[{"data":1,"prerenderedAt":15579},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":374,"course-wordcounts":15447,"ref-card-index":15503},[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":251,"blurb":376,"body":377,"description":15407,"extension":15408,"meta":15409,"module":231,"navigation":15411,"path":252,"practice":15412,"rawbody":15429,"readingTime":15430,"seo":15435,"sources":15436,"status":15443,"stem":15444,"summary":254,"topics":15445,"__hash__":15446},"course\u002F01.algorithms\u002F08.dynamic-programming\u002F04.knapsack.md","",{"type":378,"value":379,"toc":15391},"minimark",[380,605,631,636,965,1021,1272,1366,1786,1830,2417,2586,2591,2894,3827,4322,4326,4391,4597,4601,5170,5565,6010,6439,6493,6497,6552,6670,6726,6832,7299,7325,7977,8106,8110,8459,9071,9763,9767,9826,10119,10169,10293,11113,11117,11235,11263,11307,11481,12074,12180,12208,12388,12777,12871,12999,13145,13588,13655,13659,14063,14162,14272,14280,14493,14556,14599,14647,14651,15274,15387],[381,382,383,384,410,411,428,429,500,501,556,557,561,562,588,589,593,594],"p",{},"A thief breaks into a warehouse carrying a knapsack that holds at most ",[385,386,389],"span",{"className":387},[388],"katex",[385,390,394],{"className":391,"ariaHidden":393},[392],"katex-html","true",[385,395,398,403],{"className":396},[397],"base",[385,399],{"className":400,"style":402},[401],"strut","height:0.6833em;",[385,404,409],{"className":405,"style":408},[406,407],"mord","mathnormal","margin-right:0.1389em;","W"," units\nof weight. Item ",[385,412,414],{"className":413},[388],[385,415,417],{"className":416,"ariaHidden":393},[392],[385,418,420,424],{"className":419},[397],[385,421],{"className":422,"style":423},[401],"height:0.6595em;",[385,425,427],{"className":426},[406,407],"i"," weighs ",[385,430,432],{"className":431},[388],[385,433,435],{"className":434,"ariaHidden":393},[392],[385,436,438,442],{"className":437},[397],[385,439],{"className":440,"style":441},[401],"height:0.5806em;vertical-align:-0.15em;",[385,443,445,450],{"className":444},[406],[385,446,449],{"className":447,"style":448},[406,407],"margin-right:0.0269em;","w",[385,451,454],{"className":452},[453],"msupsub",[385,455,459,491],{"className":456},[457,458],"vlist-t","vlist-t2",[385,460,463,486],{"className":461},[462],"vlist-r",[385,464,468],{"className":465,"style":467},[466],"vlist","height:0.3117em;",[385,469,471,476],{"style":470},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[385,472],{"className":473,"style":475},[474],"pstrut","height:2.7em;",[385,477,483],{"className":478},[479,480,481,482],"sizing","reset-size6","size3","mtight",[385,484,427],{"className":485},[406,407,482],[385,487,490],{"className":488},[489],"vlist-s","​",[385,492,494],{"className":493},[462],[385,495,498],{"className":496,"style":497},[466],"height:0.15em;",[385,499],{}," and is worth ",[385,502,504],{"className":503},[388],[385,505,507],{"className":506,"ariaHidden":393},[392],[385,508,510,513],{"className":509},[397],[385,511],{"className":512,"style":441},[401],[385,514,516,521],{"className":515},[406],[385,517,520],{"className":518,"style":519},[406,407],"margin-right:0.0359em;","v",[385,522,524],{"className":523},[453],[385,525,527,548],{"className":526},[457,458],[385,528,530,545],{"className":529},[462],[385,531,533],{"className":532,"style":467},[466],[385,534,536,539],{"style":535},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[385,537],{"className":538,"style":475},[474],[385,540,542],{"className":541},[479,480,481,482],[385,543,427],{"className":544},[406,407,482],[385,546,490],{"className":547},[489],[385,549,551],{"className":550},[462],[385,552,554],{"className":553,"style":497},[466],[385,555],{},". Each item is taken whole or\nleft behind, with no fractions and no duplicates. Which subset maximizes the value\ncarried out without exceeding the capacity? This is the ",[558,559,560],"strong",{},"0\u002F1 knapsack problem",",\nand despite its toy framing it is one of the most important problems in\ncomputing: it is the kernel of resource allocation, budgeting, and cutting-stock\nproblems, and a canonical ",[385,563,565],{"className":564},[388],[385,566,568],{"className":567,"ariaHidden":393},[392],[385,569,571,575],{"className":570},[397],[385,572],{"className":573,"style":574},[401],"height:0.6944em;",[385,576,580],{"className":577},[578,579],"enclosing","textsc",[385,581,584],{"className":582},[406,583],"text",[385,585,587],{"className":586},[406],"NP-hard"," optimization problem whose dynamic program\nteaches a subtle lesson about what ",[590,591,592],"q",{},"polynomial"," really means.",[595,596,597],"sup",{},[598,599,604],"a",{"href":600,"ariaDescribedBy":601,"dataFootnoteRef":376,"id":603},"#user-content-fn-skiena-knap",[602],"footnote-label","user-content-fnref-skiena-knap","1",[381,606,607,608,630],{},"We will arrive at knapsack through its simpler decision cousin, ",[385,609,611],{"className":610},[388],[385,612,614],{"className":613,"ariaHidden":393},[392],[385,615,617,620],{"className":616},[397],[385,618],{"className":619,"style":574},[401],[385,621,623],{"className":622},[578,579],[385,624,626],{"className":625},[406,583],[385,627,629],{"className":628},[406],"Subset-sum",",\nwhich strips away the values and asks a plain yes\u002Fno question. The include\u002Fexclude\nrecurrence is identical, the running-time subtlety is identical, and subset-sum is\nthe cleanest place to see both.",[632,633,635],"h2",{"id":634},"subset-sum-the-decision-core","Subset-sum: the decision core",[637,638,640],"callout",{"type":639},"problem",[381,641,642,645,646,846,847,885,886,889,890,910,911,926,927,943,944,964],{},[558,643,644],{},"Input:"," a list 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of positive integers\nand a target integer ",[385,848,850],{"className":849},[388],[385,851,853,874],{"className":852,"ariaHidden":393},[392],[385,854,856,860,864,867,871],{"className":855},[397],[385,857],{"className":858,"style":859},[401],"height:0.6542em;vertical-align:-0.0391em;",[385,861,863],{"className":862},[406,407],"t",[385,865],{"className":866,"style":666},[665],[385,868,870],{"className":869},[670],">",[385,872],{"className":873,"style":666},[665],[385,875,877,881],{"className":876},[397],[385,878],{"className":879,"style":880},[401],"height:0.6444em;",[385,882,884],{"className":883},[406],"0",".\n",[558,887,888],{},"Output (simplified):"," ",[385,891,893],{"className":892},[388],[385,894,896],{"className":895,"ariaHidden":393},[392],[385,897,899,903],{"className":898},[397],[385,900],{"className":901,"style":902},[401],"height:0.625em;vertical-align:-0.1944em;",[385,904,906],{"className":905},[406,583],[385,907,909],{"className":908},[406],"yes"," if some sublist of ",[385,912,914],{"className":913},[388],[385,915,917],{"className":916,"ariaHidden":393},[392],[385,918,920,923],{"className":919},[397],[385,921],{"className":922,"style":402},[401],[385,924,661],{"className":925},[406,407]," sums to exactly\n",[385,928,930],{"className":929},[388],[385,931,933],{"className":932,"ariaHidden":393},[392],[385,934,936,940],{"className":935},[397],[385,937],{"className":938,"style":939},[401],"height:0.6151em;",[385,941,863],{"className":942},[406,407],", and ",[385,945,947],{"className":946},[388],[385,948,950],{"className":949,"ariaHidden":393},[392],[385,951,953,957],{"className":952},[397],[385,954],{"className":955,"style":956},[401],"height:0.4306em;",[385,958,960],{"className":959},[406,583],[385,961,963],{"className":962},[406],"no"," otherwise.",[381,966,967,968,1011,1012,1016,1017,1020],{},"The brute-force space is the ",[385,969,971],{"className":970},[388],[385,972,974],{"className":973,"ariaHidden":393},[392],[385,975,977,981],{"className":976},[397],[385,978],{"className":979,"style":980},[401],"height:0.6644em;",[385,982,984,987],{"className":983},[406],[385,985,767],{"className":986},[406],[385,988,990],{"className":989},[453],[385,991,993],{"className":992},[457],[385,994,996],{"className":995},[462],[385,997,999],{"className":998,"style":980},[466],[385,1000,1002,1005],{"style":1001},"top:-3.063em;margin-right:0.05em;",[385,1003],{"className":1004,"style":475},[474],[385,1006,1008],{"className":1007},[479,480,481,482],[385,1009,829],{"className":1010},[406,407,482]," sublists. To get a recurrence we shrink the\ninstance one element at a time, and the trick that supplies the ",[1013,1014,1015],"em",{},"second","\ndimension is to make the ",[1013,1018,1019],{},"target itself"," a parameter of the subproblem.",[637,1022,1024],{"type":1023},"definition",[381,1025,1026,1029,1030,1067,1068,1086,1087,1236,1237,943,1252,1271],{},[558,1027,1028],{},"Definition (Subset-sum subproblem)."," Let ",[385,1031,1033],{"className":1032},[388],[385,1034,1036],{"className":1035,"ariaHidden":393},[392],[385,1037,1039,1042,1046,1050,1053,1056,1059,1063],{"className":1038},[397],[385,1040],{"className":1041,"style":681},[401],[385,1043,1045],{"className":1044},[406,407],"A",[385,1047,1049],{"className":1048},[685],"(",[385,1051,427],{"className":1052},[406,407],[385,1054,733],{"className":1055},[732],[385,1057],{"className":1058,"style":737},[665],[385,1060,1062],{"className":1061},[406,407],"u",[385,1064,1066],{"className":1065},[844],")"," be ",[385,1069,1071],{"className":1070},[388],[385,1072,1074],{"className":1073,"ariaHidden":393},[392],[385,1075,1077,1080],{"className":1076},[397],[385,1078],{"className":1079,"style":939},[401],[385,1081,1083],{"className":1082},[406,583],[385,1084,393],{"className":1085},[406]," exactly when some sublist of the prefix\n",[385,1088,1090],{"className":1089},[388],[385,1091,1093,1126],{"className":1092,"ariaHidden":393},[392],[385,1094,1096,1099,1102,1106,1110,1113,1117,1120,1123],{"className":1095},[397],[385,1097],{"className":1098,"style":681},[401],[385,1100,661],{"className":1101},[406,407],[385,1103,1105],{"className":1104},[685],"[",[385,1107,1109],{"className":1108},[406],"1..",[385,1111,427],{"className":1112},[406,407],[385,1114,1116],{"className":1115},[844],"]",[385,1118],{"className":1119,"style":666},[665],[385,1121,671],{"className":1122},[670],[385,1124],{"className":1125,"style":666},[665],[385,1127,1129,1132,1135,1175,1178,1181,1184,1187,1190,1193,1233],{"className":1128},[397],[385,1130],{"className":1131,"style":681},[401],[385,1133,686],{"className":1134},[685],[385,1136,1138,1141],{"className":1137},[406],[385,1139,598],{"className":1140},[406,407],[385,1142,1144],{"className":1143},[453],[385,1145,1147,1167],{"className":1146},[457,458],[385,1148,1150,1164],{"className":1149},[462],[385,1151,1153],{"className":1152,"style":705},[466],[385,1154,1155,1158],{"style":708},[385,1156],{"className":1157,"style":475},[474],[385,1159,1161],{"className":1160},[479,480,481,482],[385,1162,604],{"className":1163},[406,482],[385,1165,490],{"className":1166},[489],[385,1168,1170],{"className":1169},[462],[385,1171,1173],{"className":1172,"style":497},[466],[385,1174],{},[385,1176,733],{"className":1177},[732],[385,1179],{"className":1180,"style":737},[665],[385,1182,789],{"className":1183},[788],[385,1185],{"className":1186,"style":737},[665],[385,1188,733],{"className":1189},[732],[385,1191],{"className":1192,"style":737},[665],[385,1194,1196,1199],{"className":1195},[406],[385,1197,598],{"className":1198},[406,407],[385,1200,1202],{"className":1201},[453],[385,1203,1205,1225],{"className":1204},[457,458],[385,1206,1208,1222],{"className":1207},[462],[385,1209,1211],{"className":1210,"style":467},[466],[385,1212,1213,1216],{"style":708},[385,1214],{"className":1215,"style":475},[474],[385,1217,1219],{"className":1218},[479,480,481,482],[385,1220,427],{"className":1221},[406,407,482],[385,1223,490],{"className":1224},[489],[385,1226,1228],{"className":1227},[462],[385,1229,1231],{"className":1230,"style":497},[466],[385,1232],{},[385,1234,845],{"className":1235},[844]," sums to ",[385,1238,1240],{"className":1239},[388],[385,1241,1243],{"className":1242,"ariaHidden":393},[392],[385,1244,1246,1249],{"className":1245},[397],[385,1247],{"className":1248,"style":956},[401],[385,1250,1062],{"className":1251},[406,407],[385,1253,1255],{"className":1254},[388],[385,1256,1258],{"className":1257,"ariaHidden":393},[392],[385,1259,1261,1264],{"className":1260},[397],[385,1262],{"className":1263,"style":574},[401],[385,1265,1267],{"className":1266},[406,583],[385,1268,1270],{"className":1269},[406],"false","\notherwise.",[381,1273,1274,1275,1308,1309,1361,1362,1365],{},"The answer we want is ",[385,1276,1278],{"className":1277},[388],[385,1279,1281],{"className":1280,"ariaHidden":393},[392],[385,1282,1284,1287,1290,1293,1296,1299,1302,1305],{"className":1283},[397],[385,1285],{"className":1286,"style":681},[401],[385,1288,1045],{"className":1289},[406,407],[385,1291,1049],{"className":1292},[685],[385,1294,829],{"className":1295},[406,407],[385,1297,733],{"className":1298},[732],[385,1300],{"className":1301,"style":737},[665],[385,1303,863],{"className":1304},[406,407],[385,1306,1066],{"className":1307},[844],". Now look at ",[385,1310,1312],{"className":1311},[388],[385,1313,1315],{"className":1314,"ariaHidden":393},[392],[385,1316,1318,1321],{"className":1317},[397],[385,1319],{"className":1320,"style":441},[401],[385,1322,1324,1327],{"className":1323},[406],[385,1325,598],{"className":1326},[406,407],[385,1328,1330],{"className":1329},[453],[385,1331,1333,1353],{"className":1332},[457,458],[385,1334,1336,1350],{"className":1335},[462],[385,1337,1339],{"className":1338,"style":467},[466],[385,1340,1341,1344],{"style":708},[385,1342],{"className":1343,"style":475},[474],[385,1345,1347],{"className":1346},[479,480,481,482],[385,1348,427],{"className":1349},[406,407,482],[385,1351,490],{"className":1352},[489],[385,1354,1356],{"className":1355},[462],[385,1357,1359],{"className":1358,"style":497},[466],[385,1360],{},", the last element we are\nallowed to use, and make the ",[558,1363,1364],{},"include\u002Fexclude"," decision that defines every\n0\u002F1 dynamic program:",[1367,1368,1369,1534],"ul",{},[1370,1371,1372,1428,1429,1466,1467,1482,1483,1427],"li",{},[558,1373,1374,1375,1427],{},"Exclude ",[385,1376,1378],{"className":1377},[388],[385,1379,1381],{"className":1380,"ariaHidden":393},[392],[385,1382,1384,1387],{"className":1383},[397],[385,1385],{"className":1386,"style":441},[401],[385,1388,1390,1393],{"className":1389},[406],[385,1391,598],{"className":1392},[406,407],[385,1394,1396],{"className":1395},[453],[385,1397,1399,1419],{"className":1398},[457,458],[385,1400,1402,1416],{"className":1401},[462],[385,1403,1405],{"className":1404,"style":467},[466],[385,1406,1407,1410],{"style":708},[385,1408],{"className":1409,"style":475},[474],[385,1411,1413],{"className":1412},[479,480,481,482],[385,1414,427],{"className":1415},[406,407,482],[385,1417,490],{"className":1418},[489],[385,1420,1422],{"className":1421},[462],[385,1423,1425],{"className":1424,"style":497},[466],[385,1426],{},"."," Then some sublist of the first ",[385,1430,1432],{"className":1431},[388],[385,1433,1435,1457],{"className":1434,"ariaHidden":393},[392],[385,1436,1438,1442,1445,1449,1454],{"className":1437},[397],[385,1439],{"className":1440,"style":1441},[401],"height:0.7429em;vertical-align:-0.0833em;",[385,1443,427],{"className":1444},[406,407],[385,1446],{"className":1447,"style":1448},[665],"margin-right:0.2222em;",[385,1450,1453],{"className":1451},[1452],"mbin","−",[385,1455],{"className":1456,"style":1448},[665],[385,1458,1460,1463],{"className":1459},[397],[385,1461],{"className":1462,"style":880},[401],[385,1464,604],{"className":1465},[406]," elements must already\nhit ",[385,1468,1470],{"className":1469},[388],[385,1471,1473],{"className":1472,"ariaHidden":393},[392],[385,1474,1476,1479],{"className":1475},[397],[385,1477],{"className":1478,"style":956},[401],[385,1480,1062],{"className":1481},[406,407]," on its own: ",[385,1484,1486],{"className":1485},[388],[385,1487,1489,1513],{"className":1488,"ariaHidden":393},[392],[385,1490,1492,1495,1498,1501,1504,1507,1510],{"className":1491},[397],[385,1493],{"className":1494,"style":681},[401],[385,1496,1045],{"className":1497},[406,407],[385,1499,1049],{"className":1500},[685],[385,1502,427],{"className":1503},[406,407],[385,1505],{"className":1506,"style":1448},[665],[385,1508,1453],{"className":1509},[1452],[385,1511],{"className":1512,"style":1448},[665],[385,1514,1516,1519,1522,1525,1528,1531],{"className":1515},[397],[385,1517],{"className":1518,"style":681},[401],[385,1520,604],{"className":1521},[406],[385,1523,733],{"className":1524},[732],[385,1526],{"className":1527,"style":737},[665],[385,1529,1062],{"className":1530},[406,407],[385,1532,1066],{"className":1533},[844],[1370,1535,1536,1591,1592,1644,1645,1678,1679,1427],{},[558,1537,1538,1539,1427],{},"Include ",[385,1540,1542],{"className":1541},[388],[385,1543,1545],{"className":1544,"ariaHidden":393},[392],[385,1546,1548,1551],{"className":1547},[397],[385,1549],{"className":1550,"style":441},[401],[385,1552,1554,1557],{"className":1553},[406],[385,1555,598],{"className":1556},[406,407],[385,1558,1560],{"className":1559},[453],[385,1561,1563,1583],{"className":1562},[457,458],[385,1564,1566,1580],{"className":1565},[462],[385,1567,1569],{"className":1568,"style":467},[466],[385,1570,1571,1574],{"style":708},[385,1572],{"className":1573,"style":475},[474],[385,1575,1577],{"className":1576},[479,480,481,482],[385,1578,427],{"className":1579},[406,407,482],[385,1581,490],{"className":1582},[489],[385,1584,1586],{"className":1585},[462],[385,1587,1589],{"className":1588,"style":497},[466],[385,1590],{}," Then it contributes ",[385,1593,1595],{"className":1594},[388],[385,1596,1598],{"className":1597,"ariaHidden":393},[392],[385,1599,1601,1604],{"className":1600},[397],[385,1602],{"className":1603,"style":441},[401],[385,1605,1607,1610],{"className":1606},[406],[385,1608,598],{"className":1609},[406,407],[385,1611,1613],{"className":1612},[453],[385,1614,1616,1636],{"className":1615},[457,458],[385,1617,1619,1633],{"className":1618},[462],[385,1620,1622],{"className":1621,"style":467},[466],[385,1623,1624,1627],{"style":708},[385,1625],{"className":1626,"style":475},[474],[385,1628,1630],{"className":1629},[479,480,481,482],[385,1631,427],{"className":1632},[406,407,482],[385,1634,490],{"className":1635},[489],[385,1637,1639],{"className":1638},[462],[385,1640,1642],{"className":1641,"style":497},[466],[385,1643],{}," toward the target, and the first\n",[385,1646,1648],{"className":1647},[388],[385,1649,1651,1669],{"className":1650,"ariaHidden":393},[392],[385,1652,1654,1657,1660,1663,1666],{"className":1653},[397],[385,1655],{"className":1656,"style":1441},[401],[385,1658,427],{"className":1659},[406,407],[385,1661],{"className":1662,"style":1448},[665],[385,1664,1453],{"className":1665},[1452],[385,1667],{"className":1668,"style":1448},[665],[385,1670,1672,1675],{"className":1671},[397],[385,1673],{"className":1674,"style":880},[401],[385,1676,604],{"className":1677},[406]," elements must cover the shortfall: ",[385,1680,1682],{"className":1681},[388],[385,1683,1685,1709,1737],{"className":1684,"ariaHidden":393},[392],[385,1686,1688,1691,1694,1697,1700,1703,1706],{"className":1687},[397],[385,1689],{"className":1690,"style":681},[401],[385,1692,1045],{"className":1693},[406,407],[385,1695,1049],{"className":1696},[685],[385,1698,427],{"className":1699},[406,407],[385,1701],{"className":1702,"style":1448},[665],[385,1704,1453],{"className":1705},[1452],[385,1707],{"className":1708,"style":1448},[665],[385,1710,1712,1716,1719,1722,1725,1728,1731,1734],{"className":1711},[397],[385,1713],{"className":1714,"style":1715},[401],"height:0.8389em;vertical-align:-0.1944em;",[385,1717,604],{"className":1718},[406],[385,1720,733],{"className":1721},[732],[385,1723],{"className":1724,"style":737},[665],[385,1726,1062],{"className":1727},[406,407],[385,1729],{"className":1730,"style":1448},[665],[385,1732,1453],{"className":1733},[1452],[385,1735],{"className":1736,"style":1448},[665],[385,1738,1740,1743,1783],{"className":1739},[397],[385,1741],{"className":1742,"style":681},[401],[385,1744,1746,1749],{"className":1745},[406],[385,1747,598],{"className":1748},[406,407],[385,1750,1752],{"className":1751},[453],[385,1753,1755,1775],{"className":1754},[457,458],[385,1756,1758,1772],{"className":1757},[462],[385,1759,1761],{"className":1760,"style":467},[466],[385,1762,1763,1766],{"style":708},[385,1764],{"className":1765,"style":475},[474],[385,1767,1769],{"className":1768},[479,480,481,482],[385,1770,427],{"className":1771},[406,407,482],[385,1773,490],{"className":1774},[489],[385,1776,1778],{"className":1777},[462],[385,1779,1781],{"className":1780,"style":497},[466],[385,1782],{},[385,1784,1066],{"className":1785},[844],[381,1787,1788,1789,1792,1793,1795,1796,1813,1814,1829],{},"The element can be used in ",[1013,1790,1791],{},"either"," way, so the subproblem is true if ",[1013,1794,1791],{},"\nbranch succeeds, a logical ",[385,1797,1799],{"className":1798},[388],[385,1800,1802],{"className":1801,"ariaHidden":393},[392],[385,1803,1805,1809],{"className":1804},[397],[385,1806],{"className":1807,"style":1808},[401],"height:0.5556em;",[385,1810,1812],{"className":1811},[406],"∨",". The base cases pin down the boundary: a\nbudget of ",[385,1815,1817],{"className":1816},[388],[385,1818,1820],{"className":1819,"ariaHidden":393},[392],[385,1821,1823,1826],{"className":1822},[397],[385,1824],{"className":1825,"style":880},[401],[385,1827,884],{"className":1828},[406]," is always reachable by the empty set; a positive budget is\nunreachable with no elements; a negative budget is impossible:",[385,1831,1834],{"className":1832},[1833],"katex-display",[385,1835,1837],{"className":1836},[388],[385,1838,1840,1876],{"className":1839,"ariaHidden":393},[392],[385,1841,1843,1846,1849,1852,1855,1858,1861,1864,1867,1870,1873],{"className":1842},[397],[385,1844],{"className":1845,"style":681},[401],[385,1847,1045],{"className":1848},[406,407],[385,1850,1049],{"className":1851},[685],[385,1853,427],{"className":1854},[406,407],[385,1856,733],{"className":1857},[732],[385,1859],{"className":1860,"style":737},[665],[385,1862,1062],{"className":1863},[406,407],[385,1865,1066],{"className":1866},[844],[385,1868],{"className":1869,"style":666},[665],[385,1871,671],{"className":1872},[670],[385,1874],{"className":1875,"style":666},[665],[385,1877,1879,1883],{"className":1878},[397],[385,1880],{"className":1881,"style":1882},[401],"height:6.66em;vertical-align:-3.08em;",[385,1884,1886,1989,2413],{"className":1885},[788],[385,1887,1889],{"className":1888},[685],[385,1890,1894],{"className":1891},[1892,1893],"delimsizing","mult",[385,1895,1897,1980],{"className":1896},[457,458],[385,1898,1900,1977],{"className":1899},[462],[385,1901,1904,1919,1941,1953,1965],{"className":1902,"style":1903},[466],"height:3.55em;",[385,1905,1907,1911],{"style":1906},"top:-1.366em;",[385,1908],{"className":1909,"style":1910},[474],"height:3.516em;",[385,1912,1916],{"className":1913},[1914,1915],"delimsizinginner","delim-size4",[385,1917,1918],{},"⎩",[385,1920,1922,1925],{"style":1921},"top:-1.358em;",[385,1923],{"className":1924,"style":1910},[474],[385,1926,1928],{"style":1927},"height:1.516em;width:0.8889em;",[1929,1930,1937],"svg",{"xmlns":1931,"width":1932,"height":1933,"style":1934,"viewBox":1935,"preserveAspectRatio":1936},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.8889em","1.516em","width:0.8889em","0 