560],{"className":559,"ariaHidden":421},[420],[413,561,563,567],{"className":562},[425],[413,564],{"className":565,"style":566},[429],"height:0.4306em;",[413,568,547],{"className":569},[434,521],", so a single value fits in a machine word.",[381,572,573,574,577,578,589,590],{},"The RAM is a deliberate fiction. Real multiplication is not truly constant-time\nfor arbitrarily large numbers; real memory has caches that make some accesses far\ncheaper than others. But the model is ",[385,575,576],{},"predictive",": an algorithm that is fast on\nthe RAM is, overwhelmingly, fast in practice. Skiena stresses this engineering\npayoff;",[579,580,581],"sup",{},[582,583,588],"a",{"href":584,"ariaDescribedBy":585,"dataFootnoteRef":376,"id":587},"#user-content-fn-skiena-ram",[586],"footnote-label","user-content-fnref-skiena-ram","1"," CLRS is careful to flag the places, such as bignum\narithmetic, where the constant-word assumption breaks down.",[579,591,592],{},[582,593,597],{"href":594,"ariaDescribedBy":595,"dataFootnoteRef":376,"id":596},"#user-content-fn-clrs-ram",[586],"user-content-fnref-clrs-ram","2",[599,600,602,603],"h3",{"id":601},"from-problem-to-tn","From problem to ",[413,604,606],{"className":605},[416],[413,607,609],{"className":608,"ariaHidden":421},[420],[413,610,612,615,620,623,626],{"className":611},[425],[413,613],{"className":614,"style":481},[429],[413,616,619],{"className":617,"style":618},[434,521],"margin-right:0.1389em;","T",[413,621,528],{"className":622},[527],[413,624,547],{"className":625},[434,521],[413,627,552],{"className":628},[551],[381,630,631,632,635,636,696,697,733,734,737,738,753,754,757,758,811,812,815,816,874,875,878,879,1017,1018,1054,1055,1071,1072,1071,1075,1090,1091,1142,1143,1158],{},"It helps to be precise about what we are even measuring. A\n",[398,633,634],{},"computational problem"," is just a function ",[413,637,639],{"className":638},[416],[413,640,642,665,687],{"className":641,"ariaHidden":421},[420],[413,643,645,649,653,657,662],{"className":644},[425],[413,646],{"className":647,"style":648},[429],"height:0.6833em;",[413,650,652],{"className":651,"style":618},[434,521],"P",[413,654],{"className":655,"style":656},[542],"margin-right:0.2778em;",[413,658,661],{"className":659},[660],"mrel",":",[413,663],{"className":664,"style":656},[542],[413,666,668,671,677,680,684],{"className":667},[425],[413,669],{"className":670,"style":648},[429],[413,672,676],{"className":673,"style":675},[434,674],"mathcal","margin-right:0.0738em;","I",[413,678],{"className":679,"style":656},[542],[413,681,683],{"className":682},[660],"→",[413,685],{"className":686,"style":656},[542],[413,688,690,693],{"className":689},[425],[413,691],{"className":692,"style":648},[429],[413,694,523],{"className":695,"style":522},[434,674],"\nfrom a set of possible inputs to a set of possible outputs; each element\n",[413,698,700],{"className":699},[416],[413,701,703,724],{"className":702,"ariaHidden":421},[420],[413,704,706,710,714,717,721],{"className":705},[425],[413,707],{"className":708,"style":709},[429],"height:0.7224em;vertical-align:-0.0391em;",[413,711,676],{"className":712,"style":713},[434,521],"margin-right:0.0785em;",[413,715],{"className":716,"style":656},[542],[413,718,720],{"className":719},[660],"∈",[413,722],{"className":723,"style":656},[542],[413,725,727,730],{"className":726},[425],[413,728],{"className":729,"style":648},[429],[413,731,676],{"className":732,"style":675},[434,674]," is an ",[398,735,736],{},"instance"," of ",[413,739,741],{"className":740},[416],[413,742,744],{"className":743,"ariaHidden":421},[420],[413,745,747,750],{"className":746},[425],[413,748],{"className":749,"style":648},[429],[413,751,652],{"className":752,"style":618},[434,521],". ",[385,755,756],{},"Sorting integer arrays",", for\nexample, has ",[413,759,761],{"className":760},[416],[413,762,764,783],{"className":763,"ariaHidden":421},[420],[413,765,767,770,773,776,780],{"className":766},[425],[413,768],{"className":769,"style":648},[429],[413,771,676],{"className":772,"style":675},[434,674],[413,774],{"className":775,"style":656},[542],[413,777,779],{"className":778},[660],"=",[413,781],{"className":782,"style":656},[542],[413,784,786,789],{"className":785},[425],[413,787],{"className":788,"style":481},[429],[413,790,793,799,807],{"className":791},[792],"minner",[413,794,798],{"className":795,"style":797},[527,796],"delimcenter","top:0em;","{",[413,800,803],{"className":801},[434,802],"text",[413,804,806],{"className":805},[434],"all integer arrays",[413,808,810],{"className":809,"style":797},[551,796],"}",". A ",[398,813,814],{},"size"," function\n",[413,817,819],{"className":818},[416],[413,820,822,844,862],{"className":821,"ariaHidden":421},[420],[413,823,825,829,835,838,841],{"className":824},[425],[413,826],{"className":827,"style":828},[429],"height:0.6679em;",[413,830,832],{"className":831},[532],[413,833,814],{"className":834},[434,536],[413,836],{"className":837,"style":656},[542],[413,839,661],{"className":840},[660],[413,842],{"className":843,"style":656},[542],[413,845,847,850,853,856,859],{"className":846},[425],[413,848],{"className":849,"style":648},[429],[413,851,676],{"className":852,"style":675},[434,674],[413,854],{"className":855,"style":656},[542],[413,857,683],{"className":858},[660],[413,860],{"className":861,"style":656},[542],[413,863,865,869],{"className":864},[425],[413,866],{"className":867,"style":868},[429],"height:0.6889em;",[413,870,873],{"className":871},[434,872],"mathbb","N"," records how ",[502,876,877],{},"big"," each instance\nis. For an array ",[413,880,882],{"className":881},[416],[413,883,885],{"className":884,"ariaHidden":421},[420],[413,886,888,891,895,951,956,959,963,966,969,972,1013],{"className":887},[425],[413,889],{"className":890,"style":481},[429],[413,892,894],{"className":893},[527],"⟨",[413,896,898,901],{"className":897},[434],[413,899,582],{"className":900},[434,521],[413,902,905],{"className":903},[904],"msupsub",[413,906,910,942],{"className":907},[908,909],"vlist-t","vlist-t2",[413,911,914,937],{"className":912},[913],"vlist-r",[413,915,919],{"className":916,"style":918},[917],"vlist","height:0.3011em;",[413,920,922,927],{"style":921},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[413,923],{"className":924,"style":926},[925],"pstrut","height:2.7em;",[413,928,934],{"className":929},[930,931,932,933],"sizing","reset-size6","size3","mtight",[413,935,588],{"className":936},[434,933],[413,938,941],{"className":939},[940],"vlist-s","​",[413,943,945],{"className":944},[913],[413,946,949],{"className":947,"style":948},[917],"height:0.15em;",[413,950],{},[413,952,955],{"className":953},[954],"mpunct",",",[413,957],{"className":958,"style":543},[542],[413,960,962],{"className":961},[792],"…",[413,964],{"className":965,"style":543},[542],[413,967,955],{"className":968},[954],[413,970],{"className":971,"style":543},[542],[413,973,975,978],{"className":974},[434],[413,976,582],{"className":977},[434,521],[413,979,981],{"className":980},[904],[413,982,984,1005],{"className":983},[908,909],[413,985,987,1002],{"className":986},[913],[413,988,991],{"className":989,"style":990},[917],"height:0.1514em;",[413,992,993,996],{"style":921},[413,994],{"className":995,"style":926},[925],[413,997,999],{"className":998},[930,931,932,933],[413,1000,547],{"className":1001},[434,521,933],[413,1003,941],{"className":1004},[940],[413,1006,1008],{"className":1007},[913],[413,1009,1011],{"className":1010,"style":948},[917],[413,1012],{},[413,1014,1016],{"className":1015},[551],"⟩"," we take ",[413,1019,1021],{"className":1020},[416],[413,1022,1024,1045],{"className":1023,"ariaHidden":421},[420],[413,1025,1027,1030,1036,1039,1042],{"className":1026},[425],[413,1028],{"className":1029,"style":828},[429],[413,1031,1033],{"className":1032},[532],[413,1034,814],{"className":1035},[434,536],[413,1037],{"className":1038,"style":656},[542],[413,1040,779],{"className":1041},[660],[413,1043],{"className":1044,"style":656},[542],[413,1046,1048,1051],{"className":1047},[425],[413,1049],{"className":1050,"style":566},[429],[413,1052,547],{"className":1053},[434,521],". An algorithm ",[413,1056,1058],{"className":1057},[416],[413,1059,1061],{"className":1060,"ariaHidden":421},[420],[413,1062,1064,1067],{"className":1063},[425],[413,1065],{"className":1066,"style":648},[429],[413,1068,1070],{"className":1069},[434,674],"A"," ",[398,1073,1074],{},"solves",[413,1076,1078],{"className":1077},[416],[413,1079,1081],{"className":1080,"ariaHidden":421},[420],[413,1082,1084,1087],{"className":1083},[425],[413,1085],{"className":1086,"style":648},[429],[413,1088,652],{"className":1089,"style":618},[434,521]," if ",[413,1092,1094],{"className":1093},[416],[413,1095,1097,1124],{"className":1096,"ariaHidden":421},[420],[413,1098,1100,1103,1106,1109,1112,1115,1118,1121],{"className":1099},[425],[413,1101],{"className":1102,"style":481},[429],[413,1104,1070],{"className":1105},[434,674],[413,1107,528],{"className":1108},[527],[413,1110,676],{"className":1111,"style":713},[434,521],[413,1113,552],{"className":1114},[551],[413,1116],{"className":1117,"style":656},[542],[413,1119,779],{"className":1120},[660],[413,1122],{"className":1123,"style":656},[542],[413,1125,1127,1130,1133,1136,1139],{"className":1126},[425],[413,1128],{"className":1129,"style":481},[429],[413,1131,652],{"className":1132,"style":618},[434,521],[413,1134,528],{"className":1135},[527],[413,1137,676],{"className":1138,"style":713},[434,521],[413,1140,552],{"className":1141},[551]," for every\ninstance ",[413,1144,1146],{"className":1145},[416],[413,1147,1149],{"className":1148,"ariaHidden":421},[420],[413,1150,1152,1155],{"className":1151},[425],[413,1153],{"className":1154,"style":648},[429],[413,1156,676],{"className":1157,"style":713},[434,521],".",[381,1160,1161,1162,1227,1228,1243,1244,1259],{},"Now charge ",[413,1163,1165],{"className":1164},[416],[413,1166,1168],{"className":1167,"ariaHidden":421},[420],[413,1169,1171,1175,1218,1221,1224],{"className":1170},[425],[413,1172],{"className":1173,"style":1174},[429],"height:1.1646em;vertical-align:-0.25em;",[413,1176,1178,1190],{"className":1177},[434],[413,1179,1183],{"className":1180},[1181,1182],"enclosing","textsc",[413,1184,1186],{"className":1185},[434,802],[413,1187,1189],{"className":1188},[434],"TimeCost",[413,1191,1193],{"className":1192},[904],[413,1194,1196],{"className":1195},[908],[413,1197,1199],{"className":1198},[913],[413,1200,1203],{"className":1201,"style":1202},[917],"height:0.9146em;",[413,1204,1206,1209],{"style":1205},"top:-3.1362em;margin-right:0.05em;",[413,1207],{"className":1208,"style":926},[925],[413,1210,1212],{"className":1211},[930,931,932,933],[413,1213,1215],{"className":1214},[434,933],[413,1216,1070],{"className":1217},[434,674,933],[413,1219,528],{"className":1220},[527],[413,1222,676],{"className":1223,"style":713},[434,521],[413,1225,552],{"className":1226},[551]," = the number of elementary RAM\noperations ",[413,1229,1231],{"className":1230},[416],[413,1232,1234],{"className":1233,"ariaHidden":421},[420],[413,1235,1237,1240],{"className":1236},[425],[413,1238],{"className":1239,"style":648},[429],[413,1241,1070],{"className":1242},[434,674]," performs on input ",[413,1245,1247],{"className":1246},[416],[413,1248,1250],{"className":1249,"ariaHidden":421},[420],[413,1251,1253,1256],{"className":1252},[425],[413,1254],{"className":1255,"style":648},[429],[413,1257,676],{"className":1258,"style":713},[434,521],". Inputs of the same size can cost\ndifferent amounts, so we take the worst one of each size:",[413,1261,1264],{"className":1262},[1263],"katex-display",[413,1265,1267],{"className":1266},[416],[413,1268,1270,1339],{"className":1269,"ariaHidden":421},[420],[413,1271,1273,1276,1315,1318,1321,1324,1327,1330,1333,1336],{"className":1272},[425],[413,1274],{"className":1275,"style":1174},[429],[413,1277,1279,1289],{"className":1278},[434],[413,1280,1282],{"className":1281},[1181,1182],[413,1283,1285],{"className":1284},[434,802],[413,1286,1288],{"className":1287},[434],"MaxCost",[413,1290,1292],{"className":1291},[904],[413,1293,1295],{"className":1294},[908],[413,1296,1298],{"className":1297},[913],[413,1299,1301],{"className":1300,"style":1202},[917],[413,1302,1303,1306],{"style":1205},[413,1304],{"className":1305,"style":926},[925],[413,1307,1309],{"className":1308},[930,931,932,933],[413,1310,1312],{"className":1311},[434,933],[413,1313,1070],{"className":1314},[434,674,933],[413,1316,528],{"className":1317},[527],[413,1319,547],{"className":1320},[434,521],[413,1322,552],{"className":1323},[551],[413,1325],{"className":1326,"style":656},[542],[413,1328],{"className":1329,"style":656},[542],[413,1331,779],{"className":1332},[660],[413,1334],{"className":1335,"style":656},[542],[413,1337],{"className":1338,"style":656},[542],[413,1340,1342,1346,1476,1479,1517,1520,1523,1526],{"className":1341},[425],[413,1343],{"className":1344,"style":1345},[429],"height:1.8741em;vertical-align:-0.9595em;",[413,1347,1349,1356],{"className":1348},[532],[413,1350,1352],{"className":1351},[532],[413,1353,1355],{"className":1354},[434,536],"max",[413,1357,1359],{"className":1358},[904],[413,1360,1362,1467],{"className":1361},[908,909],[413,1363,1365,1464],{"className":1364},[913],[413,1366,1369],{"className":1367,"style":1368},[917],"height:0.3448em;",[413,1370,1372,1376],{"style":1371},"top:-2.3448em;margin-right:0.05em;",[413,1373],{"className":1374,"style":1375},[925],"height:2.8272em;",[413,1377,1379],{"className":1378},[930,931,932,933],[413,1380,1382],{"className":1381},[434,933],[413,1383,1385],{"className":1384},[434,933],[413,1386,1389],{"className":1387},[1388],"mtable",[413,1390,1393],{"className":1391},[1392],"col-align-c",[413,1394,1396,1455],{"className":1395},[908,909],[413,1397,1399,1452],{"className":1398},[913],[413,1400,1403,1422],{"className":1401,"style":1402},[917],"height:1.1817em;",[413,1404,1406,1410],{"style":1405},"top:-3.2483em;",[413,1407],{"className":1408,"style":1409},[925],"height:2.75em;",[413,1411,1413,1416,1419],{"className":1412},[434,933],[413,1414,676],{"className":1415,"style":713},[434,521,933],[413,1417,720],{"className":1418},[660,933],[413,1420,676],{"className":1421,"style":675},[434,674,933],[413,1423,1425,1428],{"style":1424},"top:-2.3183em;",[413,1426],{"className":1427,"style":1409},[925],[413,1429,1431,1437,1440,1443,1446,1449],{"className":1430},[434,933],[413,1432,1434],{"className":1433},[532,933],[413,1435,814],{"className":1436},[434,536,933],[413,1438,528],{"className":1439},[527,933],[413,1441,676],{"className":1442,"style":713},[434,521,933],[413,1444,552],{"className":1445},[551,933],[413,1447,779],{"className":1448},[660,933],[413,1450,547],{"className":1451},[434,521,933],[413,1453,941],{"className":1454},[940],[413,1456,1458],{"className":1457},[913],[413,1459,1462],{"className":1460,"style":1461},[917],"height:0.6817em;",[413,1463],{},[413,1465,941],{"className":1466},[940],[413,1468,1470],{"className":1469},[913],[413,1471,1474],{"className":1472,"style":1473},[917],"height:0.9595em;",[413,1475],{},[413,1477],{"className":1478,"style":543},[542],[413,1480,1482,1491],{"className":1481},[434],[413,1483,1485],{"className":1484},[1181,1182],[413,1486,1488],{"className":1487},[434,802],[413,1489,1189],{"className":1490},[434],[413,1492,1494],{"className":1493},[904],[413,1495,1497],{"className":1496},[908],[413,1498,1500],{"className":1499},[913],[413,1501,1503],{"className":1502,"style":1202},[917],[413,1504,1505,1508],{"style":1205},[413,1506],{"className":1507,"style":926},[925],[413,1509,1511],{"className":1510},[930,931,932,933],[413,1512,1514],{"className":1513},[434,933],[413,1515,1070],{"className":1516},[434,674,933],[413,1518,528],{"className":1519},[527],[413,1521,676],{"className":1522,"style":713},[434,521],[413,1524,552],{"className":1525},[551],[413,1527,1158],{"className":1528},[434],[381,1530,1531,1532,1556,1557,1560,1561,1071,1564,1579,1580,1597,1598,1615,1616,1643],{},"When the algorithm is understood, this is exactly the function we denote\n",[413,1533,1535],{"className":1534},[416],[413,1536,1538],{"className":1537,"ariaHidden":421},[420],[413,1539,1541,1544,1547,1550,1553],{"className":1540},[425],[413,1542],{"className":1543,"style":481},[429],[413,1545,619],{"className":1546,"style":618},[434,521],[413,1548,528],{"className":1549},[527],[413,1551,547],{"className":1552},[434,521],[413,1554,552],{"className":1555},[551],", the ",[398,1558,1559],{},"running time"," as a function of the ",[398,1562,1563],{},"input size",[413,1565,1567],{"className":1566},[416],[413,1568,1570],{"className":1569,"ariaHidden":421},[420],[413,1571,1573,1576],{"className":1572},[425],[413,1574],{"className":1575,"style":566},[429],[413,1577,547],{"className":1578},[434,521],". The size is\nusually the number of elements, but sometimes the number of bits, or two\nparameters (e.g. ",[413,1581,1583],{"className":1582},[416],[413,1584,1586],{"className":1585,"ariaHidden":421},[420],[413,1587,1589,1592],{"className":1588},[425],[413,1590],{"className":1591,"style":648},[429],[413,1593,1596],{"className":1594,"style":1595},[434,521],"margin-right:0.2222em;","V"," and ",[413,1599,1601],{"className":1600},[416],[413,1602,1604],{"className":1603,"ariaHidden":421},[420],[413,1605,1607,1610],{"className":1606},[425],[413,1608],{"className":1609,"style":648},[429],[413,1611,1614],{"className":1612,"style":1613},[434,521],"margin-right:0.0576em;","E"," for a graph); choosing the right size measure is the\nfirst decision in any analysis. What we ultimately want is ",[385,1617,1618,1619],{},"a good,\nconvenient-to-understand upper bound on ",[413,1620,1622],{"className":1621},[416],[413,1623,1625],{"className":1624,"ariaHidden":421},[420],[413,1626,1628,1631,1634,1637,1640],{"className":1627},[425],[413,1629],{"className":1630,"style":481},[429],[413,1632,619],{"className":1633,"style":618},[434,521],[413,1635,528],{"className":1636},[527],[413,1638,547],{"className":1639},[434,521],[413,1641,552],{"className":1642},[551],", which is precisely what the\nnotation below provides.",[390,1645,1647],{"id":1646},"worst-average-and-best-case","Worst, average, and best case",[381,1649,1650,1651,1071,1653,1668,1669,1694,1695,1748,1749,1773,1774,1777],{},"For a fixed input ",[385,1652,814],{},[413,1654,1656],{"className":1655},[416],[413,1657,1659],{"className":1658,"ariaHidden":421},[420],[413,1660,1662,1665],{"className":1661},[425],[413,1663],{"className":1664,"style":566},[429],[413,1666,547],{"className":1667},[434,521],", different inputs of that size may cost different\namounts. Insertion sort costs ",[413,1670,1672],{"className":1671},[416],[413,1673,1675],{"className":1674,"ariaHidden":421},[420],[413,1676,1678,1681,1685,1688,1691],{"className":1677},[425],[413,1679],{"className":1680,"style":481},[429],[413,1682,1684],{"className":1683},[434],"Θ",[413,1686,528],{"className":1687},[527],[413,1689,547],{"className":1690},[434,521],[413,1692,552],{"className":1693},[551]," on a sorted array and ",[413,1696,1698],{"className":1697},[416],[413,1699,1701],{"className":1700,"ariaHidden":421},[420],[413,1702,1704,1708,1711,1714,1745],{"className":1703},[425],[413,1705],{"className":1706,"style":1707},[429],"height:1.0641em;vertical-align:-0.25em;",[413,1709,1684],{"className":1710},[434],[413,1712,528],{"className":1713},[527],[413,1715,1717,1720],{"className":1716},[434],[413,1718,547],{"className":1719},[434,521],[413,1721,1723],{"className":1722},[904],[413,1724,1726],{"className":1725},[908],[413,1727,1729],{"className":1728},[913],[413,1730,1733],{"className":1731,"style":1732},[917],"height:0.8141em;",[413,1734,1736,1739],{"style":1735},"top:-3.063em;margin-right:0.05em;",[413,1737],{"className":1738,"style":926},[925],[413,1740,1742],{"className":1741},[930,931,932,933],[413,1743,597],{"className":1744},[434,933],[413,1746,552],{"className":1747},[551]," on\na reversed one. So ",[413,1750,1752],{"className":1751},[416],[413,1753,1755],{"className":1754,"ariaHidden":421},[420],[413,1756,1758,1761,1764,1767,1770],{"className":1757},[425],[413,1759],{"className":1760,"style":481},[429],[413,1762,619],{"className":1763,"style":618},[434,521],[413,1765,528],{"className":1766},[527],[413,1768,547],{"className":1769},[434,521],[413,1771,552],{"className":1772},[551]," is not one number; it is a ",[385,1775,1776],{},"range",". We summarize it\nthree ways:",[403,1779,1780,1825,1846],{},[406,1781,1782,1071,1785,1809,1810,1158],{},[398,1783,1784],{},"Worst case",[413,1786,1788],{"className":1787},[416],[413,1789,1791],{"className":1790,"ariaHidden":421},[420],[413,1792,1794,1797,1800,1803,1806],{"className":1793},[425],[413,1795],{"className":1796,"style":481},[429],[413,1798,619],{"className":1799,"style":618},[434,521],[413,1801,528],{"className":1802},[527],[413,1804,547],{"className":1805},[434,521],[413,1807,552],{"className":1808},[551],": the maximum cost over all inputs of size ",[413,1811,1813],{"className":1812},[416],[413,1814,1816],{"className":1815,"ariaHidden":421},[420],[413,1817,1819,1822],{"className":1818},[425],[413,1820],{"className":1821,"style":566},[429],[413,1823,547],{"className":1824},[434,521],[406,1826,1827,1830,1831,1158],{},[398,1828,1829],{},"Best case",": the minimum cost over all inputs of size 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86.852",[1901],[2117,2118,2121,2122,2137,2138,2140,2141,2195,2196,2248],"figcaption",{"className":2119},[2120],"tikz-cap","For each fixed size ",[413,2123,2125],{"className":2124},[416],[413,2126,2128],{"className":2127,"ariaHidden":421},[420],[413,2129,2131,2134],{"className":2130},[425],[413,2132],{"className":2133,"style":566},[429],[413,2135,547],{"className":2136},[434,521],", the cost is a ",[385,2139,1776],{},": best at the bottom edge, worst at the top, average in between. The shaded band is the spread over all inputs of that size. Slicing at one chosen size ",[413,2142,2144],{"className":2143},[416],[413,2145,2147],{"className":2146,"ariaHidden":421},[420],[413,2148,2150,2154],{"className":2149},[425],[413,2151],{"className":2152,"style":2153},[429],"height:0.5806em;vertical-align:-0.15em;",[413,2155,2157,2160],{"className":2156},[434],[413,2158,547],{"className":2159},[434,521],[413,2161,2163],{"className":2162},[904],[413,2164,2166,2187],{"className":2165},[908,909],[413,2167,2169,2184],{"className":2168},[913],[413,2170,2172],{"className":2171,"style":918},[917],[413,2173,2174,2177],{"style":921},[413,2175],{"className":2176,"style":926},[925],[413,2178,2180],{"className":2179},[930,931,932,933],[413,2181,2183],{"className":2182},[434,933],"0",[413,2185,941],{"className":2186},[940],[413,2188,2190],{"className":2189},[913],[413,2191,2193],{"className":2192,"style":948},[917],[413,2194],{}," pins the three cases to three points on the vertical — the runtime of any single input of size ",[413,2197,2199],{"className":2198},[416],[413,2200,2202],{"className":2201,"ariaHidden":421},[420],[413,2203,2205,2208],{"className":2204},[425],[413,2206],{"className":2207,"style":2153},[429],[413,2209,2211,2214],{"className":2210},[434],[413,2212,547],{"className":2213},[434,521],[413,2215,2217],{"className":2216},[904],[413,2218,2220,2240],{"className":2219},[908,909],[413,2221,2223,2237],{"className":2222},[913],[413,2224,2226],{"className":2225,"style":918},[917],[413,2227,2228,2231],{"style":921},[413,2229],{"className":2230,"style":926},[925],[413,2232,2234],{"className":2233},[930,931,932,933],[413,2235,2183],{"className":2236},[434,933],[413,2238,941],{"className":2239},[940],[413,2241,2243],{"className":2242},[913],[413,2244,2246],{"className":2245,"style":948},[917],[413,2247],{}," lands somewhere on that segment.",[381,2250,2251,2252,2255,2256,2259],{},"We almost always report the ",[398,2253,2254],{},"worst case",". It is a ",[385,2257,2258],{},"guarantee",": the algorithm\nnever does worse, no matter how adversarial the input. The best case is nearly\nuseless as a promise, since any algorithm looks good on its luckiest input. The\naverage case is the most honest predictor of typical performance but requires us\nto commit to a distribution, and the analysis is usually harder (it often needs\nthe probabilistic tools of a later module). CLRS develops all three; Skiena\nargues that for design purposes the worst case is what you should plan\nfor.",[390,2261,2263],{"id":2262},"why-we-drop-constants-and-lower-order-terms","Why we drop constants and lower-order terms",[381,2265,2266],{},"Suppose careful counting gives the running time of some algorithm as",[413,2268,2270],{"className":2269},[1263],[413,2271,2273],{"className":2272},[416],[413,2274,2276,2303,2355,2378],{"className":2275,"ariaHidden":421},[420],[413,2277,2279,2282,2285,2288,2291,2294,2297,2300],{"className":2278},[425],[413,2280],{"className":2281,"style":481},[429],[413,2283,619],{"className":2284,"style":618},[434,521],[413,2286,528],{"className":2287},[527],[413,2289,547],{"className":2290},[434,521],[413,2292,552],{"className":2293},[551],[413,2295],{"className":2296,"style":656},[542],[413,2298,779],{"className":2299},[660],[413,2301],{"className":2302,"style":656},[542],[413,2304,2306,2310,2314,2345,2348,2352],{"className":2305},[425],[413,2307],{"className":2308,"style":2309},[429],"height:0.9474em;vertical-align:-0.0833em;",[413,2311,2313],{"className":2312},[434],"3",[413,2315,2317,2320],{"className":2316},[434],[413,2318,547],{"className":2319},[434,521],[413,2321,2323],{"className":2322},[904],[413,2324,2326],{"className":2325},[908],[413,2327,2329],{"className":2328},[913],[413,2330,2333],{"className":2331,"style":2332},[917],"height:0.8641em;",[413,2334,2336,2339],{"style":2335},"top:-3.113em;margin-right:0.05em;",[413,2337],{"className":2338,"style":926},[925],[413,2340,2342],{"className":2341},[930,931,932,933],[413,2343,597],{"className":2344},[434,933],[413,2346],{"className":2347,"style":1595},[542],[413,2349,435],{"className":2350},[2351],"mbin",[413,2353],{"className":2354,"style":1595},[542],[413,2356,2358,2362,2366,2369,2372,2375],{"className":2357},[425],[413,2359],{"className":2360,"style":2361},[429],"height:0.7278em;vertical-align:-0.0833em;",[413,2363,2365],{"className":2364},[434],"50",[413,2367,547],{"className":2368},[434,521],[413,2370],{"className":2371,"style":1595},[542],[413,2373,435],{"className":2374},[2351],[413,2376],{"className":2377,"style":1595},[542],[413,2379,2381,2385],{"className":2380},[425],[413,2382],{"className":2383,"style":2384},[429],"height:0.6444em;",[413,2386,2388],{"className":2387},[434],"200.",[381,2390,2391],{},"Two facts make most of this expression noise:",[2393,2394,2395,2659],"ol",{},[406,2396,2397,2400,2401,2416,2417,2461,2462,2499,2500,2534,2535,2570,2571,2595,2596,2613,2614,2617,2618,1158],{},[398,2398,2399],{},"The leading term dominates."," As ",[413,2402,2404],{"className":2403},[416],[413,2405,2407],{"className":2406,"ariaHidden":421},[420],[413,2408,2410,2413],{"className":2409},[425],[413,2411],{"className":2412,"style":566},[429],[413,2414,547],{"className":2415},[434,521]," grows, ",[413,2418,2420],{"className":2419},[416],[413,2421,2423],{"className":2422,"ariaHidden":421},[420],[413,2424,2426,2429,2432],{"className":2425},[425],[413,2427],{"className":2428,"style":1732},[429],[413,2430,2313],{"className":2431},[434],[413,2433,2435,2438],{"className":2434},[434],[413,2436,547],{"className":2437},[434,521],[413,2439,2441],{"className":2440},[904],[413,2442,2444],{"className":2443},[908],[413,2445,2447],{"className":2446},[913],[413,2448,2450],{"className":2449,"style":1732},[917],[413,2451,2452,2455],{"style":1735},[413,2453],{"className":2454,"style":926},[925],[413,2456,2458],{"className":2457},[930,931,932,933],[413,2459,597],{"className":2460},[434,933]," swamps ",[413,2463,2465],{"className":2464},[416],[413,2466,2468,2489],{"className":2467,"ariaHidden":421},[420],[413,2469,2471,2474,2477,2480,2483,2486],{"className":2470},[425],[413,2472],{"className":2473,"style":2361},[429],[413,2475,2365],{"className":2476},[434],[413,2478,547],{"className":2479},[434,521],[413,2481],{"className":2482,"style":1595},[542],[413,2484,435],{"className":2485},[2351],[413,2487],{"className":2488,"style":1595},[542],[413,2490,2492,2495],{"className":2491},[425],[413,2493],{"className":2494,"style":2384},[429],[413,2496,2498],{"className":2497},[434],"200",". At\n",[413,2501,2503],{"className":2502},[416],[413,2504,2506,2524],{"className":2505,"ariaHidden":421},[420],[413,2507,2509,2512,2515,2518,2521],{"className":2508},[425],[413,2510],{"className":2511,"style":566},[429],[413,2513,547],{"className":2514},[434,521],[413,2516],{"className":2517,"style":656},[542],[413,2519,779],{"className":2520},[660],[413,2522],{"className":2523,"style":656},[542],[413,2525,2527,2530],{"className":2526},[425],[413,2528],{"className":2529,"style":2384},[429],[413,2531,2533],{"className":2532},[434],"1000"," the quadratic term is ",[413,2536,2538],{"className":2537},[416],[413,2539,2541],{"className":2540,"ariaHidden":421},[420],[413,2542,2544,2548,2551,2557,2561,2567],{"className":2543},[425],[413,2545],{"className":2546,"style":2547},[429],"height:0.8389em;vertical-align:-0.1944em;",[413,2549,2313],{"className":2550},[434],[413,2552,2554],{"className":2553},[434],[413,2555,955],{"className":2556},[954],[413,2558,2560],{"className":2559},[434],"000",[413,2562,2564],{"className":2563},[434],[413,2565,955],{"className":2566},[954],[413,2568,2560],{"className":2569},[434]," and the rest is ",[413,2572,2574],{"className":2573},[416],[413,2575,2577],{"className":2576,"ariaHidden":421},[420],[413,2578,2580,2583,2586,2592],{"className":2579},[425],[413,2581],{"className":2582,"style":2547},[429],[413,2584,2365],{"className":2585},[434],[413,2587,2589],{"className":2588},[434],[413,2590,955],{"className":2591},[954],[413,2593,2498],{"className":2594},[434],",\nunder ",[413,2597,2599],{"className":2598},[416],[413,2600,2602],{"className":2601,"ariaHidden":421},[420],[413,2603,2605,2609],{"className":2604},[425],[413,2606],{"className":2607,"style":2608},[429],"height:0.8056em;vertical-align:-0.0556em;",[413,2610,2612],{"className":2611},[434],"2%",". The growth ",[385,2615,2616],{},"rate"," is governed entirely by ",[413,2619,2621],{"className":2620},[416],[413,2622,2624],{"className":2623,"ariaHidden":421},[420],[413,2625,2627,2630],{"className":2626},[425],[413,2628],{"className":2629,"style":1732},[429],[413,2631,2633,2636],{"className":2632},[434],[413,2634,547],{"className":2635},[434,521],[413,2637,2639],{"className":2638},[904],[413,2640,2642],{"className":2641},[908],[413,2643,2645],{"className":2644},[913],[413,2646,2648],{"className":2647,"style":1732},[917],[413,2649,2650,2653],{"style":1735},[413,2651],{"className":2652,"style":926},[925],[413,2654,2656],{"className":2655},[930,931,932,933],[413,2657,597],{"className":2658},[434,933],[406,2660,2661,2664,2665,2680],{},[398,2662,2663],{},"The constants are machine artifacts."," The ",[413,2666,2668],{"className":2667},[416],[413,2669,2671],{"className":2670,"ariaHidden":421},[420],[413,2672,2674,2677],{"className":2673},[425],[413,2675],{"className":2676,"style":2384},[429],[413,2678,2313],{"className":2679},[434]," depends on how many RAM\noperations our particular pseudocode spends per iteration; recompile on a\ndifferent machine and it changes. It says nothing about the algorithm's\nintrinsic scaling.",[381,2682,2683,2684,2728,2729,2770,2771,2774,2775,2806,2807,2809,2810,2853],{},"So we throw both away and say the running time is ",[398,2685,2686,2687],{},"order ",[413,2688,2690],{"className":2689},[416],[413,2691,2693],{"className":2692,"ariaHidden":421},[420],[413,2694,2696,2699],{"className":2695},[425],[413,2697],{"className":2698,"style":1732},[429],[413,2700,2702,2705],{"className":2701},[434],[413,2703,547],{"className":2704},[434,521],[413,2706,2708],{"className":2707},[904],[413,2709,2711],{"className":2710},[908],[413,2712,2714],{"className":2713},[913],[413,2715,2717],{"className":2716,"style":1732},[917],[413,2718,2719,2722],{"style":1735},[413,2720],{"className":2721,"style":926},[925],[413,2723,2725],{"className":2724},[930,931,932,933],[413,2726,597],{"className":2727},[434,933],". This is not\nsloppiness; it is the right level of abstraction. An ",[413,2730,2732],{"className":2731},[416],[413,2733,2735],{"className":2734,"ariaHidden":421},[420],[413,2736,2738,2741],{"className":2737},[425],[413,2739],{"className":2740,"style":1732},[429],[413,2742,2744,2747],{"className":2743},[434],[413,2745,547],{"className":2746},[434,521],[413,2748,2750],{"className":2749},[904],[413,2751,2753],{"className":2752},[908],[413,2754,2756],{"className":2755},[913],[413,2757,2759],{"className":2758,"style":1732},[917],[413,2760,2761,2764],{"style":1735},[413,2762],{"className":2763,"style":926},[925],[413,2765,2767],{"className":2766},[930,931,932,933],[413,2768,597],{"className":2769},[434,933]," algorithm with a tiny\nconstant still loses ",[385,2772,2773],{},"eventually"," to an ",[413,2776,2778],{"className":2777},[416],[413,2779,2781],{"className":2780,"ariaHidden":421},[420],[413,2782,2784,2788,2791,2794,2800,2803],{"className":2783},[425],[413,2785],{"className":2786,"style":2787},[429],"height:0.8889em;vertical-align:-0.1944em;",[413,2789,547],{"className":2790},[434,521],[413,2792],{"className":2793,"style":543},[542],[413,2795,2797],{"className":2796},[532],[413,2798,538],{"className":2799,"style":537},[434,536],[413,2801],{"className":2802,"style":543},[542],[413,2804,547],{"className":2805},[434,521]," algorithm with a large one,\nand ",[502,2808,2773],{}," is exactly what asymptotic analysis captures. The notation below\nmakes ",[502,2811,2686,2812],{},[413,2813,2815],{"className":2814},[416],[413,2816,2818],{"className":2817,"ariaHidden":421},[420],[413,2819,2821,2824],{"className":2820},[425],[413,2822],{"className":2823,"style":1732},[429],[413,2825,2827,2830],{"className":2826},[434],[413,2828,547],{"className":2829},[434,521],[413,2831,2833],{"className":2832},[904],[413,2834,2836],{"className":2835},[908],[413,2837,2839],{"className":2838},[913],[413,2840,2842],{"className":2841,"style":1732},[917],[413,2843,2844,2847],{"style":1735},[413,2845],{"className":2846,"style":926},[925],[413,2848,2850],{"className":2849},[930,931,932,933],[413,2851,597],{"className":2852},[434,933]," precise.",[390,2855,2857],{"id":2856},"the-asymptotic-notations","The asymptotic notations",[381,2859,2860,2861,1597,2878,2895,2896,2911,2912,1071,2915,2930,2931,1158],{},"Let ",[413,2862,2864],{"className":2863},[416],[413,2865,2867],{"className":2866,"ariaHidden":421},[420],[413,2868,2870,2873],{"className":2869},[425],[413,2871],{"className":2872,"style":2787},[429],[413,2874,2877],{"className":2875,"style":2876},[434,521],"margin-right:0.1076em;","f",[413,2879,2881],{"className":2880},[416],[413,2882,2884],{"className":2883,"ariaHidden":421},[420],[413,2885,2887,2891],{"className":2886},[425],[413,2888],{"className":2889,"style":2890},[429],"height:0.625em;vertical-align:-0.1944em;",[413,2892,1882],{"className":2893,"style":2894},[434,521],"margin-right:0.0359em;"," be functions from the positive integers to the nonnegative reals.