0 888.89 1516","xMinYMin",[1938,1939],"path",{"d":1940},"M384 0 H504 V1516 H384z M384 0 H504 V1516 H384z",[385,1942,1944,1947],{"style":1943},"top:-3.516em;",[385,1945],{"className":1946,"style":1910},[474],[385,1948,1950],{"className":1949},[1914,1915],[385,1951,1952],{},"⎨",[385,1954,1956,1959],{"style":1955},"top:-4.658em;",[385,1957],{"className":1958,"style":1910},[474],[385,1960,1961],{"style":1927},[1929,1962,1963],{"xmlns":1931,"width":1932,"height":1933,"style":1934,"viewBox":1935,"preserveAspectRatio":1936},[1938,1964],{"d":1940},[385,1966,1968,1971],{"style":1967},"top:-6.166em;",[385,1969],{"className":1970,"style":1910},[474],[385,1972,1974],{"className":1973},[1914,1915],[385,1975,1976],{},"⎧",[385,1978,490],{"className":1979},[489],[385,1981,1983],{"className":1982},[462],[385,1984,1987],{"className":1985,"style":1986},[466],"height:3.05em;",[385,1988],{},[385,1990,1992],{"className":1991},[406],[385,1993,1996,2211,2216],{"className":1994},[1995],"mtable",[385,1997,2000],{"className":1998},[1999],"col-align-l",[385,2001,2003,2202],{"className":2002},[457,458],[385,2004,2006,2199],{"className":2005},[462],[385,2007,2010,2026,2041,2056],{"className":2008,"style":2009},[466],"height:3.58em;",[385,2011,2013,2017],{"style":2012},"top:-5.58em;",[385,2014],{"className":2015,"style":2016},[474],"height:3.008em;",[385,2018,2020],{"className":2019},[406],[385,2021,2023],{"className":2022},[406,583],[385,2024,393],{"className":2025},[406],[385,2027,2029,2032],{"style":2028},"top:-3.84em;",[385,2030],{"className":2031,"style":2016},[474],[385,2033,2035],{"className":2034},[406],[385,2036,2038],{"className":2037},[406,583],[385,2039,1270],{"className":2040},[406],[385,2042,2044,2047],{"style":2043},"top:-2.1em;",[385,2045],{"className":2046,"style":2016},[474],[385,2048,2050],{"className":2049},[406],[385,2051,2053],{"className":2052},[406,583],[385,2054,1270],{"className":2055},[406],[385,2057,2059,2062],{"style":2058},"top:-0.36em;",[385,2060],{"className":2061,"style":2016},[474],[385,2063,2065,2068,2071,2074,2077,2080,2083,2086,2089,2092,2095,2098,2102,2105,2108,2111,2114,2117,2120,2123,2126,2129,2132,2135,2138,2141,2144,2147,2150,2153,2156,2196],{"className":2064},[406],[385,2066,1045],{"className":2067},[406,407],[385,2069,1049],{"className":2070},[685],[385,2072,427],{"className":2073},[406,407],[385,2075],{"className":2076,"style":1448},[665],[385,2078,1453],{"className":2079},[1452],[385,2081],{"className":2082,"style":1448},[665],[385,2084,604],{"className":2085},[406],[385,2087,733],{"className":2088},[732],[385,2090],{"className":2091,"style":737},[665],[385,2093,1062],{"className":2094},[406,407],[385,2096,1066],{"className":2097},[844],[385,2099,2101],{"className":2100},[665]," ",[385,2103],{"className":2104,"style":1448},[665],[385,2106,1812],{"className":2107},[1452],[385,2109,2101],{"className":2110},[665],[385,2112],{"className":2113,"style":1448},[665],[385,2115,1045],{"className":2116},[406,407],[385,2118,1049],{"className":2119},[685],[385,2121,427],{"className":2122},[406,407],[385,2124],{"className":2125,"style":1448},[665],[385,2127,1453],{"className":2128},[1452],[385,2130],{"className":2131,"style":1448},[665],[385,2133,604],{"className":2134},[406],[385,2136,733],{"className":2137},[732],[385,2139],{"className":2140,"style":737},[665],[385,2142],{"className":2143,"style":737},[665],[385,2145,1062],{"className":2146},[406,407],[385,2148],{"className":2149,"style":1448},[665],[385,2151,1453],{"className":2152},[1452],[385,2154],{"className":2155,"style":1448},[665],[385,2157,2159,2162],{"className":2158},[406],[385,2160,598],{"className":2161},[406,407],[385,2163,2165],{"className":2164},[453],[385,2166,2168,2188],{"className":2167},[457,458],[385,2169,2171,2185],{"className":2170},[462],[385,2172,2174],{"className":2173,"style":467},[466],[385,2175,2176,2179],{"style":708},[385,2177],{"className":2178,"style":475},[474],[385,2180,2182],{"className":2181},[479,480,481,482],[385,2183,427],{"className":2184},[406,407,482],[385,2186,490],{"className":2187},[489],[385,2189,2191],{"className":2190},[462],[385,2192,2194],{"className":2193,"style":497},[466],[385,2195],{},[385,2197,1066],{"className":2198},[844],[385,2200,490],{"className":2201},[489],[385,2203,2205],{"className":2204},[462],[385,2206,2209],{"className":2207,"style":2208},[466],"height:3.08em;",[385,2210],{},[385,2212],{"className":2213,"style":2215},[2214],"arraycolsep","width:1em;",[385,2217,2219],{"className":2218},[1999],[385,2220,2222,2405],{"className":2221},[457,458],[385,2223,2225,2402],{"className":2224},[462],[385,2226,2228,2261,2294,2348],{"className":2227,"style":2009},[466],[385,2229,2230,2233],{"style":2012},[385,2231],{"className":2232,"style":2016},[474],[385,2234,2236,2243,2246,2249,2252,2255,2258],{"className":2235},[406],[385,2237,2239],{"className":2238},[406,583],[385,2240,2242],{"className":2241},[406],"if ",[385,2244,1062],{"className":2245},[406,407],[385,2247],{"className":2248,"style":666},[665],[385,2250,671],{"className":2251},[670],[385,2253],{"className":2254,"style":666},[665],[385,2256,884],{"className":2257},[406],[385,2259,733],{"className":2260},[732],[385,2262,2263,2266],{"style":2028},[385,2264],{"className":2265,"style":2016},[474],[385,2267,2269,2275,2278,2281,2285,2288,2291],{"className":2268},[406],[385,2270,2272],{"className":2271},[406,583],[385,2273,2242],{"className":2274},[406],[385,2276,1062],{"className":2277},[406,407],[385,2279],{"className":2280,"style":666},[665],[385,2282,2284],{"className":2283},[670],"\u003C",[385,2286],{"className":2287,"style":666},[665],[385,2289,884],{"className":2290},[406],[385,2292,733],{"className":2293},[732],[385,2295,2296,2299],{"style":2043},[385,2297],{"className":2298,"style":2016},[474],[385,2300,2302,2308,2311,2314,2317,2320,2323,2330,2333,2336,2339,2342,2345],{"className":2301},[406],[385,2303,2305],{"className":2304},[406,583],[385,2306,2242],{"className":2307},[406],[385,2309,427],{"className":2310},[406,407],[385,2312],{"className":2313,"style":666},[665],[385,2315,671],{"className":2316},[670],[385,2318],{"className":2319,"style":666},[665],[385,2321,884],{"className":2322},[406],[385,2324,2326],{"className":2325},[406,583],[385,2327,2329],{"className":2328},[406]," and ",[385,2331,1062],{"className":2332},[406,407],[385,2334],{"className":2335,"style":666},[665],[385,2337,870],{"className":2338},[670],[385,2340],{"className":2341,"style":666},[665],[385,2343,884],{"className":2344},[406],[385,2346,733],{"className":2347},[732],[385,2349,2350,2353],{"style":2058},[385,2351],{"className":2352,"style":2016},[474],[385,2354,2356,2362,2365,2368,2371,2374,2377,2380,2383,2386,2389,2392,2395,2398],{"className":2355},[406],[385,2357,2359],{"className":2358},[406,583],[385,2360,2242],{"className":2361},[406],[385,2363,427],{"className":2364},[406,407],[385,2366],{"className":2367,"style":666},[665],[385,2369,870],{"className":2370},[670],[385,2372],{"className":2373,"style":666},[665],[385,2375,884],{"className":2376},[406],[385,2378,733],{"className":2379},[732],[385,2381,2101],{"className":2382},[665],[385,2384],{"className":2385,"style":737},[665],[385,2387,1062],{"className":2388},[406,407],[385,2390],{"className":2391,"style":666},[665],[385,2393,870],{"className":2394},[670],[385,2396],{"className":2397,"style":666},[665],[385,2399,2401],{"className":2400},[406],"0.",[385,2403,490],{"className":2404},[489],[385,2406,2408],{"className":2407},[462],[385,2409,2411],{"className":2410,"style":2208},[466],[385,2412],{},[385,2414],{"className":2415},[844,2416],"nulldelimiter",[381,2418,2419,2420,2423,2424,2457,2458,2510,2511,2514,2515,2530,2531,2585],{},"Every entry depends only on two entries of the ",[1013,2421,2422],{},"previous row"," (",[385,2425,2427],{"className":2426},[388],[385,2428,2430,2448],{"className":2429,"ariaHidden":393},[392],[385,2431,2433,2436,2439,2442,2445],{"className":2432},[397],[385,2434],{"className":2435,"style":1441},[401],[385,2437,427],{"className":2438},[406,407],[385,2440],{"className":2441,"style":1448},[665],[385,2443,1453],{"className":2444},[1452],[385,2446],{"className":2447,"style":1448},[665],[385,2449,2451,2454],{"className":2450},[397],[385,2452],{"className":2453,"style":880},[401],[385,2455,604],{"className":2456},[406],"): the one\ndirectly above (exclude) and the one above and ",[385,2459,2461],{"className":2460},[388],[385,2462,2464],{"className":2463,"ariaHidden":393},[392],[385,2465,2467,2470],{"className":2466},[397],[385,2468],{"className":2469,"style":441},[401],[385,2471,2473,2476],{"className":2472},[406],[385,2474,598],{"className":2475},[406,407],[385,2477,2479],{"className":2478},[453],[385,2480,2482,2502],{"className":2481},[457,458],[385,2483,2485,2499],{"className":2484},[462],[385,2486,2488],{"className":2487,"style":467},[466],[385,2489,2490,2493],{"style":708},[385,2491],{"className":2492,"style":475},[474],[385,2494,2496],{"className":2495},[479,480,481,482],[385,2497,427],{"className":2498},[406,407,482],[385,2500,490],{"className":2501},[489],[385,2503,2505],{"className":2504},[462],[385,2506,2508],{"className":2507,"style":497},[466],[385,2509],{}," columns to the left\n(include). So we fill the table ",[558,2512,2513],{},"row by row"," in increasing ",[385,2516,2518],{"className":2517},[388],[385,2519,2521],{"className":2520,"ariaHidden":393},[392],[385,2522,2524,2527],{"className":2523},[397],[385,2525],{"className":2526,"style":423},[401],[385,2528,427],{"className":2529},[406,407],", sweeping the\nbudget ",[385,2532,2534],{"className":2533},[388],[385,2535,2537,2555],{"className":2536,"ariaHidden":393},[392],[385,2538,2540,2543,2546,2549,2552],{"className":2539},[397],[385,2541],{"className":2542,"style":956},[401],[385,2544,1062],{"className":2545},[406,407],[385,2547],{"className":2548,"style":666},[665],[385,2550,671],{"className":2551},[670],[385,2553],{"className":2554,"style":666},[665],[385,2556,2558,2561,2564,2567,2570,2573,2576,2579,2582],{"className":2557},[397],[385,2559],{"className":2560,"style":1715},[401],[385,2562,884],{"className":2563},[406],[385,2565,733],{"className":2566},[732],[385,2568],{"className":2569,"style":737},[665],[385,2571,789],{"className":2572},[788],[385,2574],{"className":2575,"style":737},[665],[385,2577,733],{"className":2578},[732],[385,2580],{"className":2581,"style":737},[665],[385,2583,863],{"className":2584},[406,407]," within each row.",[2587,2588,2590],"h3",{"id":2589},"the-subset-sum-table-filled","The subset-sum table, filled",[381,2592,2593,2594,2662,2663,2697,2698,2731,2732,2747,2748,2766,2767,2782,2783,2798,2799,2802,2803,2806,2807,2859,2860,2893],{},"Take ",[385,2595,2597],{"className":2596},[388],[385,2598,2600,2618],{"className":2599,"ariaHidden":393},[392],[385,2601,2603,2606,2609,2612,2615],{"className":2602},[397],[385,2604],{"className":2605,"style":402},[401],[385,2607,661],{"className":2608},[406,407],[385,2610],{"className":2611,"style":666},[665],[385,2613,671],{"className":2614},[670],[385,2616],{"className":2617,"style":666},[665],[385,2619,2621,2624,2627,2630,2633,2636,2640,2643,2646,2650,2653,2656,2659],{"className":2620},[397],[385,2622],{"className":2623,"style":681},[401],[385,2625,686],{"className":2626},[685],[385,2628,604],{"className":2629},[406],[385,2631,733],{"className":2632},[732],[385,2634],{"className":2635,"style":737},[665],[385,2637,2639],{"className":2638},[406],"3",[385,2641,733],{"className":2642},[732],[385,2644],{"className":2645,"style":737},[665],[385,2647,2649],{"className":2648},[406],"4",[385,2651,733],{"className":2652},[732],[385,2654],{"className":2655,"style":737},[665],[385,2657,767],{"className":2658},[406],[385,2660,845],{"className":2661},[844]," and target ",[385,2664,2666],{"className":2665},[388],[385,2667,2669,2687],{"className":2668,"ariaHidden":393},[392],[385,2670,2672,2675,2678,2681,2684],{"className":2671},[397],[385,2673],{"className":2674,"style":939},[401],[385,2676,863],{"className":2677},[406,407],[385,2679],{"className":2680,"style":666},[665],[385,2682,671],{"className":2683},[670],[385,2685],{"className":2686,"style":666},[665],[385,2688,2690,2693],{"className":2689},[397],[385,2691],{"className":2692,"style":880},[401],[385,2694,2696],{"className":2695},[406],"6",". The table holds the\nboolean ",[385,2699,2701],{"className":2700},[388],[385,2702,2704],{"className":2703,"ariaHidden":393},[392],[385,2705,2707,2710,2713,2716,2719,2722,2725,2728],{"className":2706},[397],[385,2708],{"className":2709,"style":681},[401],[385,2711,1045],{"className":2712},[406,407],[385,2714,1049],{"className":2715},[685],[385,2717,427],{"className":2718},[406,407],[385,2720,733],{"className":2721},[732],[385,2723],{"className":2724,"style":737},[665],[385,2726,1062],{"className":2727},[406,407],[385,2729,1066],{"className":2730},[844],": row ",[385,2733,2735],{"className":2734},[388],[385,2736,2738],{"className":2737,"ariaHidden":393},[392],[385,2739,2741,2744],{"className":2740},[397],[385,2742],{"className":2743,"style":880},[401],[385,2745,884],{"className":2746},[406]," is ",[385,2749,2751],{"className":2750},[388],[385,2752,2754],{"className":2753,"ariaHidden":393},[392],[385,2755,2757,2760],{"className":2756},[397],[385,2758],{"className":2759,"style":939},[401],[385,2761,2763],{"className":2762},[406,583],[385,2764,393],{"className":2765},[406]," only in column ",[385,2768,2770],{"className":2769},[388],[385,2771,2773],{"className":2772,"ariaHidden":393},[392],[385,2774,2776,2779],{"className":2775},[397],[385,2777],{"className":2778,"style":880},[401],[385,2780,884],{"className":2781},[406]," (the empty set\nsums to ",[385,2784,2786],{"className":2785},[388],[385,2787,2789],{"className":2788,"ariaHidden":393},[392],[385,2790,2792,2795],{"className":2791},[397],[385,2793],{"className":2794,"style":880},[401],[385,2796,884],{"className":2797},[406],"), and each later row ORs the ",[1013,2800,2801],{},"exclude"," cell directly above with the\n",[1013,2804,2805],{},"include"," cell ",[385,2808,2810],{"className":2809},[388],[385,2811,2813],{"className":2812,"ariaHidden":393},[392],[385,2814,2816,2819],{"className":2815},[397],[385,2817],{"className":2818,"style":441},[401],[385,2820,2822,2825],{"className":2821},[406],[385,2823,598],{"className":2824},[406,407],[385,2826,2828],{"className":2827},[453],[385,2829,2831,2851],{"className":2830},[457,458],[385,2832,2834,2848],{"className":2833},[462],[385,2835,2837],{"className":2836,"style":467},[466],[385,2838,2839,2842],{"style":708},[385,2840],{"className":2841,"style":475},[474],[385,2843,2845],{"className":2844},[479,480,481,482],[385,2846,427],{"className":2847},[406,407,482],[385,2849,490],{"className":2850},[489],[385,2852,2854],{"className":2853},[462],[385,2855,2857],{"className":2856,"style":497},[466],[385,2858],{}," columns to its left. The shaded cell ",[385,2861,2863],{"className":2862},[388],[385,2864,2866],{"className":2865,"ariaHidden":393},[392],[385,2867,2869,2872,2875,2878,2881,2884,2887,2890],{"className":2868},[397],[385,2870],{"className":2871,"style":681},[401],[385,2873,1045],{"className":2874},[406,407],[385,2876,1049],{"className":2877},[685],[385,2879,2649],{"className":2880},[406],[385,2882,733],{"className":2883},[732],[385,2885],{"className":2886,"style":737},[665],[385,2888,2696],{"className":2889},[406],[385,2891,1066],{"className":2892},[844]," is the answer;\nthe two arrows show its include\u002Fexclude dependency.",[2895,2896,2900,3821],"figure",{"className":2897},[2898,2899],"tikz-figure","tikz-diagram-rendered",[1929,2901,2905],{"xmlns":1931,"width":2902,"height":2903,"viewBox":2904},"457.716","277.375","-75 -75 343.287 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129.117-64.257Q129.478-65.391 129.478-66.687Q129.478-67.307 129.399-67.898Q129.319-68.489 129.139-69.047Q128.959-69.605 128.658-70.113Q128.357-70.620 127.922-71.016Q127.887-71.051 127.887-71.086Q127.887-71.187 127.975-71.187L128.058-71.187Q128.076-71.187 128.111-71.169Q128.616-70.787 128.990-70.273Q129.363-69.759 129.601-69.181Q129.838-68.603 129.954-67.975Q130.071-67.346 130.071-66.687Q130.071-66.028 129.954-65.397Q129.838-64.767 129.598-64.182Q129.359-63.598 128.988-63.088Q128.616-62.578 128.111-62.196Q128.076-62.178 128.058-62.178",[2921],[3822,3823,3826],"figcaption",{"className":3824},[3825],"tikz-cap","Boolean subset-sum table with the answer cell and its include and exclude predecessors.",[381,3828,3829,3830,3884,3885,3922,3923,3965,3966,889,3968,4020,4021,4075,4076,4115,4116,4131,4132,889,4134,4196,4197,4305,4306,4321],{},"The answer is ",[385,3831,3833],{"className":3832},[388],[385,3834,3836,3872],{"className":3835,"ariaHidden":393},[392],[385,3837,3839,3842,3845,3848,3851,3854,3857,3860,3863,3866,3869],{"className":3838},[397],[385,3840],{"className":3841,"style":681},[401],[385,3843,1045],{"className":3844},[406,407],[385,3846,1049],{"className":3847},[685],[385,3849,2649],{"className":3850},[406],[385,3852,733],{"className":3853},[732],[385,3855],{"className":3856,"style":737},[665],[385,3858,2696],{"className":3859},[406],[385,3861,1066],{"className":3862},[844],[385,3864],{"className":3865,"style":666},[665],[385,3867,671],{"className":3868},[670],[385,3870],{"className":3871,"style":666},[665],[385,3873,3875,3878],{"className":3874},[397],[385,3876],{"className":3877,"style":939},[401],[385,3879,3881],{"className":3880},[406,583],[385,3882,393],{"className":3883},[406],", witnessed by ",[385,3886,3888],{"className":3887},[388],[385,3889,3891],{"className":3890,"ariaHidden":393},[392],[385,3892,3894,3897],{"className":3893},[397],[385,3895],{"className":3896,"style":681},[401],[385,3898,3900,3906,3909,3912,3915,3918],{"className":3899},[788],[385,3901,3905],{"className":3902,"style":3904},[685,3903],"delimcenter","top:0em;","{",[385,3907,2649],{"className":3908},[406],[385,3910,733],{"className":3911},[732],[385,3913],{"className":3914,"style":737},[665],[385,3916,767],{"className":3917},[406],[385,3919,3921],{"className":3920,"style":3904},[844,3903],"}"," or\n",[385,3924,3926],{"className":3925},[388],[385,3927,3929],{"className":3928,"ariaHidden":393},[392],[385,3930,3932,3935],{"className":3931},[397],[385,3933],{"className":3934,"style":681},[401],[385,3936,3938,3941,3944,3947,3950,3953,3956,3959,3962],{"className":3937},[788],[385,3939,3905],{"className":3940,"style":3904},[685,3903],[385,3942,604],{"className":3943},[406],[385,3945,733],{"className":3946},[732],[385,3948],{"className":3949,"style":737},[665],[385,3951,2639],{"className":3952},[406],[385,3954,733],{"className":3955},[732],[385,3957],{"className":3958,"style":737},[665],[385,3960,767],{"className":3961},[406],[385,3963,3921],{"className":3964,"style":3904},[844,3903],". Reading its two dashed predecessors: ",[1013,3967,2801],{},[385,3969,3971],{"className":3970},[388],[385,3972,3974],{"className":3973,"ariaHidden":393},[392],[385,3975,3977,3980],{"className":3976},[397],[385,3978],{"className":3979,"style":441},[401],[385,3981,3983,3986],{"className":3982},[406],[385,3984,598],{"className":3985},[406,407],[385,3987,3989],{"className":3988},[453],[385,3990,3992,4012],{"className":3991},[457,458],[385,3993,3995,4009],{"className":3994},[462],[385,3996,3998],{"className":3997,"style":705},[466],[385,3999,4000,4003],{"style":708},[385,4001],{"className":4002,"style":475},[474],[385,4004,4006],{"className":4005},[479,480,481,482],[385,4007,2649],{"className":4008},[406,482],[385,4010,490],{"className":4011},[489],[385,4013,4015],{"className":4014},[462],[385,4016,4018],{"className":4017,"style":497},[466],[385,4019],{}," asks\n",[385,4022,4024],{"className":4023},[388],[385,4025,4027,4063],{"className":4026,"ariaHidden":393},[392],[385,4028,4030,4033,4036,4039,4042,4045,4048,4051,4054,4057,4060],{"className":4029},[397],[385,4031],{"className":4032,"style":681},[401],[385,4034,1045],{"className":4035},[406,407],[385,4037,1049],{"className":4038},[685],[385,4040,2639],{"className":4041},[406],[385,4043,733],{"className":4044},[732],[385,4046],{"className":4047,"style":737},[665],[385,4049,2696],{"className":4050},[406],[385,4052,1066],{"className":4053},[844],[385,4055],{"className":4056,"style":666},[665],[385,4058,671],{"className":4059},[670],[385,4061],{"className":4062,"style":666},[665],[385,4064,4066,4069],{"className":4065},[397],[385,4067],{"className":4068,"style":574},[401],[385,4070,4072],{"className":4071},[406,583],[385,4073,1270],{"className":4074},[406]," (no subset of ",[385,4077,4079],{"className":4078},[388],[385,4080,4082],{"className":4081,"ariaHidden":393},[392],[385,4083,4085,4088,4091,4094,4097,4100,4103,4106,4109,4112],{"className":4084},[397],[385,4086],{"className":4087,"style":681},[401],[385,4089,686],{"className":4090},[685],[385,4092,604],{"className":4093},[406],[385,4095,733],{"className":4096},[732],[385,4098],{"className":4099,"style":737},[665],[385,4101,2639],{"className":4102},[406],[385,4104,733],{"className":4105},[732],[385,4107],{"className":4108,"style":737},[665],[385,4110,2649],{"className":4111},[406],[385,4113,845],{"className":4114},[844]," hits ",[385,4117,4119],{"className":4118},[388],[385,4120,4122],{"className":4121,"ariaHidden":393},[392],[385,4123,4125,4128],{"className":4124},[397],[385,4126],{"className":4127,"style":880},[401],[385,4129,2696],{"className":4130},[406],"), while\n",[1013,4133,2805],{},[385,4135,4137],{"className":4136},[388],[385,4138,4140],{"className":4139,"ariaHidden":393},[392],[385,4141,4143,4147,4187,4193],{"className":4142},[397],[385,4144],{"className":4145,"style":4146},[401],"height:0.7944em;vertical-align:-0.15em;",[385,4148,4150,4153],{"className":4149},[406],[385,4151,598],{"className":4152},[406,407],[385,4154,4156],{"className":4155},[453],[385,4157,4159,4179],{"className":4158},[457,458],[385,4160,4162,4176],{"className":4161},[462],[385,4163,4165],{"className":4164,"style":705},[466],[385,4166,4167,4170],{"style":708},[385,4168],{"className":4169,"style":475},[474],[385,4171,4173],{"className":4172},[479,480,481,482],[385,4174,2649],{"className":4175},[406,482],[385,4177,490],{"className":4178},[489],[385,4180,4182],{"className":4181},[462],[385,4183,4185],{"className":4184,"style":497},[466],[385,4186],{},[385,4188,4190],{"className":4189},[406],[385,4191,671],{"className":4192},[670],[385,4194,767],{"className":4195},[406]," asks ",[385,4198,4200],{"className":4199},[388],[385,4201,4203,4236,4257,4293],{"className":4202,"ariaHidden":393},[392],[385,4204,4206,4209,4212,4215,4218,4221,4224,4227,4230,4233],{"className":4205},[397],[385,4207],{"className":4208,"style":681},[401],[385,4210,1045],{"className":4211},[406,407],[385,4213,1049],{"className":4214},[685],[385,4216,2639],{"className":4217},[406],[385,4219,733],{"className":4220},[732],[385,4222],{"className":4223,"style":737},[665],[385,4225,2696],{"className":4226},[406],[385,4228],{"className":4229,"style":1448},[665],[385,4231,1453],{"className":4232},[1452],[385,4234],{"className":4235,"style":1448},[665],[385,4237,4239,4242,4245,4248,4251,4254],{"className":4238},[397],[385,4240],{"className":4241,"style":681},[401],[385,4243,767],{"className":4244},[406],[385,4246,1066],{"className":4247},[844],[385,4249],{"className":4250,"style":666},[665],[385,4252,671],{"className":4253},[670],[385,4255],{"className":4256,"style":666},[665],[385,4258,4260,4263,4266,4269,4272,4275,4278,4281,4284,4287,4290],{"className":4259},[397],[385,4261],{"className":4262,"style":681},[401],[385,4264,1045],{"className":4265},[406,407],[385,4267,1049],{"className":4268},[685],[385,4270,2639],{"className":4271},[406],[385,4273,733],{"className":4274},[732],[385,4276],{"className":4277,"style":737},[665],[385,4279,2649],{"className":4280},[406],[385,4282,1066],{"className":4283},[844],[385,4285],{"className":4286,"style":666},[665],[385,4288,671],{"className":4289},[670],[385,4291],{"className":4292,"style":666},[665],[385,4294,4296,4299],{"className":4295},[397],[385,4297],{"className":4298,"style":939},[401],[385,4300,4302],{"className":4301},[406,583],[385,4303,393],{"className":4304},[406],"; the ",[385,4307,4309],{"className":4308},[388],[385,4310,4312],{"className":4311,"ariaHidden":393},[392],[385,4313,4315,4318],{"className":4314},[397],[385,4316],{"className":4317,"style":1808},[401],[385,4319,1812],{"className":4320},[406]," makes\nthe cell true through the include branch.",[2587,4323,4325],{"id":4324},"subset-sum-in-pseudocode","Subset-sum in pseudocode",[4327,4328,4332],"pre",{"className":4329,"code":4330,"language":4331,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Subset-Sum}(L[1..n], t)$ — does some sublist sum to $t$?\nnumber: 1\n$A[0..n][0..t] \\gets \\text{false}$\n$A[0..n][0] \\gets \\text{true}$ \u002F\u002F empty set reaches $0$\nfor $i \\gets 1$ to $n$ do\n  for $u \\gets 1$ to $t$ do\n    if $a_i > u$ then\n      $A[i][u] \\gets A[i-1][u]$ \u002F\u002F $a_i$ too big: exclude\n    else\n      $A[i][u] \\gets A[i-1][u] \\lor A[i-1][u - a_i]$ \u002F\u002F exclude or include\nreturn $A[n][t]$\n","algorithm",[4333,4334,4335,4341,4346,4351,4356,4361,4366,4371,4376,4381,4386],"code",{"__ignoreMap":376},[385,4336,4338],{"class":4337,"line":6},"line",[385,4339,4340],{},"caption: $\\textsc{Subset-Sum}(L[1..n], t)$ — does some sublist sum to $t$?\n",[385,4342,4343],{"class":4337,"line":18},[385,4344,4345],{},"number: 1\n",[385,4347,4348],{"class":4337,"line":24},[385,4349,4350],{},"$A[0..n][0..t] \\gets \\text{false}$\n",[385,4352,4353],{"class":4337,"line":73},[385,4354,4355],{},"$A[0..n][0] \\gets \\text{true}$ \u002F\u002F empty set reaches $0$\n",[385,4357,4358],{"class":4337,"line":102},[385,4359,4360],{},"for $i \\gets 1$ to $n$ do\n",[385,4362,4363],{"class":4337,"line":108},[385,4364,4365],{},"  for $u \\gets 1$ to $t$ do\n",[385,4367,4368],{"class":4337,"line":116},[385,4369,4370],{},"    if $a_i > u$ then\n",[385,4372,4373],{"class":4337,"line":196},[385,4374,4375],{},"      $A[i][u] \\gets A[i-1][u]$ \u002F\u002F $a_i$ too big: exclude\n",[385,4377,4378],{"class":4337,"line":202},[385,4379,4380],{},"    else\n",[385,4382,4383],{"class":4337,"line":283},[385,4384,4385],{},"      $A[i][u] \\gets A[i-1][u] \\lor A[i-1][u - a_i]$ \u002F\u002F exclude or include\n",[385,4387,4388],{"class":4337,"line":333},[385,4389,4390],{},"return $A[n][t]$\n",[381,4392,4393,4394,4461,4462,4487,4488,4515,4516,4543,4544,4568,4569,4596],{},"This fills ",[385,4395,4397],{"className":4396},[388],[385,4398,4400,4422,4449],{"className":4399,"ariaHidden":393},[392],[385,4401,4403,4406,4409,4412,4415,4419],{"className":4402},[397],[385,4404],{"className":4405,"style":681},[401],[385,4407,1049],{"className":4408},[685],[385,4410,829],{"className":4411},[406,407],[385,4413],{"className":4414,"style":1448},[665],[385,4416,4418],{"className":4417},[1452],"+",[385,4420],{"className":4421,"style":1448},[665],[385,4423,4425,4428,4431,4434,4437,4440,4443,4446],{"className":4424},[397],[385,4426],{"className":4427,"style":681},[401],[385,4429,604],{"className":4430},[406],[385,4432,1066],{"className":4433},[844],[385,4435,1049],{"className":4436},[685],[385,4438,863],{"className":4439},[406,407],[385,4441],{"className":4442,"style":1448},[665],[385,4444,4418],{"className":4445},[1452],[385,4447],{"className":4448,"style":1448},[665],[385,4450,4452,4455,4458],{"className":4451},[397],[385,4453],{"className":4454,"style":681},[401],[385,4456,604],{"className":4457},[406],[385,4459,1066],{"className":4460},[844]," boolean cells in ",[385,4463,4465],{"className":4464},[388],[385,4466,4468],{"className":4467,"ariaHidden":393},[392],[385,4469,4471,4474,4478,4481,4484],{"className":4470},[397],[385,4472],{"className":4473,"style":681},[401],[385,4475,4477],{"className":4476},[406],"Θ",[385,4479,1049],{"className":4480},[685],[385,4482,604],{"className":4483},[406],[385,4485,1066],{"className":4486},[844]," apiece, for ",[385,4489,4491],{"className":4490},[388],[385,4492,4494],{"className":4493,"ariaHidden":393},[392],[385,4495,4497,4500,4503,4506,4509,4512],{"className":4496},[397],[385,4498],{"className":4499,"style":681},[401],[385,4501,4477],{"className":4502},[406],[385,4504,1049],{"className":4505},[685],[385,4507,829],{"className":4508},[406,407],[385,4510,863],{"className":4511},[406,407],[385,4513,1066],{"className":4514},[844],"\ntime and ",[385,4517,4519],{"className":4518},[388],[385,4520,4522],{"className":4521,"ariaHidden":393},[392],[385,4523,4525,4528,4531,4534,4537,4540],{"className":4524},[397],[385,4526],{"className":4527,"style":681},[401],[385,4529,4477],{"className":4530},[406],[385,4532,1049],{"className":4533},[685],[385,4535,829],{"className":4536},[406,407],[385,4538,863],{"className":4539},[406,407],[385,4541,1066],{"className":4542},[844]," space, and the space drops to ",[385,4545,4547],{"className":4546},[388],[385,4548,4550],{"className":4549,"ariaHidden":393},[392],[385,4551,4553,4556,4559,4562,4565],{"className":4552},[397],[385,4554],{"className":4555,"style":681},[401],[385,4557,4477],{"className":4558},[406],[385,4560,1049],{"className":4561},[685],[385,4563,863],{"className":4564},[406,407],[385,4566,1066],{"className":4567},[844]," because each row\nreads only the one above it. We will return to that innocent-looking ",[385,4570,4572],{"className":4571},[388],[385,4573,4575],{"className":4574,"ariaHidden":393},[392],[385,4576,4578,4581,4584,4587,4590,4593],{"className":4577},[397],[385,4579],{"className":4580,"style":681},[401],[385,4582,4477],{"className":4583},[406],[385,4585,1049],{"className":4586},[685],[385,4588,829],{"className":4589},[406,407],[385,4591,863],{"className":4592},[406,407],[385,4594,1066],{"className":4595},[844],"\nbelow, since it hides a trap. First, let us see that 0\u002F1 knapsack is the same machine\nwith values bolted on.",[632,4598,4600],{"id":4599},"the-problem-and-why-greed-fails","The problem and why greed fails",[637,4602,4603],{"type":639},[381,4604,4605,889,4607,4622,4623,4733,4734,4844,4845,885,4860,4863,4864,4931,4932,5065,5066,1427],{},[558,4606,644],{},[385,4608,4610],{"className":4609},[388],[385,4611,4613],{"className":4612,"ariaHidden":393},[392],[385,4614,4616,4619],{"className":4615},[397],[385,4617],{"className":4618,"style":956},[401],[385,4620,829],{"className":4621},[406,407]," items with positive integer weights ",[385,4624,4626],{"className":4625},[388],[385,4627,4629],{"className":4628,"ariaHidden":393},[392],[385,4630,4632,4635,4675,4678,4681,4684,4687,4690,4693],{"className":4631},[397],[385,4633],{"className":4634,"style":902},[401],[385,4636,4638,4641],{"className":4637},[406],[385,4639,449],{"className":4640,"style":448},[406,407],[385,4642,4644],{"className":4643},[453],[385,4645,4647,4667],{"className":4646},[457,458],[385,4648,4650,4664],{"className":4649},[462],[385,4651,4653],{"className":4652,"style":705},[466],[385,4654,4655,4658],{"style":470},[385,4656],{"className":4657,"style":475},[474],[385,4659,4661],{"className":4660},[479,480,481,482],[385,4662,604],{"className":4663},[406,482],[385,4665,490],{"className":4666},[489],[385,4668,4670],{"className":4669},[462],[385,4671,4673],{"className":4672,"style":497},[466],[385,4674],{},[385,4676,733],{"className":4677},[732],[385,4679],{"className":4680,"style":737},[665],[385,4682,789],{"className":4683},[788],[385,4685],{"className":4686,"style":737},[665],[385,4688,733],{"className":4689},[732],[385,4691],{"className":4692,"style":737},[665],[385,4694,4696,4699],{"className":4695},[406],[385,4697,449],{"className":4698,"style":448},[406,407],[385,4700,4702],{"className":4701},[453],[385,4703,4705,4725],{"className":4704},[457,458],[385,4706,4708,4722],{"className":4707},[462],[385,4709,4711],{"className":4710,"style":817},[466],[385,4712,4713,4716],{"style":470},[385,4714],{"className":4715,"style":475},[474],[385,4717,4719],{"className":4718},[479,480,481,482],[385,4720,829],{"className":4721},[406,407,482],[385,4723,490],{"className":4724},[489],[385,4726,4728],{"className":4727},[462],[385,4729,4731],{"className":4730,"style":497},[466],[385,4732],{}," and values\n",[385,4735,4737],{"className":4736},[388],[385,4738,4740],{"className":4739,"ariaHidden":393},[392],[385,4741,4743,4746,4786,4789,4792,4795,4798,4801,4804],{"className":4742},[397],[385,4744],{"className":4745,"style":902},[401],[385,4747,4749,4752],{"className":4748},[406],[385,4750,520],{"className":4751,"style":519},[406,407],[385,4753,4755],{"className":4754},[453],[385,4756,4758,4778],{"className":4757},[457,458],[385,4759,4761,4775],{"className":4760},[462],[385,4762,4764],{"className":4763,"style":705},[466],[385,4765,4766,4769],{"style":535},[385,4767],{"className":4768,"style":475},[474],[385,4770,4772],{"className":4771},[479,480,481,482],[385,4773,604],{"className":4774},[406,482],[385,4776,490],{"className":4777},[489],[385,4779,4781],{"className":4780},[462],[385,4782,4784],{"className":4783,"style":497},[466],[385,4785],{},[385,4787,733],{"className":4788},[732],[385,4790],{"className":4791,"style":737},[665],[385,4793,789],{"className":4794},[788],[385,4796],{"className":4797,"style":737},[665],[385,4799,733],{"className":4800},[732],[385,4802],{"className":4803,"style":737},[665],[385,4805,4807,4810],{"className":4806},[406],[385,4808,520],{"className":4809,"style":519},[406,407],[385,4811,4813],{"className":4812},[453],[385,4814,4816,4836],{"className":4815},[457,458],[385,4817,4819,4833],{"className":4818},[462],[385,4820,4822],{"className":4821,"style":817},[466],[385,4823,4824,4827],{"style":535},[385,4825],{"className":4826,"style":475},[474],[385,4828,4830],{"className":4829},[479,480,481,482],[385,4831,829],{"className":4832},[406,407,482],[385,4834,490],{"className":4835},[489],[385,4837,4839],{"className":4838},[462],[385,4840,4842],{"className":4841,"style":497},[466],[385,4843],{},", and an integer capacity ",[385,4846,4848],{"className":4847},[388],[385,4849,4851],{"className":4850,"ariaHidden":393},[392],[385,4852,4854,4857],{"className":4853},[397],[385,4855],{"className":4856,"style":402},[401],[385,4858,409],{"className":4859,"style":408},[406,407],[558,4861,4862],{},"Output:"," a subset ",[385,4865,4867],{"className":4866},[388],[385,4868,4870,4892],{"className":4869,"ariaHidden":393},[392],[385,4871,4873,4877,4882,4885,4889],{"className":4872},[397],[385,4874],{"className":4875,"style":4876},[401],"height:0.8193em;vertical-align:-0.136em;",[385,4878,4881],{"className":4879,"style":4880},[406,407],"margin-right:0.0576em;","S",[385,4883],{"className":4884,"style":666},[665],[385,4886,4888],{"className":4887},[670],"⊆",[385,4890],{"className":4891,"style":666},[665],[385,4893,4895,4898],{"className":4894},[397],[385,4896],{"className":4897,"style":681},[401],[385,4899,4901,4904,4907,4910,4913,4916,4919,4922,4925,4928],{"className":4900},[788],[385,4902,3905],{"className":4903,"style":3904},[685,3903],[385,4905,604],{"className":4906},[406],[385,4908,733],{"className":4909},[732],[385,4911],{"className":4912,"style":737},[665],[385,4914,789],{"className":4915},[788],[385,4917],{"className":4918,"style":737},[665],[385,4920,733],{"className":4921},[732],[385,4923],{"className":4924,"style":737},[665],[385,4926,829],{"className":4927},[406,407],[385,4929,3921],{"className":4930,"style":3904},[844,3903]," with ",[385,4933,4935],{"className":4934},[388],[385,4936,4938,5056],{"className":4937,"ariaHidden":393},[392],[385,4939,4941,4945,5003,5006,5046,5049,5053],{"className":4940},[397],[385,4942],{"className":4943,"style":4944},[401],"height:1.0771em;vertical-align:-0.3271em;",[385,4946,4949,4956],{"className":4947},[4948],"mop",[385,4950,4955],{"className":4951,"style":4954},[4948,4952,4953],"op-symbol","small-op","position:relative;top:0em;","∑",[385,4957,4959],{"className":4958},[453],[385,4960,4962,4994],{"className":4961},[457,458],[385,4963,4965,4991],{"className":4964},[462],[385,4966,4969],{"className":4967,"style":4968},[466],"height:0.1786em;",[385,4970,4972,4975],{"style":4971},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[385,4973],{"className":4974,"style":475},[474],[385,4976,4978],{"className":4977},[479,480,481,482],[385,4979,4981,4984,4988],{"className":4980},[406,482],[385,4982,427],{"className":4983},[406,407,482],[385,4985,4987],{"className":4986},[670,482],"∈",[385,4989,4881],{"className":4990,"style":4880},[406,407,482],[385,4992,490],{"className":4993},[489],[385,4995,4997],{"className":4996},[462],[385,4998,5001],{"className":4999,"style":5000},[466],"height:0.3271em;",[385,5002],{},[385,5004],{"className":5005,"style":737},[665],[385,5007,5009,5012],{"className":5008},[406],[385,5010,449],{"className":5011,"style":448},[406,407],[385,5013,5015],{"className":5014},[453],[385,5016,5018,5038],{"className":5017},[457,458],[385,5019,5021,5035],{"className":5020},[462],[385,5022,5024],{"className":5023,"style":467},[466],[385,5025,5026,5029],{"style":470},[385,5027],{"className":5028,"style":475},[474],[385,5030,5032],{"className":5031},[479,480,481,482],[385,5033,427],{"className":5034},[406,407,482],[385,5036,490],{"className":5037},[489],[385,5039,5041],{"className":5040},[462],[385,5042,5044],{"className":5043,"style":497},[466],[385,5045],{},[385,5047],{"className":5048,"style":666},[665],[385,5050,5052],{"className":5051},[670],"≤",[385,5054],{"className":5055,"style":666},[665],[385,5057,5059,5062],{"className":5058},[397],[385,5060],{"className":5061,"style":402},[401],[385,5063,409],{"className":5064,"style":408},[406,407]," maximizing ",[385,5067,5069],{"className":5068},[388],[385,5070,5072],{"className":5071,"ariaHidden":393},[392],[385,5073,5075,5078,5127,5130],{"className":5074},[397],[385,5076],{"className":5077,"style":4944},[401],[385,5079,5081,5084],{"className":5080},[4948],[385,5082,4955],{"className":5083,"style":4954},[4948,4952,4953],[385,5085,5087],{"className":5086},[453],[385,5088,5090,5119],{"className":5089},[457,458],[385,5091,5093,5116],{"className":5092},[462],[385,5094,5096],{"className":5095,"style":4968},[466],[385,5097,5098,5101],{"style":4971},[385,5099],{"className":5100,"style":475},[474],[385,5102,5104],{"className":5103},[479,480,481,482],[385,5105,5107,5110,5113],{"className":5106},[406,482],[385,5108,427],{"className":5109},[406,407,482],[385,5111,4987],{"className":5112},[670,482],[385,5114,4881],{"className":5115,"style":4880},[406,407,482],[385,5117,490],{"className":5118},[489],[385,5120,5122],{"className":5121},[462],[385,5123,5125],{"className":5124,"style":5000},[466],[385,5126],{},[385,5128],{"className":5129,"style":737},[665],[385,5131,5133,5136],{"className":5132},[406],[385,5134,520],{"className":5135,"style":519},[406,407],[385,5137,5139],{"className":5138},[453],[385,5140,5142,5162],{"className":5141},[457,458],[385,5143,5145,5159],{"className":5144},[462],[385,5146,5148],{"className":5147,"style":467},[466],[385,5149,5150,5153],{"style":535},[385,5151],{"className":5152,"style":475},[474],[385,5154,5156],{"className":5155},[479,480,481,482],[385,5157,427],{"className":5158},[406,407,482],[385,5160,490],{"className":5161},[489],[385,5163,5165],{"className":5164},[462],[385,5166,5168],{"className":5167,"style":497},[466],[385,5169],{},[381,5171,5172,5173,5176,5228,5229,5281,5282,5297,5298,5301,5302,5305,5306,5321,5322,5342,5343,5505,5506,5559,5560,5564],{},"Knapsack is subset-sum with two upgrades: each item carries a separate ",[558,5174,5175],{},"value",[385,5177,5179],{"className":5178},[388],[385,5180,5182],{"className":5181,"ariaHidden":393},[392],[385,5183,5185,5188],{"className":5184},[397],[385,5186],{"className":5187,"style":441},[401],[385,5189,5191,5194],{"className":5190},[406],[385,5192,520],{"className":5193,"style":519},[406,407],[385,5195,5197],{"className":5196},[453],[385,5198,5200,5220],{"className":5199},[457,458],[385,5201,5203,5217],{"className":5202},[462],[385,5204,5206],{"className":5205,"style":467},[466],[385,5207,5208,5211],{"style":535},[385,5209],{"className":5210,"style":475},[474],[385,5212,5214],{"className":5213},[479,480,481,482],[385,5215,427],{"className":5216},[406,407,482],[385,5218,490],{"className":5219},[489],[385,5221,5223],{"className":5222},[462],[385,5224,5226],{"className":5225,"style":497},[466],[385,5227],{}," as well as a weight ",[385,5230,5232],{"className":5231},[388],[385,5233,5235],{"className":5234,"ariaHidden":393},[392],[385,5236,5238,5241],{"className":5237},[397],[385,5239],{"className":5240,"style":441},[401],[385,5242,5244,5247],{"className":5243},[406],[385,5245,449],{"className":5246,"style":448},[406,407],[385,5248,5250],{"className":5249},[453],[385,5251,5253,5273],{"className":5252},[457,458],[385,5254,5256,5270],{"className":5255},[462],[385,5257,5259],{"className":5258,"style":467},[466],[385,5260,5261,5264],{"style":470},[385,5262],{"className":5263,"style":475},[474],[385,5265,5267],{"className":5266},[479,480,481,482],[385,5268,427],{"className":5269},[406,407,482],[385,5271,490],{"className":5272},[489],[385,5274,5276],{"className":5275},[462],[385,5277,5279],{"className":5278,"style":497},[466],[385,5280],{},", and the capacity ",[385,5283,5285],{"className":5284},[388],[385,5286,5288],{"className":5287,"ariaHidden":393},[392],[385,5289,5291,5294],{"className":5290},[397],[385,5292],{"className":5293,"style":402},[401],[385,5295,409],{"className":5296,"style":408},[406,407]," is an ",[1013,5299,5300],{},"upper bound"," rather\nthan an exact target, so we ",[558,5303,5304],{},"maximize"," value instead of answering yes\u002Fno. The\ninclude\u002Fexclude decision is unchanged: the ",[385,5307,5309],{"className":5308},[388],[385,5310,5312],{"className":5311,"ariaHidden":393},[392],[385,5313,5315,5318],{"className":5314},[397],[385,5316],{"className":5317,"style":1808},[401],[385,5319,1812],{"className":5320},[406]," of subset-sum becomes a ",[385,5323,5325],{"className":5324},[388],[385,5326,5328],{"className":5327,"ariaHidden":393},[392],[385,5329,5331,5334],{"className":5330},[397],[385,5332],{"className":5333,"style":956},[401],[385,5335,5337],{"className":5336},[4948],[385,5338,5341],{"className":5339},[406,5340],"mathrm","max",",\nand the boolean cell becomes a value. (Set ",[385,5344,5346],{"className":5345},[388],[385,5347,5349,5404,5459],{"className":5348,"ariaHidden":393},[392],[385,5350,5352,5355,5395,5398,5401],{"className":5351},[397],[385,5353],{"className":5354,"style":441},[401],[385,5356,5358,5361],{"className":5357},[406],[385,5359,520],{"className":5360,"style":519},[406,407],[385,5362,5364],{"className":5363},[453],[385,5365,5367,5387],{"className":5366},[457,458],[385,5368,5370,5384],{"className":5369},[462],[385,5371,5373],{"className":5372,"style":467},[466],[385,5374,5375,5378],{"style":535},[385,5376],{"className":5377,"style":475},[474],[385,5379,5381],{"className":5380},[479,480,481,482],[385,5382,427],{"className":5383},[406,407,482],[385,5385,490],{"className":5386},[489],[385,5388,5390],{"className":5389},[462],[385,5391,5393],{"className":5392,"style":497},[466],[385,5394],{},[385,5396],{"className":5397,"style":666},[665],[385,5399,671],{"className":5400},[670],[385,5402],{"className":5403,"style":666},[665],[385,5405,5407,5410,5450,5453,5456],{"className":5406},[397],[385,5408],{"className":5409,"style":441},[401],[385,5411,5413,5416],{"className":5412},[406],[385,5414,449],{"className":5415,"style":448},[406,407],[385,5417,5419],{"className":5418},[453],[385,5420,5422,5442],{"className":5421},[457,458],[385,5423,5425,5439],{"className":5424},[462],[385,5426,5428],{"className":5427,"style":467},[466],[385,5429,5430,5433],{"style":470},[385,5431],{"className":5432,"style":475},[474],[385,5434,5436],{"className":5435},[479,480,481,482],[385,5437,427],{"className":5438},[406,407,482],[385,5440,490],{"className":5441},[489],[385,5443,5445],{"className":5444},[462],[385,5446,5448],{"className":5447,"style":497},[466],[385,5449],{},[385,5451],{"className":5452,"style":666},[665],[385,5454,671],{"className":5455},[670],[385,5457],{"className":5458,"style":666},[665],[385,5460,5462,5465],{"className":5461},[397],[385,5463],{"className":5464,"style":441},[401],[385,5466,5468,5471],{"className":5467},[406],[385,5469,598],{"className":5470},[406,407],[385,5472,5474],{"className":5473},[453],[385,5475,5477,5497],{"className":5476},[457,458],[385,5478,5480,5494],{"className":5479},[462],[385,5481,5483],{"className":5482,"style":467},[466],[385,5484,5485,5488],{"style":708},[385,5486],{"className":5487,"style":475},[474],[385,5489,5491],{"className":5490},[479,480,481,482],[385,5492,427],{"className":5493},[406,407,482],[385,5495,490],{"className":5496},[489],[385,5498,5500],{"className":5499},[462],[385,5501,5503],{"className":5502,"style":497},[466],[385,5504],{}," and the answer\n",[385,5507,5509],{"className":5508},[388],[385,5510,5512,5550],{"className":5511,"ariaHidden":393},[392],[385,5513,5515,5518,5523,5526,5529,5532,5535,5538,5541,5544,5547],{"className":5514},[397],[385,5516],{"className":5517,"style":681},[401],[385,5519,5522],{"className":5520,"style":5521},[406,407],"margin-right:0.0715em;","K",[385,5524,1049],{"className":5525},[685],[385,5527,829],{"className":5528},[406,407],[385,5530,733],{"className":5531},[732],[385,5533],{"className":5534,"style":737},[665],[385,5536,863],{"className":5537},[406,407],[385,5539,1066],{"className":5540},[844],[385,5542],{"className":5543,"style":666},[665],[385,5545,671],{"className":5546},[670],[385,5548],{"className":5549,"style":666},[665],[385,5551,5553,5556],{"className":5552},[397],[385,5554],{"className":5555,"style":939},[401],[385,5557,863],{"className":5558},[406,407]," recovers subset-sum exactly, a reduction we make precise\n",[598,5561,5563],{"href":5562},"#subset-sum-as-a-special-case","below",".)",[381,5566,5567,5568,5664,5665,5668,5669,5703,5704,5770,5771,5770,5840,5907,5908,5923,5924,5940,5941,5956,5957,5972,5973,5988,5989,6005,6006,6009],{},"The tempting greedy idea, taking items in decreasing order of value-per-weight\nratio ",[385,5569,5571],{"className":5570},[388],[385,5572,5574],{"className":5573,"ariaHidden":393},[392],[385,5575,5577,5580,5620,5624],{"className":5576},[397],[385,5578],{"className":5579,"style":681},[401],[385,5581,5583,5586],{"className":5582},[406],[385,5584,520],{"className":5585,"style":519},[406,407],[385,5587,5589],{"className":5588},[453],[385,5590,5592,5612],{"className":5591},[457,458],[385,5593,5595,5609],{"className":5594},[462],[385,5596,5598],{"className":5597,"style":467},[466],[385,5599,5600,5603],{"style":535},[385,5601],{"className":5602,"style":475},[474],[385,5604,5606],{"className":5605},[479,480,481,482],[385,5607,427],{"className":5608},[406,407,482],[385,5610,490],{"className":5611},[489],[385,5613,5615],{"className":5614},[462],[385,5616,5618],{"className":5617,"style":497},[466],[385,5619],{},[385,5621,5623],{"className":5622},[406],"\u002F",[385,5625,5627,5630],{"className":5626},[406],[385,5628,449],{"className":5629,"style":448},[406,407],[385,5631,5633],{"className":5632},[453],[385,5634,5636,5656],{"className":5635},[457,458],[385,5637,5639,5653],{"className":5638},[462],[385,5640,5642],{"className":5641,"style":467},[466],[385,5643,5644,5647],{"style":470},[385,5645],{"className":5646,"style":475},[474],[385,5648,5650],{"className":5649},[479,480,481,482],[385,5651,427],{"className":5652},[406,407,482],[385,5654,490],{"className":5655},[489],[385,5657,5659],{"className":5658},[462],[385,5660,5662],{"className":5661,"style":497},[466],[385,5663],{},", is ",[1013,5666,5667],{},"wrong"," for the 0\u002F1 problem. Suppose ",[385,5670,5672],{"className":5671},[388],[385,5673,5675,5693],{"className":5674,"ariaHidden":393},[392],[385,5676,5678,5681,5684,5687,5690],{"className":5677},[397],[385,5679],{"className":5680,"style":402},[401],[385,5682,409],{"className":5683,"style":408},[406,407],[385,5685],{"className":5686,"style":666},[665],[385,5688,671],{"className":5689},[670],[385,5691],{"className":5692,"style":666},[665],[385,5694,5696,5699],{"className":5695},[397],[385,5697],{"className":5698,"style":880},[401],[385,5700,5702],{"className":5701},[406],"10"," and the\nitems are ",[385,5705,5707],{"className":5706},[388],[385,5708,5710,5728],{"className":5709,"ariaHidden":393},[392],[385,5711,5713,5716,5719,5722,5725],{"className":5712},[397],[385,5714],{"className":5715,"style":402},[401],[385,5717,1045],{"className":5718},[406,407],[385,5720],{"className":5721,"style":666},[665],[385,5723,671],{"className":5724},[670],[385,5726],{"className":5727,"style":666},[665],[385,5729,5731,5734,5737,5740,5746,5749,5752,5755,5758,5764,5767],{"className":5730},[397],[385,5732],{"className":5733,"style":681},[401],[385,5735,1049],{"className":5736},[685],[385,5738,449],{"className":5739,"style":448},[406,407],[385,5741,5743],{"className":5742},[406],[385,5744,671],{"className":5745},[670],[385,5747,2696],{"className":5748},[406],[385,5750,733],{"className":5751},[732],[385,5753],{"className":5754,"style":737},[665],[385,5756,520],{"className":5757,"style":519},[406,407],[385,5759,5761],{"className":5760},[406],[385,5762,671],{"className":5763},[670],[385,5765,5702],{"className":5766},[406],[385,5768,1066],{"className":5769},[844],", ",[385,5772,5774],{"className":5773},[388],[385,5775,5777,5797],{"className":5776,"ariaHidden":393},[392],[385,5778,5780,5783,5788,5791,5794],{"className":5779},[397],[385,5781],{"className":5782,"style":402},[401],[385,5784,5787],{"className":5785,"style":5786},[406,407],"margin-right:0.0502em;","B",[385,5789],{"className":5790,"style":666},[665],[385,5792,671],{"className":5793},[670],[385,5795],{"className":5796,"style":666},[665],[385,5798,5800,5803,5806,5809,5815,5819,5822,5825,5828,5834,5837],{"className":5799},[397],[385,5801],{"className":5802,"style":681},[401],[385,5804,1049],{"className":5805},[685],[385,5807,449],{"className":5808,"style":448},[406,407],[385,5810,5812],{"className":5811},[406],[385,5813,671],{"className":5814},[670],[385,5816,5818],{"className":5817},[406],"5",[385,5820,733],{"className":5821},[732],[385,5823],{"className":5824,"style":737},[665],[385,5826,520],{"className":5827,"style":519},[406,407],[385,5829,5831],{"className":5830},[406],[385,5832,671],{"className":5833},[670],[385,5835,2913],{"className":5836},[406],[385,5838,1066],{"className":5839},[844],[385,5841,5843],{"className":5842},[388],[385,5844,5846,5865],{"className":5845,"ariaHidden":393},[392],[385,5847,5849,5852,5856,5859,5862],{"className":5848},[397],[385,5850],{"className":5851,"style":402},[401],[385,5853,5855],{"className":5854,"style":5521},[406,407],"C",[385,5857],{"className":5858,"style":666},[665],[385,5860,671],{"className":5861},[670],[385,5863],{"className":5864,"style":666},[665],[385,5866,5868,5871,5874,5877,5883,5886,5889,5892,5895,5901,5904],{"className":5867},[397],[385,5869],{"className":5870,"style":681},[401],[385,5872,1049],{"className":5873},[685],[385,5875,449],{"className":5876,"style":448},[406,407],[385,5878,5880],{"className":5879},[406],[385,5881,671],{"className":5882},[670],[385,5884,5818],{"className":5885},[406],[385,5887,733],{"className":5888},[732],[385,5890],{"className":5891,"style":737},[665],[385,5893,520],{"className":5894,"style":519},[406,407],[385,5896,5898],{"className":5897},[406],[385,5899,671],{"className":5900},[670],[385,5902,2913],{"className":5903},[406],[385,5905,1066],{"className":5906},[844],".\nItem ",[385,5909,5911],{"className":5910},[388],[385,5912,5914],{"className":5913,"ariaHidden":393},[392],[385,5915,5917,5920],{"className":5916},[397],[385,5918],{"className":5919,"style":402},[401],[385,5921,1045],{"className":5922},[406,407]," has the best ratio (",[385,5925,5927],{"className":5926},[388],[385,5928,5930],{"className":5929,"ariaHidden":393},[392],[385,5931,5933,5936],{"className":5932},[397],[385,5934],{"className":5935,"style":880},[401],[385,5937,5939],{"className":5938},[406],"1.67"," per unit) so greed grabs it first, then\nnothing else fits, for a total value of ",[385,5942,5944],{"className":5943},[388],[385,5945,5947],{"className":5946,"ariaHidden":393},[392],[385,5948,5950,5953],{"className":5949},[397],[385,5951],{"className":5952,"style":880},[401],[385,5954,5702],{"className":5955},[406],". But taking ",[385,5958,5960],{"className":5959},[388],[385,5961,5963],{"className":5962,"ariaHidden":393},[392],[385,5964,5966,5969],{"className":5965},[397],[385,5967],{"className":5968,"style":402},[401],[385,5970,5787],{"className":5971,"style":5786},[406,407]," and ",[385,5974,5976],{"className":5975},[388],[385,5977,5979],{"className":5978,"ariaHidden":393},[392],[385,5980,5982,5985],{"className":5981},[397],[385,5983],{"className":5984,"style":402},[401],[385,5986,5855],{"className":5987,"style":5521},[406,407]," together fills the\nknapsack exactly for value ",[385,5990,5992],{"className":5991},[388],[385,5993,5995],{"className":5994,"ariaHidden":393},[392],[385,5996,5998,6001],{"className":5997},[397],[385,5999],{"className":6000,"style":880},[401],[385,6002,6004],{"className":6003},[406],"18",". Greed is fooled because a locally efficient item can\nblock a globally better combination; the 0\u002F1 problem lacks the structure that lets\ngreedy choices succeed on ",[598,6007,6008],{"href":227},"matroids",". We need to consider\nsubsets, and that calls for dynamic programming.",[2895,6011,6013,6284],{"className":6012},[2898,2899],[1929,6014,6018],{"xmlns":1931,"width":6015,"height":6016,"viewBox":6017},"348.314","135.349","-75 -75 261.235 101.512",[2906,6019,6020,6068,6118,6122,6129,6150,6180,6207,6219,6231,6234,6240,6258],{"stroke":2908,"style":2909},[2906,6021,6024,6032,6038,6044,6050,6056,6062],{"stroke":2912,"fontFamily":6022,"fontSize":6023},"cmti8","8",[2906,6025,6027],{"transform":6026},"translate(-32.313 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24.827-26.358",[2921],[3822,6285,6287,6288,6321,6322,2423,6337,6376,6377,6392,6393,5972,6408,6423,6424,1427],{"className":6286},[3825],"Why greedy fails for 0\u002F1 knapsack (",[385,6289,6291],{"className":6290},[388],[385,6292,6294,6312],{"className":6293,"ariaHidden":393},[392],[385,6295,6297,6300,6303,6306,6309],{"className":6296},[397],[385,6298],{"className":6299,"style":402},[401],[385,6301,409],{"className":6302,"style":408},[406,407],[385,6304],{"className":6305,"style":666},[665],[385,6307,671],{"className":6308},[670],[385,6310],{"className":6311,"style":666},[665],[385,6313,6315,6318],{"className":6314},[397],[385,6316],{"className":6317,"style":880},[401],[385,6319,5702],{"className":6320},[406],"). Best ratio first grabs ",[385,6323,6325],{"className":6324},[388],[385,6326,6328],{"className":6327,"ariaHidden":393},[392],[385,6329,6331,6334],{"className":6330},[397],[385,6332],{"className":6333,"style":402},[401],[385,6335,1045],{"className":6336},[406,407],[385,6338,6340],{"className":6339},[388],[385,6341,6343,6367],{"className":6342,"ariaHidden":393},[392],[385,6344,6346,6349,6352,6355,6358,6361,6364],{"className":6345},[397],[385,6347],{"className":6348,"style":681},[401],[385,6350,520],{"className":6351,"style":519},[406,407],[385,6353,5623],{"className":6354},[406],[385,6356,449],{"className":6357,"style":448},[406,407],[385,6359],{"className":6360,"style":666},[665],[385,6362,671],{"className":6363},[670],[385,6365],{"className":6366,"style":666},[665],[385,6368,6370,6373],{"className":6369},[397],[385,6371],{"className":6372,"style":880},[401],[385,6374,5939],{"className":6375},[406],") and then nothing fits: value ",[385,6378,6380],{"className":6379},[388],[385,6381,6383],{"className":6382,"ariaHidden":393},[392],[385,6384,6386,6389],{"className":6385},[397],[385,6387],{"className":6388,"style":880},[401],[385,6390,5702],{"className":6391},[406],". Taking ",[385,6394,6396],{"className":6395},[388],[385,6397,6399],{"className":6398,"ariaHidden":393},[392],[385,6400,6402,6405],{"className":6401},[397],[385,6403],{"className":6404,"style":402},[401],[385,6406,5787],{"className":6407,"style":5786},[406,407],[385,6409,6411],{"className":6410},[388],[385,6412,6414],{"className":6413,"ariaHidden":393},[392],[385,6415,6417,6420],{"className":6416},[397],[385,6418],{"className":6419,"style":402},[401],[385,6421,5855],{"className":6422,"style":5521},[406,407]," instead fills capacity exactly for value ",[385,6425,6427],{"className":6426},[388],[385,6428,6430],{"className":6429,"ariaHidden":393},[392],[385,6431,6433,6436],{"className":6432},[397],[385,6434],{"className":6435,"style":880},[401],[385,6437,6004],{"className":6438},[406],[381,6440,6441,6442,6457,6458,6492],{},"The high-ratio item ",[385,6443,6445],{"className":6444},[388],[385,6446,6448],{"className":6447,"ariaHidden":393},[392],[385,6449,6451,6454],{"className":6450},[397],[385,6452],{"className":6453,"style":402},[401],[385,6455,1045],{"className":6456},[406,407]," is a trap: it is locally efficient yet blocks the\n",[385,6459,6461],{"className":6460},[388],[385,6462,6464,6483],{"className":6463,"ariaHidden":393},[392],[385,6465,6467,6471,6474,6477,6480],{"className":6466},[397],[385,6468],{"className":6469,"style":6470},[401],"height:0.7667em;vertical-align:-0.0833em;",[385,6472,5787],{"className":6473,"style":5786},[406,407],[385,6475],{"className":6476,"style":1448},[665],[385,6478,4418],{"className":6479},[1452],[385,6481],{"className":6482,"style":1448},[665],[385,6484,6486,6489],{"className":6485},[397],[385,6487],{"className":6488,"style":402},[401],[385,6490,5855],{"className":6491,"style":5521},[406,407]," pair that packs the knapsack with no waste.",[632,6494,6496],{"id":6495},"the-subproblem-and-recurrence","The subproblem and recurrence",[381,6498,967,6499,6540,6541,6543,6544,6547,6548,6551],{},[385,6500,6502],{"className":6501},[388],[385,6503,6505],{"className":6504,"ariaHidden":393},[392],[385,6506,6508,6511],{"className":6507},[397],[385,6509],{"className":6510,"style":980},[401],[385,6512,6514,6517],{"className":6513},[406],[385,6515,767],{"className":6516},[406],[385,6518,6520],{"className":6519},[453],[385,6521,6523],{"className":6522},[457],[385,6524,6526],{"className":6525},[462],[385,6527,6529],{"className":6528,"style":980},[466],[385,6530,6531,6534],{"style":1001},[385,6532],{"className":6533,"style":475},[474],[385,6535,6537],{"className":6536},[479,480,481,482],[385,6538,829],{"className":6539},[406,407,482]," subsets. To get a recurrence we need a\nsubproblem definition that shrinks the instance one decision at a time. The key\ninsight, the source of the ",[1013,6542,1015],{}," dimension, is that we must track not only\n",[1013,6545,6546],{},"which items"," remain available but ",[1013,6549,6550],{},"how much capacity"," is left.",[637,6553,6554],{"type":1023},[381,6555,6556,1029,6559,6592,6593,6608,6609,6654,6655,1427],{},[558,6557,6558],{},"Definition (Knapsack DP state).",[385,6560,6562],{"className":6561},[388],[385,6563,6565],{"className":6564,"ariaHidden":393},[392],[385,6566,6568,6571,6574,6577,6580,6583,6586,6589],{"className":6567},[397],[385,6569],{"className":6570,"style":681},[401],[385,6572,5522],{"className":6573,"style":5521},[406,407],[385,6575,1049],{"className":6576},[685],[385,6578,427],{"className":6579},[406,407],[385,6581,733],{"className":6582},[732],[385,6584],{"className":6585,"style":737},[665],[385,6587,449],{"className":6588,"style":448},[406,407],[385,6590,1066],{"className":6591},[844]," be the maximum value achievable using only the first ",[385,6594,6596],{"className":6595},[388],[385,6597,6599],{"className":6598,"ariaHidden":393},[392],[385,6600,6602,6605],{"className":6601},[397],[385,6603],{"className":6604,"style":423},[401],[385,6606,427],{"className":6607},[406,407]," items\n",[385,6610,6612],{"className":6611},[388],[385,6613,6615],{"className":6614,"ariaHidden":393},[392],[385,6616,6618,6621],{"className":6617},[397],[385,6619],{"className":6620,"style":681},[401],[385,6622,6624,6627,6630,6633,6636,6639,6642,6645,6648,6651],{"className":6623},[788],[385,6625,3905],{"className":6626,"style":3904},[685,3903],[385,6628,604],{"className":6629},[406],[385,6631,733],{"className":6632},[732],[385,6634],{"className":6635,"style":737},[665],[385,6637,789],{"className":6638},[788],[385,6640],{"className":6641,"style":737},[665],[385,6643,733],{"className":6644},[732],[385,6646],{"className":6647,"style":737},[665],[385,6649,427],{"className":6650},[406,407],[385,6652,3921],{"className":6653,"style":3904},[844,3903]," within a weight budget of ",[385,6656,6658],{"className":6657},[388],[385,6659,6661],{"className":6660,"ariaHidden":393},[392],[385,6662,6664,6667],{"className":6663},[397],[385,6665],{"className":6666,"style":956},[401],[385,6668,449],{"className":6669,"style":448},[406,407],[381,6671,3829,6672,6705,6706,6721,6722,6725],{},[385,6673,6675],{"className":6674},[388],[385,6676,6678],{"className":6677,"ariaHidden":393},[392],[385,6679,6681,6684,6687,6690,6693,6696,6699,6702],{"className":6680},[397],[385,6682],{"className":6683,"style":681},[401],[385,6685,5522],{"className":6686,"style":5521},[406,407],[385,6688,1049],{"className":6689},[685],[385,6691,829],{"className":6692},[406,407],[385,6694,733],{"className":6695},[732],[385,6697],{"className":6698,"style":737},[665],[385,6700,409],{"className":6701,"style":408},[406,407],[385,6703,1066],{"className":6704},[844],". Now consider item ",[385,6707,6709],{"className":6708},[388],[385,6710,6712],{"className":6711,"ariaHidden":393},[392],[385,6713,6715,6718],{"className":6714},[397],[385,6716],{"className":6717,"style":423},[401],[385,6719,427],{"className":6720},[406,407],", the last one we are allowed to\nuse, and make the same include\u002Fexclude decision as in subset-sum; this is the\n",[590,6723,6724],{},"0\u002F1"," in the name.",[381,6727,6728,6746,6747,6780,6781,1427],{},[558,6729,6730,6731,1427],{},"Exclude item ",[385,6732,6734],{"className":6733},[388],[385,6735,6737],{"className":6736,"ariaHidden":393},[392],[385,6738,6740,6743],{"className":6739},[397],[385,6741],{"className":6742,"style":423},[401],[385,6744,427],{"className":6745},[406,407]," Then the best we can do is whatever the first ",[385,6748,6750],{"className":6749},[388],[385,6751,6753,6771],{"className":6752,"ariaHidden":393},[392],[385,6754,6756,6759,6762,6765,6768],{"className":6755},[397],[385,6757],{"className":6758,"style":1441},[401],[385,6760,427],{"className":6761},[406,407],[385,6763],{"className":6764,"style":1448},[665],[385,6766,1453],{"className":6767},[1452],[385,6769],{"className":6770,"style":1448},[665],[385,6772,6774,6777],{"className":6773},[397],[385,6775],{"className":6776,"style":880},[401],[385,6778,604],{"className":6779},[406]," items\nachieve within the same budget: ",[385,6782,6784],{"className":6783},[388],[385,6785,6787,6811],{"className":6786,"ariaHidden":393},[392],[385,6788,6790,6793,6796,6799,6802,6805,6808],{"className":6789},[397],[385,6791],{"className":6792,"style":681},[401],[385,6794,5522],{"className":6795,"style":5521},[406,407],[385,6797,1049],{"className":6798},[685],[385,6800,427],{"className":6801},[406,407],[385,6803],{"className":6804,"style":1448},[665],[385,6806,1453],{"className":6807},[1452],[385,6809],{"className":6810,"style":1448},[665],[385,6812,6814,6817,6820,6823,6826,6829],{"className":6813},[397],[385,6815],{"className":6816,"style":681},[401],[385,6818,604],{"className":6819},[406],[385,6821,733],{"className":6822},[732],[385,6824],{"className":6825,"style":737},[665],[385,6827,449],{"className":6828,"style":448},[406,407],[385,6830,1066],{"className":6831},[844],[381,6833,6834,6852,6853,6924,6925,6977,6978,7030,7031,7102,7103,7136,7137,1427],{},[558,6835,6836,6837,1427],{},"Include item ",[385,6838,6840],{"className":6839},[388],[385,6841,6843],{"className":6842,"ariaHidden":393},[392],[385,6844,6846,6849],{"className":6845},[397],[385,6847],{"className":6848,"style":423},[401],[385,6850,427],{"className":6851},[406,407]," This is only possible if it fits, ",[385,6854,6856],{"className":6855},[388],[385,6857,6859,6915],{"className":6858,"ariaHidden":393},[392],[385,6860,6862,6866,6906,6909,6912],{"className":6861},[397],[385,6863],{"className":6864,"style":6865},[401],"height:0.786em;vertical-align:-0.15em;",[385,6867,6869,6872],{"className":6868},[406],[385,6870,449],{"className":6871,"style":448},[406,407],[385,6873,6875],{"className":6874},[453],[385,6876,6878,6898],{"className":6877},[457,458],[385,6879,6881,6895],{"className":6880},[462],[385,6882,6884],{"className":6883,"style":467},[466],[385,6885,6886,6889],{"style":470},[385,6887],{"className":6888,"style":475},[474],[385,6890,6892],{"className":6891},[479,480,481,482],[385,6893,427],{"className":6894},[406,407,482],[385,6896,490],{"className":6897},[489],[385,6899,6901],{"className":6900},[462],[385,6902,6904],{"className":6903,"style":497},[466],[385,6905],{},[385,6907],{"className":6908,"style":666},[665],[385,6910,5052],{"className":6911},[670],[385,6913],{"className":6914,"style":666},[665],[385,6916,6918,6921],{"className":6917},[397],[385,6919],{"className":6920,"style":956},[401],[385,6922,449],{"className":6923,"style":448},[406,407],". We collect\nits value ",[385,6926,6928],{"className":6927},[388],[385,6929,6931],{"className":6930,"ariaHidden":393},[392],[385,6932,6934,6937],{"className":6933},[397],[385,6935],{"className":6936,"style":441},[401],[385,6938,6940,6943],{"className":6939},[406],[385,6941,520],{"className":6942,"style":519},[406,407],[385,6944,6946],{"className":6945},[453],[385,6947,6949,6969],{"className":6948},[457,458],[385,6950,6952,6966],{"className":6951},[462],[385,6953,6955],{"className":6954,"style":467},[466],[385,6956,6957,6960],{"style":535},[385,6958],{"className":6959,"style":475},[474],[385,6961,6963],{"className":6962},[479,480,481,482],[385,6964,427],{"className":6965},[406,407,482],[385,6967,490],{"className":6968},[489],[385,6970,6972],{"className":6971},[462],[385,6973,6975],{"className":6974,"style":497},[466],[385,6976],{}," and spend ",[385,6979,6981],{"className":6980},[388],[385,6982,6984],{"className":6983,"ariaHidden":393},[392],[385,6985,6987,6990],{"className":6986},[397],[385,6988],{"className":6989,"style":441},[401],[385,6991,6993,6996],{"className":6992},[406],[385,6994,449],{"className":6995,"style":448},[406,407],[385,6997,6999],{"className":6998},[453],[385,7000,7002,7022],{"className":7001},[457,458],[385,7003,7005,7019],{"className":7004},[462],[385,7006,7008],{"className":7007,"style":467},[466],[385,7009,7010,7013],{"style":470},[385,7011],{"className":7012,"style":475},[474],[385,7014,7016],{"className":7015},[479,480,481,482],[385,7017,427],{"className":7018},[406,407,482],[385,7020,490],{"className":7021},[489],[385,7023,7025],{"className":7024},[462],[385,7026,7028],{"className":7027,"style":497},[466],[385,7029],{}," of the budget, leaving ",[385,7032,7034],{"className":7033},[388],[385,7035,7037,7056],{"className":7036,"ariaHidden":393},[392],[385,7038,7040,7044,7047,7050,7053],{"className":7039},[397],[385,7041],{"className":7042,"style":7043},[401],"height:0.6667em;vertical-align:-0.0833em;",[385,7045,449],{"className":7046,"style":448},[406,407],[385,7048],{"className":7049,"style":1448},[665],[385,7051,1453],{"className":7052},[1452],[385,7054],{"className":7055,"style":1448},[665],[385,7057,7059,7062],{"className":7058},[397],[385,7060],{"className":7061,"style":441},[401],[385,7063,7065,7068],{"className":7064},[406],[385,7066,449],{"className":7067,"style":448},[406,407],[385,7069,7071],{"className":7070},[453],[385,7072,7074,7094],{"className":7073},[457,458],[385,7075,7077,7091],{"className":7076},[462],[385,7078,7080],{"className":7079,"style":467},[466],[385,7081,7082,7085],{"style":470},[385,7083],{"className":7084,"style":475},[474],[385,7086,7088],{"className":7087},[479,480,481,482],[385,7089,427],{"className":7090},[406,407,482],[385,7092,490],{"className":7093},[489],[385,7095,7097],{"className":7096},[462],[385,7098,7100],{"className":7099,"style":497},[466],[385,7101],{}," for the first\n",[385,7104,7106],{"className":7105},[388],[385,7107,7109,7127],{"className":7108,"ariaHidden":393},[392],[385,7110,7112,7115,7118,7121,7124],{"className":7111},[397],[385,7113],{"className":7114,"style":1441},[401],[385,7116,427],{"className":7117},[406,407],[385,7119],{"className":7120,"style":1448},[665],[385,7122,1453],{"className":7123},[1452],[385,7125],{"className":7126,"style":1448},[665],[385,7128,7130,7133],{"className":7129},[397],[385,7131],{"className":7132,"style":880},[401],[385,7134,604],{"className":7135},[406]," items: ",[385,7138,7140],{"className":7139},[388],[385,7141,7143,7199,7223,7250],{"className":7142,"ariaHidden":393},[392],[385,7144,7146,7150,7190,7193,7196],{"className":7145},[397],[385,7147],{"className":7148,"style":7149},[401],"height:0.7333em;vertical-align:-0.15em;",[385,7151,7153,7156],{"className":7152},[406],[385,7154,520],{"className":7155,"style":519},[406,407],[385,7157,7159],{"className":7158},[453],[385,7160,7162,7182],{"className":7161},[457,458],[385,7163,7165,7179],{"className":7164},[462],[385,7166,7168],{"className":7167,"style":467},[466],[385,7169,7170,7173],{"style":535},[385,7171],{"className":7172,"style":475},[474],[385,7174,7176],{"className":7175},[479,480,481,482],[385,7177,427],{"className":7178},[406,407,482],[385,7180,490],{"className":7181},[489],[385,7183,7185],{"className":7184},[462],[385,7186,7188],{"className":7187,"style":497},[466],[385,7189],{},[385,7191],{"className":7192,"style":1448},[665],[385,7194,4418],{"className":7195},[1452],[385,7197],{"className":7198,"style":1448},[665],[385,7200,7202,7205,7208,7211,7214,7217,7220],{"className":7201},[397],[385,7203],{"className":7204,"style":681},[401],[385,7206,5522],{"className":7207,"style":5521},[406,407],[385,7209,1049],{"className":7210},[685],[385,7212,427],{"className":7213},[406,407],[385,7215],{"className":7216,"style":1448},[665],[385,7218,1453],{"className":7219},[1452],[385,7221],{"className":7222,"style":1448},[665],[385,7224,7226,7229,7232,7235,7238,7241,7244,7247],{"className":7225},[397],[385,7227],{"className":7228,"style":1715},[401],[385,7230,604],{"className":7231},[406],[385,7233,733],{"className":7234},[732],[385,7236],{"className":7237,"style":737},[665],[385,7239,449],{"className":7240,"style":448},[406,407],[385,7242],{"className":7243,"style":1448},[665],[385,7245,1453],{"className":7246},[1452],[385,7248],{"className":7249,"style":1448},[665],[385,7251,7253,7256,7296],{"className":7252},[397],[385,7254],{"className":7255,"style":681},[401],[385,7257,7259,7262],{"className":7258},[406],[385,7260,449],{"className":7261,"style":448},[406,407],[385,7263,7265],{"className":7264},[453],[385,7266,7268,7288],{"className":7267},[457,458],[385,7269,7271,7285],{"className":7270},[462],[385,7272,7274],{"className":7273,"style":467},[466],[385,7275,7276,7279],{"style":470},[385,7277],{"className":7278,"style":475},[474],[385,7280,7282],{"className":7281},[479,480,481,482],[385,7283,427],{"className":7284},[406,407,482],[385,7286,490],{"className":7287},[489],[385,7289,7291],{"className":7290},[462],[385,7292,7294],{"className":7293,"style":497},[466],[385,7295],{},[385,7297,1066],{"className":7298},[844],[381,7300,7301,7302,7317,7318],{},"We take the better of the two, and if item ",[385,7303,7305],{"className":7304},[388],[385,7306,7308],{"className":7307,"ariaHidden":393},[392],[385,7309,7311,7314],{"className":7310},[397],[385,7312],{"className":7313,"style":423},[401],[385,7315,427],{"className":7316},[406,407]," does not fit, only the first\noption is available:",[595,7319,7320],{},[598,7321,767],{"href":7322,"ariaDescribedBy":7323,"dataFootnoteRef":376,"id":7324},"#user-content-fn-clrs-knap",[602],"user-content-fnref-clrs-knap",[385,7326,7328],{"className":7327},[1833],[385,7329,7331],{"className":7330},[388],[385,7332,7334,7370],{"className":7333,"ariaHidden":393},[392],[385,7335,7337,7340,7343,7346,7349,7352,7355,7358,7361,7364,7367],{"className":7336},[397],[385,7338],{"className":7339,"style":681},[401],[385,7341,5522],{"className":7342,"style":5521},[406,407],[385,7344,1049],{"className":7345},[685],[385,7347,427],{"className":7348},[406,407],[385,7350,733],{"className":7351},[732],[385,7353],{"className":7354,"style":737},[665],[385,7356,449],{"className":7357,"style":448},[406,407],[385,7359,1066],{"className":7360},[844],[385,7362],{"className":7363,"style":666},[665],[385,7365,671],{"className":7366},[670],[385,7368],{"className":7369,"style":666},[665],[385,7371,7373,7377],{"className":7372},[397],[385,7374],{"className":7375,"style":7376},[401],"height:4.92em;vertical-align:-2.21em;",[385,7378,7380,7470,7974],{"className":7379},[788],[385,7381,7383],{"className":7382},[685],[385,7384,7386],{"className":7385},[1892,1893],[385,7387,7389,7461],{"className":7388},[457,458],[385,7390,7392,7458],{"className":7391},[462],[385,7393,7396,7408,7424,7435,7447],{"className":7394,"style":7395},[466],"height:2.65em;",[385,7397,7399,7403],{"style":7398},"top:-1.9em;",[385,7400],{"className":7401,"style":7402},[474],"height:3.15em;",[385,7404,7406],{"className":7405},[1914,1915],[385,7407,1918],{},[385,7409,7411,7414],{"style":7410},"top:-1.892em;",[385,7412],{"className":7413,"style":7402},[474],[385,7415,7417],{"style":7416},"height:0.616em;width:0.8889em;",[1929,7418,7421],{"xmlns":1931,"width":1932,"height":7419,"style":1934,"viewBox":7420,"preserveAspectRatio":1936},"0.616em","0 0 888.89 616",[1938,7422],{"d":7423},"M384 0 H504 V616 H384z M384 0 H504 V616 H384z",[385,7425,7427,7430],{"style":7426},"top:-3.15em;",[385,7428],{"className":7429,"style":7402},[474],[385,7431,7433],{"className":7432},[1914,1915],[385,7434,1952],{},[385,7436,7438,7441],{"style":7437},"top:-4.292em;",[385,7439],{"className":7440,"style":7402},[474],[385,7442,7443],{"style":7416},[1929,7444,7445],{"xmlns":1931,"width":1932,"height":7419,"style":1934,"viewBox":7420,"preserveAspectRatio":1936},[1938,7446],{"d":7423},[385,7448,7450,7453],{"style":7449},"top:-4.9em;",[385,7451],{"className":7452,"style":7402},[474],[385,7454,7456],{"className":7455},[1914,1915],[385,7457,1976],{},[385,7459,490],{"className":7460},[489],[385,7462,7464],{"className":7463},[462],[385,7465,7468],{"className":7466,"style":7467},[466],"height:2.15em;",[385,7469],{},[385,7471,7473],{"className":7472},[406],[385,7474,7476,7756,7759],{"className":7475},[1995],[385,7477,7479],{"className":7478},[1999],[385,7480,7482,7747],{"className":7481},[457,458],[385,7483,7485,7744],{"className":7484},[462],[385,7486,7489,7501,7543],{"className":7487,"style":7488},[466],"height:2.71em;",[385,7490,7492,7495],{"style":7491},"top:-4.71em;",[385,7493],{"className":7494,"style":2016},[474],[385,7496,7498],{"className":7497},[406],[385,7499,884],{"className":7500},[406],[385,7502,7504,7507],{"style":7503},"top:-2.97em;",[385,7505],{"className":7506,"style":2016},[474],[385,7508,7510,7513,7516,7519,7522,7525,7528,7531,7534,7537,7540],{"className":7509},[406],[385,7511,5522],{"className":7512,"style":5521},[406,407],[385,7514,1049],{"className":7515},[685],[385,7517,427],{"className":7518},[406,407],[385,7520],{"className":7521,"style":1448},[665],[385,7523,1453],{"className":7524},[1452],[385,7526],{"className":7527,"style":1448},[665],[385,7529,604],{"className":7530},[406],[385,7532,733],{"className":7533},[732],[385,7535],{"className":7536,"style":737},[665],[385,7538,449],{"className":7539,"style":448},[406,407],[385,7541,1066],{"className":7542},[844],[385,7544,7546,7549],{"style":7545},"top:-1.23em;",[385,7547],{"className":7548,"style":2016},[474],[385,7550,7552,7558,7565,7568,7571,7574,7577,7580,7583,7586,7589,7592,7595,7598,7601,7604,7607,7647,7650,7653,7656,7659,7662,7665,7668,7671,7674,7677,7680,7683,7686,7689,7692,7695,7735,7738],{"className":7551},[406],[385,7553,7555],{"className":7554},[4948],[385,7556,5341],{"className":7557},[406,5340],[385,7559,7561],{"className":7560},[685],[385,7562,1049],{"className":7563},[1892,7564],"size1",[385,7566,5522],{"className":7567,"style":5521},[406,407],[385,7569,1049],{"className":7570},[685],[385,7572,427],{"className":7573},[406,407],[385,7575],{"className":7576,"style":1448},[665],[385,7578,1453],{"className":7579},[1452],[385,7581],{"className":7582,"style":1448},[665],[385,7584,604],{"className":7585},[406],[385,7587,733],{"className":7588},[732],[385,7590],{"className":7591,"style":737},[665],[385,7593,449],{"className":7594,"style":448},[406,407],[385,7596,1066],{"className":7597},[844],[385,7599,733],{"className":7600},[732],[385,7602,2101],{"className":7603},[665],[385,7605],{"className":7606,"style":737},[665],[385,7608,7610,7613],{"className":7609},[406],[385,7611,520],{"className":7612,"style":519},[406,407],[385,7614,7616],{"className":7615},[453],[385,7617,7619,7639],{"className":7618},[457,458],[385,7620,7622,7636],{"className":7621},[462],[385,7623,7625],{"className":7624,"style":467},[466],[385,7626,7627,7630],{"style":535},[385,7628],{"className":7629,"style":475},[474],[385,7631,7633],{"className":7632},[479,480,481,482],[385,7634,427],{"className":7635},[406,407,482],[385,7637,490],{"className":7638},[489],[385,7640,7642],{"className":7641},[462],[385,7643,7645],{"className":7644,"style":497},[466],[385,7646],{},[385,7648],{"className":7649,"style":1448},[665],[385,7651,4418],{"className":7652},[1452],[385,7654],{"className":7655,"style":1448},[665],[385,7657,5522],{"className":7658,"style":5521},[406,407],[385,7660,1049],{"className":7661},[685],[385,7663,427],{"className":7664},[406,407],[385,7666],{"className":7667,"style":1448},[665],[385,7669,1453],{"className":7670},[1452],[385,7672],{"className":7673,"style":1448},[665],[385,7675,604],{"className":7676},[406],[385,7678,733],{"className":7679},[732],[385,7681],{"className":7682,"style":737},[665],[385,7684,449],{"className":7685,"style":448},[406,407],[385,7687],{"className":7688,"style":1448},[665],[385,7690,1453],{"className":7691},[1452],[385,7693],{"className":7694,"style":1448},[665],[385,7696,7698,7701],{"className":7697},[406],[385,7699,449],{"className":7700,"style":448},[406,407],[385,7702,7704],{"className":7703},[453],[385,7705,7707,7727],{"className":7706},[457,458],[385,7708,7710,7724],{"className":7709},[462],[385,7711,7713],{"className":7712,"style":467},[466],[385,7714,7715,7718],{"style":470},[385,7716],{"className":7717,"style":475},[474],[385,7719,7721],{"className":7720},[479,480,481,482],[385,7722,427],{"className":7723},[406,407,482],[385,7725,490],{"className":7726},[489],[385,7728,7730],{"className":7729},[462],[385,7731,7733],{"className":7732,"style":497},[466],[385,7734],{},[385,7736,1066],{"className":7737},[844],[385,7739,7741],{"className":7740},[844],[385,7742,1066],{"className":7743},[1892,7564],[385,7745,490],{"className":7746},[489],[385,7748,7750],{"className":7749},[462],[385,7751,7754],{"className":7752,"style":7753},[466],"height:2.21em;",[385,7755],{},[385,7757],{"className":7758,"style":2215},[2214],[385,7760,7762],{"className":7761},[1999],[385,7763,7765,7966],{"className":7764},[457,458],[385,7766,7768,7963],{"className":7767},[462],[385,7769,7771,7825,7894],{"className":7770,"style":7488},[466],[385,7772,7773,7776],{"style":7491},[385,7774],{"className":7775,"style":2016},[474],[385,7777,7779,7785,7788,7791,7794,7797,7800,7807,7810,7813,7816,7819,7822],{"className":7778},[406],[385,7780,7782],{"className":7781},[406,583],[385,7783,2242],{"className":7784},[406],[385,7786,427],{"className":7787},[406,407],[385,7789],{"className":7790,"style":666},[665],[385,7792,671],{"className":7793},[670],[385,7795],{"className":7796,"style":666},[665],[385,7798,884],{"className":7799},[406],[385,7801,7803],{"className":7802},[406,583],[385,7804,7806],{"className":7805},[406]," or ",[385,7808,449],{"className":7809,"style":448},[406,407],[385,7811],{"className":7812,"style":666},[665],[385,7814,671],{"className":7815},[670],[385,7817],{"className":7818,"style":666},[665],[385,7820,884],{"className":7821},[406],[385,7823,733],{"className":7824},[732],[385,7826,7827,7830],{"style":7503},[385,7828],{"className":7829,"style":2016},[474],[385,7831,7833,7839,7879,7882,7885,7888,7891],{"className":7832},[406],[385,7834,7836],{"className":7835},[406,583],[385,7837,2242],{"className":7838},[406],[385,7840,7842,7845],{"className":7841},[406],[385,7843,449],{"className":7844,"style":448},[406,407],[385,7846,7848],{"className":7847},[453],[385,7849,7851,7871],{"className":7850},[457,458],[385,7852,7854,7868],{"className":7853},[462],[385,7855,7857],{"className":7856,"style":467},[466],[385,7858,7859,7862],{"style":470},[385,7860],{"className":7861,"style":475},[474],[385,7863,7865],{"className":7864},[479,480,481,482],[385,7866,427],{"className":7867},[406,407,482],[385,7869,490],{"className":7870},[489],[385,7872,7874],{"className":7873},[462],[385,7875,7877],{"className":7876,"style":497},[466],[385,7878],{},[385,7880],{"className":7881,"style":666},[665],[385,7883,870],{"className":7884},[670],[385,7886],{"className":7887,"style":666},[665],[385,7889,449],{"className":7890,"style":448},[406,407],[385,7892,733],{"className":7893},[732],[385,7895,7896,7899],{"style":7545},[385,7897],{"className":7898,"style":2016},[474],[385,7900,7902,7908,7948,7951,7954,7957,7960],{"className":7901},[406],[385,7903,7905],{"className":7904},[406,583],[385,7906,2242],{"className":7907},[406],[385,7909,7911,7914],{"className":7910},[406],[385,7912,449],{"className":7913,"style":448},[406,407],[385,7915,7917],{"className":7916},[453],[385,7918,7920,7940],{"className":7919},[457,458],[385,7921,7923,7937],{"className":7922},[462],[385,7924,7926],{"className":7925,"style":467},[466],[385,7927,7928,7931],{"style":470},[385,7929],{"className":7930,"style":475},[474],[385,7932,7934],{"className":7933},[479,480,481,482],[385,7935,427],{"className":7936},[406,407,482],[385,7938,490],{"className":7939},[489],[385,7941,7943],{"className":7942},[462],[385,7944,7946],{"className":7945,"style":497},[466],[385,7947],{},[385,7949],{"className":7950,"style":666},[665],[385,7952,5052],{"className":7953},[670],[385,7955],{"className":7956,"style":666},[665],[385,7958,449],{"className":7959,"style":448},[406,407],[385,7961,1427],{"className":7962},[406],[385,7964,490],{"className":7965},[489],[385,7967,7969],{"className":7968},[462],[385,7970,7972],{"className":7971,"style":7753},[466],[385,7973],{},[385,7975],{"className":7976},[844,2416],[381,7978,7979,7980,7995,7996,2423,7998,8031,8032,2514,8035,8050,8051,1427],{},"The base case says: with no items, or no capacity, the value is ",[385,7981,7983],{"className":7982},[388],[385,7984,7986],{"className":7985,"ariaHidden":393},[392],[385,7987,7989,7992],{"className":7988},[397],[385,7990],{"className":7991,"style":880},[401],[385,7993,884],{"className":7994},[406],". Each entry\ndepends only on entries in the ",[1013,7997,2422],{},[385,7999,8001],{"className":8000},[388],[385,8002,8004,8022],{"className":8003,"ariaHidden":393},[392],[385,8005,8007,8010,8013,8016,8019],{"className":8006},[397],[385,8008],{"className":8009,"style":1441},[401],[385,8011,427],{"className":8012},[406,407],[385,8014],{"className":8015,"style":1448},[665],[385,8017,1453],{"className":8018},[1452],[385,8020],{"className":8021,"style":1448},[665],[385,8023,8025,8028],{"className":8024},[397],[385,8026],{"className":8027,"style":880},[401],[385,8029,604],{"className":8030},[406],"), so we fill the table ",[558,8033,8034],{},"row\nby row",[385,8036,8038],{"className":8037},[388],[385,8039,8041],{"className":8040,"ariaHidden":393},[392],[385,8042,8044,8047],{"className":8043},[397],[385,8045],{"className":8046,"style":423},[401],[385,8048,427],{"className":8049},[406,407],", and within each row over all budgets ",[385,8052,8054],{"className":8053},[388],[385,8055,8057,8075],{"className":8056,"ariaHidden":393},[392],[385,8058,8060,8063,8066,8069,8072],{"className":8059},[397],[385,8061],{"className":8062,"style":956},[401],[385,8064,449],{"className":8065,"style":448},[406,407],[385,8067],{"className":8068,"style":666},[665],[385,8070,671],{"className":8071},[670],[385,8073],{"className":8074,"style":666},[665],[385,8076,8078,8082,8085,8088,8091,8094,8097,8100,8103],{"className":8077},[397],[385,8079],{"className":8080,"style":8081},[401],"height:0.8778em;vertical-align:-0.1944em;",[385,8083,884],{"className":8084},[406],[385,8086,733],{"className":8087},[732],[385,8089],{"className":8090,"style":737},[665],[385,8092,789],{"className":8093},[788],[385,8095],{"className":8096,"style":737},[665],[385,8098,733],{"className":8099},[732],[385,8101],{"className":8102,"style":737},[665],[385,8104,409],{"className":8105,"style":408},[406,407],[632,8107,8109],{"id":8108},"the-dp-table-filled","The DP table, filled",[381,8111,8112,8113,8146,8147,5770,8205,5770,8262,5770,8319,8377,8378,8411,8412,8427,8428,8443,8444,1427],{},"Take capacity ",[385,8114,8116],{"className":8115},[388],[385,8117,8119,8137],{"className":8118,"ariaHidden":393},[392],[385,8120,8122,8125,8128,8131,8134],{"className":8121},[397],[385,8123],{"className":8124,"style":402},[401],[385,8126,409],{"className":8127,"style":408},[406,407],[385,8129],{"className":8130,"style":666},[665],[385,8132,671],{"className":8133},[670],[385,8135],{"className":8136,"style":666},[665],[385,8138,8140,8143],{"className":8139},[397],[385,8141],{"className":8142,"style":880},[401],[385,8144,5818],{"className":8145},[406]," and four items: ",[385,8148,8150],{"className":8149},[388],[385,8151,8153],{"className":8152,"ariaHidden":393},[392],[385,8154,8156,8159,8162,8169,8172,8175,8181,8184,8187,8190,8193,8199,8202],{"className":8155},[397],[385,8157],{"className":8158,"style":681},[401],[385,8160,604],{"className":8161},[406],[385,8163,8165],{"className":8164},[406],[385,8166,8168],{"className":8167},[670],":",[385,8170,1049],{"className":8171},[685],[385,8173,449],{"className":8174,"style":448},[406,407],[385,8176,8178],{"className":8177},[406],[385,8179,671],{"className":8180},[670],[385,8182,604],{"className":8183},[406],[385,8185,733],{"className":8186},[732],[385,8188],{"className":8189,"style":737},[665],[385,8191,520],{"className":8192,"style":519},[406,407],[385,8194,8196],{"className":8195},[406],[385,8197,671],{"className":8198},[670],[385,8200,604],{"className":8201},[406],[385,8203,1066],{"className":8204},[844],[385,8206,8208],{"className":8207},[388],[385,8209,8211],{"className":8210,"ariaHidden":393},[392],[385,8212,8214,8217,8220,8226,8229,8232,8238,8241,8244,8247,8250,8256,8259],{"className":8213},[397],[385,8215],{"className":8216,"style":681},[401],[385,8218,767],{"className":8219},[406],[385,8221,8223],{"className":8222},[406],[385,8224,8168],{"className":8225},[670],[385,8227,1049],{"className":8228},[685],[385,8230,449],{"className":8231,"style":448},[406,407],[385,8233,8235],{"className":8234},[406],[385,8236,671],{"className":8237},[670],[385,8239,767],{"className":8240},[406],[385,8242,733],{"className":8243},[732],[385,8245],{"className":8246,"style":737},[665],[385,8248,520],{"className":8249,"style":519},[406,407],[385,8251,8253],{"className":8252},[406],[385,8254,671],{"className":8255},[670],[385,8257,2696],{"className":8258},[406],[385,8260,1066],{"className":8261},[844],[385,8263,8265],{"className":8264},[388],[385,8266,8268],{"className":8267,"ariaHidden":393},[392],[385,8269,8271,8274,8277,8283,8286,8289,8295,8298,8301,8304,8307,8313,8316],{"className":8270},[397],[385,8272],{"className":8273,"style":681},[401],[385,8275,2639],{"className":8276},[406],[385,8278,8280],{"className":8279},[406],[385,8281,8168],{"className":8282},[670],[385,8284,1049],{"className":8285},[685],[385,8287,449],{"className":8288,"style":448},[406,407],[385,8290,8292],{"className":8291},[406],[385,8293,671],{"className":8294},[670],[385,8296,2639],{"className":8297},[406],[385,8299,733],{"className":8300},[732],[385,8302],{"className":8303,"style":737},[665],[385,8305,520],{"className":8306,"style":519},[406,407],[385,8308,8310],{"className":8309},[406],[385,8311,671],{"className":8312},[670],[385,8314,5702],{"className":8315},[406],[385,8317,1066],{"className":8318},[844],[385,8320,8322],{"className":8321},[388],[385,8323,8325],{"className":8324,"ariaHidden":393},[392],[385,8326,8328,8331,8334,8340,8343,8346,8352,8355,8358,8361,8364,8370,8374],{"className":8327},[397],[385,8329],{"className":8330,"style":681},[401],[385,8332,2649],{"className":8333},[406],[385,8335,8337],{"className":8336},[406],[385,8338,8168],{"className":8339},[670],[385,8341,1049],{"className":8342},[685],[385,8344,449],{"className":8345,"style":448},[406,407],[385,8347,8349],{"className":8348},[406],[385,8350,671],{"className":8351},[670],[385,8353,5818],{"className":8354},[406],[385,8356,733],{"className":8357},[732],[385,8359],{"className":8360,"style":737},[665],[385,8362,520],{"className":8363,"style":519},[406,407],[385,8365,8367],{"className":8366},[406],[385,8368,671],{"className":8369},[670],[385,8371,8373],{"className":8372},[406],"16",[385,8375,1066],{"className":8376},[844],". The table holds ",[385,8379,8381],{"className":8380},[388],[385,8382,8384],{"className":8383,"ariaHidden":393},[392],[385,8385,8387,8390,8393,8396,8399,8402,8405,8408],{"className":8386},[397],[385,8388],{"className":8389,"style":681},[401],[385,8391,5522],{"className":8392,"style":5521},[406,407],[385,8394,1049],{"className":8395},[685],[385,8397,427],{"className":8398},[406,407],[385,8400,733],{"className":8401},[732],[385,8403],{"className":8404,"style":737},[665],[385,8406,449],{"className":8407,"style":448},[406,407],[385,8409,1066],{"className":8410},[844],";\nrow ",[385,8413,8415],{"className":8414},[388],[385,8416,8418],{"className":8417,"ariaHidden":393},[392],[385,8419,8421,8424],{"className":8420},[397],[385,8422],{"className":8423,"style":880},[401],[385,8425,884],{"className":8426},[406]," is all zeros (no items), and each later row applies the recurrence across\nbudgets ",[385,8429,8431],{"className":8430},[388],[385,8432,8434],{"className":8433,"ariaHidden":393},[392],[385,8435,8437,8440],{"className":8436},[397],[385,8438],{"className":8439,"style":880},[401],[385,8441,884],{"className":8442},[406]," through ",[385,8445,8447],{"className":8446},[388],[385,8448,8450],{"className":8449,"ariaHidden":393},[392],[385,8451,8453,8456],{"className":8452},[397],[385,8454],{"className":8455,"style":880},[401],[385,8457,5818],{"className":8458},[406],[2895,8460,8462,9067],{"className":8461},[2898,2899],[1929,8463,8467],{"xmlns":1931,"width":8464,"height":8465,"viewBox":8466},"348.576","265.972","-75 -75 261.432 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16.161-61.610",[2921],[2906,9054,9055],{"transform":9018},[1938,9056],{"d":9057,"fill":2908,"stroke":2908,"className":9058,"style":2922},"M20.218-58.630L20.134-58.630Q20.047-58.630 20.047-58.727Q20.047-58.780 20.082-58.802Q20.543-59.180 20.921-59.637Q21.299-60.094 21.602-60.619Q21.905-61.144 22.125-61.698Q22.477-62.577 22.683-63.506Q22.890-64.435 22.890-65.270Q22.890-65.675 22.831-66.059Q22.771-66.444 22.637-66.804Q22.503-67.164 22.270-67.476Q22.253-67.512 22.253-67.538Q22.253-67.639 22.341-67.639L22.424-67.639Q22.450-67.639 22.486-67.604Q22.965-67.103 23.184-66.411Q23.404-65.719 23.404-64.950Q23.404-64.075 23.189-63.174Q22.973-62.273 22.554-61.416Q22.134-60.559 21.561-59.854Q20.987-59.149 20.271-58.648Q20.236-58.630 20.218-58.630",[2921],[2906,9060,9061,9064],{"fill":3594,"stroke":3594},[1938,9062],{"fill":2912,"d":9063},"M-28.247 109.828c-37.156 0 59.634-36.621 59.634-70.702",[1938,9065],{"d":9066},"m31.387 36.62-1.35 3.585 1.35-1.18 1.35 1.18Z",[3822,9068,9070],{"className":9069},[3825],"Filled 0\u002F1 knapsack value table with include and exclude arrows into the answer.",[381,9072,9073,9074,9125,9126,9159,9160,428,9175,9209,9210,2423,9213,9265,9266,2423,9269,9484,9485,9555,9556,9571,9572,9605,9606,2423,9624,9675,9676,2423,9678,9747,9748,1427],{},"The shaded answer is ",[385,9075,9077],{"className":9076},[388],[385,9078,9080,9116],{"className":9079,"ariaHidden":393},[392],[385,9081,9083,9086,9089,9092,9095,9098,9101,9104,9107,9110,9113],{"className":9082},[397],[385,9084],{"className":9085,"style":681},[401],[385,9087,5522],{"className":9088,"style":5521},[406,407],[385,9090,1049],{"className":9091},[685],[385,9093,2649],{"className":9094},[406],[385,9096,733],{"className":9097},[732],[385,9099],{"className":9100,"style":737},[665],[385,9102,5818],{"className":9103},[406],[385,9105,1066],{"className":9106},[844],[385,9108],{"className":9109,"style":666},[665],[385,9111,671],{"className":9112},[670],[385,9114],{"className":9115,"style":666},[665],[385,9117,9119,9122],{"className":9118},[397],[385,9120],{"className":9121,"style":880},[401],[385,9123,8373],{"className":9124},[406],". The arrows trace the highlighted entry\n",[385,9127,9129],{"className":9128},[388],[385,9130,9132],{"className":9131,"ariaHidden":393},[392],[385,9133,9135,9138,9141,9144,9147,9150,9153,9156],{"className":9134},[397],[385,9136],{"className":9137,"style":681},[401],[385,9139,5522],{"className":9140,"style":5521},[406,407],[385,9142,1049],{"className":9143},[685],[385,9145,2639],{"className":9146},[406],[385,9148,733],{"className":9149},[732],[385,9151],{"className":9152,"style":737},[665],[385,9154,5818],{"className":9155},[406],[385,9157,1066],{"className":9158},[844],": item ",[385,9161,9163],{"className":9162},[388],[385,9164,9166],{"className":9165,"ariaHidden":393},[392],[385,9167,9169,9172],{"className":9168},[397],[385,9170],{"className":9171,"style":880},[401],[385,9173,2639],{"className":9174},[406],[385,9176,9178],{"className":9177},[388],[385,9179,9181,9200],{"className":9180,"ariaHidden":393},[392],[385,9182,9184,9188,9191,9194,9197],{"className":9183},[397],[385,9185],{"className":9186,"style":9187},[401],"height:0.7804em;vertical-align:-0.136em;",[385,9189,2639],{"className":9190},[406],[385,9192],{"className":9193,"style":666},[665],[385,9195,5052],{"className":9196},[670],[385,9198],{"className":9199,"style":666},[665],[385,9201,9203,9206],{"className":9202},[397],[385,9204],{"className":9205,"style":880},[401],[385,9207,5818],{"className":9208},[406],", so we compare ",[1013,9211,9212],{},"excluding it",[385,9214,9216],{"className":9215},[388],[385,9217,9219,9255],{"className":9218,"ariaHidden":393},[392],[385,9220,9222,9225,9228,9231,9234,9237,9240,9243,9246,9249,9252],{"className":9221},[397],[385,9223],{"className":9224,"style":681},[401],[385,9226,5522],{"className":9227,"style":5521},[406,407],[385,9229,1049],{"className":9230},[685],[385,9232,767],{"className":9233},[406],[385,9235,733],{"className":9236},[732],[385,9238],{"className":9239,"style":737},[665],[385,9241,5818],{"className":9242},[406],[385,9244,1066],{"className":9245},[844],[385,9247],{"className":9248,"style":666},[665],[385,9250,671],{"className":9251},[670],[385,9253],{"className":9254,"style":666},[665],[385,9256,9258,9261],{"className":9257},[397],[385,9259],{"className":9260,"style":880},[401],[385,9262,9264],{"className":9263},[406],"7",",\nthe cell directly above) against ",[1013,9267,9268],{},"including it",[385,9270,9272],{"className":9271},[388],[385,9273,9275,9330,9363,9384,9403,9439,9457,9475],{"className":9274,"ariaHidden":393},[392],[385,9276,9278,9281,9321,9324,9327],{"className":9277},[397],[385,9279],{"className":9280,"style":7149},[401],[385,9282,9284,9287],{"className":9283},[406],[385,9285,520],{"className":9286,"style":519},[406,407],[385,9288,9290],{"className":9289},[453],[385,9291,9293,9313],{"className":9292},[457,458],[385,9294,9296,9310],{"className":9295},[462],[385,9297,9299],{"className":9298,"style":705},[466],[385,9300,9301,9304],{"style":535},[385,9302],{"className":9303,"style":475},[474],[385,9305,9307],{"className":9306},[479,480,481,482],[385,9308,2639],{"className":9309},[406,482],[385,9311,490],{"className":9312},[489],[385,9314,9316],{"className":9315},[462],[385,9317,9319],{"className":9318,"style":497},[466],[385,9320],{},[385,9322],{"className":9323,"style":1448},[665],[385,9325,4418],{"className":9326},[1452],[385,9328],{"className":9329,"style":1448},[665],[385,9331,9333,9336,9339,9342,9345,9348,9351,9354,9357,9360],{"className":9332},[397],[385,9334],{"className":9335,"style":681},[401],[385,9337,5522],{"className":9338,"style":5521},[406,407],[385,9340,1049],{"className":9341},[685],[385,9343,767],{"className":9344},[406],[385,9346,733],{"className":9347},[732],[385,9349],{"className":9350,"style":737},[665],[385,9352,5818],{"className":9353},[406],[385,9355],{"className":9356,"style":1448},[665],[385,9358,1453],{"className":9359},[1452],[385,9361],{"className":9362,"style":1448},[665],[385,9364,9366,9369,9372,9375,9378,9381],{"className":9365},[397],[385,9367],{"className":9368,"style":681},[401],[385,9370,2639],{"className":9371},[406],[385,9373,1066],{"className":9374},[844],[385,9376],{"className":9377,"style":666},[665],[385,9379,671],{"className":9380},[670],[385,9382],{"className":9383,"style":666},[665],[385,9385,9387,9391,9394,9397,9400],{"className":9386},[397],[385,9388],{"className":9389,"style":9390},[401],"height:0.7278em;vertical-align:-0.0833em;",[385,9392,5702],{"className":9393},[406],[385,9395],{"className":9396,"style":1448},[665],[385,9398,4418],{"className":9399},[1452],[385,9401],{"className":9402,"style":1448},[665],[385,9404,9406,9409,9412,9415,9418,9421,9424,9427,9430,9433,9436],{"className":9405},[397],[385,9407],{"className":9408,"style":681},[401],[385,9410,5522],{"className":9411,"style":5521},[406,407],[385,9413,1049],{"className":9414},[685],[385,9416,767],{"className":9417},[406],[385,9419,733],{"className":9420},[732],[385,9422],{"className":9423,"style":737},[665],[385,9425,767],{"className":9426},[406],[385,9428,1066],{"className":9429},[844],[385,9431],{"className":9432,"style":666},[665],[385,9434,671],{"className":9435},[670],[385,9437],{"className":9438,"style":666},[665],[385,9440,9442,9445,9448,9451,9454],{"className":9441},[397],[385,9443],{"className":9444,"style":9390},[401],[385,9446,5702],{"className":9447},[406],[385,9449],{"className":9450,"style":1448},[665],[385,9452,4418],{"className":9453},[1452],[385,9455],{"className":9456,"style":1448},[665],[385,9458,9460,9463,9466,9469,9472],{"className":9459},[397],[385,9461],{"className":9462,"style":880},[401],[385,9464,2696],{"className":9465},[406],[385,9467],{"className":9468,"style":666},[665],[385,9470,671],{"className":9471},[670],[385,9473],{"className":9474,"style":666},[665],[385,9476,9478,9481],{"className":9477},[397],[385,9479],{"className":9480,"style":880},[401],[385,9482,8373],{"className":9483},[406],", reaching ",[385,9486,9488],{"className":9487},[388],[385,9489,9491,9546],{"className":9490,"ariaHidden":393},[392],[385,9492,9494,9497,9537,9540,9543],{"className":9493},[397],[385,9495],{"className":9496,"style":441},[401],[385,9498,9500,9503],{"className":9499},[406],[385,9501,449],{"className":9502,"style":448},[406,407],[385,9504,9506],{"className":9505},[453],[385,9507,9509,9529],{"className":9508},[457,458],[385,9510,9512,9526],{"className":9511},[462],[385,9513,9515],{"className":9514,"style":705},[466],[385,9516,9517,9520],{"style":470},[385,9518],{"className":9519,"style":475},[474],[385,9521,9523],{"className":9522},[479,480,481,482],[385,9524,2639],{"className":9525},[406,482],[385,9527,490],{"className":9528},[489],[385,9530,9532],{"className":9531},[462],[385,9533,9535],{"className":9534,"style":497},[466],[385,9536],{},[385,9538],{"className":9539,"style":666},[665],[385,9541,671],{"className":9542},[670],[385,9544],{"className":9545,"style":666},[665],[385,9547,9549,9552],{"className":9548},[397],[385,9550],{"className":9551,"style":880},[401],[385,9553,2639],{"className":9554},[406]," columns to the left); the include branch wins at\n",[385,9557,9559],{"className":9558},[388],[385,9560,9562],{"className":9561,"ariaHidden":393},[392],[385,9563,9565,9568],{"className":9564},[397],[385,9566],{"className":9567,"style":880},[401],[385,9569,8373],{"className":9570},[406],". One row down, ",[385,9573,9575],{"className":9574},[388],[385,9576,9578],{"className":9577,"ariaHidden":393},[392],[385,9579,9581,9584,9587,9590,9593,9596,9599,9602],{"className":9580},[397],[385,9582],{"className":9583,"style":681},[401],[385,9585,5522],{"className":9586,"style":5521},[406,407],[385,9588,1049],{"className":9589},[685],[385,9591,2649],{"className":9592},[406],[385,9594,733],{"className":9595},[732],[385,9597],{"className":9598,"style":737},[665],[385,9600,5818],{"className":9601},[406],[385,9603,1066],{"className":9604},[844]," compares ",[1013,9607,9608,9609],{},"excluding item ",[385,9610,9612],{"className":9611},[388],[385,9613,9615],{"className":9614,"ariaHidden":393},[392],[385,9616,9618,9621],{"className":9617},[397],[385,9619],{"className":9620,"style":880},[401],[385,9622,2649],{"className":9623},[406],[385,9625,9627],{"className":9626},[388],[385,9628,9630,9666],{"className":9629,"ariaHidden":393},[392],[385,9631,9633,9636,9639,9642,9645,9648,9651,9654,9657,9660,9663],{"className":9632},[397],[385,9634],{"className":9635,"style":681},[401],[385,9637,5522],{"className":9638,"style":5521},[406,407],[385,9640,1049],{"className":9641},[685],[385,9643,2639],{"className":9644},[406],[385,9646,733],{"className":9647},[732],[385,9649],{"className":9650,"style":737},[665],[385,9652,5818],{"className":9653},[406],[385,9655,1066],{"className":9656},[844],[385,9658],{"className":9659,"style":666},[665],[385,9661,671],{"className":9662},[670],[385,9664],{"className":9665,"style":666},[665],[385,9667,9669,9672],{"className":9668},[397],[385,9670],{"className":9671,"style":880},[401],[385,9673,8373],{"className":9674},[406],")\nagainst ",[1013,9677,9268],{},[385,9679,9681],{"className":9680},[388],[385,9682,9684,9702,9738],{"className":9683,"ariaHidden":393},[392],[385,9685,9687,9690,9693,9696,9699],{"className":9686},[397],[385,9688],{"className":9689,"style":9390},[401],[385,9691,8373],{"className":9692},[406],[385,9694],{"className":9695,"style":1448},[665],[385,9697,4418],{"className":9698},[1452],[385,9700],{"className":9701,"style":1448},[665],[385,9703,9705,9708,9711,9714,9717,9720,9723,9726,9729,9732,9735],{"className":9704},[397],[385,9706],{"className":9707,"style":681},[401],[385,9709,5522],{"className":9710,"style":5521},[406,407],[385,9712,1049],{"className":9713},[685],[385,9715,2639],{"className":9716},[406],[385,9718,733],{"className":9719},[732],[385,9721],{"className":9722,"style":737},[665],[385,9724,884],{"className":9725},[406],[385,9727,1066],{"className":9728},[844],[385,9730],{"className":9731,"style":666},[665],[385,9733,671],{"className":9734},[670],[385,9736],{"className":9737,"style":666},[665],[385,9739,9741,9744],{"className":9740},[397],[385,9742],{"className":9743,"style":880},[401],[385,9745,8373],{"className":9746},[406],"), a tie at ",[385,9749,9751],{"className":9750},[388],[385,9752,9754],{"className":9753,"ariaHidden":393},[392],[385,9755,9757,9760],{"className":9756},[397],[385,9758],{"className":9759,"style":880},[401],[385,9761,8373],{"className":9762},[406],[632,9764,9766],{"id":9765},"the-algorithm","The algorithm",[4327,9768,9770],{"className":4329,"code":9769,"language":4331,"meta":376,"style":376},"caption: $\\textsc{Knapsack-01}(w, v, n, W)$ — maximum value within capacity $W$\nnumber: 2\nfor $b \\gets 0$ to $W$ do\n  $K[0][b] \\gets 0$ \u002F\u002F no items: value $0$\nfor $i \\gets 1$ to $n$ do\n  for $b \\gets 0$ to $W$ do\n    $K[i][b] \\gets K[i-1][b]$ \u002F\u002F skip item $i$\n    if $w[i] \\le b$ then\n      $take \\gets v[i] + K[i-1][b - w[i]]$\n      $K[i][b] \\gets \\max(K[i][b],\\ take)$ \u002F\u002F take item $i$\nreturn $K[n][W]$\n",[4333,9771,9772,9777,9782,9787,9792,9796,9801,9806,9811,9816,9821],{"__ignoreMap":376},[385,9773,9774],{"class":4337,"line":6},[385,9775,9776],{},"caption: $\\textsc{Knapsack-01}(w, v, n, W)$ — maximum value within capacity $W$\n",[385,9778,9779],{"class":4337,"line":18},[385,9780,9781],{},"number: 2\n",[385,9783,9784],{"class":4337,"line":24},[385,9785,9786],{},"for $b \\gets 0$ to $W$ do\n",[385,9788,9789],{"class":4337,"line":73},[385,9790,9791],{},"  $K[0][b] \\gets 0$ \u002F\u002F no items: value $0$\n",[385,9793,9794],{"class":4337,"line":102},[385,9795,4360],{},[385,9797,9798],{"class":4337,"line":108},[385,9799,9800],{},"  for $b \\gets 0$ to $W$ do\n",[385,9802,9803],{"class":4337,"line":116},[385,9804,9805],{},"    $K[i][b] \\gets K[i-1][b]$ \u002F\u002F skip item $i$\n",[385,9807,9808],{"class":4337,"line":196},[385,9809,9810],{},"    if $w[i] \\le b$ then\n",[385,9812,9813],{"class":4337,"line":202},[385,9814,9815],{},"      $take \\gets v[i] + K[i-1][b - w[i]]$\n",[385,9817,9818],{"class":4337,"line":283},[385,9819,9820],{},"      $K[i][b] \\gets \\max(K[i][b],\\ take)$ \u002F\u002F take item $i$\n",[385,9822,9823],{"class":4337,"line":333},[385,9824,9825],{},"return $K[n][W]$\n",[381,9827,9828,9831,9832,9835,9836,9869,9870,9885,9886,10015,10016,10031,10032,10103,10104,1427],{},[558,9829,9830],{},"Recovering the chosen items."," As with every DP, the table holds the value; the\n",[1013,9833,9834],{},"subset"," is recovered by walking backward from ",[385,9837,9839],{"className":9838},[388],[385,9840,9842],{"className":9841,"ariaHidden":393},[392],[385,9843,9845,9848,9851,9854,9857,9860,9863,9866],{"className":9844},[397],[385,9846],{"className":9847,"style":681},[401],[385,9849,5522],{"className":9850,"style":5521},[406,407],[385,9852,1105],{"className":9853},[685],[385,9855,829],{"className":9856},[406,407],[385,9858,1116],{"className":9859},[844],[385,9861,1105],{"className":9862},[685],[385,9864,409],{"className":9865,"style":408},[406,407],[385,9867,1116],{"className":9868},[844],". At row ",[385,9871,9873],{"className":9872},[388],[385,9874,9876],{"className":9875,"ariaHidden":393},[392],[385,9877,9879,9882],{"className":9878},[397],[385,9880],{"className":9881,"style":423},[401],[385,9883,427],{"className":9884},[406,407],", if ",[385,9887,9889],{"className":9888},[388],[385,9890,9892,9970,9994],{"className":9891,"ariaHidden":393},[392],[385,9893,9895,9898,9901,9904,9907,9910,9913,9917,9920,9923,9967],{"className":9894},[397],[385,9896],{"className":9897,"style":681},[401],[385,9899,5522],{"className":9900,"style":5521},[406,407],[385,9902,1105],{"className":9903},[685],[385,9905,427],{"className":9906},[406,407],[385,9908,1116],{"className":9909},[844],[385,9911,1105],{"className":9912},[685],[385,9914,9916],{"className":9915},[406,407],"b",[385,9918,1116],{"className":9919},[844],[385,9921],{"className":9922,"style":666},[665],[385,9924,9926,9960,9964],{"className":9925},[670],[385,9927,9929],{"className":9928},[670],[385,9930,9933],{"className":9931},[406,9932],"vbox",[385,9934,9937],{"className":9935},[9936],"thinbox",[385,9938,9941,9945,9956],{"className":9939},[9940],"rlap",[385,9942],{"className":9943,"style":9944},[401],"height:0.8889em;vertical-align:-0.1944em;",[385,9946,9949],{"className":9947},[9948],"inner",[385,9950,9952],{"className":9951},[406],[385,9953,9955],{"className":9954},[670],"",[385,9957],{"className":9958},[9959],"fix",[385,9961],{"className":9962},[665,9963],"nobreak",[385,9965,671],{"className":9966},[670],[385,9968],{"className":9969,"style":666},[665],[385,9971,9973,9976,9979,9982,9985,9988,9991],{"className":9972},[397],[385,9974],{"className":9975,"style":681},[401],[385,9977,5522],{"className":9978,"style":5521},[406,407],[385,9980,1105],{"className":9981},[685],[385,9983,427],{"className":9984},[406,407],[385,9986],{"className":9987,"style":1448},[665],[385,9989,1453],{"className":9990},[1452],[385,9992],{"className":9993,"style":1448},[665],[385,9995,9997,10000,10003,10006,10009,10012],{"className":9996},[397],[385,9998],{"className":9999,"style":681},[401],[385,10001,604],{"className":10002},[406],[385,10004,1116],{"className":10005},[844],[385,10007,1105],{"className":10008},[685],[385,10010,9916],{"className":10011},[406,407],[385,10013,1116],{"className":10014},[844]," then item ",[385,10017,10019],{"className":10018},[388],[385,10020,10022],{"className":10021,"ariaHidden":393},[392],[385,10023,10025,10028],{"className":10024},[397],[385,10026],{"className":10027,"style":423},[401],[385,10029,427],{"className":10030},[406,407]," was taken: record it and drop the budget to\n",[385,10033,10035],{"className":10034},[388],[385,10036,10038,10057],{"className":10037,"ariaHidden":393},[392],[385,10039,10041,10045,10048,10051,10054],{"className":10040},[397],[385,10042],{"className":10043,"style":10044},[401],"height:0.7778em;vertical-align:-0.0833em;",[385,10046,9916],{"className":10047},[406,407],[385,10049],{"className":10050,"style":1448},[665],[385,10052,1453],{"className":10053},[1452],[385,10055],{"className":10056,"style":1448},[665],[385,10058,10060,10063],{"className":10059},[397],[385,10061],{"className":10062,"style":441},[401],[385,10064,10066,10069],{"className":10065},[406],[385,10067,449],{"className":10068,"style":448},[406,407],[385,10070,10072],{"className":10071},[453],[385,10073,10075,10095],{"className":10074},[457,458],[385,10076,10078,10092],{"className":10077},[462],[385,10079,10081],{"className":10080,"style":467},[466],[385,10082,10083,10086],{"style":470},[385,10084],{"className":10085,"style":475},[474],[385,10087,10089],{"className":10088},[479,480,481,482],[385,10090,427],{"className":10091},[406,407,482],[385,10093,490],{"className":10094},[489],[385,10096,10098],{"className":10097},[462],[385,10099,10101],{"className":10100,"style":497},[466],[385,10102],{},"; otherwise it was skipped. Continue down to row ",[385,10105,10107],{"className":10106},[388],[385,10108,10110],{"className":10109,"ariaHidden":393},[392],[385,10111,10113,10116],{"className":10112},[397],[385,10114],{"className":10115,"style":880},[401],[385,10117,884],{"className":10118},[406],[4327,10120,10122],{"className":4329,"code":10121,"language":4331,"meta":376,"style":376},"caption: $\\textsc{Knapsack-Items}(K, w, n, W)$ — recover the optimal subset\nnumber: 3\n$S \\gets \\emptyset$\n$b \\gets W$\nfor $i \\gets n$ downto $1$ do\n  if $K[i][b] \\neq K[i-1][b]$ then\n    $S \\gets S \\cup \\set{i}$ \u002F\u002F item $i$ taken\n    $b \\gets b - w[i]$\nreturn $S$\n",[4333,10123,10124,10129,10134,10139,10144,10149,10154,10159,10164],{"__ignoreMap":376},[385,10125,10126],{"class":4337,"line":6},[385,10127,10128],{},"caption: $\\textsc{Knapsack-Items}(K, w, n, W)$ — recover the optimal subset\n",[385,10130,10131],{"class":4337,"line":18},[385,10132,10133],{},"number: 3\n",[385,10135,10136],{"class":4337,"line":24},[385,10137,10138],{},"$S \\gets \\emptyset$\n",[385,10140,10141],{"class":4337,"line":73},[385,10142,10143],{},"$b \\gets W$\n",[385,10145,10146],{"class":4337,"line":102},[385,10147,10148],{},"for $i \\gets n$ downto $1$ do\n",[385,10150,10151],{"class":4337,"line":108},[385,10152,10153],{},"  if $K[i][b] \\neq K[i-1][b]$ then\n",[385,10155,10156],{"class":4337,"line":116},[385,10157,10158],{},"    $S \\gets S \\cup \\set{i}$ \u002F\u002F item $i$ taken\n",[385,10160,10161],{"class":4337,"line":196},[385,10162,10163],{},"    $b \\gets b - w[i]$\n",[385,10165,10166],{"class":4337,"line":202},[385,10167,10168],{},"return $S$\n",[381,10170,10171,10172,10205,10206,10239,10240,10292],{},"On the filled table above (",[385,10173,10175],{"className":10174},[388],[385,10176,10178,10196],{"className":10177,"ariaHidden":393},[392],[385,10179,10181,10184,10187,10190,10193],{"className":10180},[397],[385,10182],{"className":10183,"style":402},[401],[385,10185,409],{"className":10186,"style":408},[406,407],[385,10188],{"className":10189,"style":666},[665],[385,10191,671],{"className":10192},[670],[385,10194],{"className":10195,"style":666},[665],[385,10197,10199,10202],{"className":10198},[397],[385,10200],{"className":10201,"style":880},[401],[385,10203,5818],{"className":10204},[406],"), the walk starts at ",[385,10207,10209],{"className":10208},[388],[385,10210,10212],{"className":10211,"ariaHidden":393},[392],[385,10213,10215,10218,10221,10224,10227,10230,10233,10236],{"className":10214},[397],[385,10216],{"className":10217,"style":681},[401],[385,10219,5522],{"className":10220,"style":5521},[406,407],[385,10222,1105],{"className":10223},[685],[385,10225,2649],{"className":10226},[406],[385,10228,1116],{"className":10229},[844],[385,10231,1105],{"className":10232},[685],[385,10234,5818],{"className":10235},[406],[385,10237,1116],{"className":10238},[844]," and reads off\none decision per row, dropping the budget by ",[385,10241,10243],{"className":10242},[388],[385,10244,10246],{"className":10245,"ariaHidden":393},[392],[385,10247,10249,10252],{"className":10248},[397],[385,10250],{"className":10251,"style":441},[401],[385,10253,10255,10258],{"className":10254},[406],[385,10256,449],{"className":10257,"style":448},[406,407],[385,10259,10261],{"className":10260},[453],[385,10262,10264,10284],{"className":10263},[457,458],[385,10265,10267,10281],{"className":10266},[462],[385,10268,10270],{"className":10269,"style":467},[466],[385,10271,10272,10275],{"style":470},[385,10273],{"className":10274,"style":475},[474],[385,10276,10278],{"className":10277},[479,480,481,482],[385,10279,427],{"className":10280},[406,407,482],[385,10282,490],{"className":10283},[489],[385,10285,10287],{"className":10286},[462],[385,10288,10290],{"className":10289,"style":497},[466],[385,10291],{}," whenever an item is taken:",[2895,10294,10296,10766],{"className":10295},[2898,2899],[1929,10297,10301],{"xmlns":1931,"width":10298,"height":10299,"viewBox":10300},"242.586","194.451","-75 -75 181.940 145.838",[2906,10302,10303,10306,10357,10414,10470,10473,10519,10546,10599,10647,10671,10679,10687,10695],{"stroke":2908,"style":2909},[1938,10304],{"fill":2912,"d":10305},"M-65.003-49.308h96.739V-72.07h-96.74Z",[2906,10307,10308,10315,10321,10327,10333,10339,10345,10351],{"stroke":2912,"fontSize":2913},[2906,10309,10311],{"transform":10310},"translate(-42.43 2.25)",[1938,10312],{"d":10313,"fill":2908,"stroke":2908,"className":10314,"style":2922},"M-15.905-61.348Q-15.905-61.493-15.843-61.678L-15.096-63.616Q-14.986-63.915-14.986-64.134Q-14.986-64.407-15.175-64.407Q-15.527-64.407-15.762-64.046Q-15.997-63.686-16.102-63.255Q-16.120-63.172-16.195-63.172L-16.300-63.172Q-16.348-63.172-16.370-63.211Q-16.392-63.251-16.392-63.291Q-16.304-63.633-16.144-63.939Q-15.984-64.244-15.733-64.455Q-15.483-64.666-15.157-64.666Q-14.828-64.666-14.597-64.460Q-14.366-64.253-14.366-63.910Q-14.366-63.743-14.419-63.576L-15.166-61.643Q-15.272-61.357-15.285-61.120Q-15.285-61.019-15.239-60.935Q-15.193-60.852-15.087-60.852Q-14.736-60.852-14.500-61.210Q-14.265-61.568-14.160-62.003Q-14.151-62.034-14.127-62.058Q-14.103-62.082-14.068-62.082L-13.962-62.082Q-13.914-62.082-13.892-62.049Q-13.870-62.016-13.870-61.968Q-13.997-61.445-14.320-61.016Q-14.643-60.588-15.105-60.588Q-15.434-60.588-15.669-60.801Q-15.905-61.014-15.905-61.348M-14.881-66.134Q-14.881-66.332-14.718-66.485Q-14.555-66.639-14.358-66.639Q-14.208-66.639-14.107-66.543Q-14.006-66.446-14.006-66.296Q-14.006-66.090-14.164-65.940Q-14.322-65.791-14.520-65.791Q-14.670-65.791-14.775-65.888Q-14.881-65.984-14.881-66.134",[2921],[2906,10316,10317],{"transform":10310},[1938,10318],{"d":10319,"fill":2908,"stroke":2908,"className":10320,"style":2922},"M-6.968-61.832L-12.774-61.832Q-12.853-61.845-12.903-61.895Q-12.954-61.946-12.954-62.021Q-12.954-62.170-12.774-62.218L-6.968-62.218Q-6.797-62.166-6.797-62.021Q-6.797-61.867-6.968-61.832M-6.968-63.660L-12.774-63.660Q-12.954-63.690-12.954-63.849Q-12.954-63.998-12.774-64.046L-6.968-64.046Q-6.797-63.994-6.797-63.849Q-6.797-63.695-6.968-63.660",[2921],[2906,10322,10323],{"transform":10310},[1938,10324],{"d":10325,"fill":2908,"stroke":2908,"className":10326,"style":2922},"M-3.576-62.166L-6.015-62.166L-6.015-62.482L-3.189-66.630Q-3.145-66.683-3.079-66.683L-2.925-66.683Q-2.886-66.683-2.853-66.650Q-2.820-66.617-2.820-66.573L-2.820-62.482L-1.919-62.482L-1.919-62.166L-2.820-62.166L-2.820-61.300Q-2.820-61.005-1.919-61.005L-1.919-60.689L-4.472-60.689L-4.472-61.005Q-4.112-61.005-3.844-61.060Q-3.576-61.115-3.576-61.300L-3.576-62.166M-3.519-65.655L-5.681-62.482L-3.519-62.482L-3.519-65.655M-0.882-61.194Q-0.882-61.322-0.814-61.438Q-0.746-61.555-0.629-61.625Q-0.513-61.695-0.377-61.695Q-0.174-61.695-0.023-61.548Q0.129-61.401 