\nThe notations describe how ",[413,2897,2899],{"className":2898},[416],[413,2900,2902],{"className":2901,"ariaHidden":421},[420],[413,2903,2905,2908],{"className":2904},[425],[413,2906],{"className":2907,"style":2787},[429],[413,2909,2877],{"className":2910,"style":2876},[434,521]," behaves ",[385,2913,2914],{},"relative to",[413,2916,2918],{"className":2917},[416],[413,2919,2921],{"className":2920,"ariaHidden":421},[420],[413,2922,2924,2927],{"className":2923},[425],[413,2925],{"className":2926,"style":2890},[429],[413,2928,1882],{"className":2929,"style":2894},[434,521]," for all sufficiently\nlarge ",[413,2932,2934],{"className":2933},[416],[413,2935,2937],{"className":2936,"ariaHidden":421},[420],[413,2938,2940,2943],{"className":2939},[425],[413,2941],{"className":2942,"style":566},[429],[413,2944,547],{"className":2945},[434,521],[599,2947,2949],{"id":2948},"big-o-asymptotic-upper-bound","Big-O: asymptotic upper bound",[381,2951,2952],{},"State it as a clean existential:",[2954,2955,2957],"callout",{"type":2956},"definition",[381,2958,2959,1071,2962,3020,3021,3275,3276,3313,3314,1158],{},[398,2960,2961],{},"Definition (Big-O).",[413,2963,2965],{"className":2964},[416],[413,2966,2968,2995],{"className":2967,"ariaHidden":421},[420],[413,2969,2971,2974,2977,2980,2983,2986,2989,2992],{"className":2970},[425],[413,2972],{"className":2973,"style":481},[429],[413,2975,2877],{"className":2976,"style":2876},[434,521],[413,2978,528],{"className":2979},[527],[413,2981,547],{"className":2982},[434,521],[413,2984,552],{"className":2985},[551],[413,2987],{"className":2988,"style":656},[542],[413,2990,779],{"className":2991},[660],[413,2993],{"className":2994,"style":656},[542],[413,2996,2998,3001,3004,3007,3010,3013,3016],{"className":2997},[425],[413,2999],{"className":3000,"style":481},[429],[413,3002,523],{"className":3003,"style":522},[434,521],[413,3005,528],{"className":3006},[527],[413,3008,1882],{"className":3009,"style":2894},[434,521],[413,3011,528],{"className":3012},[527],[413,3014,547],{"className":3015},[434,521],[413,3017,3019],{"className":3018},[551],"))"," means\n",[413,3022,3024],{"className":3023},[416],[413,3025,3027,3055,3126,3162,3220,3248],{"className":3026,"ariaHidden":421},[420],[413,3028,3030,3034,3038,3041,3045,3048,3052],{"className":3029},[425],[413,3031],{"className":3032,"style":3033},[429],"height:0.7335em;vertical-align:-0.0391em;",[413,3035,3037],{"className":3036},[434],"∃",[413,3039],{"className":3040,"style":543},[542],[413,3042,3044],{"className":3043},[434,521],"c",[413,3046],{"className":3047,"style":656},[542],[413,3049,3051],{"className":3050},[660],">",[413,3053],{"className":3054,"style":656},[542],[413,3056,3058,3062,3065,3068,3071,3074,3077,3117,3120,3123],{"className":3057},[425],[413,3059],{"className":3060,"style":3061},[429],"height:0.8444em;vertical-align:-0.15em;",[413,3063,2183],{"className":3064},[434],[413,3066],{"className":3067,"style":656},[542],[413,3069],{"className":3070,"style":656},[542],[413,3072,3037],{"className":3073},[434],[413,3075],{"className":3076,"style":543},[542],[413,3078,3080,3083],{"className":3079},[434],[413,3081,547],{"className":3082},[434,521],[413,3084,3086],{"className":3085},[904],[413,3087,3089,3109],{"className":3088},[908,909],[413,3090,3092,3106],{"className":3091},[913],[413,3093,3095],{"className":3094,"style":918},[917],[413,3096,3097,3100],{"style":921},[413,3098],{"className":3099,"style":926},[925],[413,3101,3103],{"className":3102},[930,931,932,933],[413,3104,2183],{"className":3105},[434,933],[413,3107,941],{"className":3108},[940],[413,3110,3112],{"className":3111},[913],[413,3113,3115],{"className":3114,"style":948},[917],[413,3116],{},[413,3118],{"className":3119,"style":656},[542],[413,3121,3051],{"className":3122},[660],[413,3124],{"className":3125,"style":656},[542],[413,3127,3129,3133,3136,3139,3142,3146,3149,3152,3155,3159],{"className":3128},[425],[413,3130],{"className":3131,"style":3132},[429],"height:0.8304em;vertical-align:-0.136em;",[413,3134,2183],{"className":3135},[434],[413,3137],{"className":3138,"style":656},[542],[413,3140],{"className":3141,"style":656},[542],[413,3143,3145],{"className":3144},[434],"∀",[413,3147],{"className":3148,"style":543},[542],[413,3150,547],{"className":3151},[434,521],[413,3153],{"className":3154,"style":656},[542],[413,3156,3158],{"className":3157},[660],"≥",[413,3160],{"className":3161,"style":656},[542],[413,3163,3165,3168,3208,3211,3214,3217],{"className":3164},[425],[413,3166],{"className":3167,"style":2153},[429],[413,3169,3171,3174],{"className":3170},[434],[413,3172,547],{"className":3173},[434,521],[413,3175,3177],{"className":3176},[904],[413,3178,3180,3200],{"className":3179},[908,909],[413,3181,3183,3197],{"className":3182},[913],[413,3184,3186],{"className":3185,"style":918},[917],[413,3187,3188,3191],{"style":921},[413,3189],{"className":3190,"style":926},[925],[413,3192,3194],{"className":3193},[930,931,932,933],[413,3195,2183],{"className":3196},[434,933],[413,3198,941],{"className":3199},[940],[413,3201,3203],{"className":3202},[913],[413,3204,3206],{"className":3205,"style":948},[917],[413,3207],{},[413,3209],{"className":3210,"style":656},[542],[413,3212,661],{"className":3213},[660],[413,3215],{"className":3216,"style":656},[542],[413,3218],{"className":3219,"style":656},[542],[413,3221,3223,3226,3229,3232,3235,3238,3241,3245],{"className":3222},[425],[413,3224],{"className":3225,"style":481},[429],[413,3227,2877],{"className":3228,"style":2876},[434,521],[413,3230,528],{"className":3231},[527],[413,3233,547],{"className":3234},[434,521],[413,3236,552],{"className":3237},[551],[413,3239],{"className":3240,"style":656},[542],[413,3242,3244],{"className":3243},[660],"≤",[413,3246],{"className":3247,"style":656},[542],[413,3249,3251,3254,3257,3260,3263,3266,3269,3272],{"className":3250},[425],[413,3252],{"className":3253,"style":481},[429],[413,3255,3044],{"className":3256},[434,521],[413,3258],{"className":3259,"style":543},[542],[413,3261,1882],{"className":3262,"style":2894},[434,521],[413,3264,528],{"className":3265},[527],[413,3267,547],{"className":3268},[434,521],[413,3270,552],{"className":3271},[551],[413,3273,1158],{"className":3274},[434],"\nIn words, ",[502,3277,3278,3293,3294,1158],{},[413,3279,3281],{"className":3280},[416],[413,3282,3284],{"className":3283,"ariaHidden":421},[420],[413,3285,3287,3290],{"className":3286},[425],[413,3288],{"className":3289,"style":2787},[429],[413,3291,2877],{"className":3292,"style":2876},[434,521]," is ",[398,3295,3296,3297,3312],{},"upper bounded by ",[413,3298,3300],{"className":3299},[416],[413,3301,3303],{"className":3302,"ariaHidden":421},[420],[413,3304,3306,3309],{"className":3305},[425],[413,3307],{"className":3308,"style":2890},[429],[413,3310,1882],{"className":3311,"style":2894},[434,521],", up to a constant factor"," Roughly,\n",[413,3315,3317],{"className":3316},[416],[413,3318,3320,3340],{"className":3319,"ariaHidden":421},[420],[413,3321,3323,3326,3329,3332,3337],{"className":3322},[425],[413,3324],{"className":3325,"style":2787},[429],[413,3327,2877],{"className":3328,"style":2876},[434,521],[413,3330],{"className":3331,"style":656},[542],[413,3333,3336],{"className":3334},[660,3335],"amsrm","≼",[413,3338],{"className":3339,"style":656},[542],[413,3341,3343,3346],{"className":3342},[425],[413,3344],{"className":3345,"style":2890},[429],[413,3347,1882],{"className":3348,"style":2894},[434,521],[381,3350,3351,3352,3367,3368,3420,3421,3445,3446,3449,3450,3493],{},"The constant ",[413,3353,3355],{"className":3354},[416],[413,3356,3358],{"className":3357,"ariaHidden":421},[420],[413,3359,3361,3364],{"className":3360},[425],[413,3362],{"className":3363,"style":566},[429],[413,3365,3044],{"className":3366},[434,521]," lets us ignore multiplicative factors; the threshold ",[413,3369,3371],{"className":3370},[416],[413,3372,3374],{"className":3373,"ariaHidden":421},[420],[413,3375,3377,3380],{"className":3376},[425],[413,3378],{"className":3379,"style":2153},[429],[413,3381,3383,3386],{"className":3382},[434],[413,3384,547],{"className":3385},[434,521],[413,3387,3389],{"className":3388},[904],[413,3390,3392,3412],{"className":3391},[908,909],[413,3393,3395,3409],{"className":3394},[913],[413,3396,3398],{"className":3397,"style":918},[917],[413,3399,3400,3403],{"style":921},[413,3401],{"className":3402,"style":926},[925],[413,3404,3406],{"className":3405},[930,931,932,933],[413,3407,2183],{"className":3408},[434,933],[413,3410,941],{"className":3411},[940],[413,3413,3415],{"className":3414},[913],[413,3416,3418],{"className":3417,"style":948},[917],[413,3419],{}," lets\nus ignore small inputs where lower-order terms might still dominate. (CLRS phrases\n",[413,3422,3424],{"className":3423},[416],[413,3425,3427],{"className":3426,"ariaHidden":421},[420],[413,3428,3430,3433,3436,3439,3442],{"className":3429},[425],[413,3431],{"className":3432,"style":481},[429],[413,3434,523],{"className":3435,"style":522},[434,521],[413,3437,528],{"className":3438},[527],[413,3440,1882],{"className":3441,"style":2894},[434,521],[413,3443,552],{"className":3444},[551]," as a ",[385,3447,3448],{},"set"," of functions and adds ",[413,3451,3453],{"className":3452},[416],[413,3454,3456,3475],{"className":3455,"ariaHidden":421},[420],[413,3457,3459,3463,3466,3469,3472],{"className":3458},[425],[413,3460],{"className":3461,"style":3462},[429],"height:0.7804em;vertical-align:-0.136em;",[413,3464,2183],{"className":3465},[434],[413,3467],{"className":3468,"style":656},[542],[413,3470,3244],{"className":3471},[660],[413,3473],{"className":3474,"style":656},[542],[413,3476,3478,3481,3484,3487,3490],{"className":3477},[425],[413,3479],{"className":3480,"style":481},[429],[413,3482,2877],{"className":3483,"style":2876},[434,521],[413,3485,528],{"className":3486},[527],[413,3488,547],{"className":3489},[434,521],[413,3491,552],{"className":3492},[551]," to keep things nonnegative;\nthe two readings agree.)",[381,3495,3496,3497,3515,3516,3546,3547,3571,3572,3587,3588,3640,3641,3693],{},"The picture to keep in mind is the ",[502,3498,3499,3500],{},"for large ",[413,3501,3503],{"className":3502},[416],[413,3504,3506],{"className":3505,"ariaHidden":421},[420],[413,3507,3509,3512],{"className":3508},[425],[413,3510],{"className":3511,"style":566},[429],[413,3513,547],{"className":3514},[434,521]," sketch: the scaled curve\n",[413,3517,3519],{"className":3518},[416],[413,3520,3522],{"className":3521,"ariaHidden":421},[420],[413,3523,3525,3528,3531,3534,3537,3540,3543],{"className":3524},[425],[413,3526],{"className":3527,"style":481},[429],[413,3529,3044],{"className":3530},[434,521],[413,3532],{"className":3533,"style":543},[542],[413,3535,1882],{"className":3536,"style":2894},[434,521],[413,3538,528],{"className":3539},[527],[413,3541,547],{"className":3542},[434,521],[413,3544,552],{"className":3545},[551]," rises above ",[413,3548,3550],{"className":3549},[416],[413,3551,3553],{"className":3552,"ariaHidden":421},[420],[413,3554,3556,3559,3562,3565,3568],{"className":3555},[425],[413,3557],{"className":3558,"style":481},[429],[413,3560,2877],{"className":3561,"style":2876},[434,521],[413,3563,528],{"className":3564},[527],[413,3566,547],{"className":3567},[434,521],[413,3569,552],{"className":3570},[551]," once ",[413,3573,3575],{"className":3574},[416],[413,3576,3578],{"className":3577,"ariaHidden":421},[420],[413,3579,3581,3584],{"className":3580},[425],[413,3582],{"className":3583,"style":566},[429],[413,3585,547],{"className":3586},[434,521]," passes the threshold ",[413,3589,3591],{"className":3590},[416],[413,3592,3594],{"className":3593,"ariaHidden":421},[420],[413,3595,3597,3600],{"className":3596},[425],[413,3598],{"className":3599,"style":2153},[429],[413,3601,3603,3606],{"className":3602},[434],[413,3604,547],{"className":3605},[434,521],[413,3607,3609],{"className":3608},[904],[413,3610,3612,3632],{"className":3611},[908,909],[413,3613,3615,3629],{"className":3614},[913],[413,3616,3618],{"className":3617,"style":918},[917],[413,3619,3620,3623],{"style":921},[413,3621],{"className":3622,"style":926},[925],[413,3624,3626],{"className":3625},[930,931,932,933],[413,3627,2183],{"className":3628},[434,933],[413,3630,941],{"className":3631},[940],[413,3633,3635],{"className":3634},[913],[413,3636,3638],{"className":3637,"style":948},[917],[413,3639],{},", and stays above\nforever after. What happens to the left of ",[413,3642,3644],{"className":3643},[416],[413,3645,3647],{"className":3646,"ariaHidden":421},[420],[413,3648,3650,3653],{"className":3649},[425],[413,3651],{"className":3652,"style":2153},[429],[413,3654,3656,3659],{"className":3655},[434],[413,3657,547],{"className":3658},[434,521],[413,3660,3662],{"className":3661},[904],[413,3663,3665,3685],{"className":3664},[908,909],[413,3666,3668,3682],{"className":3667},[913],[413,3669,3671],{"className":3670,"style":918},[917],[413,3672,3673,3676],{"style":921},[413,3674],{"className":3675,"style":926},[925],[413,3677,3679],{"className":3678},[930,931,932,933],[413,3680,2183],{"className":3681},[434,933],[413,3683,941],{"className":3684},[940],[413,3686,3688],{"className":3687},[913],[413,3689,3691],{"className":3690,"style":948},[917],[413,3692],{}," is irrelevant.",[1869,3695,3697,3907],{"className":3696},[1872,1873],[1875,3698,3702],{"xmlns":1877,"width":3699,"height":3700,"viewBox":3701},"253.638","313.192","-75 -75 190.229 234.894",[1882,3703,3704,3707,3710,3717,3720,3723,3755,3793,3797,3811,3814],{"stroke":1884,"style":1885},[1887,3705],{"fill":1889,"d":3706},"M-62.27 143.953H95.065",[1887,3708],{"stroke":1889,"d":3709},"m97.065 143.953-3.2-1.6 1.2 1.6-1.2 1.6",[1882,3711,3713],{"transform":3712},"translate(162.869 2.153)",[1887,3714],{"d":3715,"fill":1884,"stroke":1884,"className":3716,"style":1902},"M-61.499 143.782Q-61.499 143.724-61.489 143.694L-60.742 140.711Q-60.668 140.432-60.668 140.223Q-60.668 139.793-60.961 139.793Q-61.274 139.793-61.425 140.166Q-61.577 140.540-61.718 141.111Q-61.718 141.141-61.748 141.158Q-61.777 141.175-61.801 141.175L-61.918 141.175Q-61.953 141.175-61.977 141.138Q-62.001 141.101-62.001 141.072Q-61.894 140.638-61.794 140.335Q-61.694 140.032-61.481 139.783Q-61.269 139.534-60.952 139.534Q-60.576 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143.777",[1901],[1882,3902,3903],{"transform":3818},[1887,3904],{"d":3905,"fill":1884,"stroke":1884,"className":3906,"style":2060},"M-20.569 145.190Q-21.131 145.190-21.461 144.903Q-21.791 144.616-21.918 144.166Q-22.046 143.716-22.046 143.151Q-22.046 142.747-21.980 142.382Q-21.914 142.017-21.751 141.721Q-21.588 141.425-21.297 141.250Q-21.005 141.074-20.569 141.074Q-20.132 141.074-19.842 141.250Q-19.552 141.425-19.388 141.720Q-19.224 142.014-19.160 142.372Q-19.095 142.729-19.095 143.151Q-19.095 143.716-19.223 144.166Q-19.350 144.616-19.677 144.903Q-20.004 145.190-20.569 145.190M-20.569 144.973Q-20.165 144.973-19.970 144.663Q-19.775 144.352-19.731 143.960Q-19.687 143.567-19.687 143.054Q-19.687 142.559-19.731 142.200Q-19.775 141.841-19.970 141.566Q-20.165 141.291-20.569 141.291Q-20.973 141.291-21.168 141.566Q-21.363 141.841-21.407 142.200Q-21.451 142.559-21.451 143.054Q-21.451 143.567-21.407 143.960Q-21.363 144.352-21.168 144.663Q-20.973 144.973-20.569 144.973",[1901],[2117,3908,3910,3911,3546,3955,3571,3979,3994,3995,1071,4000,1158],{"className":3909},[2120],"Curve ",[413,3912,3914],{"className":3913},[416],[413,3915,3917,3937],{"className":3916,"ariaHidden":421},[420],[413,3918,3920,3924,3927,3930,3934],{"className":3919},[425],[413,3921],{"className":3922,"style":3923},[429],"height:0.4445em;",[413,3925,3044],{"className":3926},[434,521],[413,3928],{"className":3929,"style":1595},[542],[413,3931,3933],{"className":3932},[2351],"⋅",[413,3935],{"className":3936,"style":1595},[542],[413,3938,3940,3943,3946,3949,3952],{"className":3939},[425],[413,3941],{"className":3942,"style":481},[429],[413,3944,1882],{"className":3945,"style":2894},[434,521],[413,3947,528],{"className":3948},[527],[413,3950,547],{"className":3951},[434,521],[413,3953,552],{"className":3954},[551],[413,3956,3958],{"className":3957},[416],[413,3959,3961],{"className":3960,"ariaHidden":421},[420],[413,3962,3964,3967,3970,3973,3976],{"className":3963},[425],[413,3965],{"className":3966,"style":481},[429],[413,3968,2877],{"className":3969,"style":2876},[434,521],[413,3971,528],{"className":3972},[527],[413,3974,547],{"className":3975},[434,521],[413,3977,552],{"className":3978},[551],[413,3980,3982],{"className":3981},[416],[413,3983,3985],{"className":3984,"ariaHidden":421},[420],[413,3986,3988,3991],{"className":3987},[425],[413,3989],{"className":3990,"style":566},[429],[413,3992,547],{"className":3993},[434,521]," passes the ",[385,3996,3997],{},[398,3998,3999],{},"threshold",[413,4001,4003],{"className":4002},[416],[413,4004,4006],{"className":4005,"ariaHidden":421},[420],[413,4007,4009,4012],{"className":4008},[425],[413,4010],{"className":4011,"style":2153},[429],[413,4013,4015,4018],{"className":4014},[434],[413,4016,547],{"className":4017},[434,521],[413,4019,4021],{"className":4020},[904],[413,4022,4024,4044],{"className":4023},[908,909],[413,4025,4027,4041],{"className":4026},[913],[413,4028,4030],{"className":4029,"style":918},[917],[413,4031,4032,4035],{"style":921},[413,4033],{"className":4034,"style":926},[925],[413,4036,4038],{"className":4037},[930,931,932,933],[413,4039,2183],{"className":4040},[434,933],[413,4042,941],{"className":4043},[940],[413,4045,4047],{"className":4046},[913],[413,4048,4050],{"className":4049,"style":948},[917],[413,4051],{},[381,4053,4054,4061,4062,4077,4078,4081,4082,4097,4098,4150,4151,4288,4289,4422,4423,4457,4458,4499,4500,4772,4773,436,4806,4876,4877,4880],{},[398,4055,4056,4057,4060],{},"The ",[502,4058,4059],{},"wanted inequality"," method."," Proofs of ",[413,4063,4065],{"className":4064},[416],[413,4066,4068],{"className":4067,"ariaHidden":421},[420],[413,4069,4071,4074],{"className":4070},[425],[413,4072],{"className":4073,"style":648},[429],[413,4075,523],{"className":4076,"style":522},[434,521],"-bounds follow a fixed recipe:\nwrite down the inequality you ",[385,4079,4080],{},"want"," to hold, then reverse-engineer constants ",[413,4083,4085],{"className":4084},[416],[413,4086,4088],{"className":4087,"ariaHidden":421},[420],[413,4089,4091,4094],{"className":4090},[425],[413,4092],{"className":4093,"style":566},[429],[413,4095,3044],{"className":4096},[434,521],"\nand ",[413,4099,4101],{"className":4100},[416],[413,4102,4104],{"className":4103,"ariaHidden":421},[420],[413,4105,4107,4110],{"className":4106},[425],[413,4108],{"className":4109,"style":2153},[429],[413,4111,4113,4116],{"className":4112},[434],[413,4114,547],{"className":4115},[434,521],[413,4117,4119],{"className":4118},[904],[413,4120,4122,4142],{"className":4121},[908,909],[413,4123,4125,4139],{"className":4124},[913],[413,4126,4128],{"className":4127,"style":918},[917],[413,4129,4130,4133],{"style":921},[413,4131],{"className":4132,"style":926},[925],[413,4134,4136],{"className":4135},[930,931,932,933],[413,4137,2183],{"className":4138},[434,933],[413,4140,941],{"className":4141},[940],[413,4143,4145],{"className":4144},[913],[413,4146,4148],{"className":4147,"style":948},[917],[413,4149],{}," that make it true. To prove ",[413,4152,4154],{"className":4153},[416],[413,4155,4157,4205,4226,4244],{"className":4156,"ariaHidden":421},[420],[413,4158,4160,4164,4167,4196,4199,4202],{"className":4159},[425],[413,4161],{"className":4162,"style":4163},[429],"height:0.8974em;vertical-align:-0.0833em;",[413,4165,2313],{"className":4166},[434],[413,4168,4170,4173],{"className":4169},[434],[413,4171,547],{"className":4172},[434,521],[413,4174,4176],{"className":4175},[904],[413,4177,4179],{"className":4178},[908],[413,4180,4182],{"className":4181},[913],[413,4183,4185],{"className":4184,"style":1732},[917],[413,4186,4187,4190],{"style":1735},[413,4188],{"className":4189,"style":926},[925],[413,4191,4193],{"className":4192},[930,931,932,933],[413,4194,597],{"className":4195},[434,933],[413,4197],{"className":4198,"style":1595},[542],[413,4200,435],{"className":4201},[2351],[413,4203],{"className":4204,"style":1595},[542],[413,4206,4208,4211,4214,4217,4220,4223],{"className":4207},[425],[413,4209],{"className":4210,"style":2361},[429],[413,4212,2365],{"className":4213},[434],[413,4215,547],{"className":4216},[434,521],[413,4218],{"className":4219,"style":1595},[542],[413,4221,435],{"className":4222},[2351],[413,4224],{"className":4225,"style":1595},[542],[413,4227,4229,4232,4235,4238,4241],{"className":4228},[425],[413,4230],{"className":4231,"style":2384},[429],[413,4233,2498],{"className":4234},[434],[413,4236],{"className":4237,"style":656},[542],[413,4239,779],{"className":4240},[660],[413,4242],{"className":4243,"style":656},[542],[413,4245,4247,4250,4253,4256,4285],{"className":4246},[425],[413,4248],{"className":4249,"style":1707},[429],[413,4251,523],{"className":4252,"style":522},[434,521],[413,4254,528],{"className":4255},[527],[413,4257,4259,4262],{"className":4258},[434],[413,4260,547],{"className":4261},[434,521],[413,4263,4265],{"className":4264},[904],[413,4266,4268],{"className":4267},[908],[413,4269,4271],{"className":4270},[913],[413,4272,4274],{"className":4273,"style":1732},[917],[413,4275,4276,4279],{"style":1735},[413,4277],{"className":4278,"style":926},[925],[413,4280,4282],{"className":4281},[930,931,932,933],[413,4283,597],{"className":4284},[434,933],[413,4286,552],{"className":4287},[551],", we want\n",[413,4290,4292],{"className":4291},[416],[413,4293,4295,4342,4363,4381],{"className":4294,"ariaHidden":421},[420],[413,4296,4298,4301,4304,4333,4336,4339],{"className":4297},[425],[413,4299],{"className":4300,"style":4163},[429],[413,4302,2313],{"className":4303},[434],[413,4305,4307,4310],{"className":4306},[434],[413,4308,547],{"className":4309},[434,521],[413,4311,4313],{"className":4312},[904],[413,4314,4316],{"className":4315},[908],[413,4317,4319],{"className":4318},[913],[413,4320,4322],{"className":4321,"style":1732},[917],[413,4323,4324,4327],{"style":1735},[413,4325],{"className":4326,"style":926},[925],[413,4328,4330],{"className":4329},[930,931,932,933],[413,4331,597],{"className":4332},[434,933],[413,4334],{"className":4335,"style":1595},[542],[413,4337,435],{"className":4338},[2351],[413,4340],{"className":4341,"style":1595},[542],[413,4343,4345,4348,4351,4354,4357,4360],{"className":4344},[425],[413,4346],{"className":4347,"style":2361},[429],[413,4349,2365],{"className":4350},[434],[413,4352,547],{"className":4353},[434,521],[413,4355],{"className":4356,"style":1595},[542],[413,4358,435],{"className":4359},[2351],[413,4361],{"className":4362,"style":1595},[542],[413,4364,4366,4369,4372,4375,4378],{"className":4365},[425],[413,4367],{"className":4368,"style":3462},[429],[413,4370,2498],{"className":4371},[434],[413,4373],{"className":4374,"style":656},[542],[413,4376,3244],{"className":4377},[660],[413,4379],{"className":4380,"style":656},[542],[413,4382,4384,4387,4390,4393],{"className":4383},[425],[413,4385],{"className":4386,"style":1732},[429],[413,4388,3044],{"className":4389},[434,521],[413,4391],{"className":4392,"style":543},[542],[413,4394,4396,4399],{"className":4395},[434],[413,4397,547],{"className":4398},[434,521],[413,4400,4402],{"className":4401},[904],[413,4403,4405],{"className":4404},[908],[413,4406,4408],{"className":4407},[913],[413,4409,4411],{"className":4410,"style":1732},[917],[413,4412,4413,4416],{"style":1735},[413,4414],{"className":4415,"style":926},[925],[413,4417,4419],{"className":4418},[930,931,932,933],[413,4420,597],{"className":4421},[434,933],". For ",[413,4424,4426],{"className":4425},[416],[413,4427,4429,4448],{"className":4428,"ariaHidden":421},[420],[413,4430,4432,4436,4439,4442,4445],{"className":4431},[425],[413,4433],{"className":4434,"style":4435},[429],"height:0.7719em;vertical-align:-0.136em;",[413,4437,547],{"className":4438},[434,521],[413,4440],{"className":4441,"style":656},[542],[413,4443,3158],{"className":4444},[660],[413,4446],{"className":4447,"style":656},[542],[413,4449,4451,4454],{"className":4450},[425],[413,4452],{"className":4453,"style":2384},[429],[413,4455,588],{"className":4456},[434]," each lower term is at most ",[413,4459,4461],{"className":4460},[416],[413,4462,4464],{"className":4463,"ariaHidden":421},[420],[413,4465,4467,4470],{"className":4466},[425],[413,4468],{"className":4469,"style":1732},[429],[413,4471,4473,4476],{"className":4472},[434],[413,4474,547],{"className":4475},[434,521],[413,4477,4479],{"className":4478},[904],[413,4480,4482],{"className":4481},[908],[413,4483,4485],{"className":4484},[913],[413,4486,4488],{"className":4487,"style":1732},[917],[413,4489,4490,4493],{"style":1735},[413,4491],{"className":4492,"style":926},[925],[413,4494,4496],{"className":4495},[930,931,932,933],[413,4497,597],{"className":4498},[434,933],", so\n",[413,4501,4503],{"className":4502},[416],[413,4504,4506,4553,4574,4592,4639,4686,4733],{"className":4505,"ariaHidden":421},[420],[413,4507,4509,4512,4515,4544,4547,4550],{"className":4508},[425],[413,4510],{"className":4511,"style":4163},[429],[413,4513,2313],{"className":4514},[434],[413,4516,4518,4521],{"className":4517},[434],[413,4519,547],{"className":4520},[434,521],[413,4522,4524],{"className":4523},[904],[413,4525,4527],{"className":4526},[908],[413,4528,4530],{"className":4529},[913],[413,4531,4533],{"className":4532,"style":1732},[917],[413,4534,4535,4538],{"style":1735},[413,4536],{"className":4537,"style":926},[925],[413,4539,4541],{"className":4540},[930,931,932,933],[413,4542,597],{"className":4543},[434,933],[413,4545],{"className":4546,"style":1595},[542],[413,4548,435],{"className":4549},[2351],[413,4551],{"className":4552,"style":1595},[542],[413,4554,4556,4559,4562,4565,4568,4571],{"className":4555},[425],[413,4557],{"className":4558,"style":2361},[429],[413,4560,2365],{"className":4561},[434],[413,4563,547],{"className":4564},[434,521],[413,4566],{"className":4567,"style":1595},[542],[413,4569,435],{"className":4570},[2351],[413,4572],{"className":4573,"style":1595},[542],[413,4575,4577,4580,4583,4586,4589],{"className":4576},[425],[413,4578],{"className":4579,"style":3462},[429],[413,4581,2498],{"className":4582},[434],[413,4584],{"className":4585,"style":656},[542],[413,4587,3244],{"className":4588},[660],[413,4590],{"className":4591,"style":656},[542],[413,4593,4595,4598,4601,4630,4633,4636],{"className":4594},[425],[413,4596],{"className":4597,"style":4163},[429],[413,4599,2313],{"className":4600},[434],[413,4602,4604,4607],{"className":4603},[434],[413,4605,547],{"className":4606},[434,521],[413,4608,4610],{"className":4609},[904],[413,4611,4613],{"className":4612},[908],[413,4614,4616],{"className":4615},[913],[413,4617,4619],{"className":4618,"style":1732},[917],[413,4620,4621,4624],{"style":1735},[413,4622],{"className":4623,"style":926},[925],[413,4625,4627],{"className":4626},[930,931,932,933],[413,4628,597],{"className":4629},[434,933],[413,4631],{"className":4632,"style":1595},[542],[413,4634,435],{"className":4635},[2351],[413,4637],{"className":4638,"style":1595},[542],[413,4640,4642,4645,4648,4677,4680,4683],{"className":4641},[425],[413,4643],{"className":4644,"style":4163},[429],[413,4646,2365],{"className":4647},[434],[413,4649,4651,4654],{"className":4650},[434],[413,4652,547],{"className":4653},[434,521],[413,4655,4657],{"className":4656},[904],[413,4658,4660],{"className":4659},[908],[413,4661,4663],{"className":4662},[913],[413,4664,4666],{"className":4665,"style":1732},[917],[413,4667,4668,4671],{"style":1735},[413,4669],{"className":4670,"style":926},[925],[413,4672,4674],{"className":4673},[930,931,932,933],[413,4675,597],{"className":4676},[434,933],[413,4678],{"className":4679,"style":1595},[542],[413,4681,435],{"className":4682},[2351],[413,4684],{"className":4685,"style":1595},[542],[413,4687,4689,4692,4695,4724,4727,4730],{"className":4688},[425],[413,4690],{"className":4691,"style":1732},[429],[413,4693,2498],{"className":4694},[434],[413,4696,4698,4701],{"className":4697},[434],[413,4699,547],{"className":4700},[434,521],[413,4702,4704],{"className":4703},[904],[413,4705,4707],{"className":4706},[908],[413,4708,4710],{"className":4709},[913],[413,4711,4713],{"className":4712,"style":1732},[917],[413,4714,4715,4718],{"style":1735},[413,4716],{"className":4717,"style":926},[925],[413,4719,4721],{"className":4720},[930,931,932,933],[413,4722,597],{"className":4723},[434,933],[413,4725],{"className":4726,"style":656},[542],[413,4728,779],{"className":4729},[660],[413,4731],{"className":4732,"style":656},[542],[413,4734,4736,4739,4743],{"className":4735},[425],[413,4737],{"className":4738,"style":1732},[429],[413,4740,4742],{"className":4741},[434],"253",[413,4744,4746,4749],{"className":4745},[434],[413,4747,547],{"className":4748},[434,521],[413,4750,4752],{"className":4751},[904],[413,4753,4755],{"className":4754},[908],[413,4756,4758],{"className":4757},[913],[413,4759,4761],{"className":4760,"style":1732},[917],[413,4762,4763,4766],{"style":1735},[413,4764],{"className":4765,"style":926},[925],[413,4767,4769],{"className":4768},[930,931,932,933],[413,4770,597],{"className":4771},[434,933],"; thus ",[413,4774,4776],{"className":4775},[416],[413,4777,4779,4797],{"className":4778,"ariaHidden":421},[420],[413,4780,4782,4785,4788,4791,4794],{"className":4781},[425],[413,4783],{"className":4784,"style":566},[429],[413,4786,3044],{"className":4787},[434,521],[413,4789],{"className":4790,"style":656},[542],[413,4792,779],{"className":4793},[660],[413,4795],{"className":4796,"style":656},[542],[413,4798,4800,4803],{"className":4799},[425],[413,4801],{"className":4802,"style":2384},[429],[413,4804,4742],{"className":4805},[434],[413,4807,4809],{"className":4808},[416],[413,4810,4812,4867],{"className":4811,"ariaHidden":421},[420],[413,4813,4815,4818,4858,4861,4864],{"className":4814},[425],[413,4816],{"className":4817,"style":2153},[429],[413,4819,4821,4824],{"className":4820},[434],[413,4822,547],{"className":4823},[434,521],[413,4825,4827],{"className":4826},[904],[413,4828,4830,4850],{"className":4829},[908,909],[413,4831,4833,4847],{"className":4832},[913],[413,4834,4836],{"className":4835,"style":918},[917],[413,4837,4838,4841],{"style":921},[413,4839],{"className":4840,"style":926},[925],[413,4842,4844],{"className":4843},[930,931,932,933],[413,4845,2183],{"className":4846},[434,933],[413,4848,941],{"className":4849},[940],[413,4851,4853],{"className":4852},[913],[413,4854,4856],{"className":4855,"style":948},[917],[413,4857],{},[413,4859],{"className":4860,"style":656},[542],[413,4862,779],{"className":4863},[660],[413,4865],{"className":4866,"style":656},[542],[413,4868,4870,4873],{"className":4869},[425],[413,4871],{"className":4872,"style":2384},[429],[413,4874,588],{"className":4875},[434],"\nwork. The same move handles ",[385,4878,4879],{},"any"," polynomial; see the theorem below.",[599,4882,4884],{"id":4883},"big-omega-asymptotic-lower-bound","Big-Omega: asymptotic lower bound",[381,4886,4887],{},"Mirror the quantifiers, flip the inequality:",[2954,4889,4890],{"type":2956},[381,4891,4892,1071,4912,3020,4969,5214,5251,5252,1158],{},[398,4893,4894,4895,4911],{},"Definition (",[413,4896,4898],{"className":4897},[416],[413,4899,4901],{"className":4900,"ariaHidden":421},[420],[413,4902,4904,4907],{"className":4903},[425],[413,4905],{"className":4906,"style":648},[429],[413,4908,4910],{"className":4909},[434],"Ω",").",[413,4913,4915],{"className":4914},[416],[413,4916,4918,4945],{"className":4917,"ariaHidden":421},[420],[413,4919,4921,4924,4927,4930,4933,4936,4939,4942],{"className":4920},[425],[413,4922],{"className":4923,"style":481},[429],[413,4925,2877],{"className":4926,"style":2876},[434,521],[413,4928,528],{"className":4929},[527],[413,4931,547],{"className":4932},[434,521],[413,4934,552],{"className":4935},[551],[413,4937],{"className":4938,"style":656},[542],[413,4940,779],{"className":4941},[660],[413,4943],{"className":4944,"style":656},[542],[413,4946,4948,4951,4954,4957,4960,4963,4966],{"className":4947},[425],[413,4949],{"className":4950,"style":481},[429],[413,4952,4910],{"className":4953},[434],[413,4955,528],{"className":4956},[527],[413,4958,1882],{"className":4959,"style":2894},[434,521],[413,4961,528],{"className":4962},[527],[413,4964,547],{"className":4965},[434,521],[413,4967,3019],{"className":4968},[551],[413,4970,4972],{"className":4971},[416],[413,4973,4975,4999,5069,5102,5160,5187],{"className":4974,"ariaHidden":421},[420],[413,4976,4978,4981,4984,4987,4990,4993,4996],{"className":4977},[425],[413,4979],{"className":4980,"style":3033},[429],[413,4982,3037],{"className":4983},[434],[413,4985],{"className":4986,"style":543},[542],[413,4988,3044],{"className":4989},[434,521],[413,4991],{"className":4992,"style":656},[542],[413,4994,3051],{"className":4995},[660],[413,4997],{"className":4998,"style":656},[542],[413,5000,5002,5005,5008,5011,5014,5017,5020,5060,5063,5066],{"className":5001},[425],[413,5003],{"className":5004,"style":3061},[429],[413,5006,2183],{"className":5007},[434],[413,5009],{"className":5010,"style":656},[542],[413,5012],{"className":5013,"style":656},[542],[413,5015,3037],{"className":5016},[434],[413,5018],{"className":5019,"style":543},[542],[413,5021,5023,5026],{"className":5022},[434],[413,5024,547],{"className":5025},[434,521],[413,5027,5029],{"className":5028},[904],[413,5030,5032,5052],{"className":5031},[908,909],[413,5033,5035,5049],{"className":5034},[913],[413,5036,5038],{"className":5037,"style":918},[917],[413,5039,5040,5043],{"style":921},[413,5041],{"className":5042,"style":926},[925],[413,5044,5046],{"className":5045},[930,931,932,933],[413,5047,2183],{"className":5048},[434,933],[413,5050,941],{"className":5051},[940],[413,5053,5055],{"className":5054},[913],[413,5056,5058],{"className":5057,"style":948},[917],[413,5059],{},[413,5061],{"className":5062,"style":656},[542],[413,5064,3051],{"className":5065},[660],[413,5067],{"className":5068,"style":656},[542],[413,5070,5072,5075,5078,5081,5084,5087,5090,5093,5096,5099],{"className":5071},[425],[413,5073],{"className":5074,"style":3132},[429],[413,5076,2183],{"className":5077},[434],[413,5079],{"className":5080,"style":656},[542],[413,5082],{"className":5083,"style":656},[542],[413,5085,3145],{"className":5086},[434],[413,5088],{"className":5089,"style":543},[542],[413,5091,547],{"className":5092},[434,521],[413,5094],{"className":5095,"style":656},[542],[413,5097,3158],{"className":5098},[660],[413,5100],{"className":5101,"style":656},[542],[413,5103,5105,5108,5148,5151,5154,5157],{"className":5104},[425],[413,5106],{"className":5107,"style":2153},[429],[413,5109,5111,5114],{"className":5110},[434],[413,5112,547],{"className":5113},[434,521],[413,5115,5117],{"className":5116},[904],[413,5118,5120,5140],{"className":5119},[908,909],[413,5121,5123,5137],{"className":5122},[913],[413,5124,5126],{"className":5125,"style":918},[917],[413,5127,5128,5131],{"style":921},[413,5129],{"className":5130,"style":926},[925],[413,5132,5134],{"className":5133},[930,931,932,933],[413,5135,2183],{"className":5136},[434,933],[413,5138,941],{"className":5139},[940],[413,5141,5143],{"className":5142},[913],[413,5144,5146],{"className":5145,"style":948},[917],[413,5147],{},[413,5149],{"className":5150,"style":656},[542],[413,5152,661],{"className":5153},[660],[413,5155],{"className":5156,"style":656},[542],[413,5158],{"className":5159,"style":656},[542],[413,5161,5163,5166,5169,5172,5175,5178,5181,5184],{"className":5162},[425],[413,5164],{"className":5165,"style":481},[429],[413,5167,2877],{"className":5168,"style":2876},[434,521],[413,5170,528],{"className":5171},[527],[413,5173,547],{"className":5174},[434,521],[413,5176,552],{"className":5177},[551],[413,5179],{"className":5180,"style":656},[542],[413,5182,3158],{"className":5183},[660],[413,5185],{"className":5186,"style":656},[542],[413,5188,5190,5193,5196,5199,5202,5205,5208,5211],{"className":5189},[425],[413,5191],{"className":5192,"style":481},[429],[413,5194,3044],{"className":5195},[434,521],[413,5197],{"className":5198,"style":543},[542],[413,5200,1882],{"className":5201,"style":2894},[434,521],[413,5203,528],{"className":5204},[527],[413,5206,547],{"className":5207},[434,521],[413,5209,552],{"className":5210},[551],[413,5212,1158],{"className":5213},[434],[502,5215,5216,5231,5232,5235,5236,1158],{},[413,5217,5219],{"className":5218},[416],[413,5220,5222],{"className":5221,"ariaHidden":421},[420],[413,5223,5225,5228],{"className":5224},[425],[413,5226],{"className":5227,"style":2787},[429],[413,5229,2877],{"className":5230,"style":2876},[434,521]," grows ",[398,5233,5234],{},"at least"," as fast as ",[413,5237,5239],{"className":5238},[416],[413,5240,5242],{"className":5241,"ariaHidden":421},[420],[413,5243,5245,5248],{"className":5244},[425],[413,5246],{"className":5247,"style":2890},[429],[413,5249,1882],{"className":5250,"style":2894},[434,521]," Roughly, ",[413,5253,5255],{"className":5254},[416],[413,5256,5258,5277],{"className":5257,"ariaHidden":421},[420],[413,5259,5261,5264,5267,5270,5274],{"className":5260},[425],[413,5262],{"className":5263,"style":2787},[429],[413,5265,2877],{"className":5266,"style":2876},[434,521],[413,5268],{"className":5269,"style":656},[542],[413,5271,5273],{"className":5272},[660,3335],"≽",[413,5275],{"className":5276,"style":656},[542],[413,5278,5280,5283],{"className":5279},[425],[413,5281],{"className":5282,"style":2890},[429],[413,5284,1882],{"className":5285,"style":2894},[434,521],[381,5287,5288,5289,5304,5305,5347,5348,1158],{},"It is the mirror image of ",[413,5290,5292],{"className":5291},[416],[413,5293,5295],{"className":5294,"ariaHidden":421},[420],[413,5296,5298,5301],{"className":5297},[425],[413,5299],{"className":5300,"style":648},[429],[413,5302,523],{"className":5303,"style":522},[434,521],": ",[413,5306,5308],{"className":5307},[416],[413,5309,5311,5329],{"className":5310,"ariaHidden":421},[420],[413,5312,5314,5317,5320,5323,5326],{"className":5313},[425],[413,5315],{"className":5316,"style":2787},[429],[413,5318,2877],{"className":5319,"style":2876},[434,521],[413,5321],{"className":5322,"style":656},[542],[413,5324,779],{"className":5325},[660],[413,5327],{"className":5328,"style":656},[542],[413,5330,5332,5335,5338,5341,5344],{"className":5331},[425],[413,5333],{"className":5334,"style":481},[429],[413,5336,523],{"className":5337,"style":522},[434,521],[413,5339,528],{"className":5340},[527],[413,5342,1882],{"className":5343,"style":2894},[434,521],[413,5345,552],{"className":5346},[551]," if and only if ",[413,5349,5351],{"className":5350},[416],[413,5352,5354,5372],{"className":5353,"ariaHidden":421},[420],[413,5355,5357,5360,5363,5366,5369],{"className":5356},[425],[413,5358],{"className":5359,"style":2890},[429],[413,5361,1882],{"className":5362,"style":2894},[434,521],[413,5364],{"className":5365,"style":656},[542],[413,5367,779],{"className":5368},[660],[413,5370],{"className":5371,"style":656},[542],[413,5373,5375,5378,5381,5384,5387],{"className":5374},[425],[413,5376],{"className":5377,"style":481},[429],[413,5379,4910],{"className":5380},[434],[413,5382,528],{"className":5383},[527],[413,5385,2877],{"className":5386,"style":2876},[434,521],[413,5388,552],{"className":5389},[551],[599,5391,5393],{"id":5392},"big-theta-asymptotic-tight-bound","Big-Theta: asymptotic tight bound",[2954,5395,5396],{"type":2956},[381,5397,5398,1071,5415,5445,5446,5470,5471,5615,5616],{},[398,5399,4894,5400,4911],{},[413,5401,5403],{"className":5402},[416],[413,5404,5406],{"className":5405,"ariaHidden":421},[420],[413,5407,5409,5412],{"className":5408},[425],[413,5410],{"className":5411,"style":648},[429],[413,5413,1684],{"className":5414},[434],[413,5416,5418],{"className":5417},[416],[413,5419,5421],{"className":5420,"ariaHidden":421},[420],[413,5422,5424,5427,5430,5433,5436,5439,5442],{"className":5423},[425],[413,5425],{"className":5426,"style":481},[429],[413,5428,1684],{"className":5429},[434],[413,5431,528],{"className":5432},[527],[413,5434,1882],{"className":5435,"style":2894},[434,521],[413,5437,528],{"className":5438},[527],[413,5440,547],{"className":5441},[434,521],[413,5443,3019],{"className":5444},[551]," is the set of functions ",[413,5447,5449],{"className":5448},[416],[413,5450,5452],{"className":5451,"ariaHidden":421},[420],[413,5453,5455,5458,5461,5464,5467],{"className":5454},[425],[413,5456],{"className":5457,"style":481},[429],[413,5459,2877],{"className":5460,"style":2876},[434,521],[413,5462,528],{"className":5463},[527],[413,5465,547],{"className":5466},[434,521],[413,5468,552],{"className":5469},[551]," for which there exist positive\nconstants ",[413,5472,5474],{"className":5473},[416],[413,5475,5477],{"className":5476,"ariaHidden":421},[420],[413,5478,5480,5483,5523,5526,5529,5569,5572,5575],{"className":5479},[425],[413,5481],{"className":5482,"style":2890},[429],[413,5484,5486,5489],{"className":5485},[434],[413,5487,3044],{"className":5488},[434,521],[413,5490,5492],{"className":5491},[904],[413,5493,5495,5515],{"className":5494},[908,909],[413,5496,5498,5512],{"className":5497},[913],[413,5499,5501],{"className":5500,"style":918},[917],[413,5502,5503,5506],{"style":921},[413,5504],{"className":5505,"style":926},[925],[413,5507,5509],{"className":5508},[930,931,932,933],[413,5510,588],{"className":5511},[434,933],[413,5513,941],{"className":5514},[940],[413,5516,5518],{"className":5517},[913],[413,5519,5521],{"className":5520,"style":948},[917],[413,5522],{},[413,5524,955],{"className":5525},[954],[413,5527],{"className":5528,"style":543},[542],[413,5530,5532,5535],{"className":5531},[434],[413,5533,3044],{"className":5534},[434,521],[413,5536,5538],{"className":5537},[904],[413,5539,5541,5561],{"className":5540},[908,909],[413,5542,5544,5558],{"className":5543},[913],[413,5545,5547],{"className":5546,"style":918},[917],[413,5548,5549,5552],{"style":921},[413,5550],{"className":5551,"style":926},[925],[413,5553,5555],{"className":5554},[930,931,932,933],[413,5556,597],{"className":5557},[434,933],[413,5559,941],{"className":5560},[940],[413,5562,5564],{"className":5563},[913],[413,5565,5567],{"className":5566,"style":948},[917],[413,5568],{},[413,5570,955],{"className":5571},[954],[413,5573],{"className":5574,"style":543},[542],[413,5576,5578,5581],{"className":5577},[434],[413,5579,547],{"className":5580},[434,521],[413,5582,5584],{"className":5583},[904],[413,5585,5587,5607],{"className":5586},[908,909],[413,5588,5590,5604],{"className":5589},[913],[413,5591,5593],{"className":5592,"style":918},[917],[413,5594,5595,5598],{"style":921},[413,5596],{"className":5597,"style":926},[925],[413,5599,5601],{"className":5600},[930,931,932,933],[413,5602,2183],{"className":5603},[434,933],[413,5605,941],{"className":5606},[940],[413,5608,5610],{"className":5609},[913],[413,5611,5613],{"className":5612,"style":948},[917],[413,5614],{}," such that\n",[413,5617,5619],{"className":5618},[416],[413,5620,5622,5640,5710,5737,5821],{"className":5621,"ariaHidden":421},[420],[413,5623,5625,5628,5631,5634,5637],{"className":5624},[425],[413,5626],{"className":5627,"style":3462},[429],[413,5629,2183],{"className":5630},[434],[413,5632],{"className":5633,"style":656},[542],[413,5635,3244],{"className":5636},[660],[413,5638],{"className":5639,"style":656},[542],[413,5641,5643,5646,5686,5689,5692,5695,5698,5701,5704,5707],{"className":5642},[425],[413,5644],{"className":5645,"style":481},[429],[413,5647,5649,5652],{"className":5648},[434],[413,5650,3044],{"className":5651},[434,521],[413,5653,5655],{"className":5654},[904],[413,5656,5658,5678],{"className":5657},[908,909],[413,5659,5661,5675],{"className":5660},[913],[413,5662,5664],{"className":5663,"style":918},[917],[413,5665,5666,5669],{"style":921},[413,5667],{"className":5668,"style":926},[925],[413,5670,5672],{"className":5671},[930,931,932,933],[413,5673,588],{"className":5674},[434,933],[413,5676,941],{"className":5677},[940],[413,5679,5681],{"className":5680},[913],[413,5682,5684],{"className":5683,"style":948},[917],[413,5685],{},[413,5687],{"className":5688,"style":543},[542],[413,5690,1882],{"className":5691,"style":2894},[434,521],[413,5693,528],{"className":5694},[527],[413,5696,547],{"className":5697},[434,521],[413,5699,552],{"className":5700},[551],[413,5702],{"className":5703,"style":656},[542],[413,5705,3244],{"className":5706},[660],[413,5708],{"className":5709,"style":656},[542],[413,5711,5713,5716,5719,5722,5725,5728,5731,5734],{"className":5712},[425],[413,5714],{"className":5715,"style":481},[429],[413,5717,2877],{"className":5718,"style":2876},[434,521],[413,5720,528],{"className":5721},[527],[413,5723,547],{"className":5724},[434,521],[413,5726,552],{"className":5727},[551],[413,5729],{"className":5730,"style":656},[542],[413,5732,3244],{"className":5733},[660],[413,5735],{"className":5736,"style":656},[542],[413,5738,5740,5743,5783,5786,5789,5792,5795,5798,5802,5809,5812,5815,5818],{"className":5739},[425],[413,5741],{"className":5742,"style":481},[429],[413,5744,5746,5749],{"className":5745},[434],[413,5747,3044],{"className":5748},[434,521],[413,5750,5752],{"className":5751},[904],[413,5753,5755,5775],{"className":5754},[908,909],[413,5756,5758,5772],{"className":5757},[913],[413,5759,5761],{"className":5760,"style":918},[917],[413,5762,5763,5766],{"style":921},[413,5764],{"className":5765,"style":926},[925],[413,5767,5769],{"className":5768},[930,931,932,933],[413,5770,597],{"className":5771},[434,933],[413,5773,941],{"className":5774},[940],[413,5776,5778],{"className":5777},[913],[413,5779,5781],{"className":5780,"style":948},[917],[413,5782],{},[413,5784],{"className":5785,"style":543},[542],[413,5787,1882],{"className":5788,"style":2894},[434,521],[413,5790,528],{"className":5791},[527],[413,5793,547],{"className":5794},[434,521],[413,5796,552],{"className":5797},[551],[413,5799],{"className":5800,"style":5801},[542],"margin-right:1em;",[413,5803,5805],{"className":5804},[434,802],[413,5806,5808],{"className":5807},[434],"for all ",[413,5810,547],{"className":5811},[434,521],[413,5813],{"className":5814,"style":656},[542],[413,5816,3158],{"className":5817},[660],[413,5819],{"className":5820,"style":656},[542],[413,5822,5824,5827,5867],{"className":5823},[425],[413,5825],{"className":5826,"style":2153},[429],[413,5828,5830,5833],{"className":5829},[434],[413,5831,547],{"className":5832},[434,521],[413,5834,5836],{"className":5835},[904],[413,5837,5839,5859],{"className":5838},[908,909],[413,5840,5842,5856],{"className":5841},[913],[413,5843,5845],{"className":5844,"style":918},[917],[413,5846,5847,5850],{"style":921},[413,5848],{"className":5849,"style":926},[925],[413,5851,5853],{"className":5852},[930,931,932,933],[413,5854,2183],{"className":5855},[434,933],[413,5857,941],{"className":5858},[940],[413,5860,5862],{"className":5861},[913],[413,5863,5865],{"className":5864,"style":948},[917],[413,5866],{},[413,5868,1158],{"className":5869},[434],[381,5871,5872,5887,5888,5903,5904,5919,5920,5923,5924,5939],{},[413,5873,5875],{"className":5874},[416],[413,5876,5878],{"className":5877,"ariaHidden":421},[420],[413,5879,5881,5884],{"className":5880},[425],[413,5882],{"className":5883,"style":648},[429],[413,5885,1684],{"className":5886},[434]," pins ",[413,5889,5891],{"className":5890},[416],[413,5892,5894],{"className":5893,"ariaHidden":421},[420],[413,5895,5897,5900],{"className":5896},[425],[413,5898],{"className":5899,"style":2787},[429],[413,5901,2877],{"className":5902,"style":2876},[434,521]," between two constant multiples of ",[413,5905,5907],{"className":5906},[416],[413,5908,5910],{"className":5909,"ariaHidden":421},[420],[413,5911,5913,5916],{"className":5912},[425],[413,5914],{"className":5915,"style":2890},[429],[413,5917,1882],{"className":5918,"style":2894},[434,521],": it grows ",[385,5921,5922],{},"exactly"," as\nfast as ",[413,5925,5927],{"className":5926},[416],[413,5928,5930],{"className":5929,"ariaHidden":421},[420],[413,5931,5933,5936],{"className":5932},[425],[413,5934],{"className":5935,"style":2890},[429],[413,5937,1882],{"className":5938,"style":2894},[434,521],", up to constants. The fundamental link is",[413,5941,5943],{"className":5942},[1263],[413,5944,5946],{"className":5945},[416],[413,5947,5949,5976,6016,6043,6102],{"className":5948,"ariaHidden":421},[420],[413,5950,5952,5955,5958,5961,5964,5967,5970,5973],{"className":5951},[425],[413,5953],{"className":5954,"style":481},[429],[413,5956,2877],{"className":5957,"style":2876},[434,521],[413,5959,528],{"className":5960},[527],[413,5962,547],{"className":5963},[434,521],[413,5965,552],{"className":5966},[551],[413,5968],{"className":5969,"style":656},[542],[413,5971,779],{"className":5972},[660],[413,5974],{"className":5975,"style":656},[542],[413,5977,5979,5982,5985,5988,5991,5994,5997,6000,6003,6006,6010,6013],{"className":5978},[425],[413,5980],{"className":5981,"style":481},[429],[413,5983,1684],{"className":5984},[434],[413,5986,528],{"className":5987},[527],[413,5989,1882],{"className":5990,"style":2894},[434,521],[413,5992,528],{"className":5993},[527],[413,5995,547],{"className":5996},[434,521],[413,5998,3019],{"className":5999},[551],[413,6001],{"className":6002,"style":656},[542],[413,6004],{"className":6005,"style":656},[542],[413,6007,6009],{"className":6008},[660],"↔",[413,6011],{"className":6012,"style":656},[542],[413,6014],{"className":6015,"style":656},[542],[413,6017,6019,6022,6025,6028,6031,6034,6037,6040],{"className":6018},[425],[413,6020],{"className":6021,"style":481},[429],[413,6023,2877],{"className":6024,"style":2876},[434,521],[413,6026,528],{"className":6027},[527],[413,6029,547],{"className":6030},[434,521],[413,6032,552],{"className":6033},[551],[413,6035],{"className":6036,"style":656},[542],[413,6038,779],{"className":6039},[660],[413,6041],{"className":6042,"style":656},[542],[413,6044,6046,6049,6052,6055,6058,6061,6064,6067,6071,6078,6081,6084,6087,6090,6093,6096,6099],{"className":6045},[425],[413,6047],{"className":6048,"style":481},[429],[413,6050,523],{"className":6051,"style":522},[434,521],[413,6053,528],{"className":6054},[527],[413,6056,1882],{"className":6057,"style":2894},[434,521],[413,6059,528],{"className":6060},[527],[413,6062,547],{"className":6063},[434,521],[413,6065,3019],{"className":6066},[551],[413,6068,6070],{"className":6069},[542]," ",[413,6072,6074],{"className":6073},[434,802],[413,6075,6077],{"className":6076},[434],"and",[413,6079,6070],{"className":6080},[542],[413,6082,2877],{"className":6083,"style":2876},[434,521],[413,6085,528],{"className":6086},[527],[413,6088,547],{"className":6089},[434,521],[413,6091,552],{"className":6092},[551],[413,6094],{"className":6095,"style":656},[542],[413,6097,779],{"className":6098},[660],[413,6100],{"className":6101,"style":656},[542],[413,6103,6105,6108,6111,6114,6117,6120,6123,6126],{"className":6104},[425],[413,6106],{"className":6107,"style":481},[429],[413,6109,4910],{"className":6110},[434],[413,6112,528],{"className":6113},[527],[413,6115,1882],{"className":6116,"style":2894},[434,521],[413,6118,528],{"className":6119},[527],[413,6121,547],{"className":6122},[434,521],[413,6124,3019],{"className":6125},[551],[413,6127,1158],{"className":6128},[434],[381,6130,6131,6132,6186,6187,1597,6269,6350,6351,6366,6367,6382,6383,6398],{},"When we say insertion sort ",[502,6133,6134,6135,6185],{},"is ",[413,6136,6138],{"className":6137},[416],[413,6139,6141],{"className":6140,"ariaHidden":421},[420],[413,6142,6144,6147,6150,6153,6182],{"className":6143},[425],[413,6145],{"className":6146,"style":1707},[429],[413,6148,1684],{"className":6149},[434],[413,6151,528],{"className":6152},[527],[413,6154,6156,6159],{"className":6155},[434],[413,6157,547],{"className":6158},[434,521],[413,6160,6162],{"className":6161},[904],[413,6163,6165],{"className":6164},[908],[413,6166,6168],{"className":6167},[913],[413,6169,6171],{"className":6170,"style":1732},[917],[413,6172,6173,6176],{"style":1735},[413,6174],{"className":6175,"style":926},[925],[413,6177,6179],{"className":6178},[930,931,932,933],[413,6180,597],{"className":6181},[434,933],[413,6183,552],{"className":6184},[551]," in the worst case,"," we mean its\nworst-case cost is sandwiched between ",[413,6188,6190],{"className":6189},[416],[413,6191,6193],{"className":6192,"ariaHidden":421},[420],[413,6194,6196,6200,6240],{"className":6195},[425],[413,6197],{"className":6198,"style":6199},[429],"height:0.9641em;vertical-align:-0.15em;",[413,6201,6203,6206],{"className":6202},[434],[413,6204,3044],{"className":6205},[434,521],[413,6207,6209],{"className":6208},[904],[413,6210,6212,6232],{"className":6211},[908,909],[413,6213,6215,6229],{"className":6214},[913],[413,6216,6218],{"className":6217,"style":918},[917],[413,6219,6220,6223],{"style":921},[413,6221],{"className":6222,"style":926},[925],[413,6224,6226],{"className":6225},[930,931,932,933],[413,6227,588],{"className":6228},[434,933],[413,6230,941],{"className":6231},[940],[413,6233,6235],{"className":6234},[913],[413,6236,6238],{"className":6237,"style":948},[917],[413,6239],{},[413,6241,6243,6246],{"className":6242},[434],[413,6244,547],{"className":6245},[434,521],[413,6247,6249],{"className":6248},[904],[413,6250,6252],{"className":6251},[908],[413,6253,6255],{"className":6254},[913],[413,6256,6258],{"className":6257,"style":1732},[917],[413,6259,6260,6263],{"style":1735},[413,6261],{"className":6262,"style":926},[925],[413,6264,6266],{"className":6265},[930,931,932,933],[413,6267,597],{"className":6268},[434,933],[413,6270,6272],{"className":6271},[416],[413,6273,6275],{"className":6274,"ariaHidden":421},[420],[413,6276,6278,6281,6321],{"className":6277},[425],[413,6279],{"className":6280,"style":6199},[429],[413,6282,6284,6287],{"className":6283},[434],[413,6285,3044],{"className":6286},[434,521],[413,6288,6290],{"className":6289},[904],[413,6291,6293,6313],{"className":6292},[908,909],[413,6294,6296,6310],{"className":6295},[913],[413,6297,6299],{"className":6298,"style":918},[917],[413,6300,6301,6304],{"style":921},[413,6302],{"className":6303,"style":926},[925],[413,6305,6307],{"className":6306},[930,931,932,933],[413,6308,597],{"className":6309},[434,933],[413,6311,941],{"className":6312},[940],[413,6314,6316],{"className":6315},[913],[413,6317,6319],{"className":6318,"style":948},[917],[413,6320],{},[413,6322,6324,6327],{"className":6323},[434],[413,6325,547],{"className":6326},[434,521],[413,6328,6330],{"className":6329},[904],[413,6331,6333],{"className":6332},[908],[413,6334,6336],{"className":6335},[913],[413,6337,6339],{"className":6338,"style":1732},[917],[413,6340,6341,6344],{"style":1735},[413,6342],{"className":6343,"style":926},[925],[413,6345,6347],{"className":6346},[930,931,932,933],[413,6348,597],{"className":6349},[434,933],", a precise,\ntwo-sided claim. When we only have an upper bound we say ",[413,6352,6354],{"className":6353},[416],[413,6355,6357],{"className":6356,"ariaHidden":421},[420],[413,6358,6360,6363],{"className":6359},[425],[413,6361],{"className":6362,"style":648},[429],[413,6364,523],{"className":6365,"style":522},[434,521],"; this is why people\nloosely write ",[413,6368,6370],{"className":6369},[416],[413,6371,6373],{"className":6372,"ariaHidden":421},[420],[413,6374,6376,6379],{"className":6375},[425],[413,6377],{"className":6378,"style":648},[429],[413,6380,523],{"className":6381,"style":522},[434,521]," even where ",[413,6384,6386],{"className":6385},[416],[413,6387,6389],{"className":6388,"ariaHidden":421},[420],[413,6390,6392,6395],{"className":6391},[425],[413,6393],{"className":6394,"style":648},[429],[413,6396,1684],{"className":6397},[434]," holds. But the distinction matters, as the\nnext two results make concrete.",[599,6400,6402],{"id":6401},"the-polynomial-theorem","The polynomial theorem",[381,6404,6405],{},"The single most useful fact for everyday analysis collapses every polynomial to\nits leading power:",[2954,6407,6409],{"type":6408},"theorem",[381,6410,6411,6414,6415,6765,6766,6837,6838,1158],{},[398,6412,6413],{},"Theorem."," If ",[413,6416,6418],{"className":6417},[416],[413,6419,6421,6448,6537,6641,6660,6719],{"className":6420,"ariaHidden":421},[420],[413,6422,6424,6427,6430,6433,6436,6439,6442,6445],{"className":6423},[425],[413,6425],{"className":6426,"style":481},[429],[413,6428,2877],{"className":6429,"style":2876},[434,521],[413,6431,528],{"className":6432},[527],[413,6434,547],{"className":6435},[434,521],[413,6437,552],{"className":6438},[551],[413,6440],{"className":6441,"style":656},[542],[413,6443,779],{"className":6444},[660],[413,6446],{"className":6447,"style":656},[542],[413,6449,6451,6455,6498,6528,6531,6534],{"className":6450},[425],[413,6452],{"className":6453,"style":6454},[429],"height:0.9991em;vertical-align:-0.15em;",[413,6456,6458,6461],{"className":6457},[434],[413,6459,582],{"className":6460},[434,521],[413,6462,6464],{"className":6463},[904],[413,6465,6467,6490],{"className":6466},[908,909],[413,6468,6470,6487],{"className":6469},[913],[413,6471,6474],{"className":6472,"style":6473},[917],"height:0.3361em;",[413,6475,6476,6479],{"style":921},[413,6477],{"className":6478,"style":926},[925],[413,6480,6482],{"className":6481},[930,931,932,933],[413,6483,6486],{"className":6484,"style":6485},[434,521,933],"margin-right:0.0315em;","k",[413,6488,941],{"className":6489},[940],[413,6491,6493],{"className":6492},[913],[413,6494,6496],{"className":6495,"style":948},[917],[413,6497],{},[413,6499,6501,6504],{"className":6500},[434],[413,6502,547],{"className":6503},[434,521],[413,6505,6507],{"className":6506},[904],[413,6508,6510],{"className":6509},[908],[413,6511,6513],{"className":6512},[913],[413,6514,6517],{"className":6515,"style":6516},[917],"height:0.8491em;",[413,6518,6519,6522],{"style":1735},[413,6520],{"className":6521,"style":926},[925],[413,6523,6525],{"className":6524},[930,931,932,933],[413,6526,6486],{"className":6527,"style":6485},[434,521,933],[413,6529],{"className":6530,"style":1595},[542],[413,6532,435],{"className":6533},[2351],[413,6535],{"className":6536,"style":1595},[542],[413,6538,6540,6544,6594,6632,6635,6638],{"className":6539},[425],[413,6541],{"className":6542,"style":6543},[429],"height:1.0574em;vertical-align:-0.2083em;",[413,6545,6547,6550],{"className":6546},[434],[413,6548,582],{"className":6549},[434,521],[413,6551,6553],{"className":6552},[904],[413,6554,6556,6585],{"className":6555},[908,909],[413,6557,6559,6582],{"className":6558},[913],[413,6560,6562],{"className":6561,"style":6473},[917],[413,6563,6564,6567],{"style":921},[413,6565],{"className":6566,"style":926},[925],[413,6568,6570],{"className":6569},[930,931,932,933],[413,6571,6573,6576,6579],{"className":6572},[434,933],[413,6574,6486],{"className":6575,"style":6485},[434,521,933],[413,6577,452],{"className":6578},[2351,933],[413,6580,588],{"className":6581},[434,933],[413,6583,941],{"className":6584},[940],[413,6586,6588],{"className":6587},[913],[413,6589,6592],{"className":6590,"style":6591},[917],"height:0.2083em;",[413,6593],{},[413,6595,6597,6600],{"className":6596},[434],[413,6598,547],{"className":6599},[434,521],[413,6601,6603],{"className":6602},[904],[413,6604,6606],{"className":6605},[908],[413,6607,6609],{"className":6608},[913],[413,6610,6612],{"className":6611,"style":6516},[917],[413,6613,6614,6617],{"style":1735},[413,6615],{"className":6616,"style":926},[925],[413,6618,6620],{"className":6619},[930,931,932,933],[413,6621,6623,6626,6629],{"className":6622},[434,933],[413,6624,6486],{"className":6625,"style":6485},[434,521,933],[413,6627,452],{"className":6628},[2351,933],[413,6630,588],{"className":6631},[434,933],[413,6633],{"className":6634,"style":1595},[542],[413,6636,435],{"className":6637},[2351],[413,6639],{"className":6640,"style":1595},[542],[413,6642,6644,6647,6651,6654,6657],{"className":6643},[425],[413,6645],{"className":6646,"style":430},[429],[413,6648,6650],{"className":6649},[792],"⋯",[413,6652],{"className":6653,"style":1595},[542],[413,6655,435],{"className":6656},[2351],[413,6658],{"className":6659,"style":1595},[542],[413,6661,6663,6667,6707,6710,6713,6716],{"className":6662},[425],[413,6664],{"className":6665,"style":6666},[429],"height:0.7333em;vertical-align:-0.15em;",[413,6668,6670,6673],{"className":6669},[434],[413,6671,582],{"className":6672},[434,521],[413,6674,6676],{"className":6675},[904],[413,6677,6679,6699],{"className":6678},[908,909],[413,6680,6682,6696],{"className":6681},[913],[413,6683,6685],{"className":6684,"style":918},[917],[413,6686,6687,6690],{"style":921},[413,6688],{"className":6689,"style":926},[925],[413,6691,6693],{"className":6692},[930,931,932,933],[413,6694,588],{"className":6695},[434,933],[413,6697,941],{"className":6698},[940],[413,6700,6702],{"className":6701},[913],[413,6703,6705],{"className":6704,"style":948},[917],[413,6706],{},[413,6708,547],{"className":6709},[434,521],[413,6711],{"className":6712,"style":1595},[542],[413,6714,435],{"className":6715},[2351],[413,6717],{"className":6718,"style":1595},[542],[413,6720,6722,6725],{"className":6721},[425],[413,6723],{"className":6724,"style":2153},[429],[413,6726,6728,6731],{"className":6727},[434],[413,6729,582],{"className":6730},[434,521],[413,6732,6734],{"className":6733},[904],[413,6735,6737,6757],{"className":6736},[908,909],[413,6738,6740,6754],{"className":6739},[913],[413,6741,6743],{"className":6742,"style":918},[917],[413,6744,6745,6748],{"style":921},[413,6746],{"className":6747,"style":926},[925],[413,6749,6751],{"className":6750},[930,931,932,933],[413,6752,2183],{"className":6753},[434,933],[413,6755,941],{"className":6756},[940],[413,6758,6760],{"className":6759},[913],[413,6761,6763],{"className":6762,"style":948},[917],[413,6764],{}," is a\npolynomial with ",[413,6767,6769],{"className":6768},[416],[413,6770,6772,6828],{"className":6771,"ariaHidden":421},[420],[413,6773,6775,6779,6819,6822,6825],{"className":6774},[425],[413,6776],{"className":6777,"style":6778},[429],"height:0.6891em;vertical-align:-0.15em;",[413,6780,6782,6785],{"className":6781},[434],[413,6783,582],{"className":6784},[434,521],[413,6786,6788],{"className":6787},[904],[413,6789,6791,6811],{"className":6790},[908,909],[413,6792,6794,6808],{"className":6793},[913],[413,6795,6797],{"className":6796,"style":6473},[917],[413,6798,6799,6802],{"style":921},[413,6800],{"className":6801,"style":926},[925],[413,6803,6805],{"className":6804},[930,931,932,933],[413,6806,6486],{"className":6807,"style":6485},[434,521,933],[413,6809,941],{"className":6810},[940],[413,6812,6814],{"className":6813},[913],[413,6815,6817],{"className":6816,"style":948},[917],[413,6818],{},[413,6820],{"className":6821,"style":656},[542],[413,6823,3051],{"className":6824},[660],[413,6826],{"className":6827,"style":656},[542],[413,6829,6831,6834],{"className":6830},[425],[413,6832],{"className":6833,"style":2384},[429],[413,6835,2183],{"className":6836},[434],", then ",[413,6839,6841],{"className":6840},[416],[413,6842,6844,6871],{"className":6843,"ariaHidden":421},[420],[413,6845,6847,6850,6853,6856,6859,6862,6865,6868],{"className":6846},[425],[413,6848],{"className":6849,"style":481},[429],[413,6851,2877],{"className":6852,"style":2876},[434,521],[413,6854,528],{"className":6855},[527],[413,6857,547],{"className":6858},[434,521],[413,6860,552],{"className":6861},[551],[413,6863],{"className":6864,"style":656},[542],[413,6866,779],{"className":6867},[660],[413,6869],{"className":6870,"style":656},[542],[413,6872,6874,6878,6881,6884,6913],{"className":6873},[425],[413,6875],{"className":6876,"style":6877},[429],"height:1.0991em;vertical-align:-0.25em;",[413,6879,1684],{"className":6880},[434],[413,6882,528],{"className":6883},[527],[413,6885,6887,6890],{"className":6886},[434],[413,6888,547],{"className":6889},[434,521],[413,6891,6893],{"className":6892},[904],[413,6894,6896],{"className":6895},[908],[413,6897,6899],{"className":6898},[913],[413,6900,6902],{"className":6901,"style":6516},[917],[413,6903,6904,6907],{"style":1735},[413,6905],{"className":6906,"style":926},[925],[413,6908,6910],{"className":6909},[930,931,932,933],[413,6911,6486],{"className":6912,"style":6485},[434,521,933],[413,6914,552],{"className":6915},[551],[381,6917,6918,6921,6922,6990,6991,7032,7033,7066],{},[385,6919,6920],{},"Upper bound"," (",[413,6923,6925],{"className":6924},[416],[413,6926,6928,6946],{"className":6927,"ariaHidden":421},[420],[413,6929,6931,6934,6937,6940,6943],{"className":6930},[425],[413,6932],{"className":6933,"style":2787},[429],[413,6935,2877],{"className":6936,"style":2876},[434,521],[413,6938],{"className":6939,"style":656},[542],[413,6941,779],{"className":6942},[660],[413,6944],{"className":6945,"style":656},[542],[413,6947,6949,6952,6955,6958,6987],{"className":6948},[425],[413,6950],{"className":6951,"style":6877},[429],[413,6953,523],{"className":6954,"style":522},[434,521],[413,6956,528],{"className":6957},[527],[413,6959,6961,6964],{"className":6960},[434],[413,6962,547],{"className":6963},[434,521],[413,6965,6967],{"className":6966},[904],[413,6968,6970],{"className":6969},[908],[413,6971,6973],{"className":6972},[913],[413,6974,6976],{"className":6975,"style":6516},[917],[413,6977,6978,6981],{"style":1735},[413,6979],{"className":6980,"style":926},[925],[413,6982,6984],{"className":6983},[930,931,932,933],[413,6985,6486],{"className":6986,"style":6485},[434,521,933],[413,6988,552],{"className":6989},[551],"). Replace every coefficient by its absolute value and\nevery lower power by ",[413,6992,6994],{"className":6993},[416],[413,6995,6997],{"className":6996,"ariaHidden":421},[420],[413,6998,7000,7003],{"className":6999},[425],[413,7001],{"className":7002,"style":6516},[429],[413,7004,7006,7009],{"className":7005},[434],[413,7007,547],{"className":7008},[434,521],[413,7010,7012],{"className":7011},[904],[413,7013,7015],{"className":7014},[908],[413,7016,7018],{"className":7017},[913],[413,7019,7021],{"className":7020,"style":6516},[917],[413,7022,7023,7026],{"style":1735},[413,7024],{"className":7025,"style":926},[925],[413,7027,7029],{"className":7028},[930,931,932,933],[413,7030,6486],{"className":7031,"style":6485},[434,521,933]," (valid for 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Pull out the leading term and bound the rest\nbelow; for 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method now finds an ",[413,8578,8580],{"className":8579},[416],[413,8581,8583],{"className":8582,"ariaHidden":421},[420],[413,8584,8586,8589],{"className":8585},[425],[413,8587],{"className":8588,"style":2153},[429],[413,8590,8592,8595],{"className":8591},[434],[413,8593,547],{"className":8594},[434,521],[413,8596,8598],{"className":8597},[904],[413,8599,8601,8621],{"className":8600},[908,909],[413,8602,8604,8618],{"className":8603},[913],[413,8605,8607],{"className":8606,"style":918},[917],[413,8608,8609,8612],{"style":921},[413,8610],{"className":8611,"style":926},[925],[413,8613,8615],{"className":8614},[930,931,932,933],[413,8616,2183],{"className":8617},[434,933],[413,8619,941],{"className":8620},[940],[413,8622,8624],{"className":8623},[913],[413,8625,8627],{"className":8626,"style":948},[917],[413,8628],{}," past which the negative tail is,\nsay, at most half of ",[413,8631,8633],{"className":8632},[416],[413,8634,8636],{"className":8635,"ariaHidden":421},[420],[413,8637,8639,8642,8682],{"className":8638},[425],[413,8640],{"className":8641,"style":6454},[429],[413,8643,8645,8648],{"className":8644},[434],[413,8646,582],{"className":8647},[434,521],[413,8649,8651],{"className":8650},[904],[413,8652,8654,8674],{"className":8653},[908,909],[413,8655,8657,8671],{"className":8656},[913],[413,8658,8660],{"className":8659,"style":6473},[917],[413,8661,8662,8665],{"style":921},[413,8663],{"className":8664,"style":926},[925],[413,8666,8668],{"className":8667},[930,931,932,933],[413,8669,6486],{"className":8670,"style":6485},[434,521,933],[413,8672,941],{"className":8673},[940],[413,8675,8677],{"className":8676},[913],[413,8678,8680],{"className":8679,"style":948},[917],[413,8681],{},[413,8683,8685,8688],{"className":8684},[434],[413,8686,547],{"className":8687},[434,521],[413,8689,8691],{"className":8690},[904],[413,8692,8694],{"className":8693},[908],[413,8695,8697],{"className":8696},[913],[413,8698,8700],{"className":8699,"style":6516},[917],[413,8701,8702,8705],{"style":1735},[413,8703],{"className":8704,"style":926},[925],[413,8706,8708],{"className":8707},[930,931,932,933],[413,8709,6486],{"className":8710,"style":6485},[434,521,933],", leaving ",[413,8713,8715],{"className":8714},[416],[413,8716,8718,8745],{"className":8717,"ariaHidden":421},[420],[413,8719,8721,8724,8727,8730,8733,8736,8739,8742],{"className":8720},[425],[413,8722],{"className":8723,"style":481},[429],[413,8725,2877],{"className":8726,"style":2876},[434,521],[413,8728,528],{"className":8729},[527],[413,8731,547],{"className":8732},[434,521],[413,8734,552],{"className":8735},[551],[413,8737],{"className":8738,"style":656},[542],[413,8740,3158],{"className":8741},[660],[413,8743],{"className":8744,"style":656},[542],[413,8746,8748,8751,8754,8757],{"className":8747},[425],[413,8749],{"className":8750,"style":6516},[429],[413,8752,3044],{"className":8753},[434,521],[413,8755],{"className":8756,"style":543},[542],[413,8758,8760,8763],{"className":8759},[434],[413,8761,547],{"className":8762},[434,521],[413,8764,8766],{"className":8765},[904],[413,8767,8769],{"className":8768},[908],[413,8770,8772],{"className":8771},[913],[413,8773,8775],{"className":8774,"style":6516},[917],[413,8776,8777,8780],{"style":1735},[413,8778],{"className":8779,"style":926},[925],[413,8781,8783],{"className":8782},[930,931,932,933],[413,8784,6486],{"className":8785,"style":6485},[434,521,933]," for all ",[413,8788,8790],{"className":8789},[416],[413,8791,8793,8811],{"className":8792,"ariaHidden":421},[420],[413,8794,8796,8799,8802,8805,8808],{"className":8795},[425],[413,8797],{"className":8798,"style":4435},[429],[413,8800,547],{"className":8801},[434,521],[413,8803],{"className":8804,"style":656},[542],[413,8806,3158],{"className":8807},[660],[413,8809],{"className":8810,"style":656},[542],[413,8812,8814,8817],{"className":8813},[425],[413,8815],{"className":8816,"style":2153},[429],[413,8818,8820,8823],{"className":8819},[434],[413,8821,547],{"className":8822},[434,521],[413,8824,8826],{"className":8825},[904],[413,8827,8829,8849],{"className":8828},[908,909],[413,8830,8832,8846],{"className":8831},[913],[413,8833,8835],{"className":8834,"style":918},[917],[413,8836,8837,8840],{"style":921},[413,8838],{"className":8839,"style":926},[925],[413,8841,8843],{"className":8842},[930,931,932,933],[413,8844,2183],{"className":8845},[434,933],[413,8847,941],{"className":8848},[940],[413,8850,8852],{"className":8851},[913],[413,8853,8855],{"className":8854,"style":948},[917],[413,8856],{},".\nTogether the two directions give ",[413,8859,8861],{"className":8860},[416],[413,8862,8864,8891],{"className":8863,"ariaHidden":421},[420],[413,8865,8867,8870,8873,8876,8879,8882,8885,8888],{"className":8866},[425],[413,8868],{"className":8869,"style":481},[429],[413,8871,2877],{"className":8872,"style":2876},[434,521],[413,8874,528],{"className":8875},[527],[413,8877,547],{"className":8878},[434,521],[413,8880,552],{"className":8881},[551],[413,8883],{"className":8884,"style":656},[542],[413,8886,779],{"className":8887},[660],[413,8889],{"className":8890,"style":656},[542],[413,8892,8894,8897,8900,8903,8932],{"className":8893},[425],[413,8895],{"className":8896,"style":6877},[429],[413,8898,1684],{"className":8899},[434],[413,8901,528],{"className":8902},[527],[413,8904,8906,8909],{"className":8905},[434],[413,8907,547],{"className":8908},[434,521],[413,8910,8912],{"className":8911},[904],[413,8913,8915],{"className":8914},[908],[413,8916,8918],{"className":8917},[913],[413,8919,8921],{"className":8920,"style":6516},[917],[413,8922,8923,8926],{"style":1735},[413,8924],{"className":8925,"style":926},[925],[413,8927,8929],{"className":8928},[930,931,932,933],[413,8930,6486],{"className":8931,"style":6485},[434,521,933],[413,8933,552],{"className":8934},[551],": drop the constants and\nlower-order terms, keep the leading power.",[599,8937,8939,8954],{"id":8938},"o-is-an-upper-bound-not-a-promise-of-tightness",[413,8940,8942],{"className":8941},[416],[413,8943,8945],{"className":8944,"ariaHidden":421},[420],[413,8946,8948,8951],{"className":8947},[425],[413,8949],{"className":8950,"style":648},[429],[413,8952,523],{"className":8953,"style":522},[434,521]," is an upper bound, not a promise of tightness",[381,8956,8957,8958,8961,8962,8989,8990,9016,9017,9020,9021,9057],{},"Consider a worked cautionary case. ",[398,8959,8960],{},"Exchange sort"," compares ",[413,8963,8965],{"className":8964},[416],[413,8966,8968],{"className":8967,"ariaHidden":421},[420],[413,8969,8971,8974,8977,8981,8985],{"className":8970},[425],[413,8972],{"className":8973,"style":481},[429],[413,8975,1070],{"className":8976},[434,521],[413,8978,8980],{"className":8979},[527],"[",[413,8982,8984],{"className":8983},[434,521],"i",[413,8986,8988],{"className":8987},[551],"]"," with\n",[413,8991,8993],{"className":8992},[416],[413,8994,8996],{"className":8995,"ariaHidden":421},[420],[413,8997,8999,9002,9005,9008,9013],{"className":8998},[425],[413,9000],{"className":9001,"style":481},[429],[413,9003,1070],{"className":9004},[434,521],[413,9006,8980],{"className":9007},[527],[413,9009,9012],{"className":9010,"style":9011},[434,521],"margin-right:0.0572em;","j",[413,9014,8988],{"className":9015},[551]," for ",[385,9018,9019],{},"every"," pair ",[413,9022,9024],{"className":9023},[416],[413,9025,9027,9047],{"className":9026,"ariaHidden":421},[420],[413,9028,9030,9034,9037,9040,9044],{"className":9029},[425],[413,9031],{"className":9032,"style":9033},[429],"height:0.6986em;vertical-align:-0.0391em;",[413,9035,8984],{"className":9036},[434,521],[413,9038],{"className":9039,"style":656},[542],[413,9041,9043],{"className":9042},[660],"\u003C",[413,9045],{"className":9046,"style":656},[542],[413,9048,9050,9054],{"className":9049},[425],[413,9051],{"className":9052,"style":9053},[429],"height:0.854em;vertical-align:-0.1944em;",[413,9055,9012],{"className":9056,"style":9011},[434,521],", so its running time satisfies",[413,9059,9061],{"className":9060},[1263],[413,9062,9064],{"className":9063},[416],[413,9065,9067,9100,9118,9227,9335,9415],{"className":9066,"ariaHidden":421},[420],[413,9068,9070,9073,9076,9079,9082,9085,9088,9091,9094,9097],{"className":9069},[425],[413,9071],{"className":9072,"style":481},[429],[413,9074,619],{"className":9075,"style":618},[434,521],[413,9077,528],{"className":9078},[527],[413,9080,547],{"className":9081},[434,521],[413,9083,552],{"className":9084},[551],[413,9086],{"className":9087,"style":656},[542],[413,9089],{"className":9090,"style":656},[542],[413,9092,3244],{"className":9093},[660],[413,9095],{"className":9096,"style":656},[542],[413,9098],{"className":9099,"style":656},[542],[413,9101,9103,9106,9109,9112,9115],{"className":9102},[425],[413,9104],{"className":9105,"style":3923},[429],[413,9107,582],{"className":9108},[434,521],[413,9110],{"className":9111,"style":1595},[542],[413,9113,3933],{"className":9114},[2351],[413,9116],{"className":9117,"style":1595},[542],[413,9119,9121,9125,9218,9221,9224],{"className":9120},[425],[413,9122],{"className":9123,"style":9124},[429],"height:2.113em;vertical-align:-0.686em;",[413,9126,9128,9132,9215],{"className":9127},[434],[413,9129],{"className":9130},[527,9131],"nulldelimiter",[413,9133,9136],{"className":9134},[9135],"mfrac",[413,9137,9139,9206],{"className":9138},[908,909],[413,9140,9142,9203],{"className":9141},[913],[413,9143,9146,9159,9170],{"className":9144,"style":9145},[917],"height:1.427em;",[413,9147,9149,9153],{"style":9148},"top:-2.314em;",[413,9150],{"className":9151,"style":9152},[925],"height:3em;",[413,9154,9156],{"className":9155},[434],[413,9157,597],{"className":9158},[434],[413,9160,9162,9165],{"style":9161},"top:-3.23em;",[413,9163],{"className":9164,"style":9152},[925],[413,9166],{"className":9167,"style":9169},[9168],"frac-line","border-bottom-width:0.04em;",[413,9171,9173,9176],{"style":9172},"top:-3.677em;",[413,9174],{"className":9175,"style":9152},[925],[413,9177,9179,9182,9185,9188,9191,9194,9197,9200],{"className":9178},[434],[413,9180,547],{"className":9181},[434,521],[413,9183,528],{"className":9184},[527],[413,9186,547],{"className":9187},[434,521],[413,9189],{"className":9190,"style":1595},[542],[413,9192,452],{"className":9193},[2351],[413,9195],{"className":9196,"style":1595},[542],[413,9198,588],{"className":9199},[434],[413,9201,552],{"className":9202},[551],[413,9204,941],{"className":9205},[940],[413,9207,9209],{"className":9208},[913],[413,9210,9213],{"className":9211,"style":9212},[917],"height:0.686em;",[413,9214],{},[413,9216],{"className":9217},[551,9131],[413,9219],{"className":9220,"style":656},[542],[413,9222,779],{"className":9223},[660],[413,9225],{"className":9226,"style":656},[542],[413,9228,9230,9234,9297,9326,9329,9332],{"className":9229},[425],[413,9231],{"className":9232,"style":9233},[429],"height:1.7936em;vertical-align:-0.686em;",[413,9235,9237,9240,9294],{"className":9236},[434],[413,9238],{"className":9239},[527,9131],[413,9241,9243],{"className":9242},[9135],[413,9244,9246,9286],{"className":9245},[908,909],[413,9247,9249,9283],{"className":9248},[913],[413,9250,9253,9264,9272],{"className":9251,"style":9252},[917],"height:1.1076em;",[413,9254,9255,9258],{"style":9148},[413,9256],{"className":9257,"style":9152},[925],[413,9259,9261],{"className":9260},[434],[413,9262,597],{"className":9263},[434],[413,9265,9266,9269],{"style":9161},[413,9267],{"className":9268,"style":9152},[925],[413,9270],{"className":9271,"style":9169},[9168],[413,9273,9274,9277],{"style":9172},[413,9275],{"className":9276,"style":9152},[925],[413,9278,9280],{"className":9279},[434],[413,9281,582],{"className":9282},[434,521],[413,9284,941],{"className":9285},[940],[413,9287,9289],{"className":9288},[913],[413,9290,9292],{"className":9291,"style":9212},[917],[413,9293],{},[413,9295],{"className":9296},[551,9131],[413,9298,9300,9303],{"className":9299},[434],[413,9301,547],{"className":9302},[434,521],[413,9304,9306],{"className":9305},[904],[413,9307,9309],{"className":9308},[908],[413,9310,9312],{"className":9311},[913],[413,9313,9315],{"className":9314,"style":2332},[917],[413,9316,9317,9320],{"style":2335},[413,9318],{"className":9319,"style":926},[925],[413,9321,9323],{"className":9322},[930,931,932,933],[413,9324,597],{"className":9325},[434,933],[413,9327],{"className":9328,"style":1595},[542],[413,9330,452],{"className":9331},[2351],[413,9333],{"className":9334,"style":1595},[542],[413,9336,9338,9341,9403,9406,9409,9412],{"className":9337},[425],[413,9339],{"className":9340,"style":9233},[429],[413,9342,9344,9347,9400],{"className":9343},[434],[413,9345],{"className":9346},[527,9131],[413,9348,9350],{"className":9349},[9135],[413,9351,9353,9392],{"className":9352},[908,909],[413,9354,9356,9389],{"className":9355},[913],[413,9357,9359,9370,9378],{"className":9358,"style":9252},[917],[413,9360,9361,9364],{"style":9148},[413,9362],{"className":9363,"style":9152},[925],[413,9365,9367],{"className":9366},[434],[413,9368,597],{"className":9369},[434],[413,9371,9372,9375],{"style":9161},[413,9373],{"className":9374,"style":9152},[925],[413,9376],{"className":9377,"style":9169},[9168],[413,9379,9380,9383],{"style":9172},[413,9381],{"className":9382,"style":9152},[925],[413,9384,9386],{"className":9385},[434],[413,9387,582],{"className":9388},[434,521],[413,9390,941],{"className":9391},[940],[413,9393,9395],{"className":9394},[913],[413,9396,9398],{"className":9397,"style":9212},[917],[413,9399],{},[413,9401],{"className":9402},[551,9131],[413,9404,547],{"className":9405},[434,521],[413,9407],{"className":9408,"style":656},[542],[413,9410,779],{"className":9411},[660],[413,9413],{"className":9414,"style":656},[542],[413,9416,9418,9422,9425,9428,9457],{"className":9417},[425],[413,9419],{"className":9420,"style":9421},[429],"height:1.1141em;vertical-align:-0.25em;",[413,9423,523],{"className":9424,"style":522},[434,521],[413,9426,528],{"className":9427},[527],[413,9429,9431,9434],{"className":9430},[434],[413,9432,547],{"className":9433},[434,521],[413,9435,9437],{"className":9436},[904],[413,9438,9440],{"className":9439},[908],[413,9441,9443],{"className":9442},[913],[413,9444,9446],{"className":9445,"style":2332},[917],[413,9447,9448,9451],{"style":2335},[413,9449],{"className":9450,"style":926},[925],[413,9452,9454],{"className":9453},[930,931,932,933],[413,9455,597],{"className":9456},[434,933],[413,9458,552],{"className":9459},[551],[381,9461,9462,9463,9513,9514,9517,9518,9595,9596,9637,9638,9641],{},"by