0.129-61.194Q0.129-60.988-0.021-60.838Q-0.170-60.689-0.377-60.689Q-0.587-60.689-0.735-60.841Q-0.882-60.992-0.882-61.194M-0.882-64.064Q-0.882-64.262-0.737-64.416Q-0.592-64.569-0.377-64.569Q-0.240-64.569-0.124-64.501Q-0.007-64.433 0.061-64.317Q0.129-64.200 0.129-64.064Q0.129-63.862-0.023-63.710Q-0.174-63.559-0.377-63.559Q-0.583-63.559-0.732-63.712Q-0.882-63.866-0.882-64.064",[2921],[2906,10328,10329],{"transform":10310},[1938,10330],{"d":10331,"fill":2908,"stroke":2908,"className":10332,"style":2922},"M7.901-60.689L5.497-60.689Q5.400-60.689 5.400-60.808Q5.400-60.869 5.433-60.937Q5.466-61.005 5.528-61.005Q6.029-61.005 6.257-61.058Q6.376-61.102 6.455-61.335L7.677-66.252Q7.694-66.340 7.694-66.384Q7.694-66.450 7.668-66.468Q7.496-66.521 6.965-66.521Q6.859-66.521 6.859-66.639Q6.859-66.723 6.899-66.780Q6.938-66.837 7.022-66.837L9.421-66.837Q9.465-66.837 9.494-66.802Q9.522-66.767 9.522-66.727Q9.522-66.648 9.491-66.584Q9.461-66.521 9.395-66.521Q8.903-66.521 8.674-66.468Q8.533-66.415 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",[385,10771,10773],{"className":10772},[388],[385,10774,10776,10812],{"className":10775,"ariaHidden":393},[392],[385,10777,10779,10782,10785,10788,10791,10794,10797,10800,10803,10806,10809],{"className":10778},[397],[385,10780],{"className":10781,"style":681},[401],[385,10783,5522],{"className":10784,"style":5521},[406,407],[385,10786,1105],{"className":10787},[685],[385,10789,2649],{"className":10790},[406],[385,10792,1116],{"className":10793},[844],[385,10795,1105],{"className":10796},[685],[385,10798,5818],{"className":10799},[406],[385,10801,1116],{"className":10802},[844],[385,10804],{"className":10805,"style":666},[665],[385,10807,671],{"className":10808},[670],[385,10810],{"className":10811,"style":666},[665],[385,10813,10815,10818],{"className":10814},[397],[385,10816],{"className":10817,"style":880},[401],[385,10819,8373],{"className":10820},[406],". At each row, ",[385,10823,10825],{"className":10824},[388],[385,10826,10828,10864,10888],{"className":10827,"ariaHidden":393},[392],[385,10829,10831,10834,10837,10840,10843,10846,10849,10852,10855,10858,10861],{"className":10830},[397],[385,10832],{"className":10833,"style":681},[401],[385,10835,5522],{"className":10836,"style":5521},[406,407],[385,10838,1105],{"className":10839},[685],[385,10841,427],{"className":10842},[406,407],[385,10844,1116],{"className":10845},[844],[385,10847,1105],{"className":10848},[685],[385,10850,9916],{"className":10851},[406,407],[385,10853,1116],{"className":10854},[844],[385,10856],{"className":10857,"style":666},[665],[385,10859,671],{"className":10860},[670],[385,10862],{"className":10863,"style":666},[665],[385,10865,10867,10870,10873,10876,10879,10882,10885],{"className":10866},[397],[385,10868],{"className":10869,"style":681},[401],[385,10871,5522],{"className":10872,"style":5521},[406,407],[385,10874,1105],{"className":10875},[685],[385,10877,427],{"className":10878},[406,407],[385,10880],{"className":10881,"style":1448},[665],[385,10883,1453],{"className":10884},[1452],[385,10886],{"className":10887,"style":1448},[665],[385,10889,10891,10894,10897,10900,10903,10906],{"className":10890},[397],[385,10892],{"className":10893,"style":681},[401],[385,10895,604],{"className":10896},[406],[385,10898,1116],{"className":10899},[844],[385,10901,1105],{"className":10902},[685],[385,10904,9916],{"className":10905},[406,407],[385,10907,1116],{"className":10908},[844]," means item ",[385,10911,10913],{"className":10912},[388],[385,10914,10916],{"className":10915,"ariaHidden":393},[392],[385,10917,10919,10922],{"className":10918},[397],[385,10920],{"className":10921,"style":423},[401],[385,10923,427],{"className":10924},[406,407]," was skipped (budget unchanged); a jump means it was taken (budget drops by ",[385,10927,10929],{"className":10928},[388],[385,10930,10932],{"className":10931,"ariaHidden":393},[392],[385,10933,10935,10938],{"className":10934},[397],[385,10936],{"className":10937,"style":441},[401],[385,10939,10941,10944],{"className":10940},[406],[385,10942,449],{"className":10943,"style":448},[406,407],[385,10945,10947],{"className":10946},[453],[385,10948,10950,10970],{"className":10949},[457,458],[385,10951,10953,10967],{"className":10952},[462],[385,10954,10956],{"className":10955,"style":467},[466],[385,10957,10958,10961],{"style":470},[385,10959],{"className":10960,"style":475},[474],[385,10962,10964],{"className":10963},[479,480,481,482],[385,10965,427],{"className":10966},[406,407,482],[385,10968,490],{"className":10969},[489],[385,10971,10973],{"className":10972},[462],[385,10974,10976],{"className":10975,"style":497},[466],[385,10977],{},"). Items ",[385,10980,10982],{"className":10981},[388],[385,10983,10985],{"className":10984,"ariaHidden":393},[392],[385,10986,10988,10991],{"className":10987},[397],[385,10989],{"className":10990,"style":880},[401],[385,10992,2639],{"className":10993},[406],[385,10995,10997],{"className":10996},[388],[385,10998,11000],{"className":10999,"ariaHidden":393},[392],[385,11001,11003,11006],{"className":11002},[397],[385,11004],{"className":11005,"style":880},[401],[385,11007,767],{"className":11008},[406]," are recovered, weight ",[385,11011,11013],{"className":11012},[388],[385,11014,11016,11034,11052],{"className":11015,"ariaHidden":393},[392],[385,11017,11019,11022,11025,11028,11031],{"className":11018},[397],[385,11020],{"className":11021,"style":9390},[401],[385,11023,2639],{"className":11024},[406],[385,11026],{"className":11027,"style":1448},[665],[385,11029,4418],{"className":11030},[1452],[385,11032],{"className":11033,"style":1448},[665],[385,11035,11037,11040,11043,11046,11049],{"className":11036},[397],[385,11038],{"className":11039,"style":880},[401],[385,11041,767],{"className":11042},[406],[385,11044],{"className":11045,"style":666},[665],[385,11047,671],{"className":11048},[670],[385,11050],{"className":11051,"style":666},[665],[385,11053,11055,11058],{"className":11054},[397],[385,11056],{"className":11057,"style":880},[401],[385,11059,5818],{"className":11060},[406],", value ",[385,11063,11065],{"className":11064},[388],[385,11066,11068,11086,11104],{"className":11067,"ariaHidden":393},[392],[385,11069,11071,11074,11077,11080,11083],{"className":11070},[397],[385,11072],{"className":11073,"style":9390},[401],[385,11075,5702],{"className":11076},[406],[385,11078],{"className":11079,"style":1448},[665],[385,11081,4418],{"className":11082},[1452],[385,11084],{"className":11085,"style":1448},[665],[385,11087,11089,11092,11095,11098,11101],{"className":11088},[397],[385,11090],{"className":11091,"style":880},[401],[385,11093,2696],{"className":11094},[406],[385,11096],{"className":11097,"style":666},[665],[385,11099,671],{"className":11100},[670],[385,11102],{"className":11103,"style":666},[665],[385,11105,11107,11110],{"className":11106},[397],[385,11108],{"className":11109,"style":880},[401],[385,11111,8373],{"className":11112},[406],[632,11114,11116],{"id":11115},"running-time-and-the-pseudo-polynomial-trap","Running time and the pseudo-polynomial trap",[381,11118,11119,11120,11186,11187,11211,11212,11234],{},"The table has ",[385,11121,11123],{"className":11122},[388],[385,11124,11126,11147,11174],{"className":11125,"ariaHidden":393},[392],[385,11127,11129,11132,11135,11138,11141,11144],{"className":11128},[397],[385,11130],{"className":11131,"style":681},[401],[385,11133,1049],{"className":11134},[685],[385,11136,829],{"className":11137},[406,407],[385,11139],{"className":11140,"style":1448},[665],[385,11142,4418],{"className":11143},[1452],[385,11145],{"className":11146,"style":1448},[665],[385,11148,11150,11153,11156,11159,11162,11165,11168,11171],{"className":11149},[397],[385,11151],{"className":11152,"style":681},[401],[385,11154,604],{"className":11155},[406],[385,11157,1066],{"className":11158},[844],[385,11160,1049],{"className":11161},[685],[385,11163,409],{"className":11164,"style":408},[406,407],[385,11166],{"className":11167,"style":1448},[665],[385,11169,4418],{"className":11170},[1452],[385,11172],{"className":11173,"style":1448},[665],[385,11175,11177,11180,11183],{"className":11176},[397],[385,11178],{"className":11179,"style":681},[401],[385,11181,604],{"className":11182},[406],[385,11184,1066],{"className":11185},[844]," entries, each filled in ",[385,11188,11190],{"className":11189},[388],[385,11191,11193],{"className":11192,"ariaHidden":393},[392],[385,11194,11196,11199,11202,11205,11208],{"className":11195},[397],[385,11197],{"className":11198,"style":681},[401],[385,11200,4477],{"className":11201},[406],[385,11203,1049],{"className":11204},[685],[385,11206,604],{"className":11207},[406],[385,11209,1066],{"className":11210},[844],", so\n",[385,11213,11215],{"className":11214},[388],[385,11216,11218],{"className":11217,"ariaHidden":393},[392],[385,11219,11221,11224],{"className":11220},[397],[385,11222],{"className":11223,"style":9944},[401],[385,11225,11227],{"className":11226},[578,579],[385,11228,11230],{"className":11229},[406,583],[385,11231,11233],{"className":11232},[406],"Knapsack-01"," runs in",[385,11236,11238],{"className":11237},[1833],[385,11239,11241],{"className":11240},[388],[385,11242,11244],{"className":11243,"ariaHidden":393},[392],[385,11245,11247,11250,11253,11256,11260],{"className":11246},[397],[385,11248],{"className":11249,"style":681},[401],[385,11251,4477],{"className":11252},[406],[385,11254,1049],{"className":11255},[685],[385,11257,11259],{"className":11258,"style":408},[406,407],"nW",[385,11261,1066],{"className":11262},[844],[381,11264,11265,11266,11290,11291,11306],{},"time and space. (Space drops to ",[385,11267,11269],{"className":11268},[388],[385,11270,11272],{"className":11271,"ariaHidden":393},[392],[385,11273,11275,11278,11281,11284,11287],{"className":11274},[397],[385,11276],{"className":11277,"style":681},[401],[385,11279,4477],{"className":11280},[406],[385,11282,1049],{"className":11283},[685],[385,11285,409],{"className":11286,"style":408},[406,407],[385,11288,1066],{"className":11289},[844]," if only the value is needed, since\neach row reads only the previous one, so scan ",[385,11292,11294],{"className":11293},[388],[385,11295,11297],{"className":11296,"ariaHidden":393},[392],[385,11298,11300,11303],{"className":11299},[397],[385,11301],{"className":11302,"style":574},[401],[385,11304,9916],{"className":11305},[406,407]," from high to low to reuse a\nsingle array.)",[381,11308,11309,11310,11389,11390,11393,11394,11409,11410,11480],{},"The descending sweep is not a detail but the whole correctness argument for the\n1-D version: when we apply an item, ",[385,11311,11313],{"className":11312},[388],[385,11314,11316,11340],{"className":11315,"ariaHidden":393},[392],[385,11317,11319,11322,11325,11328,11331,11334,11337],{"className":11318},[397],[385,11320],{"className":11321,"style":681},[401],[385,11323,5522],{"className":11324,"style":5521},[406,407],[385,11326,1105],{"className":11327},[685],[385,11329,9916],{"className":11330},[406,407],[385,11332],{"className":11333,"style":1448},[665],[385,11335,1453],{"className":11336},[1452],[385,11338],{"className":11339,"style":1448},[665],[385,11341,11343,11346,11386],{"className":11342},[397],[385,11344],{"className":11345,"style":681},[401],[385,11347,11349,11352],{"className":11348},[406],[385,11350,449],{"className":11351,"style":448},[406,407],[385,11353,11355],{"className":11354},[453],[385,11356,11358,11378],{"className":11357},[457,458],[385,11359,11361,11375],{"className":11360},[462],[385,11362,11364],{"className":11363,"style":467},[466],[385,11365,11366,11369],{"style":470},[385,11367],{"className":11368,"style":475},[474],[385,11370,11372],{"className":11371},[479,480,481,482],[385,11373,427],{"className":11374},[406,407,482],[385,11376,490],{"className":11377},[489],[385,11379,11381],{"className":11380},[462],[385,11382,11384],{"className":11383,"style":497},[466],[385,11385],{},[385,11387,1116],{"className":11388},[844]," must still hold the value from\n",[1013,11391,11392],{},"before"," this item was available, so we must reach ",[385,11395,11397],{"className":11396},[388],[385,11398,11400],{"className":11399,"ariaHidden":393},[392],[385,11401,11403,11406],{"className":11402},[397],[385,11404],{"className":11405,"style":574},[401],[385,11407,9916],{"className":11408},[406,407]," before we overwrite\n",[385,11411,11413],{"className":11412},[388],[385,11414,11416,11434],{"className":11415,"ariaHidden":393},[392],[385,11417,11419,11422,11425,11428,11431],{"className":11418},[397],[385,11420],{"className":11421,"style":10044},[401],[385,11423,9916],{"className":11424},[406,407],[385,11426],{"className":11427,"style":1448},[665],[385,11429,1453],{"className":11430},[1452],[385,11432],{"className":11433,"style":1448},[665],[385,11435,11437,11440],{"className":11436},[397],[385,11438],{"className":11439,"style":441},[401],[385,11441,11443,11446],{"className":11442},[406],[385,11444,449],{"className":11445,"style":448},[406,407],[385,11447,11449],{"className":11448},[453],[385,11450,11452,11472],{"className":11451},[457,458],[385,11453,11455,11469],{"className":11454},[462],[385,11456,11458],{"className":11457,"style":467},[466],[385,11459,11460,11463],{"style":470},[385,11461],{"className":11462,"style":475},[474],[385,11464,11466],{"className":11465},[479,480,481,482],[385,11467,427],{"className":11468},[406,407,482],[385,11470,490],{"className":11471},[489],[385,11473,11475],{"className":11474},[462],[385,11476,11478],{"className":11477,"style":497},[466],[385,11479],{}," — that is, go high to low.",[2895,11482,11484,11798],{"className":11483},[2898,2899],[1929,11485,11489],{"xmlns":1931,"width":11486,"height":11487,"viewBox":11488},"317.716","144.262","-75 -75 238.287 108.196",[2906,11490,11491,11506,11509,11516,11519,11526,11529,11534,11537,11542,11545,11550,11553,11558,11579,11598,11617,11636,11655,11674,11681,11684,11690,11693,11699,11711,11723,11734,11745],{"stroke":2908,"style":2909},[2906,11492,11493,11500],{"stroke":2912,"fontFamily":6154,"fontSize":6023},[2906,11494,11496],{"transform":11495},"translate(-63.975 2.778)",[1938,11497],{"d":11498,"fill":2908,"stroke":2908,"className":11499,"style":6031},"M2.856-45.673L2.575-45.673L2.575-50.392Q2.575-50.607 2.513-50.702Q2.450-50.798 2.333-50.819Q2.216-50.841 1.970-50.841L1.970-51.138L3.192-51.224L3.192-48.736Q3.669-49.200 4.368-49.200Q4.849-49.200 5.257-48.956Q5.665-48.712 5.901-48.298Q6.138-47.884 6.138-47.400Q6.138-47.025 5.989-46.696Q5.841-46.368 5.571-46.116Q5.302-45.864 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46.033-45.970",[2921],[2906,11781,11782],{"transform":11757},[1938,11783],{"d":11784,"fill":3594,"stroke":3594,"className":11785,"style":6031},"M58.422-47.489L52.078-47.489Q52.004-47.489 51.953-47.546Q51.903-47.603 51.903-47.673Q51.903-47.743 51.950-47.800Q51.996-47.857 52.078-47.857L58.422-47.857Q57.969-48.185 57.672-48.652Q57.375-49.118 57.270-49.657Q57.270-49.759 57.367-49.786L57.535-49.786Q57.617-49.767 57.637-49.681Q57.766-49.005 58.246-48.487Q58.727-47.970 59.391-47.778Q59.453-47.755 59.453-47.673Q59.453-47.591 59.391-47.568Q58.727-47.380 58.246-46.861Q57.766-46.341 57.637-45.665Q57.617-45.579 57.535-45.560L57.367-45.560Q57.270-45.587 57.270-45.689Q57.344-46.056 57.502-46.388Q57.660-46.720 57.893-46.997Q58.125-47.275 58.422-47.489",[2921],[2906,11787,11788],{"transform":11757},[1938,11789],{"d":11790,"fill":3594,"stroke":3594,"className":11791,"style":6031},"M64.916-45.673L63.084-45.673L63.084-45.970Q63.358-45.970 63.526-46.017Q63.694-46.064 63.694-46.232L63.694-50.392Q63.694-50.607 63.631-50.702Q63.569-50.798 63.450-50.819Q63.330-50.841 63.084-50.841L63.084-51.138L64.307-51.224L64.307-46.232Q64.307-46.064 64.475-46.017Q64.643-45.970 64.916-45.970L64.916-45.673M65.362-47.368Q65.362-47.872 65.618-48.304Q65.873-48.736 66.309-48.987Q66.744-49.239 67.244-49.239Q67.631-49.239 67.973-49.095Q68.315-48.950 68.577-48.689Q68.838-48.427 68.981-48.091Q69.123-47.755 69.123-47.368Q69.123-46.876 68.860-46.466Q68.596-46.056 68.166-45.825Q67.737-45.595 67.244-45.595Q66.752-45.595 66.319-45.827Q65.885-46.060 65.623-46.468Q65.362-46.876 65.362-47.368M67.244-45.872Q67.702-45.872 67.953-46.095Q68.205-46.318 68.293-46.669Q68.381-47.021 68.381-47.466Q68.381-47.896 68.287-48.234Q68.194-48.571 67.940-48.778Q67.686-48.986 67.244-48.986Q66.596-48.986 66.352-48.569Q66.108-48.153 66.108-47.466Q66.108-47.021 66.196-46.669Q66.284-46.318 66.535-46.095Q66.787-45.872 67.244-45.872",[2921],[2906,11793,11794],{"transform":11757},[1938,11795],{"d":11796,"fill":3594,"stroke":3594,"className":11797,"style":6031},"M70.963-45.704L69.893-48.560Q69.826-48.739 69.696-48.782Q69.565-48.825 69.307-48.825L69.307-49.122L70.987-49.122L70.987-48.825Q70.537-48.825 70.537-48.626Q70.541-48.611 70.543-48.593Q70.545-48.575 70.545-48.560L71.338-46.466L72.049-48.376Q72.014-48.470 72.014-48.515Q72.014-48.560 71.979-48.560Q71.912-48.739 71.782-48.782Q71.651-48.825 71.397-48.825L71.397-49.122L72.987-49.122L72.987-48.825Q72.537-48.825 72.537-48.626Q72.541-48.607 72.543-48.589Q72.545-48.571 72.545-48.560L73.377-46.345L74.131-48.345Q74.155-48.403 74.155-48.474Q74.155-48.634 74.018-48.730Q73.881-48.825 73.713-48.825L73.713-49.122L75.100-49.122L75.100-48.825Q74.866-48.825 74.688-48.698Q74.510-48.571 74.428-48.345L73.444-45.704Q73.389-45.595 73.276-45.595L73.217-45.595Q73.104-45.595 73.061-45.704L72.201-47.978L71.346-45.704Q71.307-45.595 71.186-45.595L71.131-45.595Q71.018-45.595 70.963-45.704",[2921],[3822,11799,11801,11802,2423,11817,11859,11860,11912,11913,12023,12024,12057,12058,12073],{"className":11800},[3825],"0\u002F1 knapsack on one rolling array, applying item ",[385,11803,11805],{"className":11804},[388],[385,11806,11808],{"className":11807,"ariaHidden":393},[392],[385,11809,11811,11814],{"className":11810},[397],[385,11812],{"className":11813,"style":880},[401],[385,11815,767],{"className":11816},[406],[385,11818,11820],{"className":11819},[388],[385,11821,11823],{"className":11822,"ariaHidden":393},[392],[385,11824,11826,11829,11832,11838,11841,11844,11847,11850,11856],{"className":11825},[397],[385,11827],{"className":11828,"style":1715},[401],[385,11830,449],{"className":11831,"style":448},[406,407],[385,11833,11835],{"className":11834},[406],[385,11836,671],{"className":11837},[670],[385,11839,767],{"className":11840},[406],[385,11842,733],{"className":11843},[732],[385,11845],{"className":11846,"style":737},[665],[385,11848,520],{"className":11849,"style":519},[406,407],[385,11851,11853],{"className":11852},[406],[385,11854,671],{"className":11855},[670],[385,11857,2696],{"className":11858},[406],") with the budget swept ",[385,11861,11863],{"className":11862},[388],[385,11864,11866,11884,11903],{"className":11865,"ariaHidden":393},[392],[385,11867,11869,11872,11875,11878,11881],{"className":11868},[397],[385,11870],{"className":11871,"style":574},[401],[385,11873,9916],{"className":11874},[406,407],[385,11876],{"className":11877,"style":666},[665],[385,11879,671],{"className":11880},[670],[385,11882],{"className":11883,"style":666},[665],[385,11885,11887,11890,11893,11896,11900],{"className":11886},[397],[385,11888],{"className":11889,"style":880},[401],[385,11891,5818],{"className":11892},[406],[385,11894],{"className":11895,"style":666},[665],[385,11897,11899],{"className":11898},[670],"→",[385,11901],{"className":11902,"style":666},[665],[385,11904,11906,11909],{"className":11905},[397],[385,11907],{"className":11908,"style":880},[401],[385,11910,767],{"className":11911},[406]," (descending). Each update ",[385,11914,11916],{"className":11915},[388],[385,11917,11919,11947,11995],{"className":11918,"ariaHidden":393},[392],[385,11920,11922,11925,11928,11931,11934,11937,11940,11944],{"className":11921},[397],[385,11923],{"className":11924,"style":681},[401],[385,11926,5522],{"className":11927,"style":5521},[406,407],[385,11929,1105],{"className":11930},[685],[385,11932,9916],{"className":11933},[406,407],[385,11935,1116],{"className":11936},[844],[385,11938],{"className":11939,"style":666},[665],[385,11941,11943],{"className":11942},[670],"←",[385,11945],{"className":11946,"style":666},[665],[385,11948,11950,11953,11959,11962,11965,11968,11971,11974,11977,11980,11983,11986,11989,11992],{"className":11949},[397],[385,11951],{"className":11952,"style":681},[401],[385,11954,11956],{"className":11955},[4948],[385,11957,5341],{"className":11958},[406,5340],[385,11960,1049],{"className":11961},[685],[385,11963,5522],{"className":11964,"style":5521},[406,407],[385,11966,1105],{"className":11967},[685],[385,11969,9916],{"className":11970},[406,407],[385,11972,1116],{"className":11973},[844],[385,11975,733],{"className":11976},[732],[385,11978],{"className":11979,"style":737},[665],[385,11981],{"className":11982,"style":737},[665],[385,11984,2696],{"className":11985},[406],[385,11987],{"className":11988,"style":1448},[665],[385,11990,4418],{"className":11991},[1452],[385,11993],{"className":11994,"style":1448},[665],[385,11996,11998,12001,12004,12007,12010,12016,12019],{"className":11997},[397],[385,11999],{"className":12000,"style":681},[401],[385,12002,5522],{"className":12003,"style":5521},[406,407],[385,12005,1105],{"className":12006},[685],[385,12008,9916],{"className":12009},[406,407],[385,12011,12013],{"className":12012},[406],[385,12014,1453],{"className":12015},[406],[385,12017,767],{"className":12018},[406],[385,12020,12022],{"className":12021},[844],"])"," reads ",[385,12025,12027],{"className":12026},[388],[385,12028,12030],{"className":12029,"ariaHidden":393},[392],[385,12031,12033,12036,12039,12042,12045,12051,12054],{"className":12032},[397],[385,12034],{"className":12035,"style":681},[401],[385,12037,5522],{"className":12038,"style":5521},[406,407],[385,12040,1105],{"className":12041},[685],[385,12043,9916],{"className":12044},[406,407],[385,12046,12048],{"className":12047},[406],[385,12049,1453],{"className":12050},[406],[385,12052,767],{"className":12053},[406],[385,12055,1116],{"className":12056},[844]," before it is overwritten, so item ",[385,12059,12061],{"className":12060},[388],[385,12062,12064],{"className":12063,"ariaHidden":393},[392],[385,12065,12067,12070],{"className":12066},[397],[385,12068],{"className":12069,"style":880},[401],[385,12071,767],{"className":12072},[406]," is used at most once.",[381,12075,12076,12077,12156,12157,12172,12173,12176,12179],{},"Sweeping the other way, low to high, would let ",[385,12078,12080],{"className":12079},[388],[385,12081,12083,12107],{"className":12082,"ariaHidden":393},[392],[385,12084,12086,12089,12092,12095,12098,12101,12104],{"className":12085},[397],[385,12087],{"className":12088,"style":681},[401],[385,12090,5522],{"className":12091,"style":5521},[406,407],[385,12093,1105],{"className":12094},[685],[385,12096,9916],{"className":12097},[406,407],[385,12099],{"className":12100,"style":1448},[665],[385,12102,1453],{"className":12103},[1452],[385,12105],{"className":12106,"style":1448},[665],[385,12108,12110,12113,12153],{"className":12109},[397],[385,12111],{"className":12112,"style":681},[401],[385,12114,12116,12119],{"className":12115},[406],[385,12117,449],{"className":12118,"style":448},[406,407],[385,12120,12122],{"className":12121},[453],[385,12123,12125,12145],{"className":12124},[457,458],[385,12126,12128,12142],{"className":12127},[462],[385,12129,12131],{"className":12130,"style":467},[466],[385,12132,12133,12136],{"style":470},[385,12134],{"className":12135,"style":475},[474],[385,12137,12139],{"className":12138},[479,480,481,482],[385,12140,427],{"className":12141},[406,407,482],[385,12143,490],{"className":12144},[489],[385,12146,12148],{"className":12147},[462],[385,12149,12151],{"className":12150,"style":497},[466],[385,12152],{},[385,12154,1116],{"className":12155},[844]," already