the polynomial theorem. But ",[413,9464,9466],{"className":9465},[416],[413,9467,9469],{"className":9468,"ariaHidden":421},[420],[413,9470,9472,9475,9478,9481,9510],{"className":9471},[425],[413,9473],{"className":9474,"style":1707},[429],[413,9476,523],{"className":9477,"style":522},[434,521],[413,9479,528],{"className":9480},[527],[413,9482,9484,9487],{"className":9483},[434],[413,9485,547],{"className":9486},[434,521],[413,9488,9490],{"className":9489},[904],[413,9491,9493],{"className":9492},[908],[413,9494,9496],{"className":9495},[913],[413,9497,9499],{"className":9498,"style":1732},[917],[413,9500,9501,9504],{"style":1735},[413,9502],{"className":9503,"style":926},[925],[413,9505,9507],{"className":9506},[930,931,932,933],[413,9508,597],{"className":9509},[434,933],[413,9511,552],{"className":9512},[551]," also implies the ",[385,9515,9516],{},"true but useless","\nstatement ",[413,9519,9521],{"className":9520},[416],[413,9522,9524,9551],{"className":9523,"ariaHidden":421},[420],[413,9525,9527,9530,9533,9536,9539,9542,9545,9548],{"className":9526},[425],[413,9528],{"className":9529,"style":481},[429],[413,9531,619],{"className":9532,"style":618},[434,521],[413,9534,528],{"className":9535},[527],[413,9537,547],{"className":9538},[434,521],[413,9540,552],{"className":9541},[551],[413,9543],{"className":9544,"style":656},[542],[413,9546,779],{"className":9547},[660],[413,9549],{"className":9550,"style":656},[542],[413,9552,9554,9557,9560,9563,9592],{"className":9553},[425],[413,9555],{"className":9556,"style":1707},[429],[413,9558,523],{"className":9559,"style":522},[434,521],[413,9561,528],{"className":9562},[527],[413,9564,9566,9569],{"className":9565},[434],[413,9567,547],{"className":9568},[434,521],[413,9570,9572],{"className":9571},[904],[413,9573,9575],{"className":9574},[908],[413,9576,9578],{"className":9577},[913],[413,9579,9581],{"className":9580,"style":1732},[917],[413,9582,9583,9586],{"style":1735},[413,9584],{"className":9585,"style":926},[925],[413,9587,9589],{"className":9588},[930,931,932,933],[413,9590,2313],{"className":9591},[434,933],[413,9593,552],{"className":9594},[551],": a correct upper bound need not be tight. So how loose\ncan we go? Not below ",[413,9597,9599],{"className":9598},[416],[413,9600,9602],{"className":9601,"ariaHidden":421},[420],[413,9603,9605,9608],{"className":9604},[425],[413,9606],{"className":9607,"style":1732},[429],[413,9609,9611,9614],{"className":9610},[434],[413,9612,547],{"className":9613},[434,521],[413,9615,9617],{"className":9616},[904],[413,9618,9620],{"className":9619},[908],[413,9621,9623],{"className":9622},[913],[413,9624,9626],{"className":9625,"style":1732},[917],[413,9627,9628,9631],{"style":1735},[413,9629],{"className":9630,"style":926},[925],[413,9632,9634],{"className":9633},[930,931,932,933],[413,9635,597],{"className":9636},[434,933],". The number of pairs is itself a ",[385,9639,9640],{},"lower"," bound on the\nwork:",[413,9643,9645],{"className":9644},[1263],[413,9646,9648],{"className":9647},[416],[413,9649,9651,9684,9709,9785,9883,9988],{"className":9650,"ariaHidden":421},[420],[413,9652,9654,9657,9660,9663,9666,9669,9672,9675,9678,9681],{"className":9653},[425],[413,9655],{"className":9656,"style":481},[429],[413,9658,619],{"className":9659,"style":618},[434,521],[413,9661,528],{"className":9662},[527],[413,9664,547],{"className":9665},[434,521],[413,9667,552],{"className":9668},[551],[413,9670],{"className":9671,"style":656},[542],[413,9673],{"className":9674,"style":656},[542],[413,9676,3158],{"className":9677},[660],[413,9679],{"className":9680,"style":656},[542],[413,9682],{"className":9683,"style":656},[542],[413,9685,9687,9690,9693,9697,9700,9703,9706],{"className":9686},[425],[413,9688],{"className":9689,"style":481},[429],[413,9691,7113],{"className":9692},[434],[413,9694,9696],{"className":9695,"style":1613},[434,521],"S",[413,9698,7113],{"className":9699},[434],[413,9701],{"className":9702,"style":656},[542],[413,9704,779],{"className":9705},[660],[413,9707],{"className":9708,"style":656},[542],[413,9710,9712,9716,9776,9779,9782],{"className":9711},[425],[413,9713],{"className":9714,"style":9715},[429],"height:2.4em;vertical-align:-0.95em;",[413,9717,9719,9725,9770],{"className":9718},[434],[413,9720,9722],{"className":9721,"style":797},[527,796],[413,9723,528],{"className":9724},[7416,932],[413,9726,9728],{"className":9727},[9135],[413,9729,9731,9762],{"className":9730},[908,909],[413,9732,9734,9759],{"className":9733},[913],[413,9735,9737,9748],{"className":9736,"style":9252},[917],[413,9738,9739,9742],{"style":9148},[413,9740],{"className":9741,"style":9152},[925],[413,9743,9745],{"className":9744},[434],[413,9746,597],{"className":9747},[434],[413,9749,9750,9753],{"style":9172},[413,9751],{"className":9752,"style":9152},[925],[413,9754,9756],{"className":9755},[434],[413,9757,547],{"className":9758},[434,521],[413,9760,941],{"className":9761},[940],[413,9763,9765],{"className":9764},[913],[413,9766,9768],{"className":9767,"style":9212},[917],[413,9769],{},[413,9771,9773],{"className":9772,"style":797},[551,796],[413,9774,552],{"className":9775},[7416,932],[413,9777],{"className":9778,"style":656},[542],[413,9780,779],{"className":9781},[660],[413,9783],{"className":9784,"style":656},[542],[413,9786,9788,9791,9874,9877,9880],{"className":9787},[425],[413,9789],{"className":9790,"style":9124},[429],[413,9792,9794,9797,9871],{"className":9793},[434],[413,9795],{"className":9796},[527,9131],[413,9798,9800],{"className":9799},[9135],[413,9801,9803,9863],{"className":9802},[908,909],[413,9804,9806,9860],{"className":9805},[913],[413,9807,9809,9820,9828],{"className":9808,"style":9145},[917],[413,9810,9811,9814],{"style":9148},[413,9812],{"className":9813,"style":9152},[925],[413,9815,9817],{"className":9816},[434],[413,9818,597],{"className":9819},[434],[413,9821,9822,9825],{"style":9161},[413,9823],{"className":9824,"style":9152},[925],[413,9826],{"className":9827,"style":9169},[9168],[413,9829,9830,9833],{"style":9172},[413,9831],{"className":9832,"style":9152},[925],[413,9834,9836,9839,9842,9845,9848,9851,9854,9857],{"className":9835},[434],[413,9837,547],{"className":9838},[434,521],[413,9840,528],{"className":9841},[527],[413,9843,547],{"className":9844},[434,521],[413,9846],{"className":9847,"style":1595},[542],[413,9849,452],{"className":9850},[2351],[413,9852],{"className":9853,"style":1595},[542],[413,9855,588],{"className":9856},[434],[413,9858,552],{"className":9859},[551],[413,9861,941],{"className":9862},[940],[413,9864,9866],{"className":9865},[913],[413,9867,9869],{"className":9868,"style":9212},[917],[413,9870],{},[413,9872],{"className":9873},[551,9131],[413,9875],{"className":9876,"style":656},[542],[413,9878,779],{"className":9879},[660],[413,9881],{"className":9882,"style":656},[542],[413,9884,9886,9890,9979,9982,9985],{"className":9885},[425],[413,9887],{"className":9888,"style":9889},[429],"height:2.1771em;vertical-align:-0.686em;",[413,9891,9893,9896,9976],{"className":9892},[434],[413,9894],{"className":9895},[527,9131],[413,9897,9899],{"className":9898},[9135],[413,9900,9902,9968],{"className":9901},[908,909],[413,9903,9905,9965],{"className":9904},[913],[413,9906,9909,9920,9928],{"className":9907,"style":9908},[917],"height:1.4911em;",[413,9910,9911,9914],{"style":9148},[413,9912],{"className":9913,"style":9152},[925],[413,9915,9917],{"className":9916},[434],[413,9918,597],{"className":9919},[434],[413,9921,9922,9925],{"style":9161},[413,9923],{"className":9924,"style":9152},[925],[413,9926],{"className":9927,"style":9169},[9168],[413,9929,9930,9933],{"style":9172},[413,9931],{"className":9932,"style":9152},[925],[413,9934,9936],{"className":9935},[434],[413,9937,9939,9942],{"className":9938},[434],[413,9940,547],{"className":9941},[434,521],[413,9943,9945],{"className":9944},[904],[413,9946,9948],{"className":9947},[908],[413,9949,9951],{"className":9950},[913],[413,9952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forces ",[413,10063,10065],{"className":10064},[416],[413,10066,10068,10135],{"className":10067,"ariaHidden":421},[420],[413,10069,10071,10074,10077,10080,10083,10086,10089,10132],{"className":10070},[425],[413,10072],{"className":10073,"style":481},[429],[413,10075,619],{"className":10076,"style":618},[434,521],[413,10078,528],{"className":10079},[527],[413,10081,547],{"className":10082},[434,521],[413,10084,552],{"className":10085},[551],[413,10087],{"className":10088,"style":656},[542],[413,10090,10092,10125,10129],{"className":10091},[660],[413,10093,10095],{"className":10094},[660],[413,10096,10099],{"className":10097},[434,10098],"vbox",[413,10100,10103],{"className":10101},[10102],"thinbox",[413,10104,10107,10110,10121],{"className":10105},[10106],"rlap",[413,10108],{"className":10109,"style":2787},[429],[413,10111,10114],{"className":10112},[10113],"inner",[413,10115,10117],{"className":10116},[434],[413,10118,10120],{"className":10119},[660],"",[413,10122],{"className":10123},[10124],"fix",[413,10126],{"className":10127},[542,10128],"nobreak",[413,10130,779],{"className":10131},[660],[413,10133],{"className":10134,"style":656},[542],[413,10136,10138,10141,10144,10147,10180],{"className":10137},[425],[413,10139],{"className":10140,"style":1707},[429],[413,10142,523],{"className":10143,"style":522},[434,521],[413,10145,528],{"className":10146},[527],[413,10148,10150,10153],{"className":10149},[434],[413,10151,547],{"className":10152},[434,521],[413,10154,10156],{"className":10155},[904],[413,10157,10159],{"className":10158},[908],[413,10160,10162],{"className":10161},[913],[413,10163,10165],{"className":10164,"style":1732},[917],[413,10166,10167,10170],{"style":1735},[413,10168],{"className":10169,"style":926},[925],[413,10171,10173],{"className":10172},[930,931,932,933],[413,10174,10176],{"className":10175},[434,933],[413,10177,10179],{"className":10178},[434,933],"1.9",[413,10181,552],{"className":10182},[551],"; indeed ",[413,10185,10187],{"className":10186},[416],[413,10188,10190,10308,10426],{"className":10189,"ariaHidden":421},[420],[413,10191,10193,10197,10299,10302,10305],{"className":10192},[425],[413,10194],{"className":10195,"style":10196},[429],"height:1.3629em;vertical-align:-0.345em;",[413,10198,10200,10203,10296],{"className":10199},[434],[413,10201],{"className":10202},[527,9131],[413,10204,10206],{"className":10205},[9135],[413,10207,10209,10287],{"className":10208},[908,909],[413,10210,10212,10284],{"className":10211},[913],[413,10213,10216,10231,10239],{"className":10214,"style":10215},[917],"height:1.0179em;",[413,10217,10219,10222],{"style":10218},"top:-2.655em;",[413,10220],{"className":10221,"style":9152},[925],[413,10223,10225],{"className":10224},[930,931,932,933],[413,10226,10228],{"className":10227},[434,933],[413,10229,597],{"className":10230},[434,933],[413,10232,10233,10236],{"style":9161},[413,10234],{"className":10235,"style":9152},[925],[413,10237],{"className":10238,"style":9169},[9168],[413,10240,10242,10245],{"style":10241},"top:-3.394em;",[413,10243],{"className":10244,"style":9152},[925],[413,10246,10248],{"className":10247},[930,931,932,933],[413,10249,10251],{"className":10250},[434,933],[413,10252,10254,10257],{"className":10253},[434,933],[413,10255,547],{"className":10256},[434,521,933],[413,10258,10260],{"className":10259},[904],[413,10261,10263],{"className":10262},[908],[413,10264,10266],{"className":10265},[913],[413,10267,10270],{"className":10268,"style":10269},[917],"height:0.8913em;",[413,10271,10273,10277],{"style":10272},"top:-2.931em;margin-right:0.0714em;",[413,10274],{"className":10275,"style":10276},[925],"height:2.5em;",[413,10278,10281],{"className":10279},[930,10280,7417,933],"reset-size3",[413,10282,597],{"className":10283},[434,933],[413,10285,941],{"className":10286},[940],[413,10288,10290],{"className":10289},[913],[413,10291,10294],{"className":10292,"style":10293},[917],"height:0.345em;",[413,10295],{},[413,10297],{"className":10298},[551,9131],[413,10300],{"className":10301,"style":1595},[542],[413,10303,452],{"className":10304},[2351],[413,10306],{"className":10307,"style":1595},[542],[413,10309,10311,10315,10384,10387,10423],{"className":10310},[425],[413,10312],{"className":10313,"style":10314},[429],"height:1.0404em;vertical-align:-0.345em;",[413,10316,10318,10321,10381],{"className":10317},[434],[413,10319],{"className":10320},[527,9131],[413,10322,10324],{"className":10323},[9135],[413,10325,10327,10373],{"className":10326},[908,909],[413,10328,10330,10370],{"className":10329},[913],[413,10331,10334,10348,10356],{"className":10332,"style":10333},[917],"height:0.6954em;",[413,10335,10336,10339],{"style":10218},[413,10337],{"className":10338,"style":9152},[925],[413,10340,10342],{"className":10341},[930,931,932,933],[413,10343,10345],{"className":10344},[434,933],[413,10346,597],{"className":10347},[434,933],[413,10349,10350,10353],{"style":9161},[413,10351],{"className":10352,"style":9152},[925],[413,10354],{"className":10355,"style":9169},[9168],[413,10357,10358,10361],{"style":10241},[413,10359],{"className":10360,"style":9152},[925],[413,10362,10364],{"className":10363},[930,931,932,933],[413,10365,10367],{"className":10366},[434,933],[413,10368,547],{"className":10369},[434,521,933],[413,10371,941],{"className":10372},[940],[413,10374,10376],{"className":10375},[913],[413,10377,10379],{"className":10378,"style":10293},[917],[413,10380],{},[413,10382],{"className":10383},[551,9131],[413,10385],{"className":10386,"style":656},[542],[413,10388,10390,10417,10420],{"className":10389},[660],[413,10391,10393],{"className":10392},[660],[413,10394,10396],{"className":10395},[434,10098],[413,10397,10399],{"className":10398},[10102],[413,10400,10402,10405,10414],{"className":10401},[10106],[413,10403],{"className":10404,"style":2787},[429],[413,10406,10408],{"className":10407},[10113],[413,10409,10411],{"className":10410},[434],[413,10412,10120],{"className":10413},[660],[413,10415],{"className":10416},[10124],[413,10418],{"className":10419},[542,10128],[413,10421,779],{"className":10422},[660],[413,10424],{"className":10425,"style":656},[542],[413,10427,10429,10432,10435,10438,10467],{"className":10428},[425],[413,10430],{"className":10431,"style":6877},[429],[413,10433,523],{"className":10434,"style":522},[434,521],[413,10436,528],{"className":10437},[527],[413,10439,10441,10444],{"className":10440},[434],[413,10442,547],{"className":10443},[434,521],[413,10445,10447],{"className":10446},[904],[413,10448,10450],{"className":10449},[908],[413,10451,10453],{"className":10452},[913],[413,10454,10456],{"className":10455,"style":6516},[917],[413,10457,10458,10461],{"style":1735},[413,10459],{"className":10460,"style":926},[925],[413,10462,10464],{"className":10463},[930,931,932,933],[413,10465,6486],{"className":10466,"style":6485},[434,521,933],[413,10468,552],{"className":10469},[551],[398,10471,4879],{},[413,10473,10475],{"className":10474},[416],[413,10476,104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No constant ",[413,10507,10509],{"className":10508},[416],[413,10510,10512],{"className":10511,"ariaHidden":421},[420],[413,10513,10515,10518],{"className":10514},[425],[413,10516],{"className":10517,"style":566},[429],[413,10519,3044],{"className":10520},[434,521]," can keep ",[413,10523,10525],{"className":10524},[416],[413,10526,10528],{"className":10527,"ariaHidden":421},[420],[413,10529,10531,10534],{"className":10530},[425],[413,10532],{"className":10533,"style":10196},[429],[413,10535,10537,10540,10625],{"className":10536},[434],[413,10538],{"className":10539},[527,9131],[413,10541,10543],{"className":10542},[9135],[413,10544,10546,10617],{"className":10545},[908,909],[413,10547,10549,10614],{"className":10548},[913],[413,10550,10552,10566,10574],{"className":10551,"style":10215},[917],[413,10553,10554,10557],{"style":10218},[413,10555],{"className":10556,"style":9152},[925],[413,10558,10560],{"className":10559},[930,931,932,933],[413,10561,10563],{"className":10562},[434,933],[413,10564,597],{"className":10565},[434,933],[413,10567,10568,10571],{"style":9161},[413,10569],{"className":10570,"style":9152},[925],[413,10572],{"className":10573,"style":9169},[9168],[413,10575,10576,10579],{"style":10241},[413,10577],{"className":10578,"style":9152},[925],[413,10580,10582],{"className":10581},[930,931,932,933],[413,10583,10585],{"className":10584},[434,933],[413,10586,10588,10591],{"className":10587},[434,933],[413,10589,547],{"className":10590},[434,521,933],[413,10592,10594],{"className":10593},[904],[413,10595,10597],{"className":10596},[908],[413,10598,10600],{"className":10599},[913],[413,10601,10603],{"className":10602,"style":10269},[917],[413,10604,10605,10608],{"style":10272},[413,10606],{"className":10607,"style":10276},[925],[413,10609,10611],{"className":10610},[930,10280,7417,933],[413,10612,597],{"className":10613},[434,933],[413,10615,941],{"className":10616},[940],[413,10618,10620],{"className":10619},[913],[413,10621,10623],{"className":10622,"style":10293},[917],[413,10624],{},[413,10626],{"className":10627},[551,9131]," under\n",[413,10630,10632],{"className":10631},[416],[413,10633,10635],{"className":10634,"ariaHidden":421},[420],[413,10636,10638,10641,10644,10647],{"className":10637},[425],[413,10639],{"className":10640,"style":1732},[429],[413,10642,3044],{"className":10643},[434,521],[413,10645],{"className":10646,"style":543},[542],[413,10648,10650,10653],{"className":10649},[434],[413,10651,547],{"className":10652},[434,521],[413,10654,10656],{"className":10655},[904],[413,10657,10659],{"className":10658},[908],[413,10660,10662],{"className":10661},[913],[413,10663,10665],{"className":10664,"style":1732},[917],[413,10666,10667,10670],{"style":1735},[413,10668],{"className":10669,"style":926},[925],[413,10671,10673],{"className":10672},[930,931,932,933],[413,10674,10676],{"className":10675},[434,933],[413,10677,10179],{"className":10678},[434,933],[413,10680,10682],{"className":10681},[416],[413,10683,10685],{"className":10684,"ariaHidden":421},[420],[413,10686,10688,10691],{"className":10687},[425],[413,10689],{"className":10690,"style":566},[429],[413,10692,547],{"className":10693},[434,521]," is large enough, because ",[413,10696,10698],{"className":10697},[416],[413,10699,10701,10810],{"className":10700,"ariaHidden":421},[420],[413,10702,10704,10707,10801,10804,10807],{"className":10703},[425],[413,10705],{"className":10706,"style":10196},[429],[413,10708,10710,10713,10798],{"className":10709},[434],[413,10711],{"className":10712},[527,9131],[413,10714,10716],{"className":10715},[9135],[413,10717,10719,10790],{"className":10718},[908,909],[413,10720,10722,10787],{"className":10721},[913],[413,10723,10725,10739,10747],{"className":10724,"style":10215},[917],[413,10726,10727,10730],{"style":10218},[413,10728],{"className":10729,"style":9152},[925],[413,10731,10733],{"className":10732},[930,931,932,933],[413,10734,10736],{"className":10735},[434,933],[413,10737,597],{"className":10738},[434,933],[413,10740,10741,10744],{"style":9161},[413,10742],{"className":10743,"style":9152},[925],[413,10745],{"className":10746,"style":9169},[9168],[413,10748,10749,10752],{"style":10241},[413,10750],{"className":10751,"style":9152},[925],[413,10753,10755],{"className":10754},[930,931,932,933],[413,10756,10758],{"className":10757},[434,933],[413,10759,10761,10764],{"className":10760},[434,933],[413,10762,547],{"className":10763},[434,521,933],[413,10765,10767],{"className":10766},[904],[413,10768,10770],{"className":10769},[908],[413,10771,10773],{"className":10772},[913],[413,10774,10776],{"className":10775,"style":10269},[917],[413,10777,10778,10781],{"style":10272},[413,10779],{"className":10780,"style":10276},[925],[413,10782,10784],{"className":10783},[930,10280,7417,933],[413,10785,597],{"className":10786},[434,933],[413,10788,941],{"className":10789},[940],[413,10791,10793],{"className":10792},[913],[413,10794,10796],{"className":10795,"style":10293},[917],[413,10797],{},[413,10799],{"className":10800},[551,9131],[413,10802],{"className":10803,"style":656},[542],[413,10805,3051],{"className":10806},[660],[413,10808],{"className":10809,"style":656},[542],[413,10811,10813,10816,10819,10822],{"className":10812},[425],[413,10814],{"className":10815,"style":1732},[429],[413,10817,3044],{"className":10818},[434,521],[413,10820],{"className":10821,"style":543},[542],[413,10823,10825,10828],{"className":10824},[434],[413,10826,547],{"className":10827},[434,521],[413,10829,10831],{"className":10830},[904],[413,10832,10834],{"className":10833},[908],[413,10835,10837],{"className":10836},[913],[413,10838,10840],{"className":10839,"style":1732},[917],[413,10841,10842,10845],{"style":1735},[413,10843],{"className":10844,"style":926},[925],[413,10846,10848],{"className":10847},[930,931,932,933],[413,10849,10851],{"className":10850},[434,933],[413,10852,10179],{"className":10853},[434,933],"\nwhenever ",[413,10856,10858],{"className":10857},[416],[413,10859,10861,10880],{"className":10860,"ariaHidden":421},[420],[413,10862,10864,10868,10871,10874,10877],{"className":10863},[425],[413,10865],{"className":10866,"style":10867},[429],"height:0.5782em;vertical-align:-0.0391em;",[413,10869,547],{"className":10870},[434,521],[413,10872],{"className":10873,"style":656},[542],[413,10875,3051],{"className":10876},[660],[413,10878],{"className":10879,"style":656},[542],[413,10881,10883,10886,10889,10892,10895],{"className":10882},[425],[413,10884],{"className":10885,"style":1707},[429],[413,10887,528],{"className":10888},[527],[413,10890,597],{"className":10891},[434],[413,10893,3044],{"className":10894},[434,521],[413,10896,10898,10901],{"className":10897},[551],[413,10899,552],{"className":10900},[551],[413,10902,10904],{"className":10903},[904],[413,10905,10907],{"className":10906},[908],[413,10908,10910],{"className":10909},[913],[413,10911,10913],{"className":10912,"style":1732},[917],[413,10914,10915,10918],{"style":1735},[413,10916],{"className":10917,"style":926},[925],[413,10919,10921],{"className":10920},[930,931,932,933],[413,10922,10924],{"className":10923},[434,933],[413,10925,1911],{"className":10926},[434,933],". The moral: ",[413,10929,10931],{"className":10930},[416],[413,10932,10934],{"className":10933,"ariaHidden":421},[420],[413,10935,10937,10940],{"className":10936},[425],[413,10938],{"className":10939,"style":648},[429],[413,10941,523],{"className":10942,"style":522},[434,521]," alone tells you the cost is ",[385,10945,10946],{},"no worse\nthan"," something; only matching it with ",[413,10949,10951],{"className":10950},[416],[413,10952,10954],{"className":10953,"ariaHidden":421},[420],[413,10955,10957,10960],{"className":10956},[425],[413,10958],{"className":10959,"style":648},[429],[413,10961,4910],{"className":10962},[434]," (i.e. proving ",[413,10965,10967],{"className":10966},[416],[413,10968,10970],{"className":10969,"ariaHidden":421},[420],[413,10971,10973,10976],{"className":10972},[425],[413,10974],{"className":10975,"style":648},[429],[413,10977,1684],{"className":10978},[434],") certifies\nyou have found the true growth rate.",[381,10981,10982,10983,11033,11034,11068,11069,11119,11120,11153,11154,11157,11158,11191,11192,11195,11196,11211,11212,11245,11246,1158],{},"Picture the exponents on a line. Exchange sort's cost ",[413,10984,10986],{"className":10985},[416],[413,10987,10989],{"className":10988,"ariaHidden":421},[420],[413,10990,10992,10995,10998,11001,11030],{"className":10991},[425],[413,10993],{"className":10994,"style":1707},[429],[413,10996,1684],{"className":10997},[434],[413,10999,528],{"className":11000},[527],[413,11002,11004,11007],{"className":11003},[434],[413,11005,547],{"className":11006},[434,521],[413,11008,11010],{"className":11009},[904],[413,11011,11013],{"className":11012},[908],[413,11014,11016],{"className":11015},[913],[413,11017,11019],{"className":11018,"style":1732},[917],[413,11020,11021,11024],{"style":1735},[413,11022],{"className":11023,"style":926},[925],[413,11025,11027],{"className":11026},[930,931,932,933],[413,11028,597],{"className":11029},[434,933],[413,11031,552],{"className":11032},[551]," sits at\n",[413,11035,11037],{"className":11036},[416],[413,11038,11040,11059],{"className":11039,"ariaHidden":421},[420],[413,11041,11043,11047,11050,11053,11056],{"className":11042},[425],[413,11044],{"className":11045,"style":11046},[429],"height:0.6944em;",[413,11048,6486],{"className":11049,"style":6485},[434,521],[413,11051],{"className":11052,"style":656},[542],[413,11054,779],{"className":11055},[660],[413,11057],{"className":11058,"style":656},[542],[413,11060,11062,11065],{"className":11061},[425],[413,11063],{"className":11064,"style":2384},[429],[413,11066,597],{"className":11067},[434],". Every ",[413,11070,11072],{"className":11071},[416],[413,11073,11075],{"className":11074,"ariaHidden":421},[420],[413,11076,11078,11081,11084,11087,11116],{"className":11077},[425],[413,11079],{"className":11080,"style":6877},[429],[413,11082,523],{"className":11083,"style":522},[434,521],[413,11085,528],{"className":11086},[527],[413,11088,11090,11093],{"className":11089},[434],[413,11091,547],{"className":11092},[434,521],[413,11094,11096],{"className":11095},[904],[413,11097,11099],{"className":11098},[908],[413,11100,11102],{"className":11101},[913],[413,11103,11105],{"className":11104,"style":6516},[917],[413,11106,11107,11110],{"style":1735},[413,11108],{"className":11109,"style":926},[925],[413,11111,11113],{"className":11112},[930,931,932,933],[413,11114,6486],{"className":11115,"style":6485},[434,521,933],[413,11117,552],{"className":11118},[551]," with ",[413,11121,11123],{"className":11122},[416],[413,11124,11126,11144],{"className":11125,"ariaHidden":421},[420],[413,11127,11129,11132,11135,11138,11141],{"className":11128},[425],[413,11130],{"className":11131,"style":3132},[429],[413,11133,6486],{"className":11134,"style":6485},[434,521],[413,11136],{"className":11137,"style":656},[542],[413,11139,3158],{"className":11140},[660],[413,11142],{"className":11143,"style":656},[542],[413,11145,11147,11150],{"className":11146},[425],[413,11148],{"className":11149,"style":2384},[429],[413,11151,597],{"className":11152},[434]," is a ",[385,11155,11156],{},"valid"," upper bound, but only ",[413,11159,11161],{"className":11160},[416],[413,11162,11164,11182],{"className":11163,"ariaHidden":421},[420],[413,11165,11167,11170,11173,11176,11179],{"className":11166},[425],[413,11168],{"className":11169,"style":11046},[429],[413,11171,6486],{"className":11172,"style":6485},[434,521],[413,11174],{"className":11175,"style":656},[542],[413,11177,779],{"className":11178},[660],[413,11180],{"className":11181,"style":656},[542],[413,11183,11185,11188],{"className":11184},[425],[413,11186],{"className":11187,"style":2384},[429],[413,11189,597],{"className":11190},[434],"\nis ",[385,11193,11194],{},"tight","; the ",[413,11197,11199],{"className":11198},[416],[413,11200,11202],{"className":11201,"ariaHidden":421},[420],[413,11203,11205,11208],{"className":11204},[425],[413,11206],{"className":11207,"style":648},[429],[413,11209,4910],{"className":11210},[434]," side forbids any upper bound with ",[413,11213,11215],{"className":11214},[416],[413,11216,11218,11236],{"className":11217,"ariaHidden":421},[420],[413,11219,11221,11224,11227,11230,11233],{"className":11220},[425],[413,11222],{"className":11223,"style":3033},[429],[413,11225,6486],{"className":11226,"style":6485},[434,521],[413,11228],{"className":11229,"style":656},[542],[413,11231,9043],{"className":11232},[660],[413,11234],{"className":11235,"style":656},[542],[413,11237,11239,11242],{"className":11238},[425],[413,11240],{"className":11241,"style":2384},[429],[413,11243,597],{"className":11244},[434],", walling off\nthe left. Where the two bounds meet is ",[413,11247,11249],{"className":11248},[416],[413,11250,11252],{"className":11251,"ariaHidden":421},[420],[413,11253,11255,11258],{"className":11254},[425],[413,11256],{"className":11257,"style":648},[429],[413,11259,1684],{"className":11260},[434],[1869,11262,11264,11598],{"className":11263},[1872,1873],[1875,11265,11269],{"xmlns":1877,"width":11266,"height":11267,"viewBox":11268},"362.116","144.573","-75 -75 271.587 108.430",[1882,11270,11271,11275,11278,11281,11284,11291,11294,11301,11304,11324,11327,11334,11337,11356,11359,11366,11369,11410,11524,11589],{"stroke":1884,"style":1885},[1887,11272],{"fill":11273,"stroke":1889,"d":11274},"var(--tk-soft-warn)","M-51.377-18.78v-10.242h85.358v10.243Zm85.358-10.242",[1887,11276],{"fill":1939,"stroke":1889,"d":11277},"M33.981-18.78v-10.242h142.264v10.243Zm142.264-10.242",[1887,11279],{"fill":1889,"d":11280},"M-57.067-23.9h243.094",[1887,11282],{"d":11283},"m189.902-23.9-5.613-2.112 1.838 2.112-1.838 2.111Z",[1882,11285,11287],{"transform":11286},"translate(245.381 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",[413,11603,11605],{"className":11604},[416],[413,11606,11608],{"className":11607,"ariaHidden":421},[420],[413,11609,11611,11614,11617,11620,11649],{"className":11610},[425],[413,11612],{"className":11613,"style":1707},[429],[413,11615,1684],{"className":11616},[434],[413,11618,528],{"className":11619},[527],[413,11621,11623,11626],{"className":11622},[434],[413,11624,547],{"className":11625},[434,521],[413,11627,11629],{"className":11628},[904],[413,11630,11632],{"className":11631},[908],[413,11633,11635],{"className":11634},[913],[413,11636,11638],{"className":11637,"style":1732},[917],[413,11639,11640,11643],{"style":1735},[413,11641],{"className":11642,"style":926},[925],[413,11644,11646],{"className":11645},[930,931,932,933],[413,11647,597],{"className":11648},[434,933],[413,11650,552],{"className":11651},[551]," cost, by exponent. Every ",[413,11654,11656],{"className":11655},[416],[413,11657,11659],{"className":11658,"ariaHidden":421},[420],[413,11660,11662,11665,11668,11671,11700],{"className":11661},[425],[413,11663],{"className":11664,"style":6877},[429],[413,11666,523],{"className":11667,"style":522},[434,521],[413,11669,528],{"className":11670},[527],[413,11672,11674,11677],{"className":11673},[434],[413,11675,547],{"className":11676},[434,521],[413,11678,11680],{"className":11679},[904],[413,11681,11683],{"className":11682},[908],[413,11684,11686],{"className":11685},[913],[413,11687,11689],{"className":11688,"style":6516},[917],[413,11690,11691,11694],{"style":1735},[413,11692],{"className":11693,"style":926},[925],[413,11695,11697],{"className":11696},[930,931,932,933],[413,11698,6486],{"className":11699,"style":6485},[434,521,933],[413,11701,552],{"className":11702},[551],[413,11704,11706],{"className":11705},[416],[413,11707,11709,11727],{"className":11708,"ariaHidden":421},[420],[413,11710,11712,11715,11718,11721,11724],{"className":11711},[425],[413,11713],{"className":11714,"style":3132},[429],[413,11716,6486],{"className":11717,"style":6485},[434,521],[413,11719],{"className":11720,"style":656},[542],[413,11722,3158],{"className":11723},[660],[413,11725],{"className":11726,"style":656},[542],[413,11728,11730,11733],{"className":11729},[425],[413,11731],{"className":11732,"style":2384},[429],[413,11734,597],{"className":11735},[434]," holds but only ",[413,11738,11740],{"className":11739},[416],[413,11741,11743,11761],{"className":11742,"ariaHidden":421},[420],[413,11744,11746,11749,11752,11755,11758],{"className":11745},[425],[413,11747],{"className":11748,"style":11046},[429],[413,11750,6486],{"className":11751,"style":6485},[434,521],[413,11753],{"className":11754,"style":656},[542],[413,11756,779],{"className":11757},[660],[413,11759],{"className":11760,"style":656},[542],[413,11762,11764,11767],{"className":11763},[425],[413,11765],{"className":11766,"style":2384},[429],[413,11768,597],{"className":11769},[434]," is tight; ",[413,11772,11774],{"className":11773},[416],[413,11775,11777],{"className":11776,"ariaHidden":421},[420],[413,11778,11780,11783,11786,11789,11818],{"className":11779},[425],[413,11781],{"className":11782,"style":1707},[429],[413,11784,4910],{"className":11785},[434],[413,11787,528],{"className":11788},[527],[413,11790,11792,11795],{"className":11791},[434],[413,11793,547],{"className":11794},[434,521],[413,11796,11798],{"className":11797},[904],[413,11799,11801],{"className":11800},[908],[413,11802,11804],{"className":11803},[913],[413,11805,11807],{"className":11806,"style":1732},[917],[413,11808,11809,11812],{"style":1735},[413,11810],{"className":11811,"style":926},[925],[413,11813,11815],{"className":11814},[930,931,932,933],[413,11816,597],{"className":11817},[434,933],[413,11819,552],{"className":11820},[551]," rules out the whole region below ",[413,11823,11825],{"className":11824},[416],[413,11826,11828,11846],{"className":11827,"ariaHidden":421},[420],[413,11829,11831,11834,11837,11840,11843],{"className":11830},[425],[413,11832],{"className":11833,"style":11046},[429],[413,11835,6486],{"className":11836,"style":6485},[434,521],[413,11838],{"className":11839,"style":656},[542],[413,11841,779],{"className":11842},[660],[413,11844],{"className":11845,"style":656},[542],[413,11847,11849,11852],{"className":11848},[425],[413,11850],{"className":11851,"style":2384},[429],[413,11853,597],{"className":11854},[434],". The bounds pinch shut exactly at ",[413,11857,11859],{"className":11858},[416],[413,11860,11862],{"className":11861,"ariaHidden":421},[420],[413,11863,11865,11868,11871,11874,11903],{"className":11864},[425],[413,11866],{"className":11867,"style":1707},[429],[413,11869,1684],{"className":11870},[434],[413,11872,528],{"className":11873},[527],[413,11875,11877,11880],{"className":11876},[434],[413,11878,547],{"className":11879},[434,521],[413,11881,11883],{"className":11882},[904],[413,11884,11886],{"className":11885},[908],[413,11887,11889],{"className":11888},[913],[413,11890,11892],{"className":11891,"style":1732},[917],[413,11893,11894,11897],{"style":1735},[413,11895],{"className":11896,"style":926},[925],[413,11898,11900],{"className":11899},[930,931,932,933],[413,11901,597],{"className":11902},[434,933],[413,11904,552],{"className":11905},[551],[599,11907,11909],{"id":11908},"little-o-and-little-omega-strict-bounds","Little-o and little-omega: strict bounds",[381,11911,11912,1597,11927,11942,11943,1597,11958,11973,11974,11977,11978,11981],{},[413,11913,11915],{"className":11914},[416],[413,11916,11918],{"className":11917,"ariaHidden":421},[420],[413,11919,11921,11924],{"className":11920},[425],[413,11922],{"className":11923,"style":648},[429],[413,11925,523],{"className":11926,"style":522},[434,521],[413,11928,11930],{"className":11929},[416],[413,11931,11933],{"className":11932,"ariaHidden":421},[420],[413,11934,11936,11939],{"className":11935},[425],[413,11937],{"className":11938,"style":648},[429],[413,11940,4910],{"className":11941},[434]," allow ",[413,11944,11946],{"className":11945},[416],[413,11947,11949],{"className":11948,"ariaHidden":421},[420],[413,11950,11952,11955],{"className":11951},[425],[413,11953],{"className":11954,"style":2787},[429],[413,11956,2877],{"className":11957,"style":2876},[434,521],[413,11959,11961],{"className":11960},[416],[413,11962,11964],{"className":11963,"ariaHidden":421},[420],[413,11965,11967,11970],{"className":11966},[425],[413,11968],{"className":11969,"style":2890},[429],[413,11971,1882],{"className":11972,"style":2894},[434,521]," to grow at the ",[385,11975,11976],{},"same"," rate. The lowercase\nversions forbid that; they assert a ",[385,11979,11980],{},"strict"," gap.",[2954,11983,11984],{"type":2956},[381,11985,11986,1071,11989,12047,12048,3293,12072,12102,12103,12105,12106,12139,12140,11119,12192,12249,12250,12320,12321],{},[398,11987,11988],{},"Definition (Little-o).",[413,11990,11992],{"className":11991},[416],[413,11993,11995,12022],{"className":11994,"ariaHidden":421},[420],[413,11996,11998,12001,12004,12007,12010,12013,12016,12019],{"className":11997},[425],[413,11999],{"className":12000,"style":481},[429],[413,12002,2877],{"className":12003,"style":2876},[434,521],[413,12005,528],{"className":12006},[527],[413,12008,547],{"className":12009},[434,521],[413,12011,552],{"className":12012},[551],[413,12014],{"className":12015,"style":656},[542],[413,12017,779],{"className":12018},[660],[413,12020],{"className":12021,"style":656},[542],[413,12023,12025,12028,12032,12035,12038,12041,12044],{"className":12024},[425],[413,12026],{"className":12027,"style":481},[429],[413,12029,12031],{"className":12030},[434,521],"o",[413,12033,528],{"className":12034},[527],[413,12036,1882],{"className":12037,"style":2894},[434,521],[413,12039,528],{"className":12040},[527],[413,12042,547],{"className":12043},[434,521],[413,12045,3019],{"className":12046},[551]," means ",[413,12049,12051],{"className":12050},[416],[413,12052,12054],{"className":12053,"ariaHidden":421},[420],[413,12055,12057,12060,12063,12066,12069],{"className":12056},[425],[413,12058],{"className":12059,"style":481},[429],[413,12061,2877],{"className":12062,"style":2876},[434,521],[413,12064,528],{"className":12065},[527],[413,12067,547],{"className":12068},[434,521],[413,12070,552],{"className":12071},[551],[398,12073,12074,12075,12077,12078],{},"less than ",[385,12076,4879],{}," constant times ",[413,12079,12081],{"className":12080},[416],[413,12082,12084],{"className":12083,"ariaHidden":421},[420],[413,12085,12087,12090,12093,12096,12099],{"className":12086},[425],[413,12088],{"className":12089,"style":481},[429],[413,12091,1882],{"className":12092,"style":2894},[434,521],[413,12094,528],{"className":12095},[527],[413,12097,547],{"className":12098},[434,521],[413,12100,552],{"className":12101},[551],": for\n",[398,12104,9019],{}," constant ",[413,12107,12109],{"className":12108},[416],[413,12110,12112,12130],{"className":12111,"ariaHidden":421},[420],[413,12113,12115,12118,12121,12124,12127],{"className":12114},[425],[413,12116],{"className":12117,"style":10867},[429],[413,12119,3044],{"className":12120},[434,521],[413,12122],{"className":12123,"style":656},[542],[413,12125,3051],{"className":12126},[660],[413,12128],{"className":12129,"style":656},[542],[413,12131,12133,12136],{"className":12132},[425],[413,12134],{"className":12135,"style":2384},[429],[413,12137,2183],{"className":12138},[434]," there is an ",[413,12141,12143],{"className":12142},[416],[413,12144,12146],{"className":12145,"ariaHidden":421},[420],[413,12147,12149,12152],{"className":12148},[425],[413,12150],{"className":12151,"style":2153},[429],[413,12153,12155,12158],{"className":12154},[434],[413,12156,547],{"className":12157},[434,521],[413,12159,12161],{"className":12160},[904],[413,12162,12164,12184],{"className":12163},[908,909],[413,12165,12167,12181],{"className":12166},[913],[413,12168,12170],{"className":12169,"style":918},[917],[413,12171,12172,12175],{"style":921},[413,12173],{"className":12174,"style":926},[925],[413,12176,12178],{"className":12177},[930,931,932,933],[413,12179,2183],{"className":12180},[434,933],[413,12182,941],{"className":12183},[940],[413,12185,12187],{"className":12186},[913],[413,12188,12190],{"className":12189,"style":948},[917],[413,12191],{},[413,12193,12195],{"className":12194},[416],[413,12196,12198,12225],{"className":12197,"ariaHidden":421},[420],[413,12199,12201,12204,12207,12210,12213,12216,12219,12222],{"className":12200},[425],[413,12202],{"className":12203,"style":481},[429],[413,12205,2877],{"className":12206,"style":2876},[434,521],[413,12208,528],{"className":12209},[527],[413,12211,547],{"className":12212},[434,521],[413,12214,552],{"className":12215},[551],[413,12217],{"className":12218,"style":656},[542],[413,12220,9043],{"className":12221},[660],[413,12223],{"className":12224,"style":656},[542],[413,12226,12228,12231,12234,12237,12240,12243,12246],{"className":12227},[425],[413,12229],{"className":12230,"style":481},[429],[413,12232,3044],{"className":12233},[434,521],[413,12235],{"className":12236,"style":543},[542],[413,12238,1882],{"className":12239,"style":2894},[434,521],[413,12241,528],{"className":12242},[527],[413,12244,547],{"className":12245},[434,521],[413,12247,552],{"className":12248},[551]," for all\n",[413,12251,12253],{"className":12252},[416],[413,12254,12256,12274],{"className":12255,"ariaHidden":421},[420],[413,12257,12259,12262,12265,12268,12271],{"className":12258},[425],[413,12260],{"className":12261,"style":4435},[429],[413,12263,547],{"className":12264},[434,521],[413,12266],{"className":12267,"style":656},[542],[413,12269,3158],{"className":12270},[660],[413,12272],{"className":12273,"style":656},[542],[413,12275,12277,12280],{"className":12276},[425],[413,12278],{"className":12279,"style":2153},[429],[413,12281,12283,12286],{"className":12282},[434],[413,12284,547],{"className":12285},[434,521],[413,12287,12289],{"className":12288},[904],[413,12290,12292,12312],{"className":12291},[908,909],[413,12293,12295,12309],{"className":12294},[913],[413,12296,12298],{"className":12297,"style":918},[917],[413,12299,12300,12303],{"style":921},[413,12301],{"className":12302,"style":926},[925],[413,12304,12306],{"className":12305},[930,931,932,933],[413,12307,2183],{"className":12308},[434,933],[413,12310,941],{"className":12311},[940],[413,12313,12315],{"className":12314},[913],[413,12316,12318],{"className":12317,"style":948},[917],[413,12319],{},". There is also a slick limit form,\n",[413,12322,12324],{"className":12323},[416],[413,12325,12327,12354,12393,12556],{"className":12326,"ariaHidden":421},[420],[413,12328,12330,12333,12336,12339,12342,12345,12348,12351],{"className":12329},[425],[413,12331],{"className":12332,"style":481},[429],[413,12334,2877],{"className":12335,"style":2876},[434,521],[413,12337,528],{"className":12338},[527],[413,12340,547],{"className":12341},[434,521],[413,12343,552],{"className":12344},[551],[413,12346],{"className":12347,"style":656},[542],[413,12349,779],{"className":12350},[660],[413,12352],{"className":12353,"style":656},[542],[413,12355,12357,12360,12363,12366,12369,12372,12375,12378,12381,12384,12387,12390],{"className":12356},[425],[413,12358],{"className":12359,"style":481},[429],[413,12361,12031],{"className":12362},[434,521],[413,12364,528],{"className":12365},[527],[413,12367,1882],{"className":12368,"style":2894},[434,521],[413,12370,528],{"className":12371},[527],[413,12373,547],{"className":12374},[434,521],[413,12376,3019],{"className":12377},[551],[413,12379],{"className":12380,"style":656},[542],[413,12382],{"className":12383,"style":656},[542],[413,12385,6009],{"className":12386},[660],[413,12388],{"className":12389,"style":656},[542],[413,12391],{"className":12392,"style":656},[542],[413,12394,12396,12400,12455,12458,12547,12550,12553],{"className":12395},[425],[413,12397],{"className":12398,"style":12399},[429],"height:1.53em;vertical-align:-0.52em;",[413,12401,12403,12410],{"className":12402},[532],[413,12404,12406],{"className":12405},[532],[413,12407,12409],{"className":12408},[434,536],"lim",[413,12411,12413],{"className":12412},[904],[413,12414,12416,12447],{"className":12415},[908,909],[413,12417,12419,12444],{"className":12418},[913],[413,12420,12422],{"className":12421,"style":990},[917],[413,12423,12425,12428],{"style":12424},"top:-2.55em;margin-right:0.05em;",[413,12426],{"className":12427,"style":926},[925],[413,12429,12431],{"className":12430},[930,931,932,933],[413,12432,12434,12437,12440],{"className":12433},[434,933],[413,12435,547],{"className":12436},[434,521,933],[413,12438,683],{"className":12439},[660,933],[413,12441,12443],{"className":12442},[434,933],"∞",[413,12445,941],{"className":12446},[940],[413,12448,12450],{"className":12449},[913],[413,12451,12453],{"className":12452,"style":948},[917],[413,12454],{},[413,12456],{"className":12457,"style":543},[542],[413,12459,12461,12464,12544],{"className":12460},[434],[413,12462],{"className":12463},[527,9131],[413,12465,12467],{"className":12466},[9135],[413,12468,12470,12535],{"className":12469},[908,909],[413,12471,12473,12532],{"className":12472},[913],[413,12474,12477,12500,12508],{"className":12475,"style":12476},[917],"height:1.01em;",[413,12478,12479,12482],{"style":10218},[413,12480],{"className":12481,"style":9152},[925],[413,12483,12485],{"className":12484},[930,931,932,933],[413,12486,12488,12491,12494,12497],{"className":12487},[434,933],[413,12489,1882],{"className":12490,"style":2894},[434,521,933],[413,12492,528],{"className":12493},[527,933],[413,12495,547],{"className":12496},[434,521,933],[413,12498,552],{"className":12499},[551,933],[413,12501,12502,12505],{"style":9161},[413,12503],{"className":12504,"style":9152},[925],[413,12506],{"className":12507,"style":9169},[9168],[413,12509,12511,12514],{"style":12510},"top:-3.485em;",[413,12512],{"className":12513,"style":9152},[925],[413,12515,12517],{"className":12516},[930,931,932,933],[413,12518,12520,12523,12526,12529],{"className":12519},[434,933],[413,12521,2877],{"className":12522,"style":2876},[434,521,933],[413,12524,528],{"className":12525},[527,933],[413,12527,547],{"className":12528},[434,521,933],[413,12530,552],{"className":12531},[551,933],[413,12533,941],{"className":12534},[940],[413,12536,12538],{"className":12537},[913],[413,12539,12542],{"className":12540,"style":12541},[917],"height:0.52em;",[413,12543],{},[413,12545],{"className":12546},[551,9131],[413,12548],{"className":12549,"style":656},[542],[413,12551,779],{"className":12552},[660],[413,12554],{"className":12555,"style":656},[542],[413,12557,12559,12562],{"className":12558},[425],[413,12560],{"className":12561,"style":2384},[429],[413,12563,12565],{"className":12564},[434],"0.",[381,12567,12568,12569,12047,12611,12626,12627,12630,12631,12646,12647,1597,12715,12766,12767,12825,12826,13182,13183,13198,13199,13214,13215,13267,13268,13346],{},"So ",[413,12570,12572],{"className":12571},[416],[413,12573,12575,12593],{"className":12574,"ariaHidden":421},[420],[413,12576,12578,12581,12584,12587,12590],{"className":12577},[425],[413,12579],{"className":12580,"style":2787},[429],[413,12582,2877],{"className":12583,"style":2876},[434,521],[413,12585],{"className":12586,"style":656},[542],[413,12588,779],{"className":12589},[660],[413,12591],{"className":12592,"style":656},[542],[413,12594,12596,12599,12602,12605,12608],{"className":12595},[425],[413,12597],{"className":12598,"style":481},[429],[413,12600,12031],{"className":12601},[434,521],[413,12603,528],{"className":12604},[527],[413,12606,1882],{"className":12607,"style":2894},[434,521],[413,12609,552],{"className":12610},[551],[413,12612,12614],{"className":12613},[416],[413,12615,12617],{"className":12616,"ariaHidden":421},[420],[413,12618,12620,12623],{"className":12619},[425],[413,12621],{"className":12622,"style":2787},[429],[413,12624,2877],{"className":12625,"style":2876},[434,521]," becomes ",[385,12628,12629],{},"negligible"," compared to ",[413,12632,12634],{"className":12633},[416],[413,12635,12637],{"className":12636,"ariaHidden":421},[420],[413,12638,12640,12643],{"className":12639},[425],[413,12641],{"className":12642,"style":2890},[429],[413,12644,1882],{"className":12645,"style":2894},[434,521],": e.g.\n",[413,12648,12650],{"className":12649},[416],[413,12651,12653,12671],{"className":12652,"ariaHidden":421},[420],[413,12654,12656,12659,12662,12665,12668],{"className":12655},[425],[413,12657],{"className":12658,"style":566},[429],[413,12660,547],{"className":12661},[434,521],[413,12663],{"className":12664,"style":656},[542],[413,12666,779],{"className":12667},[660],[413,12669],{"className":12670,"style":656},[542],[413,12672,12674,12677,12680,12683,12712],{"className":12673},[425],[413,12675],{"className":12676,"style":1707},[429],[413,12678,12031],{"className":12679},[434,521],[413,12681,528],{"className":12682},[527],[413,12684,12686,12689],{"className":12685},[434],[413,12687,547],{"className":12688},[434,521],[413,12690,12692],{"className":12691},[904],[413,12693,12695],{"className":12694},[908],[413,12696,12698],{"className":12697},[913],[413,12699,12701],{"className":12700,"style":1732},[917],[413,12702,12703,12706],{"style":1735},[413,12704],{"className":12705,"style":926},[925],[413,12707,12709],{"className":12708},[930,931,932,933],[413,12710,597],{"className":12711},[434,933],[413,12713,552],{"className":12714},[551],[413,12716,12718],{"className":12717},[416],[413,12719,12721,12748],{"className":12720,"ariaHidden":421},[420],[413,12722,12724,12727,12733,12736,12739,12742,12745],{"className":12723},[425],[413,12725],{"className":12726,"style":2787},[429],[413,12728,12730],{"className":12729},[532],[413,12731,538],{"className":12732,"style":537},[434,536],[413,12734],{"className":12735,"style":543},[542],[413,12737,547],{"className":12738},[434,521],[413,12740],{"className":12741,"style":656},[542],[413,12743,779],{"className":12744},[660],[413,12746],{"className":12747,"style":656},[542],[413,12749,12751,12754,12757,12760,12763],{"className":12750},[425],[413,12752],{"className":12753,"style":481},[429],[413,12755,12031],{"className":12756},[434,521],[413,12758,528],{"className":12759},[527],[413,12761,547],{"className":12762},[434,521],[413,12764,552],{"className":12765},[551],". Symmetrically, ",[413,12768,12770],{"className":12769},[416],[413,12771,12773,12800],{"className":12772,"ariaHidden":421},[420],[413,12774,12776,12779,12782,12785,12788,12791,12794,12797],{"className":12775},[425],[413,12777],{"className":12778,"style":481},[429],[413,12780,2877],{"className":12781,"style":2876},[434,521],[413,12783,528],{"className":12784},[527],[413,12786,547],{"className":12787},[434,521],[413,12789,552],{"className":12790},[551],[413,12792],{"className":12793,"style":656},[542],[413,12795,779],{"className":12796},[660],[413,12798],{"className":12799,"style":656},[542],[413,12801,12803,12806,12810,12813,12816,12819,12822],{"className":12802},[425],[413,12804],{"className":12805,"style":481},[429],[413,12807,12809],{"className":12808,"style":2894},[434,521],"ω",[413,12811,528],{"className":12812},[527],[413,12814,1882],{"className":12815,"style":2894},[434,521],[413,12817,528],{"className":12818},[527],[413,12820,547],{"className":12821},[434,521],[413,12823,3019],{"className":12824},[551]," is defined\nby the reciprocal limit,\n",[413,12827,12829],{"className":12828},[416],[413,12830,12832,12988,13014,13170],{"className":12831,"ariaHidden":421},[420],[413,12833,12835,12838,12890,12893,12979,12982,12985],{"className":12834},[425],[413,12836],{"className":12837,"style":12399},[429],[413,12839,12841,12847],{"className":12840},[532],[413,12842,12844],{"className":12843},[532],[413,12845,12409],{"className":12846},[434,536],[413,12848,12850],{"className":12849},[904],[413,12851,12853,12882],{"className":12852},[908,909],[413,12854,12856,12879],{"className":12855},[913],[413,12857,12859],{"className":12858,"style":990},[917],[413,12860,12861,12864],{"style":12424},[413,12862],{"className":12863,"style":926},[925],[413,12865,12867],{"className":12866},[930,931,932,933],[413,12868,12870,12873,12876],{"className":12869},[434,933],[413,12871,547],{"className":12872},[434,521,933],[413,12874,683],{"className":12875},[660,933],[413,12877,12443],{"className":12878},[434,933],[413,12880,941],{"className":12881},[940],[413,12883,12885],{"className":12884},[913],[413,12886,12888],{"className":12887,"style":948},[917],[413,12889],{},[413,12891],{"className":12892,"style":543},[542],[413,12894,12896,12899,12976],{"className":12895},[434],[413,12897],{"className":12898},[527,9131],[413,12900,12902],{"className":12901},[9135],[413,12903,12905,12968],{"className":12904},[908,909],[413,12906,12908,12965],{"className":12907},[913],[413,12909,12911,12934,12942],{"className":12910,"style":12476},[917],[413,12912,12913,12916],{"style":10218},[413,12914],{"className":12915,"style":9152},[925],[413,12917,12919],{"className":12918},[930,931,932,933],[413,12920,12922,12925,12928,12931],{"className":12921},[434,933],[413,12923,2877],{"className":12924,"style":2876},[434,521,933],[413,12926,528],{"className":12927},[527,933],[413,12929,547],{"className":12930},[434,521,933],[413,12932,552],{"className":12933},[551,933],[413,12935,12936,12939],{"style":9161},[413,12937],{"className":12938,"style":9152},[925],[413,12940],{"className":12941,"style":9169},[9168],[413,12943,12944,12947],{"style":12510},[413,12945],{"className":12946,"style":9152},[925],[413,12948,12950],{"className":12949},[930,931,932,933],[413,12951,12953,12956,12959,12962],{"className":12952},[434,933],[413,12954,1882],{"className":12955,"style":2894},[434,521,933],[413,12957,528],{"className":12958},[527,933],[413,12960,547],{"className":12961},[434,521,933],[413,12963,552],{"className":12964},[551,933],[413,12966,941],{"className":12967},[940],[413,12969,12971],{"className":12970},[913],[413,12972,12974],{"className":12973,"style":12541},[917],[413,12975],{},[413,12977],{"className":12978},[551,9131],[413,12980],{"className":12981,"style":656},[542],[413,12983,779],{"className":12984},[660],[413,12986],{"className":12987,"style":656},[542],[413,12989,12991,12995,12998,13001,13004,13008,13011],{"className":12990},[425],[413,12992],{"className":12993,"style":12994},[429],"height:0.6684em;vertical-align:-0.024em;",[413,12996,2183],{"className":12997},[434],[413,12999],{"className":13000,"style":5801},[542],[413,13002],{"className":13003,"style":656},[542],[413,13005,13007],{"className":13006},[660],"⟺",[413,13009],{"className":13010,"style":5801},[542],[413,13012],{"className":13013,"style":656},[542],[413,13015,13017,13020,13072,13075,13161,13164,13167],{"className":13016},[425],[413,13018],{"className":13019,"style":12399},[429],[413,13021,13023,13029],{"className":13022},[532],[413,13024,13026],{"className":13025},[532],[413,13027,12409],{"className":13028},[434,536],[413,13030,13032],{"className":13031},[904],[413,13033,13035,13064],{"className":13034},[908,909],[413,13036,13038,13061],{"className":13037},[913],[413,13039,13041],{"className":13040,"style":990},[917],[413,13042,13043,13046],{"style":12424},[413,13044],{"className":13045,"style":926},[925],[413,13047,13049],{"className":13048},[930,931,932,933],[413,13050,13052,13055,13058],{"className":13051},[434,933],[413,13053,547],{"className":13054},[434,521,933],[413,13056,683],{"className":13057},[660,933],[413,13059,12443],{"className":13060},[434,933],[413,13062,941],{"className":13063},[940],[413,13065,13067],{"className":13066},[913],[413,13068,13070],{"className":13069,"style":948},[917],[413,13071],{},[413,13073],{"className":13074,"style":543},[542],[413,13076,13078,13081,13158],{"className":13077},[434],[413,13079],{"className":13080},[527,9131],[413,13082,13084],{"className":13083},[9135],[413,13085,13087,13150],{"className":13086},[908,909],[413,13088,13090,13147],{"className":13089},[913],[413,13091,13093,13116,13124],{"className":13092,"style":12476},[917],[413,13094,13095,13098],{"style":10218},[413,13096],{"className":13097,"style":9152},[925],[413,13099,13101],{"className":13100},[930,931,932,933],[413,13102,13104,13107,13110,13113],{"className":13103},[434,933],[413,13105,1882],{"className":13106,"style":2894},[434,521,933],[413,13108,528],{"className":13109},[527,933],[413,13111,547],{"className":13112},[434,521,9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",[413,13184,13186],{"className":13185},[416],[413,13187,13189],{"className":13188,"ariaHidden":421},[420],[413,13190,13192,13195],{"className":13191},[425],[413,13193],{"className":13194,"style":2787},[429],[413,13196,2877],{"className":13197,"style":2876},[434,521]," dominates ",[413,13200,13202],{"className":13201},[416],[413,13203,13205],{"className":13204,"ariaHidden":421},[420],[413,13206,13208,13211],{"className":13207},[425],[413,13209],{"className":13210,"style":2890},[429],[413,13212,1882],{"className":13213,"style":2894},[434,521],". A useful analogy from CLRS: ",[413,13216,13218],{"className":13217},[416],[413,13219,13221],{"className":13220,"ariaHidden":421},[420],[413,13222,13224,13228,13231,13234,13237,13240,13243,13246,13249,13252,13255,13258,13261,13264],{"className":13223},[425],[413,13225],{"className":13226,"style":13227},[429],"height:0.8778em;vertical-align:-0.1944em;",[413,13229,523],{"className":13230,"style":522},[434,521],[413,13232,955],{"className":13233},[954],[413,13235],{"className":13236,"style":543},[542],[413,13238,4910],{"className":13239},[434],[413,13241,955],{"className":13242},[954],[413,13244],{"className":13245,"style":543},[542],[413,13247,1684],{"className":13248},[434],[413,13250,955],{"className":13251},[954],[413,13253],{"className":13254,"style":543},[542],[413,13256,12031],{"className":13257},[434,521],[413,13259,955],{"className":13260},[954],[413,13262],{"className":13263,"style":543},[542],[413,13265,12809],{"className":13266,"style":2894},[434,521],"\nare to functions as ",[413,13269,13271],{"className":13270},[416],[413,13272,13274,13283,13299,13315,13331],{"className":13273,"ariaHidden":421},[420],[413,13275,13277,13280],{"className":13276},[425],[413,13278],{"className":13279,"style":4435},[429],[413,13281,3244],{"className":13282},[660],[413,13284,13286,13290,13293,13296],{"className":13285},[425],[413,13287],{"className":13288,"style":13289},[429],"height:0.8304em;vertical-align:-0.1944em;",[413,13291,955],{"className":13292},[954],[413,13294],{"className":13295,"style":656},[542],[413,13297,3158],{"className":13298},[660],[413,13300,13302,13306,13309,13312],{"className":13301},[425],[413,13303],{"className":13304,"style":13305},[429],"height:0.5613em;vertical-align:-0.1944em;",[413,13307,955],{"className":13308},[954],[413,13310],{"className":13311,"style":656},[542],[413,13313,779],{"className":13314},[660],[413,13316,13318,13322,13325,13328],{"className":13317},[425],[413,13319],{"className":13320,"style":13321},[429],"height:0.7335em;vertical-align:-0.1944em;",[413,13323,955],{"className":13324},[954],[413,13326],{"className":13327,"style":656},[542],[413,13329,9043],{"className":13330},[660],[413,13332,13334,13337,13340,13343],{"className":13333},[425],[413,13335],{"className":13336,"style":13321},[429],[413,13338,955],{"className":13339},[954],[413,13341],{"className":13342,"style":656},[542],[413,13344,3051],{"className":13345},[660]," are to numbers.",[1869,13348,13350,13548],{"className":13349},[1872,1873],[1875,13351,13355],{"xmlns":1877,"width":13352,"height":13353,"viewBox":13354},"585.944","167.335","-75 -75 439.458 125.501",[1882,13356,13357,13364,13385,13388,13396,13399,13406,13409,13416,13419,13426,13429,13436,13448,13451,13458,13461,13468,13471,13478,13490,13499,13507,13515,13523,13531,13540],{"stroke":1884,"style":1885},[1882,13358,13360],{"transform":13359},"translate(-88.76 2.683)",[1887,13361],{"d":13362,"fill":1884,"stroke":1884,"className":13363,"style":2041},"M64.602-49.508L62.746-49.508L62.746-49.805Q63.020-49.805 63.188-49.852Q63.356-49.899 63.356-50.067L63.356-52.203Q63.356-52.418 63.293-52.514Q63.231-52.610 63.112-52.631Q62.993-52.653 62.746-52.653L62.746-52.949L63.938-53.035L63.938-52.301Q64.051-52.516 64.245-52.684Q64.438-52.852 64.676-52.944Q64.914-53.035 65.168-53.035Q66.336-53.035 66.336-51.957L66.336-50.067Q66.336-49.899 66.506-49.852Q66.676-49.805 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79.403-50.469",[1901],[1882,13541,13542],{"fill":11412,"stroke":11412},[1882,13543,13545],{"transform":13544},"translate(266.284 88.04)",[1887,13546],{"d":13538,"fill":11412,"stroke":11412,"className":13547,"style":2041},[1901],[2117,13549,13551,13552,13603,13604,13679,13680,13682,13683,1597,13698,13713],{"className":13550},[2120],"Each asymptotic notation mirrors a comparison on numbers: ",[413,13553,13555],{"className":13554},[416],[413,13556,13558],{"className":13557,"ariaHidden":421},[420],[413,13559,13561,13564,13567,13570,13573,13576,13579,13582,13585,13588,13591,13594,13597,13600],{"className":13560},[425],[413,13562],{"className":13563,"style":13227},[429],[413,13565,12031],{"className":13566},[434,521],[413,13568,955],{"className":13569},[954],[413,13571],{"className":13572,"style":543},[542],[413,13574,523],{"className":13575,"style":522},[434,521],[413,13577,955],{"className":13578},[954],[413,13580],{"className":13581,"style":543},[542],[413,13583,1684],{"className":13584},[434],[413,13586,955],{"className":13587},[954],[413,13589],{"className":13590,"style":543},[542],[413,13592,4910],{"className":13593},[434],[413,13595,955],{"className":13596},[954],[413,13598],{"className":13599,"style":543},[542],[413,13601,12809],{"className":13602,"style":2894},[434,521]," line up with ",[413,13605,13607],{"className":13606},[416],[413,13608,13610,13619,13634,13649,13664],{"className":13609,"ariaHidden":421},[420],[413,13611,13613,13616],{"className":13612},[425],[413,13614],{"className":13615,"style":10867},[429],[413,13617,9043],{"className":13618},[660],[413,13620,13622,13625,13628,13631],{"className":13621},[425],[413,13623],{"className":13624,"style":13289},[429],[413,13626,955],{"className":13627},[954],[413,13629],{"className":13630,"style":656},[542],[413,13632,3244],{"className":13633},[660],[413,13635,13637,13640,13643,13646],{"className":13636},[425],[413,13638],{"className":13639,"style":13305},[429],[413,13641,955],{"className":13642},[954],[413,13644],{"className":13645,"style":656},[542],[413,13647,779],{"className":13648},[660],[413,13650,13652,13655,13658,13661],{"className":13651},[425],[413,13653],{"className":13654,"style":13289},[429],[413,13656,955],{"className":13657},[954],[413,13659],{"className":13660,"style":656},[542],[413,13662,3158],{"className":13663},[660],[413,13665,13667,13670,13673,13676],{"className":13666},[425],[413,13668],{"className":13669,"style":13321},[429],[413,13671,955],{"className":13672},[954],[413,13674],{"className":13675,"style":656},[542],[413,13677,3051],{"className":13678},[660],". The little-o and little-omega ends are ",[385,13681,11980],{}," (they forbid equal growth), exactly as ",[413,13684,13686],{"className":13685},[416],[413,13687,13689],{"className":13688,"ariaHidden":421},[420],[413,13690,13692,13695],{"className":13691},[425],[413,13693],{"className":13694,"style":10867},[429],[413,13696,9043],{"className":13697},[660],[413,13699,13701],{"className":13700},[416],[413,13702,13704],{"className":13703,"ariaHidden":421},[420],[413,13705,13707,13710],{"className":13706},[425],[413,13708],{"className":13709,"style":10867},[429],[413,13711,3051],{"className":13712},[660]," are strict.",[390,13715,13717],{"id":13716},"comparing-functions-with-limits","Comparing functions with limits",[381,13719,13720,13721,13760],{},"The limit of the ratio ",[413,13722,13724],{"className":13723},[416],[413,13725,13727],{"className":13726,"ariaHidden":421},[420],[413,13728,13730,13733,13736,13739,13742,13745,13748,13751,13754,13757],{"className":13729},[425],[413,13731],{"className":13732,"style":481},[429],[413,13734,2877],{"className":13735,"style":2876},[434,521],[413,13737,528],{"className":13738},[527],[413,13740,547],{"className":13741},[434,521],[413,13743,552],{"className":13744},[551],[413,13746,485],{"className":13747},[434],[413,13749,1882],{"className":13750,"style":2894},[434,521],[413,13752,528],{"className":13753},[527],[413,13755,547],{"className":13756},[434,521],[413,13758,552],{"className":13759},[551]," is how you rank growth rates:",[413,13762,13764],{"className":13763},[1263],[413,13765,13767],{"className":13766},[416],[413,13768,13770,13922],{"className":13769,"ariaHidden":421},[420],[413,13771,13773,13777,13829,13832,13913,13916,13919],{"className":13772},[425],[413,13774],{"className":13775,"style":13776},[429],"height:2.363em;vertical-align:-0.936em;",[413,13778,13780,13786],{"className":13779},[532],[413,13781,13783],{"className":13782},[532],[413,13784,12409],{"className":13785},[434,536],[413,13787,13789],{"className":13788},[904],[413,13790,13792,13821],{"className":13791},[908,909],[413,13793,13795,13818],{"className":13794},[913],[413,13796,13798],{"className":13797,"style":990},[917],[413,13799,13800,13803],{"style":12424},[413,13801],{"className":13802,"style":926},[925],[413,13804,13806],{"className":13805},[930,931,932,933],[413,13807,13809,13812,13815],{"className":13808},[434,933],[413,13810,547],{"className":13811},[434,521,933],[413,13813,683],{"className":13814},[660,933],[413,13816,12443],{"className":13817},[434,933],[413,13819,941],{"className":13820},[940],[413,13822,13824],{"className":13823},[913],[413,13825,13827],{"className":13826,"style":948},[917],[413,13828],{},[413,13830],{"className":13831,"style":543},[542],[413,13833,13835,13838,13910],{"className":13834},[434],[413,13836],{"className":13837},[527,9131],[413,13839,13841],{"className":13840},[9135],[413,13842,13844,13901],{"className":13843},[908,909],[413,13845,13847,13898],{"className":13846},[913],[413,13848,13850,13870,13878],{"className":13849,"style":9145},[917],[413,13851,13852,13855],{"style":9148},[413,13853],{"className":13854,"style":9152},[925],[413,13856,13858,13861,13864,13867],{"className":13857},[434],[413,13859,1882],{"className":13860,"style":2894},[434,521],[413,13862,528],{"className":13863},[527],[413,13865,547],{"className":13866},[434,521],[413,13868,552],{"className":13869},[551],[413,13871,13872,13875],{"style":9161},[413,13873],{"className":13874,"style":9152},[925],[413,13876],{"className":13877,"style":9169},[9168],[413,13879,13880,13883],{"style":9172},[413,13881],{"className":13882,"style":9152},[925],[413,13884,13886,13889,13892,13895],{"className":13885},[434],[413,13887,2877],{"className":13888,"style":2876},[434,521],[413,13890,528],{"className":13891},[527],[413,13893,547],{"className":13894},[434,521],[413,13896,552],{"className":13897},[551],[413,13899,941],{"className":13900},[940],[413,13902,13904],{"className":13903},[913],[413,13905,13908],{"className":13906,"style":13907},[917],"height:0.936em;",[413,13909],{},[413,13911],{"className":13912},[551,9131],[413,13914],{"className":13915,"style":656},[542],[413,13917,779],{"className":13918},[660],[413,13920],{"className":13921,"style":656},[542],[413,13923,13925,13929],{"className":13924},[425],[413,13926],{"className":13927,"style":13928},[429],"height:4.32em;vertical-align:-1.91em;",[413,13930,13932,14031,14346],{"className":13931},[792],[413,13933,13935],{"className":13934},[527],[413,13936,13939],{"className":13937},[7416,13938],"mult",[413,13940,13942,14022],{"className":13941},[908,909],[413,13943,13945,14019],{"className":13944},[913],[413,13946,13949,13964,13983,13995,14007],{"className":13947,"style":13948},[917],"height:2.35em;",[413,13950,13952,13956],{"style":13951},"top:-2.2em;",[413,13953],{"className":13954,"style":13955},[925],"height:3.15em;",[413,13957,13961],{"className":13958},[13959,13960],"delimsizinginner","delim-size4",[413,13962,13963],{},"⎩",[413,13965,13967,13970],{"style":13966},"top:-2.192em;",[413,13968],{"className":13969,"style":13955},[925],[413,13971,13973],{"style":13972},"height:0.316em;width:0.8889em;",[1875,13974,13980],{"xmlns":1877,"width":13975,"height":13976,"style":13977,"viewBox":13978,"preserveAspectRatio":13979},"0.8889em","0.316em","width:0.8889em","0 0 888.89 316","xMinYMin",[1887,13981],{"d":13982},"M384 0 H504 V316 H384z M384 0 H504 V316 H384z",[413,13984,13986,13989],{"style":13985},"top:-3.15em;",[413,13987],{"className":13988,"style":13955},[925],[413,13990,13992],{"className":13991},[13959,13960],[413,13993,13994],{},"⎨",[413,13996,13998,14001],{"style":13997},"top:-4.292em;",[413,13999],{"className":14000,"style":13955},[925],[413,14002,14003],{"style":13972},[1875,14004,14005],{"xmlns":1877,"width":13975,"height":13976,"style":13977,"viewBox":13978,"preserveAspectRatio":13979},[1887,14006],{"d":13982},[413,14008,14010,14013],{"style":14009},"top:-4.6em;",[413,14011],{"className":14012,"style":13955},[925],[413,14014,14016],{"className":14015},[13959,13960],[413,14017,14018],{},"⎧",[413,14020,941],{"className":14021},[940],[413,14023,14025],{"className":14024},[913],[413,14026,14029],{"className":14027,"style":14028},[917],"height:1.85em;",[413,14030],{},[413,14032,14034],{"className":14033},[434],[413,14035,14037,14127,14132],{"className":14036},[1388],[413,14038,14041],{"className":14039},[14040],"col-align-l",[413,14042,14044,14118],{"className":14043},[908,909],[413,14045,14047,14115],{"className":14046},[913],[413,14048,14051,14064,14103],{"className":14049,"style":14050},[917],"height:2.41em;",[413,14052,14054,14058],{"style":14053},"top:-4.41em;",[413,14055],{"className":14056,"style":14057},[925],"height:3.008em;",[413,14059,14061],{"className":14060},[434],[413,14062,2183],{"className":14063},[434],[413,14065,14067,14070],{"style":14066},"top:-2.97em;",[413,14068],{"className":14069,"style":14057},[925],[413,14071,14073,14076,14079,14082,14085,14088,14091,14094,14097,14100],{"className":14072},[434],[413,14074,3044],{"className":14075},[434,521],[413,14077],{"className":14078,"style":656},[542],[413,14080,720],{"className":14081},[660],[413,14083],{"className":14084,"style":656},[542],[413,14086,528],{"className":14087},[527],[413,14089,2183],{"className":14090},[434],[413,14092,955],{"className":14093},[954],[413,14095],{"className":14096,"style":543},[542],[413,14098,12443],{"className":14099},[434],[413,14101,552],{"className":14102},[551],[413,14104,14106,14109],{"style":14105},"top:-1.53em;",[413,14107],{"className":14108,"style":14057},[925],[413,14110,14112],{"className":14111},[434],[413,14113,12443],{"className":14114},[434],[413,14116,941],{"className":14117},[940],[413,14119,14121],{"className":14120},[913],[413,14122,14125],{"className":14123,"style":14124},[917],"height:1.91em;",[413,14126],{},[413,14128],{"className":14129,"style":14131},[14130],"arraycolsep","width:1em;",[413,14133,14135],{"className":14134},[14040],[413,14136,14138,14338],{"className":14137},[908,909],[413,14139,14141,14335],{"className":14140},[913],[413,14142,14144,14220,14258],{"className":14143,"style":14050},[917],[413,14145,14146,14149],{"style":14053},[413,14147],{"className":14148,"style":14057},[925],[413,14150,14152,14156,14159,14162,14165,14168,14171,14174,14177,14180,14183,14186,14189,14196,14199,14202,14205,14208,14211,14214,14217],{"className":14151},[434],[413,14153,14155],{"className":14154},[660],"⇒",[413,14157],{"className":14158,"style":656},[542],[413,14160,2877],{"className":14161,"style":2876},[434,521],[413,14163],{"className":14164,"style":656},[542],[413,14166,779],{"className":14167},[660],[413,14169],{"className":14170,"style":656},[542],[413,14172,12031],{"className":14173},[434,521],[413,14175,528],{"className":14176},[527],[413,14178,1882],{"className":14179,"style":2894},[434,521],[413,14181,552],{"className":14182},[551],[413,14184,6070],{"className":14185},[542],[413,14187,528],{"className":14188},[527],[413,14190,14192],{"className":14191},[434,802],[413,14193,14195],{"className":14194},[434],"so also ",[413,14197,2877],{"className":14198,"style":2876},[434,521],[413,14200],{"className":14201,"style":656},[542],[413,14203,779],{"className":14204},[660],[413,14206],{"className":14207,"style":656},[542],[413,14209,523],{"className":14210,"style":522},[434,521],[413,14212,528],{"className":14213},[527],[413,14215,1882],{"className":14216,"style":2894},[434,521],[413,14218,3019],{"className":14219},[551],[413,14221,14222,14225],{"style":14066},[413,14223],{"className":14224,"style":14057},[925],[413,14226,14228,14231,14234,14237,14240,14243,14246,14249,14252,14255],{"className":14227},[434],[413,14229,14155],{"className":14230},[660],[413,14232],{"className":14233,"style":656},[542],[413,14235,2877],{"className":14236,"style":2876},[434,521],[413,14238],{"className":14239,"style":656},[542],[413,14241,779],{"className":14242},[660],[413,14244],{"className":14245,"style":656},[542],[413,14247,1684],{"className":14248},[434],[413,14250,528],{"className":14251},[527],[413,14253,1882],{"className":14254,"style":2894},[434,521],[413,14256,552],{"className":14257},[551],[413,14259,14260,14263],{"style":14105},[413,14261],{"className":14262,"style":14057},[925],[413,14264,14266,14269,14272,14275,14278,14281,14284,14287,14290,14293,14296,14299,14302,14308,14311,14314,14317,14320,14323,14326,14329,14332],{"className":14265},[434],[413,14267,14155],{"className":14268},[660],[413,14270],{"className":14271,"style":656},[542],[413,14273,2877],{"className":14274,"style":2876},[434,521],[413,14276],{"className":14277,"style":656},[542],[413,14279,779],{"className":14280},[660],[413,14282],{"className":14283,"style":656},[542],[413,14285,12809],{"className":14286,"style":2894},[434,521],[413,14288,528],{"className":14289},[527],[413,14291,1882],{"className":14292,"style":2894},[434,521],[413,14294,552],{"className":14295},[551],[413,14297,6070],{"className":14298},[542],[413,14300,528],{"className":14301},[527],[413,14303,14305],{"className":14304},[434,802],[413,14306,14195],{"className":14307},[434],[413,14309,2877],{"className":14310,"style":2876},[434,521],[413,14312],{"className":14313,"style":656},[542],[413,14315,779],{"className":14316},[660],[413,14318],{"className":14319,"style":656},[542],[413,14321,4910],{"className":14322},[434],[413,14324,528],{"className":14325},[527],[413,14327,1882],{"className":14328,"style":2894},[434,521],[413,14330,3019],{"className":14331},[551],[413,14333,1158],{"className":14334},[434],[413,14336,941],{"className":14337},[940],[413,14339,14341],{"className":14340},[913],[413,14342,14344],{"className":14343,"style":14124},[917],[413,14345],{},[413,14347],{"className":14348},[551,9131],[381,14350,14351,14352,1597,14394,14448,14449,14491,14492,14593,14594,14596,14597,14599,14600,14607,14608,14702,14703,14718,14719,14789,14790,14793,14794,14802,14803,14806,14807,14846],{},"To compare, say, ",[413,14353,14355],{"className":14354},[416],[413,14356,14358],{"className":14357,"ariaHidden":421},[420],[413,14359,14361,14365],{"className":14360},[425],[413,14362],{"className":14363,"style":14364},[429],"height:0.6644em;",[413,14366,14368,14371],{"className":14367},[434],[413,14369,547],{"className":14370},[434,521],[413,14372,14374],{"className":14373},[904],[413,14375,14377],{"className":14376},[908],[413,14378,14380],{"className":14379},[913],[413,14381,14383],{"className":14382,"style":14364},[917],[413,14384,14385,14388],{"style":1735},[413,14386],{"className":14387,"style":926},[925],[413,14389,14391],{"className":14390},[930,931,932,933],[413,14392,582],{"className":14393},[434,521,933],[413,14395,14397],{"className":14396},[416],[413,14398,14400],{"className":14399,"ariaHidden":421},[420],[413,14401,14403,14407,14442,14445],{"className":14402},[425],[413,14404],{"className":14405,"style":14406},[429],"height:1.1279em;vertical-align:-0.1944em;",[413,14408,14410,14416],{"className":14409},[532],[413,14411,14413],{"className":14412},[532],[413,14414,538],{"className":14415,"style":537},[434,536],[413,14417,14419],{"className":14418},[904],[413,14420,14422],{"className":14421},[908],[413,14423,14425],{"className":14424},[913],[413,14426,14429],{"className":14427,"style":14428},[917],"height:0.9334em;",[413,14430,14432,14435],{"style":14431},"top:-3.1473em;margin-right:0.05em;",[413,14433],{"className":14434,"style":926},[925],[413,14436,14438],{"className":14437},[930,931,932,933],[413,14439,14441],{"className":14440},[434,521,933],"b",[413,14443],{"className":14444,"style":543},[542],[413,14446,547],{"className":14447},[434,521]," for any constants ",[413,14450,14452],{"className":14451},[416],[413,14453,14455,14482],{"className":14454,"ariaHidden":421},[420],[413,14456,14458,14461,14464,14467,14470,14473,14476,14479],{"className":14457},[425],[413,14459],{"className":14460,"style":2787},[429],[413,14462,582],{"className":14463},[434,521],[413,14465,955],{"className":14466},[954],[413,14468],{"className":14469,"style":543},[542],[413,14471,14441],{"className":14472},[434,521],[413,14474],{"className":14475,"style":656},[542],[413,14477,3051],{"className":14478},[660],[413,14480],{"className":14481,"style":656},[542],[413,14483,14485,14488],{"className":14484},[425],[413,14486],{"className":14487,"style":2384},[429],[413,14489,2183],{"className":14490},[434],", one shows the\nratio ",[413,14493,14495],{"className":14494},[416],[413,14496,14498,14584],{"className":14497,"ariaHidden":421},[420],[413,14499,14501,14505,14537,14540,14543,14546,14575,14578,14581],{"className":14500},[425],[413,14502],{"className":14503,"style":14504},[429],"height:1.1834em;vertical-align:-0.25em;",[413,14506,14508,14514],{"className":14507},[532],[413,14509,14511],{"className":14510},[532],[413,14512,538],{"className":14513,"style":537},[434,536],[413,14515,14517],{"className":14516},[904],[413,14518,14520],{"className":14519},[908],[413,14521,14523],{"className":14522},[913],[413,14524,14526],{"className":14525,"style":14428},[917],[413,14527,14528,14531],{"style":14431},[413,14529],{"className":14530,"style":926},[925],[413,14532,14534],{"className":14533},[930,931,932,933],[413,14535,14441],{"className":14536},[434,521,933],[413,14538],{"className":14539,"style":543},[542],[413,14541,547],{"className":14542},[434,521],[413,14544,485],{"className":14545},[434],[413,14547,14549,14552],{"className":14548},[434],[413,14550,547],{"className":14551},[434,521],[413,14553,14555],{"className":14554},[904],[413,14556,14558],{"className":14557},[908],[413,14559,14561],{"className":14560},[913],[413,14562,14564],{"className":14563,"style":14364},[917],[413,14565,14566,14569],{"style":1735},[413,14567],{"className":14568,"style":926},[925],[413,14570,14572],{"className":14571},[930,931,932,933],[413,14573,582],{"className":14574},[434,521,933],[413,14576],{"className":14577,"style":656},[542],[413,14579,683],{"className":14580},[660],[413,14582],{"className":14583,"style":656},[542],[413,14585,14587,14590],{"className":14586},[425],[413,14588],{"className":14589,"style":2384},[429],[413,14591,2183],{"className":14592},[434],", so ",[398,14595,9019],{}," polynomial beats ",[398,14598,9019],{},"\npolylogarithm.",[579,14601,14602],{},[582,14603,2313],{"href":14604,"ariaDescribedBy":14605,"dataFootnoteRef":376,"id":14606},"#user-content-fn-erickson-growth",[586],"user-content-fnref-erickson-growth"," Likewise ",[413,14609,14611],{"className":14610},[416],[413,14612,14614,14658],{"className":14613,"ariaHidden":421},[420],[413,14615,14617,14620,14649,14652,14655],{"className":14616},[425],[413,14618],{"className":14619,"style":6516},[429],[413,14621,14623,14626],{"className":14622},[434],[413,14624,547],{"className":14625},[434,521],[413,14627,14629],{"className":14628},[904],[413,14630,14632],{"className":14631},[908],[413,14633,14635],{"className":14634},[913],[413,14636,14638],{"className":14637,"style":6516},[917],[413,14639,14640,14643],{"style":1735},[413,14641],{"className":14642,"style":926},[925],[413,14644,14646],{"className":14645},[930,931,932,933],[413,14647,6486],{"className":14648,"style":6485},[434,521,933],[413,14650],{"className":14651,"style":656},[542],[413,14653,779],{"className":14654},[660],[413,14656],{"className":14657,"style":656},[542],[413,14659,14661,14664,14667,14670,14699],{"className":14660},[425],[413,14662],{"className":14663,"style":481},[429],[413,14665,12031],{"className":14666},[434,521],[413,14668,528],{"className":14669},[527],[413,14671,14673,14676],{"className":14672},[434],[413,14674,597],{"className":14675},[434],[413,14677,14679],{"className":14678},[904],[413,14680,14682],{"className":14681},[908],[413,14683,14685],{"className":14684},[913],[413,14686,14688],{"className":14687,"style":14364},[917],[413,14689,14690,14693],{"style":1735},[413,14691],{"className":14692,"style":926},[925],[413,14694,14696],{"className":14695},[930,931,932,933],[413,14697,547],{"className":14698},[434,521,933],[413,14700,552],{"className":14701},[551]," for every constant ",[413,14704,14706],{"className":14705},[416],[413,14707,14709],{"className":14708,"ariaHidden":421},[420],[413,14710,14712,14715],{"className":14711},[425],[413,14713],{"className":14714,"style":11046},[429],[413,14716,6486],{"className":14717,"style":6485},[434,521],": exponentials\ndominate all polynomials. A useful fact for sums: ",[413,14720,14722],{"className":14721},[416],[413,14723,14725,14756],{"className":14724,"ariaHidden":421},[420],[413,14726,14728,14731,14737,14740,14743,14747,14750,14753],{"className":14727},[425],[413,14729],{"className":14730,"style":481},[429],[413,14732,14734],{"className":14733},[532],[413,14735,538],{"className":14736,"style":537},[434,536],[413,14738,528],{"className":14739},[527],[413,14741,547],{"className":14742},[434,521],[413,14744,14746],{"className":14745},[551],"!)",[413,14748],{"className":14749,"style":656},[542],[413,14751,779],{"className":14752},[660],[413,14754],{"className":14755,"style":656},[542],[413,14757,14759,14762,14765,14768,14771,14774,14780,14783,14786],{"className":14758},[425],[413,14760],{"className":14761,"style":481},[429],[413,14763,1684],{"className":14764},[434],[413,14766,528],{"className":14767},[527],[413,14769,547],{"className":14770},[434,521],[413,14772],{"className":14773,"style":543},[542],[413,14775,14777],{"className":14776},[532],[413,14778,538],{"className":14779,"style":537},[434,536],[413,14781],{"className":14782,"style":543},[542],[413,14784,547],{"className":14785},[434,521],[413,14787,552],{"className":14788},[551],"\n(by ",[398,14791,14792],{},"Stirling's approximation","),",[579,14795,14796],{},[582,14797,14801],{"href":14798,"ariaDescribedBy":14799,"dataFootnoteRef":376,"id":14800},"#user-content-fn-clrs-stirling",[586],"user-content-fnref-clrs-stirling","4"," which is why ",[582,14804,14805],{"href":62},"comparison sorts"," that do\n",[413,14808,14810],{"className":14809},[416],[413,14811,14813],{"className":14812,"ariaHidden":421},[420],[413,14814,14816,14819,14822,14825,14828,14831,14837,14840,14843],{"className":14815},[425],[413,14817],{"className":14818,"style":481},[429],[413,14820,1684],{"className":14821},[434],[413,14823,528],{"className":14824},[527],[413,14826,547],{"className":14827},[434,521],[413,14829],{"className":14830,"style":543},[542],[413,14832,14834],{"className":14833},[532],[413,14835,538],{"className":14836,"style":537},[434,536],[413,14838],{"className":14839,"style":543},[542],[413,14841,547],{"className":14842},[434,521],[413,14844,552],{"className":14845},[551]," work are optimal in a sense we will prove later.",[381,14848,14849,14877],{},[398,14850,14851,14852,14876],{},"Why we write ",[413,14853,14855],{"className":14854},[416],[413,14856,14858],{"className":14857,"ariaHidden":421},[420],[413,14859,14861,14864,14870,14873],{"className":14860},[425],[413,14862],{"className":14863,"style":2787},[429],[413,14865,14867],{"className":14866},[532],[413,14868,538],{"className":14869,"style":537},[434,536],[413,14871],{"className":14872,"style":543},[542],[413,14874,547],{"className":14875},[434,521]," with no base."," Inside asymptotic notation the base is\nirrelevant, because changing base only multiplies by a constant:",[413,14879,14881],{"className":14880},[1263],[413,14882,14884],{"className":14883},[416],[413,14885,14887,14955,15128,15214],{"className":14886,"ariaHidden":421},[420],[413,14888,14890,14894,14940,14943,14946,14949,14952],{"className":14889},[425],[413,14891],{"className":14892,"style":14893},[429],"height:0.9386em;vertical-align:-0.2441em;",[413,14895,14897,14903],{"className":14896},[532],[413,14898,14900],{"className":14899},[532],[413,14901,538],{"className":14902,"style":537},[434,536],[413,14904,14906],{"className":14905},[904],[413,14907,14909,14931],{"className":14908},[908,909],[413,14910,14912,14928],{"className":14911},[913],[413,14913,14916],{"className":14914,"style":14915},[917],"height:0.242em;",[413,14917,14919,14922],{"style":14918},"top:-2.4559em;margin-right:0.05em;",[413,14920],{"className":14921,"style":926},[925],[413,14923,14925],{"className":14924},[930,931,932,933],[413,14926,14441],{"className":14927},[434,521,933],[413,14929,941],{"className":14930},[940],[413,14932,14934],{"className":14933},[913],[413,14935,14938],{"className":14936,"style":14937},[917],"height:0.2441em;",[413,14939],{},[413,14941],{"className":14942,"style":543},[542],[413,14944,547],{"className":14945},[434,521],[413,14947],{"className":14948,"style":656},[542],[413,14950,779],{"className":14951},[660],[413,14953],{"className":14954,"style":656},[542],[413,14956,14958,14962,15119,15122,15125],{"className":14957},[425],[413,14959],{"className":14960,"style":14961},[429],"height:2.3016em;vertical-align:-0.9301em;",[413,14963,14965,14968,15116],{"className":14964},[434],[413,14966],{"className":14967},[527,9131],[413,14969,14971],{"className":14970},[9135],[413,14972,14974,15107],{"className":14973},[908,909],[413,14975,14977,15104],{"className":14976},[913],[413,14978,14981,15039,15047],{"className":14979,"style":14980},[917],"height:1.3714em;",[413,14982,14983,14986],{"style":9148},[413,14984],{"className":14985,"style":9152},[925],[413,14987,14989,15033,15036],{"className":14988},[434],[413,14990,14992,14998],{"className":14991},[532],[413,14993,14995],{"className":14994},[532],[413,14996,538],{"className":14997,"style":537},[434,536],[413,14999,15001],{"className":15000},[904],[413,15002,15004,15025],{"className":15003},[908,909],[413,15005,15007,15022],{"className":15006},[913],[413,15008,15011],{"className":15009,"style":15010},[917],"height:0.207em;",[413,15012,15013,15016],{"style":14918},[413,15014],{"className":15015,"style":926},[925],[413,15017,15019],{"className":15018},[930,931,932,933],[413,15020,597],{"className":15021},[434,933],[413,15023,941],{"className":15024},[940],[413,15026,15028],{"className":15027},[913],[413,15029,15031],{"className":15030,"style":14937},[917],[413,15032],{},[413,15034],{"className":15035,"style":543},[542],[413,15037,14441],{"className":15038},[434,521],[413,15040,15041,15044],{"style":9161},[413,15042],{"className":15043,"style":9152},[925],[413,15045],{"className":15046,"style":9169},[9168],[413,15048,15049,15052],{"style":9172},[413,15050],{"className":15051,"style":9152},[925],[413,15053,15055,15098,15101],{"className":15054},[434],[413,15056,15058,15064],{"className":15057},[532],[413,15059,15061],{"className":15060},[532],[413,15062,538],{"className":15063,"style":537},[434,536],[413,15065,15067],{"className":15066},[904],[413,15068,15070,15090],{"className":15069},[908,909],[413,15071,15073,15087],{"className":15072},[913],[413,15074,15076],{"className":15075,"style":15010},[917],[413,15077,15078,15081],{"style":14918},[413,15079],{"className":15080,"style":926},[925],[413,15082,15084],{"className":15083},[930,931,932,933],[413,15085,597],{"className":15086},[434,933],[413,15088,941],{"className":15089},[940],[413,15091,15093],{"className":15092},[913],[413,15094,15096],{"className":15095,"style":14937},[917],[413,15097],{},[413,15099],{"className":15100,"style":543},[542],[413,15102,547],{"className":15103},[434,521],[413,15105,941],{"className":15106},[940],[413,15108,15110],{"className":15109},[913],[413,15111,15114],{"className":15112,"style":15113},[917],"height:0.9301em;",[413,15115],{},[413,15117],{"className":15118},[551,9131],[413,15120],{"className":15121,"style":656},[542],[413,15123,779],{"className":15124},[660],[413,15126],{"className":15127,"style":656},[542],[413,15129,15131,15134,15137,15140,15183,15186,15189,15192,15195,15202,15205,15208,15211],{"className":15130},[425],[413,15132],{"className":15133,"style":481},[429],[413,15135,1684],{"className":15136},[434],[413,15138,528],{"className":15139},[527],[413,15141,15143,15149],{"className":15142},[532],[413,15144,15146],{"className":15145},[532],[413,15147,538],{"className":15148,"style":537},[434,536],[413,15150,15152],{"className":15151},[904],[413,15153,15155,15175],{"className":15154},[908,909],[413,15156,15158,15172],{"className":15157},[913],[413,15159,15161],{"className":15160,"style":15010},[917],[413,15162,15163,15166],{"style":14918},[413,15164],{"className":15165,"style":926},[925],[413,15167,15169],{"className":15168},[930,931,932,933],[413,15170,597],{"className":15171},[434,933],[413,15173,941],{"className":15174},[940],[413,15176,15178],{"className":15177},[913],[413,15179,15181],{"className":15180,"style":14937},[917],[413,15182],{},[413,15184],{"className":15185,"style":543},[542],[413,15187,547],{"className":15188},[434,521],[413,15190,552],{"className":15191},[551],[413,15193],{"className":15194,"style":5801},[542],[413,15196,15198],{"className":15197},[434,802],[413,15199,15201],{"className":15200},[434],"for every constant ",[413,15203,14441],{"className":15204},[434,521],[413,15206],{"className":15207,"style":656},[542],[413,15209,3051],{"className":15210},[660],[413,15212],{"className":15213,"style":656},[542],[413,15215,15217,15220],{"className":15216},[425],[413,15218],{"className":15219,"style":2384},[429],[413,15221,15223],{"className":15222},[434],"1.",[381,15225,12568,15226,436,15287,15351,15352,15377,15378,15393,15394,15427,15428,15496,15497,1597,15512,15527],{},[413,15227,15229],{"className":15228},[416],[413,15230,15232],{"className":15231,"ariaHidden":421},[420],[413,15233,15235,15238,15281,15284],{"className":15234},[425],[413,15236],{"className":15237,"style":14893},[429],[413,15239,15241,15247],{"className":15240},[532],[413,15242,15244],{"className":15243},[532],[413,15245,538],{"className":15246,"style":537},[434,536],[413,15248,15250],{"className":15249},[904],[413,15251,15253,15273],{"className":15252},[908,909],[413,15254,15256,15270],{"className":15255},[913],[413,15257,15259],{"className":15258,"style":15010},[917],[413,15260,15261,15264],{"style":14918},[413,15262],{"className":15263,"style":926},[925],[413,15265,15267],{"className":15266},[930,931,932,933],[413,15268,597],{"className":15269},[434,933],[413,15271,941],{"className":15272},[940],[413,15274,15276],{"className":15275},[913],[413,15277,15279],{"className":15278,"style":14937},[917],[413,15280],{},[413,15282],{"className":15283,"style":543},[542],[413,15285,547],{"className":15286},[434,521],[413,15288,15290],{"className":15289},[416],[413,15291,15293],{"className":15292,"ariaHidden":421},[420],[413,15294,15296,15299,15345,15348],{"className":15295},[425],[413,15297],{"className":15298,"style":14893},[429],[413,15300,15302,15308],{"className":15301},[532],[413,15303,15305],{"className":15304},[532],[413,15306,538],{"className":15307,"style":537},[434,536],[413,15309,15311],{"className":15310},[904],[413,15312,15314,15337],{"className":15313},[908,909],[413,15315,15317,15334],{"className":15316},[913],[413,15318,15320],{"className":15319,"style":15010},[917],[413,15321,15322,15325],{"style":14918},[413,15323],{"className":15324,"style":926},[925],[413,15326,15328],{"className":15327},[930,931,932,933],[413,15329,15331],{"className":15330},[434,933],[413,15332,1911],{"className":15333},[434,933],[413,15335,941],{"className":15336},[940],[413,15338,15340],{"className":15339},[913],[413,15341,15343],{"className":15342,"style":14937},[917],[413,15344],{},[413,15346],{"className":15347,"style":543},[542],[413,15349,547],{"className":15350},[434,521],", and ",[413,15353,15355],{"className":15354},[416],[413,15356,15358],{"className":15357,"ariaHidden":421},[420],[413,15359,15361,15364,15371,15374],{"className":15360},[425],[413,15362],{"className":15363,"style":11046},[429],[413,15365,15367],{"className":15366},[532],[413,15368,15370],{"className":15369},[434,536],"ln",[413,15372],{"className":15373,"style":543},[542],[413,15375,547],{"className":15376},[434,521]," all live in the same ",[413,15379,15381],{"className":15380},[416],[413,15382,15384],{"className":15383,"ariaHidden":421},[420],[413,15385,15387,15390],{"className":15386},[425],[413,15388],{"className":15389,"style":648},[429],[413,15391,1684],{"className":15392},[434],"-class, and\n",[413,15395,15397],{"className":15396},[416],[413,15398,15400],{"className":15399,"ariaHidden":421},[420],[413,15401,15403,15406,15409,15412,15418,15421,15424],{"className":15402},[425],[413,15404],{"className":15405,"style":481},[429],[413,15407,523],{"className":15408,"style":522},[434,521],[413,15410,528],{"className":15411},[527],[413,15413,15415],{"className":15414},[532],[413,15416,538],{"className":15417,"style":537},[434,536],[413,15419],{"className":15420,"style":543},[542],[413,15422,547],{"className":15423},[434,521],[413,15425,552],{"className":15426},[551]," means the same thing whichever base you had in mind. (The constant\n",[413,15429,15431],{"className":15430},[416],[413,15432,15434],{"className":15433,"ariaHidden":421},[420],[413,15435,15437,15440,15444,15447,15490,15493],{"className":15436},[425],[413,15438],{"className":15439,"style":481},[429],[413,15441,15443],{"className":15442},[434],"1\u002F",[413,15445],{"className":15446,"style":543},[542],[413,15448,15450,15456],{"className":15449},[532],[413,15451,15453],{"className":15452},[532],[413,15454,538],{"className":15455,"style":537},[434,536],[413,15457,15459],{"className":15458},[904],[413,15460,15462,15482],{"className":15461},[908,909],[413,15463,15465,15479],{"className":15464},[913],[413,15466,15468],{"className":15467,"style":15010},[917],[413,15469,15470,15473],{"style":14918},[413,15471],{"className":15472,"style":926},[925],[413,15474,15476],{"className":15475},[930,931,932,933],[413,15477,597],{"className":15478},[434,933],[413,15480,941],{"className":15481},[940],[413,15483,15485],{"className":15484},[913],[413,15486,15488],{"className":15487,"style":14937},[917],[413,15489],{},[413,15491],{"className":15492,"style":543},[542],[413,15494,14441],{"className":15495},[434,521]," is exactly the multiplicative factor ",[413,15498,15500],{"className":15499},[416],[413,15501,15503],{"className":15502,"ariaHidden":421},[420],[413,15504,15506,15509],{"className":15505},[425],[413,15507],{"className":15508,"style":648},[429],[413,15510,523],{"className":15511,"style":522},[434,521],[413,15513,15515],{"className":15514},[416],[413,15516,15518],{"className":15517,"ariaHidden":421},[420],[413,15519,15521,15524],{"className":15520},[425],[413,15522],{"className":15523,"style":648},[429],[413,15525,4910],{"className":15526},[434]," are designed to\nabsorb.)",[390,15529,15531],{"id":15530},"the-growth-hierarchy","The growth hierarchy",[381,15533,15534],{},"The functions algorithm designers meet most often, in strictly increasing order\nof growth, are:",[15536,15537,15538,15555],"table",{},[15539,15540,15541],"thead",{},[15542,15543,15544,15549,15552],"tr",{},[15545,15546,15548],"th",{"align":15547},"left","Class",[15545,15550,15551],{"align":15547},"Name",[15545,15553,15554],{"align":15547},"A typical algorithm",[15556,15557,15558,15592,15635,15669,15718,15778,15838,15898],"tbody",{},[15542,15559,15560,15587,15589],{},[15561,15562,15563],"td",{"align":15547},[413,15564,15566],{"className":15565},[416],[413,15567,15569],{"className":15568,"ariaHidden":421},[420],[413,15570,15572,15575,15578,15581,15584],{"className":15571},[425],[413,15573],{"className":15574,"style":481},[429],[413,15576,1684],{"className":15577},[434],[413,15579,528],{"className":15580},[527],[413,15582,588],{"className":15583},[434],[413,15585,552],{"className":15586},[551],[15561,15588,489],{"align":15547},[15561,15590,15591],{"align":15547},"array index, hash lookup (expected)",[15542,15593,15594,15629,15632],{},[15561,15595,15596],{"align":15547},[413,15597,15599],{"className":15598},[416],[413,15600,15602],{"className":15601,"ariaHidden":421},[420],[413,15603,15605,15608,15611,15614,15620,15623,15626],{"className":15604},[425],[413,15606],{"className":15607,"style":481},[429],[413,15609,1684],{"className":15610},[434],[413,15612,528],{"className":15613},[527],[413,15615,15617],{"className":15616},[532],[413,15618,538],{"className":15619,"style":537},[434,536],[413,15621],{"className":15622,"style":543},[542],[413,15624,547],{"className":15625},[434,521],[413,15627,552],{"className":15628},[551],[15561,15630,15631],{"align":15547},"logarithmic",[15561,15633,15634],{"align":15547},"binary search",[15542,15636,15637,15663,15666],{},[15561,15638,15639],{"align":15547},[413,15640,15642],{"className":15641},[416],[413,15643,15645],{"className":15644,"ariaHidden":421},[420],[413,15646,15648,15651,15654,15657,15660],{"className":15647},[425],[413,15649],{"className":15650,"style":481},[429],[413,15652,1684],{"className":15653},[434],[413,15655,528],{"className":15656},[527],[413,15658,547],{"className":15659},[434,521],[413,15661,552],{"className":15662},[551],[15561,15664,15665],{"align":15547},"linear",[15561,15667,15668],{"align":15547},"scan \u002F find max",[15542,15670,15671,15712,15715],{},[15561,15672,15673],{"align":15547},[413,15674,15676],{"className":15675},[416],[413,15677,15679],{"className":15678,"ariaHidden":421},[420],[413,15680,15682,15685,15688,15691,15694,15697,15703,15706,15709],{"className":15681},[425],[413,15683],{"className":15684,"style":481},[429],[413,15686,1684],{"className":15687},[434],[413,15689,528],{"className":15690},[527],[413,15692,547],{"className":15693},[434,521],[413,15695],{"className":15696,"style":543},[542],[413,15698,15700],{"className":15699},[532],[413,15701,538],{"className":15702,"style":537},[434,536],[413,15704],{"className":15705,"style":543},[542],[413,15707,547],{"className":15708},[434,521],[413,15710,552],{"className":15711},[551],[15561,15713,15714],{"align":15547},"linearithmic",[15561,15716,15717],{"align":15547},"merge sort, heapsort",[15542,15719,15720,15772,15775],{},[15561,15721,15722],{"align":15547},[413,15723,15725],{"className":15724},[416],[413,15726,15728],{"className":15727,"ariaHidden":421},[420],[413,15729,15731,15734,15737,15740,15769],{"className":15730},[425],[413,15732],{"className":15733,"style":1707},[429],[413,15735,1684],{"className":15736},[434],[413,15738,528],{"className":15739},[527],[413,15741,15743,15746],{"className":15742},[434],[413,15744,547],{"className":15745},[434,521],[413,15747,15749],{"className":15748},[904],[413,15750,15752],{"className":15751},[908],[413,15753,15755],{"className":15754},[913],[413,15756,15758],{"className":15757,"style":1732},[917],[413,15759,15760,15763],{"style":1735},[413,15761],{"className":15762,"style":926},[925],[413,15764,15766],{"className":15765},[930,931,932,933],[413,15767,597],{"className":15768},[434,933],[413,15770,552],{"className":15771},[551],[15561,15773,15774],{"align":15547},"quadratic",[15561,15776,15777],{"align":15547},"insertion sort (worst case)",[15542,15779,15780,15832,15835],{},[15561,15781,15782],{"align":15547},[413,15783,15785],{"className":15784},[416],[413,15786,15788],{"className":15787,"ariaHidden":421},[420],[413,15789,15791,15794,15797,15800,15829],{"className":15790},[425],[413,15792],{"className":15793,"style":1707},[429],[413,15795,1684],{"className":15796},[434],[413,15798,528],{"className":15799},[527],[413,15801,15803,15806],{"className":15802},[434],[413,15804,547],{"className":15805},[434,521],[413,15807,15809],{"className":15808},[904],[413,15810,15812],{"className":15811},[908],[413,15813,15815],{"className":15814},[913],[413,15816,15818],{"className":15817,"style":1732},[917],[413,15819,15820,15823],{"style":1735},[413,15821],{"className":15822,"style":926},[925],[413,15824,15826],{"className":15825},[930,931,932,933],[413,15827,2313],{"className":15828},[434,933],[413,15830,552],{"className":15831},[551],[15561,15833,15834],{"align":15547},"cubic",[15561,15836,15837],{"align":15547},"naive matrix multiply",[15542,15839,15840,15892,15895],{},[15561,15841,15842],{"align":15547},[413,15843,15845],{"className":15844},[416],[413,15846,15848],{"className":15847,"ariaHidden":421},[420],[413,15849,15851,15854,15857,15860,15889],{"className":15850},[425],[413,15852],{"className":15853,"style":481},[429],[413,15855,1684],{"className":15856},[434],[413,15858,528],{"className":15859},[527],[413,15861,15863,15866],{"className":15862},[434],[413,15864,597],{"className":15865},[434],[413,15867,15869],{"className":15868},[904],[413,15870,15872],{"className":15871},[908],[413,15873,15875],{"className":15874},[913],[413,15876,15878],{"className":15877,"style":14364},[917],[413,15879,15880,15883],{"style":1735},[413,15881],{"className":15882,"style":926},[925],[413,15884,15886],{"className":15885},[930,931,932,933],[413,15887,547],{"className":15888},[434,521,933],[413,15890,552],{"className":15891},[551],[15561,15893,15894],{"align":15547},"exponential",[15561,15896,15897],{"align":15547},"subset enumeration",[15542,15899,15900,15926,15929],{},[15561,15901,15902],{"align":15547},[413,15903,15905],{"className":15904},[416],[413,15906,15908],{"className":15907,"ariaHidden":421},[420],[413,15909,15911,15914,15917,15920,15923],{"className":15910},[425],[413,15912],{"className":15913,"style":481},[429],[413,15915,1684],{"className":15916},[434],[413,15918,528],{"className":15919},[527],[413,15921,547],{"className":15922},[434,521],[413,15924,14746],{"className":15925},[551],[15561,15927,15928],{"align":15547},"factorial",[15561,15930,15931],{"align":15547},"brute-force permutations (TSP)",[381,15933,15934,15935,15950,15951,15954,15955,15996,15997,16012,16013,1597,16016,16018,16019,16027,16028,16062,16063,1158],{},"Each row's growth dwarfs every row above it for large 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The practical\ndividing line Skiena draws is between ",[398,15952,15953],{},"polynomial"," time (",[413,15956,15958],{"className":15957},[416],[413,15959,15961],{"className":15960,"ariaHidden":421},[420],[413,15962,15964,15967],{"className":15963},[425],[413,15965],{"className":15966,"style":14364},[429],[413,15968,15970,15973],{"className":15969},[434],[413,15971,547],{"className":15972},[434,521],[413,15974,15976],{"className":15975},[904],[413,15977,15979],{"className":15978},[908],[413,15980,15982],{"className":15981},[913],[413,15983,15985],{"className":15984,"style":14364},[917],[413,15986,15987,15990],{"style":1735},[413,15988],{"className":15989,"style":926},[925],[413,15991,15993],{"className":15992},[930,931,932,933],[413,15994,3044],{"className":15995},[434,521,933]," for a constant\n",[413,15998,16000],{"className":15999},[416],[413,16001,16003],{"className":16002,"ariaHidden":421},[420],[413,16004,16006,16009],{"className":16005},[425],[413,16007],{"className":16008,"style":566},[429],[413,16010,3044],{"className":16011},[434,521],"), generally considered ",[502,16014,16015],{},"tractable,",[398,16017,15894],{}," time, which becomes\nhopeless very fast.",[579,16020,16021],{},[582,16022,16026],{"href":16023,"ariaDescribedBy":16024,"dataFootnoteRef":376,"id":16025},"#user-content-fn-skiena-growth",[586],"user-content-fnref-skiena-growth","5"," A factorial-time algorithm that handles ",[413,16029,16031],{"className":16030},[416],[413,16032,16034,16052],{"className":16033,"ariaHidden":421},[420],[413,16035,16037,16040,16043,16046,16049],{"className":16036},[425],[413,16038],{"className":16039,"style":566},[429],[413,16041,547],{"className":16042},[434,521],[413,16044],{"className":16045,"style":656},[542],[413,16047,779],{"className":16048},[660],[413,16050],{"className":16051,"style":656},[542],[413,16053,16055,16058],{"className":16054},[425],[413,16056],{"className":16057,"style":2384},[429],[413,16059,16061],{"className":16060},[434],"12"," in\na second needs years at ",[413,16064,16066],{"className":16065},[416],[413,16067,16069,16087],{"className":16068,"ariaHidden":421},[420],[413,16070,16072,16075,16078,16081,16084],{"className":16071},[425],[413,16073],{"className":16074,"style":566},[429],[413,16076,547],{"className":16077},[434,521],[413,16079],{"className":16080,"style":656},[542],[413,16082,779],{"className":16083},[660],[413,16085],{"className":16086,"style":656},[542],[413,16088,16090,16093],{"className":16089},[425],[413,16091],{"className":16092,"style":2384},[429],[413,16094,16096],{"className":16095},[434],"18",[381,16098,16099,16100,661],{},"A sketch of the curves makes the separation clear. Even with a generous\nconstant on the slower-growing function, the faster one wins past some 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124.619",[1901],[2117,16280,16282,16283,436,16307,436,16322,15351,16363,1158],{"className":16281},[2120],"Growth-rate curves for ",[413,16284,16286],{"className":16285},[416],[413,16287,16289],{"className":16288,"ariaHidden":421},[420],[413,16290,16292,16295,16301,16304],{"className":16291},[425],[413,16293],{"className":16294,"style":2787},[429],[413,16296,16298],{"className":16297},[532],[413,16299,538],{"className":16300,"style":537},[434,536],[413,16302],{"className":16303,"style":543},[542],[413,16305,547],{"className":16306},[434,521],[413,16308,16310],{"className":16309},[416],[413,16311,16313],{"className":16312,"ariaHidden":421},[420],[413,16314,16316,16319],{"className":16315},[425],[413,16317],{"className":16318,"style":566},[429],[413,16320,547],{"className":16321},[434,521],[413,16323,16325],{"className":16324},[416],[413,16326,16328],{"className":16327,"ariaHidden":421},[420],[413,16329,16331,16334],{"className":16330},[425],[413,16332],{"className":16333,"style":1732},[429],[413,16335,16337,16340],{"className":16336},[434],[413,16338,547],{"className":16339},[434,521],[413,16341,16343],{"className":16342},[904],[413,16344,16346],{"className":16345},[908],[413,16347,16349],{"className":16348},[913],[413,16350,16352],{"className":16351,"style":1732},[917],[413,16353,16354,16357],{"style":1735},[413,16355],{"className":16356,"style":926},[925],[413,16358,16360],{"className":16359},[930,931,932,933],[413,16361,597],{"className":16362},[434,933],[413,16364,16366],{"className":16365},[416],[413,16367,16369],{"className":16368,"ariaHidden":421},[420],[413,16370,16372,16375],{"className":16371},[425],[413,16373],{"className":16374,"style":14364},[429],[413,16376,16378,16381],{"className":16377},[434],[413,16379,597],{"className":16380},[434],[413,16382,16384],{"className":16383},[904],[413,16385,16387],{"className":16386},[908],[413,16388,16390],{"className":16389},[913],[413,16391,16393],{"className":16392,"style":14364},[917],[413,16394,16395,16398],{"style":1735},[413,16396],{"className":16397,"style":926},[925],[413,16399,16401],{"className":16400},[930,931,932,933],[413,16402,547],{"className":16403},[434,521,933],[390,16405,16407],{"id":16406},"reading-the-cost-off-loops","Reading the cost off loops",[381,16409,16410],{},"Most analysis reduces to counting how many times each line runs. A few rules\ncover the common cases.",[381,16412,16413,16416,16417,16432,16433,16457,16458,16475,16476,16526,16527,16657],{},[398,16414,16415],{},"Sequential blocks add; we keep the max."," If block ",[413,16418,16420],{"className":16419},[416],[413,16421,16423],{"className":16422,"ariaHidden":421},[420],[413,16424,16426,16429],{"className":16425},[425],[413,16427],{"className":16428,"style":648},[429],[413,16430,1070],{"className":16431},[434,521]," costs ",[413,16434,16436],{"className":16435},[416],[413,16437,16439],{"className":16438,"ariaHidden":421},[420],[413,16440,16442,16445,16448,16451,16454],{"className":16441},[425],[413,16443],{"className":16444,"style":481},[429],[413,16446,1684],{"className":16447},[434],[413,16449,528],{"className":16450},[527],[413,16452,547],{"className":16453},[434,521],[413,16455,552],{"className":16456},[551]," and is\nfollowed by block ",[413,16459,16461],{"className":16460},[416],[413,16462,16464],{"className":16463,"ariaHidden":421},[420],[413,16465,16467,16470],{"className":16466},[425],[413,16468],{"className":16469,"style":648},[429],[413,16471,16474],{"className":16472,"style":16473},[434,521],"margin-right:0.0502em;","B"," costing ",[413,16477,16479],{"className":16478},[416],[413,16480,16482],{"className":16481,"ariaHidden":421},[420],[413,16483,16485,16488,16491,16494,16523],{"className":16484},[425],[413,16486],{"className":16487,"style":1707},[429],[413,16489,1684],{"className":16490},[434],[413,16492,528],{"className":16493},[527],[413,16495,16497,16500],{"className":16496},[434],[413,16498,547],{"className":16499},[434,521],[413,16501,16503],{"className":16502},[904],[413,16504,16506],{"className":16505},[908],[413,16507,16509],{"className":16508},[913],[413,16510,16512],{"className":16511,"style":1732},[917],[413,16513,16514,16517],{"style":1735},[413,16515],{"className":16516,"style":926},[925],[413,16518,16520],{"className":16519},[930,931,932,933],[413,16521,597],{"className":16522},[434,933],[413,16524,552],{"className":16525},[551],", the total is\n",[413,16528,16530],{"className":16529},[416],[413,16531,16533,16560,16613],{"className":16532,"ariaHidden":421},[420],[413,16534,16536,16539,16542,16545,16548,16551,16554,16557],{"className":16535},[425],[413,16537],{"className":16538,"style":481},[429],[413,16540,1684],{"className":16541},[434],[413,16543,528],{"className":16544},[527],[413,16546,547],{"className":16547},[434,521],[413,16549,552],{"className":16550},[551],[413,16552],{"className":16553,"style":1595},[542],[413,16555,435],{"className":16556},[2351],[413,16558],{"className":16559,"style":1595},[542],[413,16561,16563,16566,16569,16572,16601,16604,16607,16610],{"className":16562},[425],[413,16564],{"className":16565,"style":1707},[429],[413,16567,1684],{"className":16568},[434],[413,16570,528],{"className":16571},[527],[413,16573,16575,16578],{"className":16574},[434],[413,16576,547],{"className":16577},[434,521],[413,16579,16581],{"className":16580},[904],[413,16582,16584],{"className":16583},[908],[413,16585,16587],{"className":16586},[913],[413,16588,16590],{"className":16589,"style":1732},[917],[413,16591,16592,16595],{"style":1735},[413,16593],{"className":16594,"style":926},[925],[413,16596,16598],{"className":16597},[930,931,932,933],[413,16599,597],{"className":16600},[434,933],[413,16602,552],{"className":16603},[551],[413,16605],{"className":16606,"style":656},[542],[413,16608,779],{"className":16609},[660],[413,16611],{"className":16612,"style":656},[542],[413,16614,16616,16619,16622,16625,16654],{"className":16615},[425],[413,16617],{"className":16618,"style":1707},[429],[413,16620,1684],{"className":16621},[434],[413,16623,528],{"className":16624},[527],[413,16626,16628,16631],{"className":16627},[434],[413,16629,547],{"className":16630},[434,521],[413,16632,16634],{"className":16633},[904],[413,16635,16637],{"className":16636},[908],[413,16638,16640],{"className":16639},[913],[413,16641,16643],{"className":16642,"style":1732},[917],[413,16644,16645,16648],{"style":1735},[413,16646],{"className":16647,"style":926},[925],[413,16649,16651],{"className":16650},[930,931,932,933],[413,16652,597],{"className":16653},[434,933],[413,16655,552],{"className":16656},[551],": the larger term absorbs the smaller.",[381,16659,16660,16663,16664,16697],{},[398,16661,16662],{},"A simple loop multiplies the body by the iteration count."," This nested\nfragment runs the constant-time body ",[413,16665,16667],{"className":16666},[416],[413,16668,16670,16688],{"className":16669,"ariaHidden":421},[420],[413,16671,16673,16676,16679,16682,16685],{"className":16672},[425],[413,16674],{"className":16675,"style":3923},[429],[413,16677,547],{"className":16678},[434,521],[413,16680],{"className":16681,"style":1595},[542],[413,16683,3933],{"className":16684},[2351],[413,16686],{"className":16687,"style":1595},[542],[413,16689,16691,16694],{"className":16690},[425],[413,16692],{"className":16693,"style":566},[429],[413,16695,547],{"className":16696},[434,521]," times:",[16699,16700,16704],"pre",{"className":16701,"code":16702,"language":16703,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: Counting work in nested loops\nfor $i \\gets 1$ to $n$ do\n  for $j \\gets 1$ to $n$ do\n    $c \\gets c + A[i] \\cdot B[j]$ \u002F\u002F $\\Theta(1)$ body, $n^2$ times\n","algorithm",[16705,16706,16707,16713,16718,16723],"code",{"__ignoreMap":376},[413,16708,16710],{"class":16709,"line":6},"line",[413,16711,16712],{},"caption: Counting work in nested loops\n",[413,16714,16715],{"class":16709,"line":18},[413,16716,16717],{},"for $i \\gets 1$ to $n$ do\n",[413,16719,16720],{"class":16709,"line":24},[413,16721,16722],{},"  for $j \\gets 1$ to $n$ do\n",[413,16724,16725],{"class":16709,"line":73},[413,16726,16727],{},"    $c \\gets c + A[i] \\cdot B[j]$ \u002F\u002F $\\Theta(1)$ body, $n^2$ times\n",[381,16729,16730,16731,1158],{},"so its cost is ",[413,16732,16734],{"className":16733},[416],[413,16735,16737],{"className":16736,"ariaHidden":421},[420],[413,16738,16740,16743,16746,16749,16778],{"className":16739},[425],[413,16741],{"className":16742,"style":1707},[429],[413,16744,1684],{"className":16745},[434],[413,16747,528],{"className":16748},[527],[413,16750,16752,16755],{"className":16751},[434],[413,16753,547],{"className":16754},[434,521],[413,16756,16758],{"className":16757},[904],[413,16759,16761],{"className":16760},[908],[413,16762,16764],{"className":16763},[913],[413,16765,16767],{"className":16766,"style":1732},[917],[413,16768,16769,16772],{"style":1735},[413,16770],{"className":16771,"style":926},[925],[413,16773,16775],{"className":16774},[930,931,932,933],[413,16776,597],{"className":16777},[434,933],[413,16779,552],{"className":16780},[551],[381,16782,16783,16786,16787,16849,16850,17092,17093,17175],{},[398,16784,16785],{},"When the inner bound depends on the outer index, sum a series."," If the inner\nloop runs ",[413,16788,16790],{"className":16789},[416],[413,16791,16793,16823],{"className":16792,"ariaHidden":421},[420],[413,16794,16796,16799,16807,16810,16813,16816,16820],{"className":16795},[425],[413,16797],{"className":16798,"style":2787},[429],[413,16800,16802],{"className":16801},[434,802],[413,16803,16806],{"className":16804},[434,16805],"textbf","for",[413,16808,6070],{"className":16809},[542],[413,16811,9012],{"className":16812,"style":9011},[434,521],[413,16814],{"className":16815,"style":656},[542],[413,16817,16819],{"className":16818},[660],"←",[413,16821],{"className":16822,"style":656},[542],[413,16824,16826,16830,16833,16836,16843,16846],{"className":16825},[425],[413,16827],{"className":16828,"style":16829},[429],"height:0.6595em;",[413,16831,588],{"className":16832},[434],[413,16834,6070],{"className":16835},[542],[413,16837,16839],{"className":16838},[434,802],[413,16840,16842],{"className":16841},[434,16805],"to",[413,16844,6070],{"className":16845},[542],[413,16847,8984],{"className":16848},[434,521],", the body executes\n",[413,16851,16853],{"className":16852},[416],[413,16854,16856,16949,17048],{"className":16855,"ariaHidden":421},[420],[413,16857,16859,16863,16934,16937,16940,16943,16946],{"className":16858},[425],[413,16860],{"className":16861,"style":16862},[429],"height:1.104em;vertical-align:-0.2997em;",[413,16864,16866,16873],{"className":16865},[532],[413,16867,16872],{"className":16868,"style":16871},[532,16869,16870],"op-symbol","small-op","position:relative;top:0em;","∑",[413,16874,16876],{"className":16875},[904],[413,16877,16879,16925],{"className":16878},[908,909],[413,16880,16882,16922],{"className":16881},[913],[413,16883,16886,16907],{"className":16884,"style":16885},[917],"height:0.8043em;",[413,16887,16889,16892],{"style":16888},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[413,16890],{"className":16891,"style":926},[925],[413,16893,16895],{"className":16894},[930,931,932,933],[413,16896,16898,16901,16904],{"className":16897},[434,933],[413,16899,8984],{"className":16900},[434,521,933],[413,16902,779],{"className":16903},[660,933],[413,16905,588],{"className":16906},[434,933],[413,16908,16910,16913],{"style":16909},"top:-3.2029em;margin-right:0.05em;",[413,16911],{"className":16912,"style":926},[925],[413,16914,16916],{"className":16915},[930,931,932,933],[413,16917,16919],{"className":16918},[434,933],[413,16920,547],{"className":16921},[434,521,933],[413,16923,941],{"className":16924},[940],[413,16926,16928],{"className":16927},[913],[413,16929,16932],{"className":16930,"style":16931},[917],"height:0.2997em;",[413,16933],{},[413,16935],{"className":16936,"style":543},[542],[413,16938,8984],{"className":16939},[434,521],[413,16941],{"className":16942,"style":656},[542],[413,16944,779],{"className":16945},[660],[413,16947],{"className":16948,"style":656},[542],[413,16950,16952,16956,17039,17042,17045],{"className":16951},[425],[413,16953],{"className":16954,"style":16955},[429],"height:1.355em;vertical-align:-0.345em;",[413,16957,16959,16962,17036],{"className":16958},[434],[413,16960],{"className":16961},[527,9131],[413,16963,16965],{"className":16964},[9135],[413,16966,16968,17028],{"className":16967},[908,909],[413,16969,16971,17025],{"className":16970},[913],[413,16972,16974,16988,16996],{"className":16973,"style":12476},[917],[413,16975,16976,16979],{"style":10218},[413,16977],{"className":16978,"style":9152},[925],[413,16980,16982],{"className":16981},[930,931,932,933],[413,16983,16985],{"className":16984},[434,933],[413,16986,597],{"className":16987},[434,933],[413,16989,16990,16993],{"style":9161},[413,16991],{"className":16992,"style":9152},[925],[413,16994],{"className":16995,"style":9169},[9168],[413,16997,16998,17001],{"style":12510},[413,16999],{"className":17000,"style":9152},[925],[413,17002,17004],{"className":17003},[930,931,932,933],[413,17005,17007,17010,17013,17016,17019,17022],{"className":17006},[434,933],[413,17008,547],{"className":17009},[434,521,933],[413,17011,528],{"className":17012},[527,933],[413,17014,547],{"className":17015},[434,521,933],[413,17017,435],{"className":17018},[2351,933],[413,17020,588],{"className":17021},[434,933],[413,17023,552],{"className":17024},[551,933],[413,17026,941],{"className":17027},[940],[413,17029,17031],{"className":17030},[913],[413,17032,17034],{"className":17033,"style":10293},[917],[413,17035],{},[413,17037],{"className":17038},[551,9131],[413,17040],{"className":17041,"style":656},[542],[413,17043,779],{"className":17044},[660],[413,17046],{"className":17047,"style":656},[542],[413,17049,17051,17054,17057,17060,17089],{"className":17050},[425],[413,17052],{"className":17053,"style":1707},[429],[413,17055,1684],{"className":17056},[434],[413,17058,528],{"className":17059},[527],[413,17061,17063,17066],{"className":17062},[434],[413,17064,547],{"className":17065},[434,521],[413,17067,17069],{"className":17068},[904],[413,17070,17072],{"className":17071},[908],[413,17073,17075],{"className":17074},[913],[413,17076,17078],{"className":17077,"style":1732},[917],[413,17079,17080,17083],{"style":1735},[413,17081],{"className":17082,"style":926},[925],[413,17084,17086],{"className":17085},[930,931,932,933],[413,17087,597],{"className":17088},[434,933],[413,17090,552],{"className":17091},[551]," times — still quadratic,\nbecause triangular work is half of square work, and the constant ",[413,17094,17096],{"className":17095},[416],[413,17097,17099],{"className":17098,"ariaHidden":421},[420],[413,17100,17102,17106],{"className":17101},[425],[413,17103],{"className":17104,"style":17105},[429],"height:1.1901em;vertical-align:-0.345em;",[413,17107,17109,17112,17172],{"className":17108},[434],[413,17110],{"className":17111},[527,9131],[413,17113,17115],{"className":17114},[9135],[413,17116,17118,17164],{"className":17117},[908,909],[413,17119,17121,17161],{"className":17120},[913],[413,17122,17125,17139,17147],{"className":17123,"style":17124},[917],"height:0.8451em;",[413,17126,17127,17130],{"style":10218},[413,17128],{"className":17129,"style":9152},[925],[413,17131,17133],{"className":17132},[930,931,932,933],[413,17134,17136],{"className":17135},[434,933],[413,17137,597],{"className":17138},[434,933],[413,17140,17141,17144],{"style":9161},[413,17142],{"className":17143,"style":9152},[925],[413,17145],{"className":17146,"style":9169},[9168],[413,17148,17149,17152],{"style":10241},[413,17150],{"className":17151,"style":9152},[925],[413,17153,17155],{"className":17154},[930,931,932,933],[413,17156,17158],{"className":17157},[434,933],[413,17159,588],{"className":17160},[434,933],[413,17162,941],{"className":17163},[940],[413,17165,17167],{"className":17166},[913],[413,17168,17170],{"className":17169,"style":10293},[917],[413,17171],{},[413,17173],{"className":17174},[551,9131],"\nvanishes. This is exactly the shape of insertion sort's worst case.",[381,17177,17178,17181,17182,17234,17235,17268,17269,17330,17331,17346,17347,17384],{},[398,17179,17180],{},"When the loop variable is scaled, take a logarithm."," A loop that does\n",[413,17183,17185],{"className":17184},[416],[413,17186,17188,17206,17225],{"className":17187,"ariaHidden":421},[420],[413,17189,17191,17194,17197,17200,17203],{"className":17190},[425],[413,17192],{"className":17193,"style":16829},[429],[413,17195,8984],{"className":17196},[434,521],[413,17198],{"className":17199,"style":656},[542],[413,17201,16819],{"className":17202},[660],[413,17204],{"className":17205,"style":656},[542],[413,17207,17209,17213,17216,17219,17222],{"className":17208},[425],[413,17210],{"className":17211,"style":17212},[429],"height:0.7429em;vertical-align:-0.0833em;",[413,17214,8984],{"className":17215},[434,521],[413,17217],{"className":17218,"style":1595},[542],[413,17220,468],{"className":17221},[2351],[413,17223],{"className":17224,"style":1595},[542],[413,17226,17228,17231],{"className":17227},[425],[413,17229],{"className":17230,"style":2384},[429],[413,17232,597],{"className":17233},[434]," until ",[413,17236,17238],{"className":17237},[416],[413,17239,17241,17259],{"className":17240,"ariaHidden":421},[420],[413,17242,17244,17247,17250,17253,17256],{"className":17243},[425],[413,17245],{"className":17246,"style":9033},[429],[413,17248,8984],{"className":17249},[434,521],[413,17251],{"className":17252,"style":656},[542],[413,17254,3051],{"className":17255},[660],[413,17257],{"className":17258,"style":656},[542],[413,17260,17262,17265],{"className":17261},[425],[413,17263],{"className":17264,"style":566},[429],[413,17266,547],{"className":17267},[434,521]," runs about ",[413,17270,17272],{"className":17271},[416],[413,17273,17275],{"className":17274,"ariaHidden":421},[420],[413,17276,17278,17281,17324,17327],{"className":17277},[425],[413,17279],{"className":17280,"style":14893},[429],[413,17282,17284,17290],{"className":17283},[532],[413,17285,17287],{"className":17286},[532],[413,17288,538],{"className":17289,"style":537},[434,536],[413,17291,17293],{"className":17292},[904],[413,17294,17296,17316],{"className":17295},[908,909],[413,17297,17299,17313],{"className":17298},[913],[413,17300,17302],{"className":17301,"style":15010},[917],[413,17303,17304,17307],{"style":14918},[413,17305],{"className":17306,"style":926},[925],[413,17308,17310],{"className":17309},[930,931,932,933],[413,17311,597],{"className":17312},[434,933],[413,17314,941],{"className":17315},[940],[413,17317,17319],{"className":17318},[913],[413,17320,17322],{"className":17321,"style":14937},[917],[413,17323],{},[413,17325],{"className":17326,"style":543},[542],[413,17328,547],{"className":17329},[434,521]," times, since ",[413,17332,17334],{"className":17333},[416],[413,17335,17337],{"className":17336,"ariaHidden":421},[420],[413,17338,17340,17343],{"className":17339},[425],[413,17341],{"className":17342,"style":16829},[429],[413,17344,8984],{"className":17345},[434,521],"\ndoubles each pass. This is the source of every logarithm in algorithm analysis:\n",[398,17348,17349,17350,17383],{},"repeatedly halving (or doubling) gives ",[413,17351,17353],{"className":17352},[416],[413,17354,17356],{"className":17355,"ariaHidden":421},[420],[413,17357,17359,17362,17365,17368,17374,17377,17380],{"className":17358},[425],[413,17360],{"className":17361,"style":481},[429],[413,17363,1684],{"className":17364},[434],[413,17366,528],{"className":17367},[527],[413,17369,17371],{"className":17370},[532],[413,17372,538],{"className":17373,"style":537},[434,536],[413,17375],{"className":17376,"style":543},[542],[413,17378,547],{"className":17379},[434,521],[413,17381,552],{"className":17382},[551]," steps."," Binary search\nand balanced-tree depth are the canonical examples.",[381,17386,17387,17388,17391,17392,17433,17434,17449,17450,17500,17501,17562],{},"The three loop shapes read off as ",[385,17389,17390],{},"area"," (or, for doubling, as the number of\nrungs). A square grid of iterations is ",[413,17393,17395],{"className":17394},[416],[413,17396,17398],{"className":17397,"ariaHidden":421},[420],[413,17399,17401,17404],{"className":17400},[425],[413,17402],{"className":17403,"style":1732},[429],[413,17405,17407,17410],{"className":17406},[434],[413,17408,547],{"className":17409},[434,521],[413,17411,17413],{"className":17412},[904],[413,17414,17416],{"className":17415},[908],[413,17417,17419],{"className":17418},[913],[413,17420,17422],{"className":17421,"style":1732},[917],[413,17423,17424,17427],{"style":1735},[413,17425],{"className":17426,"style":926},[925],[413,17428,17430],{"className":17429},[930,931,932,933],[413,17431,597],{"className":17432},[434,933]," work; bounding the inner loop by the\nouter index ",[413,17435,17437],{"className":17436},[416],[413,17438,17440],{"className":17439,"ariaHidden":421},[420],[413,17441,17443,17446],{"className":17442},[425],[413,17444],{"className":17445,"style":16829},[429],[413,17447,8984],{"className":17448},[434,521]," fills only the triangle below the diagonal, half as much but still\n",[413,17451,17453],{"className":17452},[416],[413,17454,17456],{"className":17455,"ariaHidden":421},[420],[413,17457,17459,17462,17465,17468,17497],{"className":17458},[425],[413,17460],{"className":17461,"style":1707},[429],[413,17463,1684],{"className":17464},[434],[413,17466,528],{"className":17467},[527],[413,17469,17471,17474],{"className":17470},[434],[413,17472,547],{"className":17473},[434,521],[413,17475,17477],{"className":17476},[904],[413,17478,17480],{"className":17479},[908],[413,17481,17483],{"className":17482},[913],[413,17484,17486],{"className":17485,"style":1732},[917],[413,17487,17488,17491],{"style":1735},[413,17489],{"className":17490,"style":926},[925],[413,17492,17494],{"className":17493},[930,931,932,933],[413,17495,597],{"className":17496},[434,933],[413,17498,552],{"className":17499},[551],"; and a doubling index touches just the 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powers of two.",[1869,17564,17566,18000],{"className":17565},[1872,1873],[1875,17567,17571],{"xmlns":1877,"width":17568,"height":17569,"viewBox":17570},"462.272","193.844","-75 -75 346.704 145.383",[1882,17572,17573,17576,17581,17583,17586,17589,17596,17599,17602,17609,17654,17674,17719,17722,17725,17728,17731,17734,17740,17743,17746,17752,17791,17826,17871,17874,17877,17883,17886,17894,17902,17910,17917,17924,17931,17938,17955],{"stroke":1884,"style":1885},[1887,17574],{"fill":1939,"stroke":1889,"d":17575},"M-60.548 23.704v-68.286H7.74v68.286ZM7.74-44.582",[1887,17577],{"fill":1889,"stroke":17578,"d":17579,"style":17580},"var(--tk-soft-neutral)","M-60.548 23.704H7.74m-68.287-11.381H7.74M-60.548.943H7.74M-60.548-10.44H7.74m-68.287-11.38H7.74m-68.287-11.382H7.74m-68.287-11.38H7.74m-68.287 68.285v-68.286m11.381 68.286v-68.286m11.381 68.286v-68.286m11.381 68.286v-68.286m11.38 68.286v-68.286m11.382 68.286v-68.286m11.38 68.286v-68.286m.002 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23.528",[1901],[1882,17995,17996],{"transform":17959},[1887,17997],{"d":17998,"fill":1884,"stroke":1884,"className":17999,"style":2041},"M-22.442 25.704L-22.524 25.704Q-22.560 25.704-22.585 25.675Q-22.610 25.645-22.610 25.606Q-22.610 25.556-22.579 25.536Q-22.192 25.200-21.909 24.751Q-21.626 24.302-21.460 23.802Q-21.294 23.302-21.220 22.784Q-21.145 22.267-21.145 21.704Q-21.145 21.134-21.220 20.618Q-21.294 20.102-21.460 19.606Q-21.626 19.110-21.905 18.663Q-22.185 18.216-22.579 17.872Q-22.610 17.852-22.610 17.802Q-22.610 17.763-22.585 17.733Q-22.560 17.704-22.524 17.704L-22.442 17.704Q-22.431 17.704-22.421 17.706Q-22.411 17.708-22.403 17.712Q-21.790 18.169-21.388 18.804Q-20.985 19.438-20.790 20.184Q-20.595 20.931-20.595 21.704Q-20.595 22.477-20.790 23.224Q-20.985 23.970-21.388 24.604Q-21.790 25.239-22.403 25.696Q-22.415 25.696-22.423 25.698Q-22.431 25.700-22.442 25.704",[1901],[2117,18001,18003,18004,18045,18046,18091,18092,18201,18202,18263],{"className":18002},[2120],"Reading loop cost as area. A square grid of iterations is ",[413,18005,18007],{"className":18006},[416],[413,18008,18010],{"className":18009,"ariaHidden":421},[420],[413,18011,18013,18016],{"className":18012},[425],[413,18014],{"className":18015,"style":1732},[429],[413,18017,18019,18022],{"className":18018},[434],[413,18020,547],{"className":18021},[434,521],[413,18023,18025],{"className":18024},[904],[413,18026,18028],{"className":18027},[908],[413,18029,18031],{"className":18030},[913],[413,18032,18034],{"className":18033,"style":1732},[917],[413,18035,18036,18039],{"style":1735},[413,18037],{"className":18038,"style":926},[925],[413,18040,18042],{"className":18041},[930,931,932,933],[413,18043,597],{"className":18044},[434,933]," work; an inner bound ",[413,18047,18049],{"className":18048},[416],[413,18050,18052,18070],{"className":18051,"ariaHidden":421},[420],[413,18053,18055,18058,18061,18064,18067],{"className":18054},[425],[413,18056],{"className":18057,"style":9053},[429],[413,18059,9012],{"className":18060,"style":9011},[434,521],[413,18062],{"className":18063,"style":656},[542],[413,18065,16819],{"className":18066},[660],[413,18068],{"className":18069,"style":656},[542],[413,18071,18073,18076,18079,18082,18085,18088],{"className":18072},[425],[413,18074],{"className":18075,"style":16829},[429],[413,18077,588],{"className":18078},[434],[413,18080],{"className":18081,"style":543},[542],[413,18083,962],{"className":18084},[792],[413,18086],{"className":18087,"style":543},[542],[413,18089,8984],{"className":18090},[434,521]," fills only the triangle, ",[413,18093,18095],{"className":18094},[416],[413,18096,18098],{"className":18097,"ariaHidden":421},[420],[413,18099,18101,18104,18172],{"className":18100},[425],[413,18102],{"className":18103,"style":17105},[429],[413,18105,18107,18110,18169],{"className":18106},[434],[413,18108],{"className":18109},[527,9131],[413,18111,18113],{"className":18112},[9135],[413,18114,18116,18161],{"className":18115},[908,909],[413,18117,18119,18158],{"className":18118},[913],[413,18120,18122,18136,18144],{"className":18121,"style":17124},[917],[413,18123,18124,18127],{"style":10218},[413,18125],{"className":18126,"style":9152},[925],[413,18128,18130],{"className":18129},[930,931,932,933],[413,18131,18133],{"className":18132},[434,933],[413,18134,597],{"className":18135},[434,933],[413,18137,18138,18141],{"style":9161},[413,18139],{"className":18140,"style":9152},[925],[413,18142],{"className":18143,"style":9169},[9168],[413,18145,18146,18149],{"style":10241},[413,18147],{"className":18148,"style":9152},[925],[413,18150,18152],{"className":18151},[930,931,932,933],[413,18153,18155],{"className":18154},[434,933],[413,18156,588],{"className":18157},[434,933],[413,18159,941],{"className":18160},[940],[413,18162,18164],{"className":18163},[913],[413,18165,18167],{"className":18166,"style":10293},[917],[413,18168],{},[413,18170],{"className":18171},[551,9131],[413,18173,18175,18178],{"className":18174},[434],[413,18176,547],{"className":18177},[434,521],[413,18179,18181],{"className":18180},[904],[413,18182,18184],{"className":18183},[908],[413,18185,18187],{"className":18186},[913],[413,18188,18190],{"className":18189,"style":1732},[917],[413,18191,18192,18195],{"style":1735},[413,18193],{"className":18194,"style":926},[925],[413,18196,18198],{"className":18197},[930,931,932,933],[413,18199,597],{"className":18200},[434,933],"; a doubling index visits just ",[413,18203,18205],{"className":18204},[416],[413,18206,18208],{"className":18207,"ariaHidden":421},[420],[413,18209,18211,18214,18257,18260],{"className":18210},[425],[413,18212],{"className":18213,"style":14893},[429],[413,18215,18217,18223],{"className":18216},[532],[413,18218,18220],{"className":18219},[532],[413,18221,538],{"className":18222,"style":537},[434,536],[413,18224,18226],{"className":18225},[904],[413,18227,18229,18249],{"className":18228},[908,909],[413,18230,18232,18246],{"className":18231},[913],[413,18233,18235],{"className":18234,"style":15010},[917],[413,18236,18237,18240],{"style":14918},[413,18238],{"className":18239,"style":926},[925],[413,18241,18243],{"className":18242},[930,931,932,933],[413,18244,597],{"className":18245},[434,933],[413,18247,941],{"className":18248},[940],[413,18250,18252],{"className":18251},[913],[413,18253,18255],{"className":18254,"style":14937},[917],[413,18256],{},[413,18258],{"className":18259,"style":543},[542],[413,18261,547],{"className":18262},[434,521]," rungs.",[381,18265,18266,18267,18282,18283,18286,18287,18348,18349,18391,18392,18431,18432,18435,18436,18657,18658,1158],{},"Two more loop shapes round out the catalog, and both read off as areas just like\nthe three above. An outer loop of ",[413,18268,18270],{"className":18269},[416],[413,18271,18273],{"className":18272,"ariaHidden":421},[420],[413,18274,18276,18279],{"className":18275},[425],[413,18277],{"className":18278,"style":566},[429],[413,18280,547],{"className":18281},[434,521]," passes wrapped around a ",[385,18284,18285],{},"doubling"," inner\nloop does ",[413,18288,18290],{"className":18289},[416],[413,18291,18293],{"className":18292,"ariaHidden":421},[420],[413,18294,18296,18299,18342,18345],{"className":18295},[425],[413,18297],{"className":18298,"style":14893},[429],[413,18300,18302,18308],{"className":18301},[532],[413,18303,18305],{"className":18304},[532],[413,18306,538],{"className":18307,"style":537},[434,536],[413,18309,18311],{"className":18310},[904],[413,18312,18314,18334],{"className":18313},[908,909],[413,18315,18317,18331],{"className":18316},[913],[413,18318,18320],{"className":18319,"style":15010},[917],[413,18321,18322,18325],{"style":14918},[413,18323],{"className":18324,"style":926},[925],[413,18326,18328],{"className":18327},[930,931,932,933],[413,18329,597],{"className":18330},[434,933],[413,18332,941],{"className":18333},[940],[413,18335,18337],{"className":18336},[913],[413,18338,18340],{"className":18339,"style":14937},[917],[413,18341],{},[413,18343],{"className":18344,"style":543},[542],[413,18346,547],{"className":18347},[434,521]," work per pass — an ",[413,18350,18352],{"className":18351},[416],[413,18353,18355,18373],{"className":18354,"ariaHidden":421},[420],[413,18356,18358,18361,18364,18367,18370],{"className":18357},[425],[413,18359],{"className":18360,"style":430},[429],[413,18362,547],{"className":18363},[434,521],[413,18365],{"className":18366,"style":1595},[542],[413,18368,468],{"className":18369},[2351],[413,18371],{"className":18372,"style":1595},[542],[413,18374,18376,18379,18385,18388],{"className":18375},[425],[413,18377],{"className":18378,"style":2787},[429],[413,18380,18382],{"className":18381},[532],[413,18383,538],{"className":18384,"style":537},[434,536],[413,18386],{"className":18387,"style":543},[542],[413,18389,547],{"className":18390},[434,521]," grid, ",[413,18393,18395],{"className":18394},[416],[413,18396,18398],{"className":18397,"ariaHidden":421},[420],[413,18399,18401,18404,18407,18410,18413,18416,18422,18425,18428],{"className":18400},[425],[413,18402],{"className":18403,"style":481},[429],[413,18405,1684],{"className":18406},[434],[413,18408,528],{"className":18409},[527],[413,18411,547],{"className":18412},[434,521],[413,18414],{"className":18415,"style":543},[542],[413,18417,18419],{"className":18418},[532],[413,18420,538],{"className":18421,"style":537},[434,536],[413,18423],{"className":18424,"style":543},[542],[413,18426,547],{"className":18427},[434,521],[413,18429,552],{"className":18430},[551],".\nAnd a loop whose live problem ",[385,18433,18434],{},"halves"," every pass while doing linear work on what\nremains sums a geometric series 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A linear loop wrapping a doubling inner loop fills an ",[413,19007,19009],{"className":19008},[416],[413,19010,19012,19030],{"className":19011,"ariaHidden":421},[420],[413,19013,19015,19018,19021,19024,19027],{"className":19014},[425],[413,19016],{"className":19017,"style":430},[429],[413,19019,547],{"className":19020},[434,521],[413,19022],{"className":19023,"style":1595},[542],[413,19025,468],{"className":19026},[2351],[413,19028],{"className":19029,"style":1595},[542],[413,19031,19033,19036,19042,19045],{"className":19032},[425],[413,19034],{"className":19035,"style":2787},[429],[413,19037,19039],{"className":19038},[532],[413,19040,538],{"className":19041,"style":537},[434,536],[413,19043],{"className":19044,"style":543},[542],[413,19046,547],{"className":19047},[434,521]," grid (",[413,19050,19052],{"className":19051},[416],[413,19053,19055],{"className":19054,"ariaHidden":421},[420],[413,19056,19058,19061,19064,19067,19070,19073,19079,19082,19085],{"className":19057},[425],[413,19059],{"className":19060,"style":481},[429],[413,19062,1684],{"className":19063},[434],[413,19065,528],{"className":19066},[527],[413,19068,547],{"className":19069},[434,521],[413,19071],{"className":19072,"style":543},[542],[413,19074,19076],{"className":19075},[532],[413,19077,538],{"className":19078,"style":537},[434,536],[413,19080],{"className":19081,"style":543},[542],[413,19083,547],{"className":19084},[434,521],[413,19086,552],{"className":19087},[551],"); a loop that halves its live problem each pass stacks bars ",[413,19090,19092],{"className":19091},[416],[413,19093,19095],{"className":19094,"ariaHidden":421},[420],[413,19096,19098,19101,19104,19107,19110,19178,19181,19184,19252,19255,19258],{"className":19097},[425],[413,19099],{"className":19100,"style":10314},[429],[413,19102,547],{"className":19103},[434,521],[413,19105,955],{"className":19106},[954],[413,19108],{"className":19109,"style":543},[542],[413,19111,19113,19116,19175],{"className":19112},[434],[413,19114],{"className":19115},[527,9131],[413,19117,19119],{"className":19118},[9135],[413,19120,19122,19167],{"className":19121},[908,909],[413,19123,19125,19164],{"className":19124},[913],[413,19126,19128,19142,19150],{"className":19127,"style":10333},[917],[413,19129,19130,19133],{"style":10218},[413,19131],{"className":19132,"style":9152},[925],[413,19134,19136],{"className":19135},[930,931,932,933],[413,19137,19139],{"className":19138},[434,933],[413,19140,597],{"className":19141},[434,933],[413,19143,19144,19147],{"style":9161},[413,19145],{"className":19146,"style":9152},[925],[413,19148],{"className":19149,"style":9169},[9168],[413,19151,19152,19155],{"style":10241},[413,19153],{"className":19154,"style":9152},[925],[413,19156,19158],{"className":19157},[930,931,932,933],[413,19159,19161],{"className":19160},[434,933],[413,19162,547],{"className":19163},[434,521,933],[413,19165,941],{"className":19166},[940],[413,19168,19170],{"className":19169},[913],[413,19171,19173],{"className":19172,"style":10293},[917],[413,19174],{},[413,19176],{"className":19177},[551,9131],[413,19179,955],{"className":19180},[954],[413,19182],{"className":19183,"style":543},[542],[413,19185,19187,19190,19249],{"className":19186},[434],[413,19188],{"className":19189},[527,9131],[413,19191,19193],{"className":19192},[9135],[413,19194,19196,19241],{"className":19195},[908,909],[413,19197,19199,19238],{"className":19198},[913],[413,19200,19202,19216,19224],{"className":19201,"style":10333},[917],[413,19203,19204,19207],{"style":10218},[413,19205],{"className":19206,"style":9152},[925],[413,19208,19210],{"className":19209},[930,931,932,933],[413,19211,19213],{"className":19212},[434,933],[413,19214,14801],{"className":19215},[434,933],[413,19217,19218,19221],{"style":9161},[413,19219],{"className":19220,"style":9152},[925],[413,19222],{"className":19223,"style":9169},[9168],[413,19225,19226,19229],{"style":10241},[413,19227],{"className":19228,"style":9152},[925],[413,19230,19232],{"className":19231},[930,931,932,933],[413,19233,19235],{"className":19234},[434,933],[413,19236,547],{"className":19237},[434,521,933],[413,19239,941],{"className":19240},[940],[413,19242,19244],{"className":19243},[913],[413,19245,19247],{"className":19246,"style":10293},[917],[413,19248],{},[413,19250],{"className":19251},[551,9131],[413,19253,955],{"className":19254},[954],[413,19256],{"className":19257,"style":543},[542],[413,19259,962],{"className":19260},[792]," that sum to ",[413,19263,19265],{"className":19264},[416],[413,19266,19268],{"className":19267,"ariaHidden":421},[420],[413,19269,19271,19274,19277],{"className":19270},[425],[413,19272],{"className":19273,"style":2384},[429],[413,19275,597],{"className":19276},[434],[413,19278,547],{"className":19279},[434,521],[413,19281,19283],{"className":19282},[416],[413,19284,19286],{"className":19285,"ariaHidden":421},[420],[413,19287,19289,19292,19295,19298,19301],{"className":19288},[425],[413,19290],{"className":19291,"style":481},[429],[413,19293,1684],{"className":19294},[434],[413,19296,528],{"className":19297},[527],[413,19299,547],{"className":19300},[434,521],[413,19302,552],{"className":19303},[551],[381,19305,4056,19306,19309,19310,19313],{},[582,19307,19308],{"href":23},"recurrences"," that arise when a loop is\nreplaced by ",[385,19311,19312],{},"recursion",", a function calling smaller copies of itself, need their\nown machinery, which is the subject of the next lesson.",[390,19315,19317],{"id":19316},"takeaways","Takeaways",[403,19319,19320,19351,19357,19363,19579,19709],{},[406,19321,4056,19322,19325,19326,19350],{},[398,19323,19324],{},"RAM model"," charges constant time per primitive operation and lets us\nmeasure running time as a function ",[413,19327,19329],{"className":19328},[416],[413,19330,19332],{"className":19331,"ariaHidden":421},[420],[413,19333,19335,19338,19341,19344,19347],{"className":19334},[425],[413,19336],{"className":19337,"style":481},[429],[413,19339,619],{"className":19340,"style":618},[434,521],[413,19342,528],{"className":19343},[527],[413,19345,547],{"className":19346},[434,521],[413,19348,552],{"className":19349},[551]," of input size, machine-independently.",[406,19352,19353,19354,19356],{},"Report the ",[398,19355,2254],{}," by default: it is a guarantee. Best case promises\nnothing; average case is honest but needs a distribution.",[406,19358,19359,19362],{},[398,19360,19361],{},"Drop constants and lower-order terms."," They are machine artifacts and noise;\nthe leading term's growth rate is what scales.",[406,19364,19365,19380,19381,19396,19397,19412,19413,1597,19428,19443,19444,4097,19537,1158],{},[413,19366,19368],{"className":19367},[416],[413,19369,19371],{"className":19370,"ariaHidden":421},[420],[413,19372,19374,19377],{"className":19373},[425],[413,19375],{"className":19376,"style":648},[429],[413,19378,523],{"className":19379,"style":522},[434,521]," is an upper bound, ",[413,19382,19384],{"className":19383},[416],[413,19385,19387],{"className":19386,"ariaHidden":421},[420],[413,19388,19390,19393],{"className":19389},[425],[413,19391],{"className":19392,"style":648},[429],[413,19394,4910],{"className":19395},[434]," a lower bound, ",[413,19398,19400],{"className":19399},[416],[413,19401,19403],{"className":19402,"ariaHidden":421},[420],[413,19404,19406,19409],{"className":19405},[425],[413,19407],{"className":19408,"style":648},[429],[413,19410,1684],{"className":19411},[434]," a tight (two-sided)\nbound; ",[413,19414,19416],{"className":19415},[416],[413,19417,19419],{"className":19418,"ariaHidden":421},[420],[413,19420,19422,19425],{"className":19421},[425],[413,19423],{"className":19424,"style":566},[429],[413,19426,12031],{"className":19427},[434,521],[413,19429,19431],{"className":19430},[416],[413,19432,19434],{"className":19433,"ariaHidden":421},[420],[413,19435,19437,19440],{"className":19436},[425],[413,19438],{"className":19439,"style":566},[429],[413,19441,12809],{"className":19442,"style":2894},[434,521]," are their strict versions. ",[413,19445,19447],{"className":19446},[416],[413,19448,19450,19468,19501,19519],{"className":19449,"ariaHidden":421},[420],[413,19451,19453,19456,19459,19462,19465],{"className":19452},[425],[413,19454],{"className":19455,"style":2787},[429],[413,19457,2877],{"className":19458,"style":2876},[434,521],[413,19460],{"className":19461,"style":656},[542],[413,19463,779],{"className":19464},[660],[413,19466],{"className":19467,"style":656},[542],[413,19469,19471,19474,19477,19480,19483,19486,19489,19492,19495,19498],{"className":19470},[425],[413,19472],{"className":19473,"style":481},[429],[413,19475,1684],{"className":19476},[434],[413,19478,528],{"className":19479},[527],[413,19481,1882],{"className":19482,"style":2894},[434,521],[413,19484,552],{"className":19485},[551],[413,19487],{"className":19488,"style":656},[542],[413,19490],{"className":19491,"style":656},[542],[413,19493,6009],{"className":19494},[660],[413,19496],{"className":19497,"style":656},[542],[413,19499],{"className":19500,"style":656},[542],[413,19502,19504,19507,19510,19513,19516],{"className":19503},[425],[413,19505],{"className":19506,"style":2787},[429],[413,19508,2877],{"className":19509,"style":2876},[434,521],[413,19511],{"className":19512,"style":656},[542],[413,19514,779],{"className":19515},[660],[413,19517],{"className":19518,"style":656},[542],[413,19520,19522,19525,19528,19531,19534],{"className":19521},[425],[413,19523],{"className":19524,"style":481},[429],[413,19526,523],{"className":19527,"style":522},[434,521],[413,19529,528],{"className":19530},[527],[413,19532,1882],{"className":19533,"style":2894},[434,521],[413,19535,552],{"className":19536},[551],[413,19538,19540],{"className":19539},[416],[413,19541,19543,19561],{"className":19542,"ariaHidden":421},[420],[413,19544,19546,19549,19552,19555,19558],{"className":19545},[425],[413,19547],{"className":19548,"style":2787},[429],[413,19550,2877],{"className":19551,"style":2876},[434,521],[413,19553],{"className":19554,"style":656},[542],[413,19556,779],{"className":19557},[660],[413,19559],{"className":19560,"style":656},[542],[413,19562,19564,19567,19570,19573,19576],{"className":19563},[425],[413,19565],{"className":19566,"style":481},[429],[413,19568,4910],{"className":19569},[434],[413,19571,528],{"className":19572},[527],[413,19574,1882],{"className":19575,"style":2894},[434,521],[413,19577,552],{"className":19578},[551],[406,19580,19581,19582,19630,19631,19646,19647,19662,19663,436,19678,19693,19694,1158],{},"The limit ",[413,19583,19585],{"className":19584},[416],[413,19586,19588],{"className":19587,"ariaHidden":421},[420],[413,19589,19591,19594,19600,19603,19606,19609,19612,19615,19618,19621,19624,19627],{"className":19590},[425],[413,19592],{"className":19593,"style":481},[429],[413,19595,19597],{"className":19596},[532],[413,19598,12409],{"className":19599},[434,536],[413,19601],{"className":19602,"style":543},[542],[413,19604,2877],{"className":19605,"style":2876},[434,521],[413,19607,528],{"className":19608},[527],[413,19610,547],{"className":19611},[434,521],[413,19613,552],{"className":19614},[551],[413,19616,485],{"className":19617},[434],[413,19619,1882],{"className":19620,"style":2894},[434,521],[413,19622,528],{"className":19623},[527],[413,19625,547],{"className":19626},[434,521],[413,19628,552],{"className":19629},[551]," ranks two functions: ",[413,19632,19634],{"className":19633},[416],[413,19635,19637],{"className":19636,"ariaHidden":421},[420],[413,19638,19640,19643],{"className":19639},[425],[413,19641],{"className":19642,"style":2384},[429],[413,19644,2183],{"className":19645},[434],", a constant, or ",[413,19648,19650],{"className":19649},[416],[413,19651,19653],{"className":19652,"ariaHidden":421},[420],[413,19654,19656,19659],{"className":19655},[425],[413,19657],{"className":19658,"style":566},[429],[413,19660,12443],{"className":19661},[434],"\ngive ",[413,19664,19666],{"className":19665},[416],[413,19667,19669],{"className":19668,"ariaHidden":421},[420],[413,19670,19672,19675],{"className":19671},[425],[413,19673],{"className":19674,"style":566},[429],[413,19676,12031],{"className":19677},[434,521],[413,19679,19681],{"className":19680},[416],[413,19682,19684],{"className":19683,"ariaHidden":421},[420],[413,19685,19687,19690],{"className":19686},[425],[413,19688],{"className":19689,"style":648},[429],[413,19691,1684],{"className":19692},[434],", or ",[413,19695,19697],{"className":19696},[416],[413,19698,19700],{"className":19699,"ariaHidden":421},[420],[413,19701,19703,19706],{"className":19702},[425],[413,19704],{"className":19705,"style":566},[429],[413,19707,12809],{"className":19708,"style":2894},[434,521],[406,19710,19711,19712,19919],{},"Memorize the hierarchy ",[413,19713,19715],{"className":19714},[416],[413,19716,19718,19738,19765,19783,19816,19861,19906],{"className":19717,"ariaHidden":421},[420],[413,19719,19721,19725,19728,19731,19735],{"className":19720},[425],[413,19722],{"className":19723,"style":19724},[429],"height:0.6835em;vertical-align:-0.0391em;",[413,19726,588],{"className":19727},[434],[413,19729],{"className":19730,"style":656},[542],[413,19732,19734],{"className":19733},[660],"≺",[413,19736],{"className":19737,"style":656},[542],[413,19739,19741,19744,19750,19753,19756,19759,19762],{"className":19740},[425],[413,19742],{"className":19743,"style":2787},[429],[413,19745,19747],{"className":19746},[532],[413,19748,538],{"className":19749,"style":537},[434,536],[413,19751],{"className":19752,"style":543},[542],[413,19754,547],{"className":19755},[434,521],[413,19757],{"className":19758,"style":656},[542],[413,19760,19734],{"className":19761},[660],[413,19763],{"className":19764,"style":656},[542],[413,19766,19768,19771,19774,19777,19780],{"className":19767},[425],[413,19769],{"className":19770,"style":10867},[429],[413,19772,547],{"className":19773},[434,521],[413,19775],{"className":19776,"style":656},[542],[413,19778,19734],{"className":19779},[660],[413,19781],{"className":19782,"style":656},[542],[413,19784,19786,19789,19792,19795,19801,19804,19807,19810,19813],{"className":19785},[425],[413,19787],{"className":19788,"style":2787},[429],[413,19790,547],{"className":19791},[434,521],[413,19793],{"className":19794,"style":543},[542],[413,19796,19798],{"className":19797},[532],[413,19799,538],{"className":19800,"style":537},[434,536],[413,19802],{"className":19803,"style":543},[542],[413,19805,547],{"className":19806},[434,521],[413,19808],{"className":19809,"style":656},[542],[413,19811,19734],{"className":19812},[660],[413,19814],{"className":19815,"style":656},[542],[413,19817,19819,19823,19852,19855,19858],{"className":19818},[425],[413,19820],{"className":19821,"style":19822},[429],"height:0.8532em;vertical-align:-0.0391em;",[413,19824,19826,19829],{"className":19825},[434],[413,19827,547],{"className":19828},[434,521],[413,19830,19832],{"className":19831},[904],[413,19833,19835],{"className":19834},[908],[413,19836,19838],{"className":19837},[913],[413,19839,19841],{"className":19840,"style":1732},[917],[413,19842,19843,19846],{"style":1735},[413,19844],{"className":19845,"style":926},[925],[413,19847,19849],{"className":19848},[930,931,932,933],[413,19850,597],{"className":19851},[434,933],[413,19853],{"className":19854,"style":656},[542],[413,19856,19734],{"className":19857},[660],[413,19859],{"className":19860,"style":656},[542],[413,19862,19864,19868,19897,19900,19903],{"className":19863},[425],[413,19865],{"className":19866,"style":19867},[429],"height:0.7035em;vertical-align:-0.0391em;",[413,19869,19871,19874],{"className":19870},[434],[413,19872,597],{"className":19873},[434],[413,19875,19877],{"className":19876},[904],[413,19878,19880],{"className":19879},[908],[413,19881,19883],{"className":19882},[913],[413,19884,19886],{"className":19885,"style":14364},[917],[413,19887,19888,19891],{"style":1735},[413,19889],{"className":19890,"style":926},[925],[413,19892,19894],{"className":19893},[930,931,932,933],[413,19895,547],{"className":19896},[434,521,933],[413,19898],{"className":19899,"style":656},[542],[413,19901,19734],{"className":19902},[660],[413,19904],{"className":19905,"style":656},[542],[413,19907,19909,19912,19915],{"className":19908},[425],[413,19910],{"className":19911,"style":11046},[429],[413,19913,547],{"className":19914},[434,521],[413,19916,19918],{"className":19917},[551],"!","; the polynomial\u002Fexponential boundary is the line between tractable and\nhopeless.",[19921,19922,19925,19930],"section",{"className":19923,"dataFootnotes":376},[19924],"footnotes",[390,19926,19929],{"className":19927,"id":586},[19928],"sr-only","Footnotes",[2393,19931,19932,19946,19958,19973,20054],{},[406,19933,19935,19938,19939],{"id":19934},"user-content-fn-skiena-ram",[398,19936,19937],{},"Skiena",", §2 — Algorithm Analysis: the RAM model as a machine-independent abstraction whose engineering payoff is predicting real-world speed. ",[582,19940,19945],{"href":19941,"ariaLabel":19942,"className":19943,"dataFootnoteBackref":376},"#user-content-fnref-skiena-ram","Back to reference 1",[19944],"data-footnote-backref","↩",[406,19947,19949,19952,19953],{"id":19948},"user-content-fn-clrs-ram",[398,19950,19951],{},"CLRS",", Ch. 3 — Characterizing Running Times: the RAM model's constant-word assumption and where it breaks down (e.g. bignum arithmetic). ",[582,19954,19945],{"href":19955,"ariaLabel":19956,"className":19957,"dataFootnoteBackref":376},"#user-content-fnref-clrs-ram","Back to reference 2",[19944],[406,19959,19961,436,19964,19967,19968],{"id":19960},"user-content-fn-erickson-growth",[398,19962,19963],{},"Erickson",[385,19965,19966],{},"Algorithms",", Appendix — Solving Recurrences (analysis throughout): ranking growth rates by the limit of the ratio, so every polynomial dominates every polylogarithm. ",[582,19969,19945],{"href":19970,"ariaLabel":19971,"className":19972,"dataFootnoteBackref":376},"#user-content-fnref-erickson-growth","Back to reference 3",[19944],[406,19974,19976,19978,19979,20048,20049],{"id":19975},"user-content-fn-clrs-stirling",[398,19977,19951],{},", Ch. 3 — Characterizing Running Times: ",[413,19980,19982],{"className":19981},[416],[413,19983,19985,20015],{"className":19984,"ariaHidden":421},[420],[413,19986,19988,19991,19997,20000,20003,20006,20009,20012],{"className":19987},[425],[413,19989],{"className":19990,"style":481},[429],[413,19992,19994],{"className":19993},[532],[413,19995,538],{"className":19996,"style":537},[434,536],[413,19998,528],{"className":19999},[527],[413,20001,547],{"className":20002},[434,521],[413,20004,14746],{"className":20005},[551],[413,20007],{"className":20008,"style":656},[542],[413,20010,779],{"className":20011},[660],[413,20013],{"className":20014,"style":656},[542],[413,20016,20018,20021,20024,20027,20030,20033,20039,20042,20045],{"className":20017},[425],[413,20019],{"className":20020,"style":481},[429],[413,20022,1684],{"className":20023},[434],[413,20025,528],{"className":20026},[527],[413,20028,547],{"className":20029},[434,521],[413,20031],{"className":20032,"style":543},[542],[413,20034,20036],{"className":20035},[532],[413,20037,538],{"className":20038,"style":537},[434,536],[413,20040],{"className":20041,"style":543},[542],[413,20043,547],{"className":20044},[434,521],[413,20046,552],{"className":20047},[551]," via Stirling's approximation. ",[582,20050,19945],{"href":20051,"ariaLabel":20052,"className":20053,"dataFootnoteBackref":376},"#user-content-fnref-clrs-stirling","Back to reference 4",[19944],[406,20055,20057,20059,20060],{"id":20056},"user-content-fn-skiena-growth",[398,20058,19937],{},", §2 — Algorithm Analysis: the polynomial-vs-exponential dividing line between tractable and hopeless running times. ",[582,20061,19945],{"href":20062,"ariaLabel":20063,"className":20064,"dataFootnoteBackref":376},"#user-content-fnref-skiena-growth","Back to reference 5",[19944],[20066,20067,20068],"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":20070},[20071,20075,20076,20077,20086,20087,20088,20089,20090],{"id":392,"depth":18,"text":393,"children":20072},[20073],{"id":601,"depth":24,"text":20074},"From problem to T(n)",{"id":1646,"depth":18,"text":1647},{"id":2262,"depth":18,"text":2263},{"id":2856,"depth":18,"text":2857,"children":20078},[20079,20080,20081,20082,20083,20085],{"id":2948,"depth":24,"text":2949},{"id":4883,"depth":24,"text":4884},{"id":5392,"depth":24,"text":5393},{"id":6401,"depth":24,"text":6402},{"id":8938,"depth":24,"text":20084},"O is an upper bound, not a promise of tightness",{"id":11908,"depth":24,"text":11909},{"id":13716,"depth":18,"text":13717},{"id":15530,"depth":18,"text":15531},{"id":16406,"depth":18,"text":16407},{"id":19316,"depth":18,"text":19317},{"id":586,"depth":18,"text":19929},"In the previous lesson we saw the same algorithm, insertion sort, cost\nquadratically many comparisons on one input and linearly many on another, all on\nthe same machine. To compare algorithms as algorithms, independent of the\nhardware and the particular input, we need two things: a model of computation\nabstract enough to ignore the machine, and a notation coarse enough to ignore\nconstants. This lesson supplies both.","md",{"moduleNumber":6,"lessonNumber":18,"order":20094},102,true,[20097,20101,20104,20107,20110],{"title":20098,"slug":20099,"difficulty":20100},"Best Time to Buy and Sell Stock","best-time-to-buy-and-sell-stock","Easy",{"title":20102,"slug":20103,"difficulty":20100},"Contains Duplicate","contains-duplicate",{"title":20105,"slug":20106,"difficulty":20100},"Majority Element","majority-element",{"title":20108,"slug":20109,"difficulty":20100},"Squares of a Sorted Array","squares-of-a-sorted-array",{"title":20111,"slug":20112,"difficulty":20113},"Maximum Subarray","maximum-subarray","Medium","---\ntitle: Asymptotic Analysis\nmodule: Foundations\nmoduleNumber: 1\nlessonNumber: 2\norder: 102\nsummary: >\n  We measure an algorithm's running time as a function of its input size, then\n  strip away machine-specific constants and lower-order terms to compare\n  algorithms cleanly. This lesson defines the RAM model and the $O$, $\\Omega$,\n  $\\Theta$, $o$, and $\\omega$ notations, and shows how to read the cost of loops\n  off the page.\ntopics: [Asymptotic Analysis]\nsources:\n  - book: CLRS\n    ref: \"Ch. 3 — Characterizing Running Times\"\n  - book: Skiena\n    ref: \"§2 — Algorithm Analysis\"\n  - book: Erickson\n    ref: \"Appendix — Solving Recurrences; analysis throughout\"\npractice:\n  - title: 'Best Time to Buy and Sell Stock'\n    slug: best-time-to-buy-and-sell-stock\n    difficulty: Easy\n  - title: 'Contains Duplicate'\n    slug: contains-duplicate\n    difficulty: Easy\n  - title: 'Majority Element'\n    slug: majority-element\n    difficulty: Easy\n  - title: 'Squares of a Sorted Array'\n    slug: squares-of-a-sorted-array\n    difficulty: Easy\n  - title: 'Maximum Subarray'\n    slug: maximum-subarray\n    difficulty: Medium\n---\n\nIn the previous lesson we saw the same algorithm, insertion sort, cost\nquadratically many comparisons on one input and linearly many on another, all on\nthe same machine. To compare algorithms _as algorithms_, independent of the\nhardware and the particular input, we need two things: a model of computation\nabstract enough to ignore the machine, and a notation coarse enough to ignore\nconstants. This lesson supplies both.\n\n## The RAM model of computation\n\nWe analyze algorithms against an idealized machine, the **random-access machine\n(RAM)**. It has the properties all three books assume, usually tacitly:\n\n- Instructions execute one at a time, no concurrency.\n- The basic operations, namely arithmetic ($+$, $-$, $\\times$, $\u002F$),\n  comparisons, data movement (load, store, copy), and control flow, each take a\n  **constant** amount of time.\n- Memory is an unbounded array of cells, and accessing any cell by its index\n  costs the same constant (this is what _random access_ means).\n- Each cell holds an integer or float of \"reasonable\" size, roughly $O(\\log n)$\n  bits for an input of size $n$, so a single value fits in a machine word.\n\nThe RAM is a deliberate fiction. Real multiplication is not truly constant-time\nfor arbitrarily large numbers; real memory has caches that make some accesses far\ncheaper than others. But the model is _predictive_: an algorithm that is fast on\nthe RAM is, overwhelmingly, fast in practice. Skiena stresses this engineering\npayoff;[^skiena-ram] CLRS is careful to flag the places, such as bignum\narithmetic, where the constant-word assumption breaks down.[^clrs-ram]\n\n### From problem to $T(n)$\n\nIt helps to be precise about what we are even measuring. A\n**computational problem** is just a function $P : \\mathcal{I} \\to \\mathcal{O}$\nfrom a set of possible inputs to a set of possible outputs; each element\n$I \\in \\mathcal{I}$ is an **instance** of $P$. _Sorting integer arrays_, for\nexample, has $\\mathcal{I} = \\set{\\text{all integer arrays}}$. A **size** function\n$\\operatorname{size} : \\mathcal{I} \\to \\mathbb{N}$ records how \"big\" each instance\nis. For an array $\\langle a_1, \\dots, a_n\\rangle$ we take $\\operatorname{size} =\nn$. An algorithm $\\mathcal{A}$ **solves** $P$ if $\\mathcal{A}(I) = P(I)$ for every\ninstance $I$.\n\nNow charge $\\textsc{TimeCost}^{\\mathcal{A}}(I)$ = the number of elementary RAM\noperations $\\mathcal{A}$ performs on input $I$. Inputs of the same size can cost\ndifferent amounts, so we take the worst one of each size:\n\n$$\n\\textsc{MaxCost}^{\\mathcal{A}}(n) \\;=\\; \\max_{\\substack{I \\in \\mathcal{I} \\\\ \\operatorname{size}(I) = n}} \\textsc{TimeCost}^{\\mathcal{A}}(I).\n$$\n\nWhen the algorithm is understood, this is exactly the function we denote\n$T(n)$, the **running time** as a function of the **input size** $n$. The size is\nusually the number of elements, but sometimes the number of bits, or two\nparameters (e.g. $V$ and $E$ for a graph); choosing the right size measure is the\nfirst decision in any analysis. What we ultimately want is _a good,\nconvenient-to-understand upper bound on $T(n)$_, which is precisely what the\nnotation below provides.\n\n## Worst, average, and best case\n\nFor a fixed input _size_ $n$, different inputs of that size may cost different\namounts. Insertion sort costs $\\Theta(n)$ on a sorted array and $\\Theta(n^2)$ on\na reversed one. So $T(n)$ is not one number; it is a _range_. We summarize it\nthree ways:\n\n- **Worst case** $T(n)$: the maximum cost over all inputs of size $n$.\n- **Best case**: the minimum cost over all inputs of size $n$.\n- **Average case**: the expected cost over a probability distribution on\n  inputs of size $n$ (usually the uniform distribution).\n\n$$\n% caption: For each fixed size $n$, the cost is a _range_: best at the bottom edge, worst\n%          at the top, average in between. The shaded band is the spread over all inputs\n%          of that size. Slicing at one chosen size $n_0$ pins the three cases to three\n%          points on the vertical — the runtime of any single input of size $n_0$ lands\n%          somewhere on that segment.\n\\begin{tikzpicture}[scale=1.0,>=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[->] (0,0) -- (6.4,0) node[right] {$n$};\n  \\draw[->] (0,0) -- (0,5.0) node[above] {$T(n)$};\n  % spread between best (bottom) and worst (top); average runs strictly between\n  \\fill[acc!15]\n    plot[domain=0:5.4,smooth] (\\x,{0.30*\\x + 0.09*\\x*\\x})\n    -- plot[domain=5.4:0,smooth] (\\x,{0.30*\\x})\n    -- cycle;\n  \\draw[domain=0:5.4,smooth,thick] plot (\\x,{0.30*\\x + 0.09*\\x*\\x}) node[right] {worst};\n  \\draw[domain=0:5.4,smooth,thick,dashed] plot (\\x,{0.30*\\x + 0.045*\\x*\\x}) node[right] {average};\n  \\draw[domain=0:5.4,smooth,thick] plot (\\x,{0.30*\\x}) node[right] {best};\n  % slice at one fixed size n_0: the three cases are three points on it\n  \\draw[dotted] (4.0,0) -- (4.0,2.64);\n  \\foreach \\fy in {1.2,1.92,2.64} \\fill (4.0,\\fy) circle (1.3pt);\n  \\draw[\u003C->, acc] (4.35,1.2) -- (4.35,2.64) node[midway, right, font=\\footnotesize, fill=white, inner sep=1pt] {range at $n_0$};\n  \\node[below] at (4.0,0) {$n_0$};\n  \\node[acc, font=\\footnotesize, fill=white, inner sep=1.5pt] at (2.55,0.9) {spread over inputs};\n\\end{tikzpicture}\n$$\n\nWe almost always report the **worst case**. It is a _guarantee_: the algorithm\nnever does worse, no matter how adversarial the input. The best case is nearly\nuseless as a promise, since any algorithm looks good on its luckiest input. The\naverage case is the most honest predictor of typical performance but requires us\nto commit to a distribution, and the analysis is usually harder (it often needs\nthe probabilistic tools of a later module). CLRS develops all three; Skiena\nargues that for design purposes the worst case is what you should plan\nfor.\n\n## Why we drop constants and lower-order terms\n\nSuppose careful counting gives the running time of some algorithm as\n$$\nT(n) = 3n^2 + 50n + 200.\n$$\nTwo facts make most of this expression noise:\n\n1. **The leading term dominates.** As $n$ grows, $3n^2$ swamps $50n + 200$. At\n   $n = 1000$ the quadratic term is $3{,}000{,}000$ and the rest is $50{,}200$,\n   under $2\\%$. The growth _rate_ is governed entirely by $n^2$.\n2. **The constants are machine artifacts.** The $3$ depends on how many RAM\n   operations our particular pseudocode spends per iteration; recompile on a\n   different machine and it changes. It says nothing about the algorithm's\n   intrinsic scaling.\n\nSo we throw both away and say the running time is **order $n^2$**. This is not\nsloppiness; it is the right level of abstraction. An $n^2$ algorithm with a tiny\nconstant still loses _eventually_ to an $n \\log n$ algorithm with a large one,\nand \"eventually\" is exactly what asymptotic analysis captures. The notation below\nmakes \"order $n^2$\" precise.\n\n## The asymptotic notations\n\nLet $f$ and $g$ be functions from the positive integers to the nonnegative reals.\nThe notations describe how $f$ behaves _relative to_ $g$ for all sufficiently\nlarge $n$.\n\n### Big-O: asymptotic upper bound\n\nState it as a clean existential:\n\n> **Definition (Big-O).** $f(n) = O(g(n))$ means\n> $$\\exists\\, c > 0 \\;\\; \\exists\\, n_0 > 0 \\;\\; \\forall\\, n \\ge n_0 : \\; f(n) \\le c\\,g(n).$$\n> In words, \"$f$ is **upper bounded by $g$, up to a constant factor**.\" Roughly,\n> $f \\preccurlyeq g$.\n\nThe constant $c$ lets us ignore multiplicative factors; the threshold $n_0$ lets\nus ignore small inputs where lower-order terms might still dominate. (CLRS phrases\n$O(g)$ as a _set_ of functions and adds $0 \\le f(n)$ to keep things nonnegative;\nthe two readings agree.)\n\nThe picture to keep in mind is the \"for large $n$\" sketch: the scaled curve\n$c\\,g(n)$ rises above $f(n)$ once $n$ passes the threshold $n_0$, and stays above\nforever after. What happens to the left of $n_0$ is irrelevant.\n\n$$\n% caption: Curve $c \\cdot g(n)$ rises above $f(n)$ once $n$ passes the _**threshold**_\n%          $n_0$.\n\\begin{tikzpicture}[scale=1.0,>=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[->] (0,0) -- (5.6,0) node[right] {$n$};\n  \\draw[->] (0,0) -- (0,7.3) node[above] {};\n  % f(n): grows, but is dominated past n0\n  \\draw[domain=0:5,smooth,thick,black] plot (\\x, {0.45*\\x + 0.10*\\x*\\x})\n    node[right] {$f(n)$};\n  % c*g(n): the dominating bound — crosses f(n) exactly at n0 = 2.5 (y = 1.75)\n  \\draw[domain=0:5,smooth,thick,acc] plot (\\x, {0.28*\\x*\\x})\n    node[above right] {$c\\,g(n)$};\n  % crossover threshold n0: f(2.5) = cg(2.5) = 1.75, so the marker sits on the meet\n  \\draw[dashed] (2.5,0) -- (2.5,1.75);\n  \\node[below] at (2.5,0) {$n_0$};\n  \\fill (2.5,1.75) circle (1.4pt);\n  \\node[align=center] at (4.2,0.55) {\\footnotesize $f(n)\\le c\\,g(n)$\\\\[-2pt]\\footnotesize for $n\\ge n_0$};\n\\end{tikzpicture}\n$$\n\n**The \"wanted inequality\" method.** Proofs of $O$-bounds follow a fixed recipe:\nwrite down the inequality you _want_ to hold, then reverse-engineer constants $c$\nand $n_0$ that make it true. To prove $3n^2 + 50n + 200 = O(n^2)$, we want\n$3n^2 + 50n + 200 \\le c\\,n^2$. For $n \\ge 1$ each lower term is at most $n^2$, so\n$3n^2 + 50n + 200 \\le 3n^2 + 50n^2 + 200n^2 = 253n^2$; thus $c = 253$, $n_0 = 1$\nwork. The same move handles _any_ polynomial; see the theorem below.\n\n### Big-Omega: asymptotic lower bound\n\nMirror the quantifiers, flip the inequality:\n\n> **Definition ($\\Omega$).** $f(n) = \\Omega(g(n))$ means\n> $$\\exists\\, c > 0 \\;\\; \\exists\\, n_0 > 0 \\;\\; \\forall\\, n \\ge n_0 : \\; f(n) \\ge c\\,g(n).$$\n> \"$f$ grows **at least** as fast as $g$.\" Roughly, $f \\succcurlyeq g$.\n\nIt is the mirror image of $O$: $f = O(g)$ if and only if $g = \\Omega(f)$.\n\n### Big-Theta: asymptotic tight bound\n\n> **Definition ($\\Theta$).** $\\Theta(g(n))$ is the set of functions $f(n)$ for which there exist positive\n> constants $c_1, c_2, n_0$ such that\n> $$0 \\le c_1\\,g(n) \\le f(n) \\le c_2\\,g(n) \\quad\\text{for all } n \\ge n_0.$$\n\n$\\Theta$ pins $f$ between two constant multiples of $g$: it grows _exactly_ as\nfast as $g$, up to constants. The fundamental link is\n\n$$\nf(n) = \\Theta(g(n)) \\iff f(n) = O(g(n)) \\ \\text{and}\\ f(n) = \\Omega(g(n)).\n$$\n\nWhen we say insertion sort \"is $\\Theta(n^2)$ in the worst case,\" we mean its\nworst-case cost is sandwiched between $c_1 n^2$ and $c_2 n^2$, a precise,\ntwo-sided claim. When we only have an upper bound we say $O$; this is why people\nloosely write $O$ even where $\\Theta$ holds. But the distinction matters, as the\nnext two results make concrete.\n\n### The polynomial theorem\n\nThe single most useful fact for everyday analysis collapses every polynomial to\nits leading power:\n\n> **Theorem.** If $f(n) = a_k n^k + a_{k-1}n^{k-1} + \\cdots + a_1 n + a_0$ is a\n> polynomial with $a_k > 0$, then $f(n) = \\Theta(n^k)$.\n\n_Upper bound_ ($f = O(n^k)$). Replace every coefficient by its absolute value and\nevery lower power by $n^k$ (valid for $n \\ge 1$):\n\n$$\nf(n) \\le |a_k|n^k + |a_{k-1}|n^k + \\cdots + |a_0|n^k = \\big(|a_k| + \\cdots + |a_0|\\big)\\,n^k = c\\,n^k.\n$$\n\nSo define $c := |a_k| + \\cdots + |a_0|$ and pick $n_0 = 1$.\n\n_Lower bound_ ($f = \\Omega(n^k)$). Pull out the leading term and bound the rest\nbelow; for $n \\ge 1$,\n\n$$\nf(n) \\ge a_k n^k - \\big(|a_{k-1}|n^{k-1} + \\cdots + |a_0|\\big) \\ge a_k n^k - \\big(|a_{k-1}| + \\cdots + |a_0|\\big)\\,n^{k-1}.\n$$\n\nThe \"wanted inequality\" method now finds an $n_0$ past which the negative tail is,\nsay, at most half of $a_k n^k$, leaving $f(n) \\ge c\\,n^k$ for all $n \\ge n_0$.\nTogether the two directions give $f(n) = \\Theta(n^k)$: drop the constants and\nlower-order terms, keep the leading power.\n\n### $O$ is an upper bound, not a promise of tightness\n\nConsider a worked cautionary case. **Exchange sort** compares $A[i]$ with\n$A[j]$ for _every_ pair $i \u003C j$, so its running time satisfies\n\n$$\nT(n) \\;\\le\\; a\\cdot\\frac{n(n-1)}{2} = \\frac{a}{2}n^2 - \\frac{a}{2}n = O(n^2)\n$$\n\nby the polynomial theorem. But $O(n^2)$ also implies the _true but useless_\nstatement $T(n) = O(n^3)$: a correct upper bound need not be tight. So how loose\ncan we go? Not below $n^2$. The number of pairs is itself a _lower_ bound on the\nwork:\n\n$$\nT(n) \\;\\ge\\; |S| = \\binom{n}{2} = \\frac{n(n-1)}{2} = \\frac{n^2}{2} - \\frac{n}{2}.\n$$\n\nThis forces $T(n) \\neq O(n^{1.9})$; indeed $\\tfrac{n^2}{2} - \\tfrac{n}{2} \\neq\nO(n^k)$ for **any** $k \u003C 2$. No constant $c$ can keep $\\tfrac{n^2}{2}$ under\n$c\\,n^{1.9}$ once $n$ is large enough, because $\\tfrac{n^2}{2} > c\\,n^{1.9}$\nwhenever $n > (2c)^{10}$. The moral: $O$ alone tells you the cost is _no worse\nthan_ something; only matching it with $\\Omega$ (i.e. proving $\\Theta$) certifies\nyou have found the true growth rate.\n\nPicture the exponents on a line. Exchange sort's cost $\\Theta(n^2)$ sits at\n$k = 2$. Every $O(n^k)$ with $k \\ge 2$ is a _valid_ upper bound, but only $k = 2$\nis _tight_; the $\\Omega$ side forbids any upper bound with $k \u003C 2$, walling off\nthe left. Where the two bounds meet is $\\Theta$.\n\n$$\n% caption: Valid bounds for exchange sort's $\\Theta(n^2)$ cost, by exponent. Every\n%          $O(n^k)$ with $k\\ge 2$ holds but only $k=2$ is tight; $\\Omega(n^2)$ rules out\n%          the whole region below $k=2$. The bounds pinch shut exactly at $\\Theta(n^2)$.\n\\begin{tikzpicture}[scale=1.0,>={Stealth[length=2.4mm]}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-0.5,-1.9) rectangle (8.6,1.7);\n  % forbidden region k\u003C2 (Omega wall)\n  \\fill[red!12] (0,-0.18) rectangle (3.0,0.18);\n  % valid-but-loose O region k>2\n  \\fill[acc!15] (3.0,-0.18) rectangle (8.0,0.18);\n  \\draw[->] (-0.2,0) -- (8.5,0) node[right] {$k$};\n  \\foreach \\x\u002F\\t in {1\u002F1, 1.9\u002F1.9, 3\u002F2, 5\u002F2.5, 7\u002F3} {\n    \\draw (\\x,0.13) -- (\\x,-0.13);\n    \\node[below=1.5mm, font=\\footnotesize] at (\\x,-0.13) {$\\t$};\n  }\n  % tight point at k=2\n  \\fill[acc] (3.0,0) circle (2.2pt);\n  \\node[acc, font=\\footnotesize, align=center, above=3mm] at (3.0,0.13) {$\\Theta(n^2)$\\\\[-1pt]tight};\n  % labels\n  \\node[font=\\footnotesize, red!75!black, align=center, below=4mm] at (1.45,-0.5) {ruled out by $\\Omega(n^2)$\\\\(no $O(n^{k}),\\,k\u003C2$)};\n  \\node[font=\\footnotesize, acc, align=center] at (5.7,1.15) {valid $O(n^k)$ but loose};\n  \\draw[acc, ->] (5.7,0.92) -- (5.5,0.22);\n\\end{tikzpicture}\n$$\n\n### Little-o and little-omega: strict bounds\n\n$O$ and $\\Omega$ allow $f$ and $g$ to grow at the _same_ rate. The lowercase\nversions forbid that; they assert a _strict_ gap.\n\n> **Definition (Little-o).** $f(n) = o(g(n))$ means $f(n)$ is **less than *any* constant times $g(n)$**: for\n> **every** constant $c > 0$ there is an $n_0$ with $f(n) \u003C c\\,g(n)$ for all\n> $n \\ge n_0$. There is also a slick limit form,\n> $$f(n) = o(g(n)) \\iff \\lim_{n\\to\\infty} \\frac{f(n)}{g(n)} = 0.$$\n\nSo $f = o(g)$ means $f$ becomes _negligible_ compared to $g$: e.g.\n$n = o(n^2)$ and $\\log n = o(n)$. Symmetrically, $f(n) = \\omega(g(n))$ is defined\nby the reciprocal limit,\n$$\\lim_{n\\to\\infty} \\frac{g(n)}{f(n)} = 0 \\quad\\Longleftrightarrow\\quad \\lim_{n\\to\\infty} \\frac{f(n)}{g(n)} = \\infty,$$\nso $f$ dominates $g$. A useful analogy from CLRS: $O, \\Omega, \\Theta, o, \\omega$\nare to functions as $\\le, \\ge, =, \u003C, >$ are to numbers.\n\n$$\n% caption: Each asymptotic notation mirrors a comparison on numbers:\n%          $o,O,\\Theta,\\Omega,\\omega$ line up with $\u003C,\\le,=,\\ge,>$. The little-o and\n%          little-omega ends are _strict_ (they forbid equal growth), exactly as $\u003C$ and\n%          $>$ are strict.\n\\begin{tikzpicture}[\n    nb\u002F.style={draw, rounded corners, minimum width=15mm, minimum height=8mm, font=\\small, align=center},\n    lbl\u002F.style={font=\\footnotesize},\n    >={Stealth[length=2.4mm]}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-4.5,-3.4) rectangle (10.5,0.8);\n  \\node[lbl, anchor=east] at (-1.9,0) {notation};\n  \\node[lbl, anchor=east] at (-1.9,-1.9) {analogue on $\\mathbb{R}$};\n  \\node[nb] (o)  at (0.3,0) {$o$};\n  \\node[nb] (O)  at (2.65,0) {$O$};\n  \\node[nb] (Th) at (5.0,0) {$\\Theta$};\n  \\node[nb] (Om) at (7.35,0) {$\\Omega$};\n  \\node[nb] (w)  at (9.7,0) {$\\omega$};\n  \\node[nb, fill=acc!15] (lt) at (0.3,-1.9) {$\u003C$};\n  \\node[nb] (le) at (2.65,-1.9) {$\\le$};\n  \\node[nb] (eq) at (5.0,-1.9) {$=$};\n  \\node[nb] (ge) at (7.35,-1.9) {$\\ge$};\n  \\node[nb, fill=acc!15] (gt) at (9.7,-1.9) {$>$};\n  \\draw[->, black!55] (o) -- (lt);\n  \\draw[->, black!55] (O) -- (le);\n  \\draw[->, black!55] (Th) -- (eq);\n  \\draw[->, black!55] (Om) -- (ge);\n  \\draw[->, black!55] (w) -- (gt);\n  \\node[lbl, red!75!black, align=center] at (0.3,-3.0) {strict};\n  \\node[lbl, red!75!black, align=center] at (9.7,-3.0) {strict};\n\\end{tikzpicture}\n$$\n\n## Comparing functions with limits\n\nThe limit of the ratio $f(n)\u002Fg(n)$ is how you rank growth rates:\n\n$$\n\\lim_{n\\to\\infty} \\frac{f(n)}{g(n)} =\n\\begin{cases}\n0 & \\Rightarrow f = o(g) \\ (\\text{so also } f = O(g)) \\\\\nc \\in (0,\\infty) & \\Rightarrow f = \\Theta(g) \\\\\n\\infty & \\Rightarrow f = \\omega(g) \\ (\\text{so also } f = \\Omega(g)).\n\\end{cases}\n$$\n\nTo compare, say, $n^a$ and $\\log^b n$ for any constants $a, b > 0$, one shows the\nratio $\\log^b n \u002F n^a \\to 0$, so **every** polynomial beats **every**\npolylogarithm.[^erickson-growth] Likewise $n^k = o(2^n)$ for every constant $k$: exponentials\ndominate all polynomials. A useful fact for sums: $\\log(n!) = \\Theta(n \\log n)$\n(by **Stirling's approximation**),[^clrs-stirling] which is why [comparison sorts](\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds) that do\n$\\Theta(n \\log n)$ work are optimal in a sense we will prove later.\n\n**Why we write $\\log n$ with no base.** Inside asymptotic notation the base is\nirrelevant, because changing base only multiplies by a constant:\n$$\n\\log_b n = \\frac{\\log_2 n}{\\log_2 b} = \\Theta(\\log_2 n) \\quad\\text{for every constant } b > 1.\n$$\nSo $\\log_2 n$, $\\log_{10} n$, and $\\ln n$ all live in the same $\\Theta$-class, and\n$O(\\log n)$ means the same thing whichever base you had in mind. (The constant\n$1\u002F\\log_2 b$ is exactly the multiplicative factor $O$ and $\\Omega$ are designed to\nabsorb.)\n\n## The growth hierarchy\n\nThe functions algorithm designers meet most often, in strictly increasing order\nof growth, are:\n\n| Class | Name | A typical algorithm |\n| :--- | :--- | :--- |\n| $\\Theta(1)$ | constant | array index, hash lookup (expected) |\n| $\\Theta(\\log n)$ | logarithmic | binary search |\n| $\\Theta(n)$ | linear | scan \u002F find max |\n| $\\Theta(n \\log n)$ | linearithmic | merge sort, heapsort |\n| $\\Theta(n^2)$ | quadratic | insertion sort (worst case) |\n| $\\Theta(n^3)$ | cubic | naive matrix multiply |\n| $\\Theta(2^n)$ | exponential | subset enumeration |\n| $\\Theta(n!)$ | factorial | brute-force permutations (TSP) |\n\nEach row's growth dwarfs every row above it for large $n$. The practical\ndividing line Skiena draws is between **polynomial** time ($n^c$ for a constant\n$c$), generally considered \"tractable,\" and **exponential** time, which becomes\nhopeless very fast.[^skiena-growth] A factorial-time algorithm that handles $n = 12$ in\na second needs years at $n = 18$.\n\nA sketch of the curves makes the separation clear. Even with a generous\nconstant on the slower-growing function, the faster one wins past some crossover\n$n_0$:\n\n$$\n% caption: Growth-rate curves for $\\log n$, $n$, $n^2$, and $2^n$.\n\\begin{tikzpicture}[scale=0.85]\n  \\draw[->] (0,0) -- (5.6,0) node[right] {$n$};\n  \\draw[->] (0,0) -- (0,7.6) node[above] {$T(n)$};\n  \\draw[domain=0.05:5,smooth,thick] plot (\\x, {0.55*ln(\\x+1)})\n    node[right] {$\\log n$};\n  \\draw[domain=0:5,smooth,thick] plot (\\x, {0.45*\\x})\n    node[right] {$n$};\n  \\draw[domain=0:4.5,smooth,thick] plot (\\x, {0.22*\\x*\\x})\n    node[right] {$n^2$};\n  % 2^n: starts at the origin (exp(x)-1), crosses n^2 around n=3.6, then pulls\n  % clear above it — the taller y-axis shows the exponential racing away.\n  \\draw[domain=0:4.5,smooth,thick] plot (\\x, {0.08*(exp(\\x)-1)})\n    node[above] {$2^n$};\n\\end{tikzpicture}\n$$\n\n## Reading the cost off loops\n\nMost analysis reduces to counting how many times each line runs. A few rules\ncover the common cases.\n\n**Sequential blocks add; we keep the max.** If block $A$ costs $\\Theta(n)$ and is\nfollowed by block $B$ costing $\\Theta(n^2)$, the total is\n$\\Theta(n) + \\Theta(n^2) = \\Theta(n^2)$: the larger term absorbs the smaller.\n\n**A simple loop multiplies the body by the iteration count.** This nested\nfragment runs the constant-time body $n \\cdot n$ times:\n\n```algorithm\ncaption: Counting work in nested loops\nfor $i \\gets 1$ to $n$ do\n  for $j \\gets 1$ to $n$ do\n    $c \\gets c + A[i] \\cdot B[j]$ \u002F\u002F $\\Theta(1)$ body, $n^2$ times\n```\n\nso its cost is $\\Theta(n^2)$.\n\n**When the inner bound depends on the outer index, sum a series.** If the inner\nloop runs $\\textbf{for}\\ j \\gets 1\\ \\textbf{to}\\ i$, the body executes\n$\\sum_{i=1}^{n} i = \\frac{n(n+1)}{2} = \\Theta(n^2)$ times — still quadratic,\nbecause triangular work is half of square work, and the constant $\\tfrac12$\nvanishes. This is exactly the shape of insertion sort's worst case.\n\n**When the loop variable is scaled, take a logarithm.** A loop that does\n$i \\gets i \\times 2$ until $i > n$ runs about $\\log_2 n$ times, since $i$\ndoubles each pass. This is the source of every logarithm in algorithm analysis:\n**repeatedly halving (or doubling) gives $\\Theta(\\log n)$ steps.** Binary search\nand balanced-tree depth are the canonical examples.\n\nThe three loop shapes read off as _area_ (or, for doubling, as the number of\nrungs). A square grid of iterations is $n^2$ work; bounding the inner loop by the\nouter index $i$ fills only the triangle below the diagonal, half as much but still\n$\\Theta(n^2)$; and a doubling index touches just the $\\log_2 n$ powers of two.\n\n$$\n% caption: Reading loop cost as area. A square grid of iterations is $n^2$ work; an inner\n%          bound $j \\gets 1 \\dots i$ fills only the triangle, $\\tfrac12 n^2$; a doubling\n%          index visits just $\\log_2 n$ rungs.\n\\begin{tikzpicture}[scale=1.0,>=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  % --- panel 1: full square, n^2 ---\n  \\begin{scope}[shift={(0,0)}]\n    \\fill[acc!15] (0,0) rectangle (2.4,2.4);\n    \\draw[step=0.4,black!35,very thin] (0,0) grid (2.4,2.4);\n    \\draw[thick] (0,0) rectangle (2.4,2.4);\n    \\draw[->] (0,2.4) -- (0,2.9) node[above] {$i$};\n    \\draw[->] (2.4,0) -- (2.9,0) node[right] {$j$};\n    \\node at (1.2,-0.55) {\\footnotesize $j \\gets 1 \\dots n$};\n    \\node[fill=white, inner sep=1.5pt] at (1.2,1.2) {$n^2$};\n    \\node[align=center] at (1.2,-1.1) {\\footnotesize square loop\\\\[-2pt]\\footnotesize $\\Theta(n^2)$};\n  \\end{scope}\n  % --- panel 2: triangle, 1\u002F2 n^2 ---\n  \\begin{scope}[shift={(4.1,0)}]\n    \\fill[acc!22] (0,0) -- (2.4,2.4) -- (0,2.4) -- cycle;\n    \\draw[step=0.4,black!35,very thin] (0,0) grid (2.4,2.4);\n    \\draw[thick] (0,0) rectangle (2.4,2.4);\n    \\draw[thick] (0,0) -- (2.4,2.4);\n    \\draw[->] (0,2.4) -- (0,2.9) node[above] {$i$};\n    \\draw[->] (2.4,0) -- (2.9,0) node[right] {$j$};\n    \\node at (1.2,-0.55) {\\footnotesize $j \\gets 1 \\dots i$};\n    \\node[fill=white, inner sep=1.5pt] at (0.75,1.7) {$\\tfrac12 n^2$};\n    \\node[align=center] at (1.2,-1.1) {\\footnotesize triangular loop\\\\[-2pt]\\footnotesize $\\Theta(n^2)$};\n  \\end{scope}\n  % --- panel 3: doubling, log n ---\n  \\begin{scope}[shift={(8.2,0)}]\n    \\draw[->] (0,0) -- (3.0,0) node[right] {$i$};\n    \\foreach \\x in {0.2,0.5,1.1,2.3} {\n      \\fill[acc] (\\x,0.18) circle (1.6pt);\n    }\n    \\draw[acc,thick,->] (0.2,0.18) to[bend left=30] (0.5,0.18);\n    \\draw[acc,thick,->] (0.5,0.18) to[bend left=30] (1.1,0.18);\n    \\draw[acc,thick,->] (1.1,0.18) to[bend left=30] (2.3,0.18);\n    \\node[below] at (0.2,0) {\\footnotesize $1$};\n    \\node[below] at (0.5,0) {\\footnotesize $2$};\n    \\node[below] at (1.1,0) {\\footnotesize $4$};\n    \\node[below] at (2.3,0) {\\footnotesize $8$};\n    \\node[acc] at (0.8,0.7) {\\footnotesize $\\times 2$};\n    \\node[align=center] at (1.4,-1.1) {\\footnotesize doubling loop\\\\[-2pt]\\footnotesize $\\Theta(\\log n)$};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nTwo more loop shapes round out the catalog, and both read off as areas just like\nthe three above. An outer loop of $n$ passes wrapped around a _doubling_ inner\nloop does $\\log_2 n$ work per pass — an $n \\times \\log n$ grid, $\\Theta(n\\log n)$.\nAnd a loop whose live problem _halves_ every pass while doing linear work on what\nremains sums a geometric series $n + \\tfrac n2 + \\tfrac n4 + \\cdots = 2n$, so the\nwhole stack of shrinking bars is no taller than two of the first — $\\Theta(n)$.\n\n$$\n% caption: Two more loop costs as area. A linear loop wrapping a doubling inner loop fills\n%          an $n \\times \\log n$ grid ($\\Theta(n\\log n)$); a loop that halves its live\n%          problem each pass stacks bars $n, \\tfrac n2, \\tfrac n4, \\dots$ that sum to $2n$\n%          ($\\Theta(n)$).\n\\begin{tikzpicture}[scale=1.0,>=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  % --- panel 1: n x log n grid, n log n ---\n  \\begin{scope}[shift={(0,0)}]\n    \\fill[acc!15] (0,0) rectangle (3.2,1.2);\n    \\draw[step=0.4,black!35,very thin] (0,0) grid (3.2,1.2);\n    \\draw[thick] (0,0) rectangle (3.2,1.2);\n    \\draw[->] (0,1.2) -- (0,1.7) node[above] {\\footnotesize $\\log n$};\n    \\draw[->] (3.2,0) -- (3.7,0) node[right] {$i$};\n    \\node[fill=white, inner sep=1.5pt] at (1.6,0.6) {$n\\log n$};\n    \\node at (1.6,-0.5) {\\footnotesize $i \\gets 1 \\dots n$};\n    \\node[align=center] at (1.6,-1.05) {\\footnotesize doubling inner loop\\\\[-2pt]\\footnotesize $\\Theta(n\\log n)$};\n  \\end{scope}\n  % --- panel 2: halving bars, geometric sum = 2n -> Theta(n) ---\n  \\begin{scope}[shift={(5.4,0)}]\n    \\fill[acc!22] (0,0)   rectangle (0.5,2.4);\n    \\fill[acc!22] (0.6,0) rectangle (1.1,1.2);\n    \\fill[acc!22] (1.2,0) rectangle (1.7,0.6);\n    \\fill[acc!22] (1.8,0) rectangle (2.3,0.3);\n    \\draw[thick,->] (0,0) -- (2.9,0) node[right] {\\footnotesize pass};\n    \\node[above, font=\\footnotesize] at (0.25,2.4) {$n$};\n    \\node[above, font=\\footnotesize] at (0.85,1.2) {$\\tfrac n2$};\n    \\node[above, font=\\footnotesize] at (1.45,0.6) {$\\tfrac n4$};\n    \\node[above, font=\\footnotesize] at (2.05,0.3) {$\\tfrac n8$};\n    \\node[align=center] at (1.15,-0.78) {\\footnotesize halving work, sum $=2n$\\\\[-2pt]\\footnotesize $\\Theta(n)$};\n  \\end{scope}\n\\end{tikzpicture}\n$$\n\nThe [recurrences](\u002Falgorithms\u002Ffoundations\u002Frecurrences) that arise when a loop is\nreplaced by _recursion_, a function calling smaller copies of itself, need their\nown machinery, which is the subject of the next lesson.\n\n## Takeaways\n\n- The **RAM model** charges constant time per primitive operation and lets us\n  measure running time as a function $T(n)$ of input size, machine-independently.\n- Report the **worst case** by default: it is a guarantee. Best case promises\n  nothing; average case is honest but needs a distribution.\n- **Drop constants and lower-order terms.** They are machine artifacts and noise;\n  the leading term's growth rate is what scales.\n- $O$ is an upper bound, $\\Omega$ a lower bound, $\\Theta$ a tight (two-sided)\n  bound; $o$ and $\\omega$ are their strict versions. $f = \\Theta(g) \\iff f = O(g)$\n  and $f = \\Omega(g)$.\n- The limit $\\lim f(n)\u002Fg(n)$ ranks two functions: $0$, a constant, or $\\infty$\n  give $o$, $\\Theta$, or $\\omega$.\n- Memorize the hierarchy $1 \\prec \\log n \\prec n \\prec n\\log n \\prec n^2 \\prec 2^n\n  \\prec n!$; the polynomial\u002Fexponential boundary is the line between tractable and\n  hopeless.\n\n[^skiena-ram]: **Skiena**, §2 — Algorithm Analysis: the RAM model as a machine-independent abstraction whose engineering payoff is predicting real-world speed.\n[^clrs-ram]: **CLRS**, Ch. 3 — Characterizing Running Times: the RAM model's constant-word assumption and where it breaks down (e.g. bignum arithmetic).\n[^clrs-stirling]: **CLRS**, Ch. 3 — Characterizing Running Times: $\\log(n!) = \\Theta(n\\log n)$ via Stirling's approximation.\n[^skiena-growth]: **Skiena**, §2 — Algorithm Analysis: the polynomial-vs-exponential dividing line between tractable and hopeless running times.\n[^erickson-growth]: **Erickson**, _Algorithms_, Appendix — Solving Recurrences (analysis throughout): ranking growth rates by the limit of the ratio, so every polynomial dominates every polylogarithm.\n",{"text":20116,"minutes":20117,"time":20118,"words":20119},"11 min read",10.535,632100,2107,{"title":16,"description":20091},[20122,20124,20126],{"book":19951,"ref":20123},"Ch. 3 — Characterizing Running Times",{"book":19937,"ref":20125},"§2 — Algorithm Analysis",{"book":19963,"ref":20127},"Appendix — Solving Recurrences; analysis 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