include item\n",[385,12158,12160],{"className":12159},[388],[385,12161,12163],{"className":12162,"ariaHidden":393},[392],[385,12164,12166,12169],{"className":12165},[397],[385,12167],{"className":12168,"style":880},[401],[385,12170,767],{"className":12171},[406],", silently packing it twice — exactly the reuse that the\n",[598,12174,12175],{"href":257},"unbounded knapsack",[1013,12177,12178],{},"wants",", and that the ascending sweep there deliberately permits.",[381,12181,12182,12183,12207],{},"That ",[385,12184,12186],{"className":12185},[388],[385,12187,12189],{"className":12188,"ariaHidden":393},[392],[385,12190,12192,12195,12198,12201,12204],{"className":12191},[397],[385,12193],{"className":12194,"style":681},[401],[385,12196,4477],{"className":12197},[406],[385,12199,1049],{"className":12200},[685],[385,12202,11259],{"className":12203,"style":408},[406,407],[385,12205,1066],{"className":12206},[844]," looks polynomial, and here lies one of the most\nimportant subtleties in all of algorithms.",[381,12209,12210,12220,12221,12248,12249,12273,12274,12307,12308,943,12346,12364,12365,12380,12381,12384,12385,1427],{},[558,12211,12212,12213,889,12216,12219],{},"Is this ",[1013,12214,12215],{},"really",[590,12217,12218],{},"polynomial time","?"," It is tempting to call ",[385,12222,12224],{"className":12223},[388],[385,12225,12227],{"className":12226,"ariaHidden":393},[392],[385,12228,12230,12233,12236,12239,12242,12245],{"className":12229},[397],[385,12231],{"className":12232,"style":681},[401],[385,12234,4477],{"className":12235},[406],[385,12237,1049],{"className":12238},[685],[385,12240,829],{"className":12241},[406,407],[385,12243,863],{"className":12244},[406,407],[385,12246,1066],{"className":12247},[844]," (or\n",[385,12250,12252],{"className":12251},[388],[385,12253,12255],{"className":12254,"ariaHidden":393},[392],[385,12256,12258,12261,12264,12267,12270],{"className":12257},[397],[385,12259],{"className":12260,"style":681},[401],[385,12262,4477],{"className":12263},[406],[385,12265,1049],{"className":12266},[685],[385,12268,11259],{"className":12269,"style":408},[406,407],[385,12271,1066],{"className":12272},[844],") polynomial: the input is a list of ",[385,12275,12277],{"className":12276},[388],[385,12278,12280,12298],{"className":12279,"ariaHidden":393},[392],[385,12281,12283,12286,12289,12292,12295],{"className":12282},[397],[385,12284],{"className":12285,"style":7043},[401],[385,12287,829],{"className":12288},[406,407],[385,12290],{"className":12291,"style":1448},[665],[385,12293,4418],{"className":12294},[1452],[385,12296],{"className":12297,"style":1448},[665],[385,12299,12301,12304],{"className":12300},[397],[385,12302],{"className":12303,"style":880},[401],[385,12305,604],{"className":12306},[406]," numbers, so surely its\nsize is ",[385,12309,12311],{"className":12310},[388],[385,12312,12314,12328],{"className":12313,"ariaHidden":393},[392],[385,12315,12317,12321,12325],{"className":12316},[397],[385,12318],{"className":12319,"style":12320},[401],"height:0.4831em;",[385,12322,12324],{"className":12323},[670],"≈",[385,12326],{"className":12327,"style":666},[665],[385,12329,12331,12334,12337,12340,12343],{"className":12330},[397],[385,12332],{"className":12333,"style":681},[401],[385,12335,4477],{"className":12336},[406],[385,12338,1049],{"className":12339},[685],[385,12341,829],{"className":12342},[406,407],[385,12344,1066],{"className":12345},[844],[385,12347,12349],{"className":12348},[388],[385,12350,12352],{"className":12351,"ariaHidden":393},[392],[385,12353,12355,12358,12361],{"className":12354},[397],[385,12356],{"className":12357,"style":939},[401],[385,12359,829],{"className":12360},[406,407],[385,12362,863],{"className":12363},[406,407]," is polynomial in ",[385,12366,12368],{"className":12367},[388],[385,12369,12371],{"className":12370,"ariaHidden":393},[392],[385,12372,12374,12377],{"className":12373},[397],[385,12375],{"className":12376,"style":956},[401],[385,12378,829],{"className":12379},[406,407],". That accounting is\nwrong, and seeing why is what matters here. We have to count the input's size ",[1013,12382,12383],{},"in\nbits",", not in numbers. This is the same bit-length accounting that underlies\n",[598,12386,12387],{"href":17},"asymptotic analysis",[637,12389,12391],{"type":12390},"remark",[381,12392,12393,12396,12397,12449,12450,12465,12466,12600,12601,12697,12698,12701,12702,12717,12718,12760,12761,12776],{},[558,12394,12395],{},"Remark (An honest size parameter)."," Suppose each ",[385,12398,12400],{"className":12399},[388],[385,12401,12403],{"className":12402,"ariaHidden":393},[392],[385,12404,12406,12409],{"className":12405},[397],[385,12407],{"className":12408,"style":441},[401],[385,12410,12412,12415],{"className":12411},[406],[385,12413,598],{"className":12414},[406,407],[385,12416,12418],{"className":12417},[453],[385,12419,12421,12441],{"className":12420},[457,458],[385,12422,12424,12438],{"className":12423},[462],[385,12425,12427],{"className":12426,"style":467},[466],[385,12428,12429,12432],{"style":708},[385,12430],{"className":12431,"style":475},[474],[385,12433,12435],{"className":12434},[479,480,481,482],[385,12436,427],{"className":12437},[406,407,482],[385,12439,490],{"className":12440},[489],[385,12442,12444],{"className":12443},[462],[385,12445,12447],{"className":12446,"style":497},[466],[385,12448],{}," fits in ",[385,12451,12453],{"className":12452},[388],[385,12454,12456],{"className":12455,"ariaHidden":393},[392],[385,12457,12459,12462],{"className":12458},[397],[385,12460],{"className":12461,"style":574},[401],[385,12463,9916],{"className":12464},[406,407]," bits, so\n",[385,12467,12469],{"className":12468},[388],[385,12470,12472,12490,12545,12591],{"className":12471,"ariaHidden":393},[392],[385,12473,12475,12478,12481,12484,12487],{"className":12474},[397],[385,12476],{"className":12477,"style":9187},[401],[385,12479,884],{"className":12480},[406],[385,12482],{"className":12483,"style":666},[665],[385,12485,5052],{"className":12486},[670],[385,12488],{"className":12489,"style":666},[665],[385,12491,12493,12496,12536,12539,12542],{"className":12492},[397],[385,12494],{"className":12495,"style":6865},[401],[385,12497,12499,12502],{"className":12498},[406],[385,12500,598],{"className":12501},[406,407],[385,12503,12505],{"className":12504},[453],[385,12506,12508,12528],{"className":12507},[457,458],[385,12509,12511,12525],{"className":12510},[462],[385,12512,12514],{"className":12513,"style":467},[466],[385,12515,12516,12519],{"style":708},[385,12517],{"className":12518,"style":475},[474],[385,12520,12522],{"className":12521},[479,480,481,482],[385,12523,427],{"className":12524},[406,407,482],[385,12526,490],{"className":12527},[489],[385,12529,12531],{"className":12530},[462],[385,12532,12534],{"className":12533,"style":497},[466],[385,12535],{},[385,12537],{"className":12538,"style":666},[665],[385,12540,5052],{"className":12541},[670],[385,12543],{"className":12544,"style":666},[665],[385,12546,12548,12552,12582,12585,12588],{"className":12547},[397],[385,12549],{"className":12550,"style":12551},[401],"height:0.9324em;vertical-align:-0.0833em;",[385,12553,12555,12558],{"className":12554},[406],[385,12556,767],{"className":12557},[406],[385,12559,12561],{"className":12560},[453],[385,12562,12564],{"className":12563},[457],[385,12565,12567],{"className":12566},[462],[385,12568,12571],{"className":12569,"style":12570},[466],"height:0.8491em;",[385,12572,12573,12576],{"style":1001},[385,12574],{"className":12575,"style":475},[474],[385,12577,12579],{"className":12578},[479,480,481,482],[385,12580,9916],{"className":12581},[406,407,482],[385,12583],{"className":12584,"style":1448},[665],[385,12586,1453],{"className":12587},[1452],[385,12589],{"className":12590,"style":1448},[665],[385,12592,12594,12597],{"className":12593},[397],[385,12595],{"className":12596,"style":880},[401],[385,12598,604],{"className":12599},[406],", and likewise ",[385,12602,12604],{"className":12603},[388],[385,12605,12607,12625,12644,12688],{"className":12606,"ariaHidden":393},[392],[385,12608,12610,12613,12616,12619,12622],{"className":12609},[397],[385,12611],{"className":12612,"style":9187},[401],[385,12614,884],{"className":12615},[406],[385,12617],{"className":12618,"style":666},[665],[385,12620,5052],{"className":12621},[670],[385,12623],{"className":12624,"style":666},[665],[385,12626,12628,12632,12635,12638,12641],{"className":12627},[397],[385,12629],{"className":12630,"style":12631},[401],"height:0.7719em;vertical-align:-0.136em;",[385,12633,863],{"className":12634},[406,407],[385,12636],{"className":12637,"style":666},[665],[385,12639,5052],{"className":12640},[670],[385,12642],{"className":12643,"style":666},[665],[385,12645,12647,12650,12679,12682,12685],{"className":12646},[397],[385,12648],{"className":12649,"style":12551},[401],[385,12651,12653,12656],{"className":12652},[406],[385,12654,767],{"className":12655},[406],[385,12657,12659],{"className":12658},[453],[385,12660,12662],{"className":12661},[457],[385,12663,12665],{"className":12664},[462],[385,12666,12668],{"className":12667,"style":12570},[466],[385,12669,12670,12673],{"style":1001},[385,12671],{"className":12672,"style":475},[474],[385,12674,12676],{"className":12675},[479,480,481,482],[385,12677,9916],{"className":12678},[406,407,482],[385,12680],{"className":12681,"style":1448},[665],[385,12683,1453],{"className":12684},[1452],[385,12686],{"className":12687,"style":1448},[665],[385,12689,12691,12694],{"className":12690},[397],[385,12692],{"className":12693,"style":880},[401],[385,12695,604],{"className":12696},[406],". Then the ",[1013,12699,12700],{},"entire","\ninput — the ",[385,12703,12705],{"className":12704},[388],[385,12706,12708],{"className":12707,"ariaHidden":393},[392],[385,12709,12711,12714],{"className":12710},[397],[385,12712],{"className":12713,"style":956},[401],[385,12715,829],{"className":12716},[406,407]," integers plus the target — is written in about ",[385,12719,12721],{"className":12720},[388],[385,12722,12724,12745],{"className":12723,"ariaHidden":393},[392],[385,12725,12727,12730,12733,12736,12739,12742],{"className":12726},[397],[385,12728],{"className":12729,"style":681},[401],[385,12731,1049],{"className":12732},[685],[385,12734,829],{"className":12735},[406,407],[385,12737],{"className":12738,"style":1448},[665],[385,12740,4418],{"className":12741},[1452],[385,12743],{"className":12744,"style":1448},[665],[385,12746,12748,12751,12754,12757],{"className":12747},[397],[385,12749],{"className":12750,"style":681},[401],[385,12752,604],{"className":12753},[406],[385,12755,1066],{"className":12756},[844],[385,12758,9916],{"className":12759},[406,407]," bits.\nThat, not ",[385,12762,12764],{"className":12763},[388],[385,12765,12767],{"className":12766,"ariaHidden":393},[392],[385,12768,12770,12773],{"className":12769},[397],[385,12771],{"className":12772,"style":956},[401],[385,12774,829],{"className":12775},[406,407]," alone, is the honest size parameter.",[381,12778,12779,12780,12795,12796,12811,12812,733],{},"Now rewrite the running time against ",[385,12781,12783],{"className":12782},[388],[385,12784,12786],{"className":12785,"ariaHidden":393},[392],[385,12787,12789,12792],{"className":12788},[397],[385,12790],{"className":12791,"style":574},[401],[385,12793,9916],{"className":12794},[406,407],". Since ",[385,12797,12799],{"className":12798},[388],[385,12800,12802],{"className":12801,"ariaHidden":393},[392],[385,12803,12805,12808],{"className":12804},[397],[385,12806],{"className":12807,"style":939},[401],[385,12809,863],{"className":12810},[406,407]," can be as large as ",[385,12813,12815],{"className":12814},[388],[385,12816,12818,12862],{"className":12817,"ariaHidden":393},[392],[385,12819,12821,12824,12853,12856,12859],{"className":12820},[397],[385,12822],{"className":12823,"style":12551},[401],[385,12825,12827,12830],{"className":12826},[406],[385,12828,767],{"className":12829},[406],[385,12831,12833],{"className":12832},[453],[385,12834,12836],{"className":12835},[457],[385,12837,12839],{"className":12838},[462],[385,12840,12842],{"className":12841,"style":12570},[466],[385,12843,12844,12847],{"style":1001},[385,12845],{"className":12846,"style":475},[474],[385,12848,12850],{"className":12849},[479,480,481,482],[385,12851,9916],{"className":12852},[406,407,482],[385,12854],{"className":12855,"style":1448},[665],[385,12857,1453],{"className":12858},[1452],[385,12860],{"className":12861,"style":1448},[665],[385,12863,12865,12868],{"className":12864},[397],[385,12866],{"className":12867,"style":880},[401],[385,12869,604],{"className":12870},[406],[385,12872,12874],{"className":12873},[1833],[385,12875,12877],{"className":12876},[388],[385,12878,12880,12916,12949],{"className":12879,"ariaHidden":393},[392],[385,12881,12883,12886,12889,12892,12895,12898,12901,12904,12907,12910,12913],{"className":12882},[397],[385,12884],{"className":12885,"style":681},[401],[385,12887,4477],{"className":12888},[406],[385,12890,1049],{"className":12891},[685],[385,12893,829],{"className":12894},[406,407],[385,12896,863],{"className":12897},[406,407],[385,12899,1066],{"className":12900},[844],[385,12902],{"className":12903,"style":666},[665],[385,12905],{"className":12906,"style":666},[665],[385,12908,671],{"className":12909},[670],[385,12911],{"className":12912,"style":666},[665],[385,12914],{"className":12915,"style":666},[665],[385,12917,12919,12923,12926,12930,12936,12939,12942,12946],{"className":12918},[397],[385,12920],{"className":12921,"style":12922},[401],"height:1.2em;vertical-align:-0.35em;",[385,12924,4477],{"className":12925},[406],[385,12927],{"className":12928,"style":12929},[665],"margin-right:-0.1667em;",[385,12931,12933],{"className":12932},[685],[385,12934,1049],{"className":12935},[1892,7564],[385,12937,829],{"className":12938},[406,407],[385,12940],{"className":12941,"style":1448},[665],[385,12943,12945],{"className":12944},[1452],"⋅",[385,12947],{"className":12948,"style":1448},[665],[385,12950,12952,12956,12990,12996],{"className":12951},[397],[385,12953],{"className":12954,"style":12955},[401],"height:1.2491em;vertical-align:-0.35em;",[385,12957,12959,12962],{"className":12958},[406],[385,12960,767],{"className":12961},[406],[385,12963,12965],{"className":12964},[453],[385,12966,12968],{"className":12967},[457],[385,12969,12971],{"className":12970},[462],[385,12972,12975],{"className":12973,"style":12974},[466],"height:0.8991em;",[385,12976,12978,12981],{"style":12977},"top:-3.113em;margin-right:0.05em;",[385,12979],{"className":12980,"style":475},[474],[385,12982,12984],{"className":12983},[479,480,481,482],[385,12985,12987],{"className":12986},[406,482],[385,12988,9916],{"className":12989},[406,407,482],[385,12991,12993],{"className":12992},[844],[385,12994,1066],{"className":12995},[1892,7564],[385,12997,733],{"className":12998},[732],[381,13000,13001,13002,13005,13006,13021,13022,13037,13038,13041,13042,13045,13046,13049,13050,13052,13053,13069,13070,13092,13093,1427,13134,13141,13142,1427],{},"which is ",[558,13003,13004],{},"exponential"," in ",[385,13007,13009],{"className":13008},[388],[385,13010,13012],{"className":13011,"ariaHidden":393},[392],[385,13013,13015,13018],{"className":13014},[397],[385,13016],{"className":13017,"style":574},[401],[385,13019,9916],{"className":13020},[406,407],", the number of bits of a single input integer.\nDoubling the bits used to write the target squares ",[385,13023,13025],{"className":13024},[388],[385,13026,13028],{"className":13027,"ariaHidden":393},[392],[385,13029,13031,13034],{"className":13030},[397],[385,13032],{"className":13033,"style":939},[401],[385,13035,863],{"className":13036},[406,407]," and thus squares the\nrunning time. An algorithm whose time is polynomial in the ",[1013,13039,13040],{},"numeric value"," of an\ninput but exponential in its ",[1013,13043,13044],{},"encoded length"," is called ",[558,13047,13048],{},"pseudo-polynomial",": it\ncounts as ",[590,13051,592],{}," only if we dishonestly pretend an integer ",[385,13054,13056],{"className":13055},[388],[385,13057,13059],{"className":13058,"ariaHidden":393},[392],[385,13060,13062,13065],{"className":13061},[397],[385,13063],{"className":13064,"style":956},[401],[385,13066,13068],{"className":13067},[406,407],"s"," has size\n",[385,13071,13073],{"className":13072},[388],[385,13074,13076],{"className":13075,"ariaHidden":393},[392],[385,13077,13079,13082,13086,13089],{"className":13078},[397],[385,13080],{"className":13081,"style":681},[401],[385,13083,13085],{"className":13084},[406],"∣",[385,13087,13068],{"className":13088},[406,407],[385,13090,13085],{"className":13091},[406]," rather than its true size ",[385,13094,13096],{"className":13095},[388],[385,13097,13099],{"className":13098,"ariaHidden":393},[392],[385,13100,13102,13105,13108,13111,13119,13122,13125,13128,13131],{"className":13101},[397],[385,13103],{"className":13104,"style":681},[401],[385,13106,4477],{"className":13107},[406],[385,13109,1049],{"className":13110},[685],[385,13112,13114],{"className":13113},[4948],[385,13115,13118],{"className":13116,"style":13117},[406,5340],"margin-right:0.0139em;","log",[385,13120],{"className":13121,"style":737},[665],[385,13123,13085],{"className":13124},[406],[385,13126,13068],{"className":13127},[406,407],[385,13129,13085],{"className":13130},[406],[385,13132,1066],{"className":13133},[844],[595,13135,13136],{},[598,13137,2639],{"href":13138,"ariaDescribedBy":13139,"dataFootnoteRef":376,"id":13140},"#user-content-fn-erickson-knap",[602],"user-content-fnref-erickson-knap"," Such running\ntimes are central to\n",[598,13143,13144],{"href":369},"coping with hardness",[637,13146,13147],{"type":12390},[381,13148,13149,13152,13153,13156,13157,13399,13400,13498,13499,13502,13503,889,13544,13565,13566,1427],{},[558,13150,13151],{},"Remark (Pseudo-polynomial complexity)."," Can we get a ",[1013,13154,13155],{},"real"," polynomial-time algorithm? That would mean running time\n",[385,13158,13160],{"className":13159},[388],[385,13161,13163,13200,13240],{"className":13162,"ariaHidden":393},[392],[385,13164,13166,13169,13173,13176,13179,13182,13185,13188,13191,13194,13197],{"className":13165},[397],[385,13167],{"className":13168,"style":681},[401],[385,13170,13172],{"className":13171,"style":408},[406,407],"T",[385,13174,1049],{"className":13175},[685],[385,13177,829],{"className":13178},[406,407],[385,13180,733],{"className":13181},[732],[385,13183],{"className":13184,"style":737},[665],[385,13186,9916],{"className":13187},[406,407],[385,13189,1066],{"className":13190},[844],[385,13192],{"className":13193,"style":666},[665],[385,13195,671],{"className":13196},[670],[385,13198],{"className":13199,"style":666},[665],[385,13201,13203,13206,13213,13216,13219,13222,13225,13228,13231,13234,13237],{"className":13202},[397],[385,13204],{"className":13205,"style":681},[401],[385,13207,13209],{"className":13208},[4948],[385,13210,13212],{"className":13211,"style":13117},[406,5340],"poly",[385,13214,1049],{"className":13215},[685],[385,13217,829],{"className":13218},[406,407],[385,13220,733],{"className":13221},[732],[385,13223],{"className":13224,"style":737},[665],[385,13226,9916],{"className":13227},[406,407],[385,13229,1066],{"className":13230},[844],[385,13232],{"className":13233,"style":666},[665],[385,13235,671],{"className":13236},[670],[385,13238],{"className":13239,"style":666},[665],[385,13241,13243,13246,13249,13252,13327,13396],{"className":13242},[397],[385,13244],{"className":13245,"style":681},[401],[385,13247,4477],{"className":13248},[406],[385,13250,1049],{"className":13251},[685],[385,13253,13255,13258],{"className":13254},[406],[385,13256,829],{"className":13257},[406,407],[385,13259,13261],{"className":13260},[453],[385,13262,13264],{"className":13263},[457],[385,13265,13267],{"className":13266},[462],[385,13268,13270],{"className":13269,"style":980},[466],[385,13271,13272,13275],{"style":1001},[385,13273],{"className":13274,"style":475},[474],[385,13276,13278],{"className":13277},[479,480,481,482],[385,13279,13281],{"className":13280},[406,482],[385,13282,13284,13288],{"className":13283},[406,482],[385,13285,13287],{"className":13286},[406,407,482],"c",[385,13289,13291],{"className":13290},[453],[385,13292,13294,13318],{"className":13293},[457,458],[385,13295,13297,13315],{"className":13296},[462],[385,13298,13301],{"className":13299,"style":13300},[466],"height:0.3173em;",[385,13302,13304,13308],{"style":13303},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[385,13305],{"className":13306,"style":13307},[474],"height:2.5em;",[385,13309,13312],{"className":13310},[479,13311,7564,482],"reset-size3",[385,13313,604],{"className":13314},[406,482],[385,13316,490],{"className":13317},[489],[385,13319,13321],{"className":13320},[462],[385,13322,13325],{"className":13323,"style":13324},[466],"height:0.143em;",[385,13326],{},[385,13328,13330,13333],{"className":13329},[406],[385,13331,9916],{"className":13332},[406,407],[385,13334,13336],{"className":13335},[453],[385,13337,13339],{"className":13338},[457],[385,13340,13342],{"className":13341},[462],[385,13343,13345],{"className":13344,"style":980},[466],[385,13346,13347,13350],{"style":1001},[385,13348],{"className":13349,"style":475},[474],[385,13351,13353],{"className":13352},[479,480,481,482],[385,13354,13356],{"className":13355},[406,482],[385,13357,13359,13362],{"className":13358},[406,482],[385,13360,13287],{"className":13361},[406,407,482],[385,13363,13365],{"className":13364},[453],[385,13366,13368,13388],{"className":13367},[457,458],[385,13369,13371,13385],{"className":13370},[462],[385,13372,13374],{"className":13373,"style":13300},[466],[385,13375,13376,13379],{"style":13303},[385,13377],{"className":13378,"style":13307},[474],[385,13380,13382],{"className":13381},[479,13311,7564,482],[385,13383,767],{"className":13384},[406,482],[385,13386,490],{"className":13387},[489],[385,13389,13391],{"className":13390},[462],[385,13392,13394],{"className":13393,"style":13324},[466],[385,13395],{},[385,13397,1066],{"className":13398},[844]," for some\nconstants ",[385,13401,13403],{"className":13402},[388],[385,13404,13406],{"className":13405,"ariaHidden":393},[392],[385,13407,13409,13412,13452,13455,13458],{"className":13408},[397],[385,13410],{"className":13411,"style":902},[401],[385,13413,13415,13418],{"className":13414},[406],[385,13416,13287],{"className":13417},[406,407],[385,13419,13421],{"className":13420},[453],[385,13422,13424,13444],{"className":13423},[457,458],[385,13425,13427,13441],{"className":13426},[462],[385,13428,13430],{"className":13429,"style":705},[466],[385,13431,13432,13435],{"style":708},[385,13433],{"className":13434,"style":475},[474],[385,13436,13438],{"className":13437},[479,480,481,482],[385,13439,604],{"className":13440},[406,482],[385,13442,490],{"className":13443},[489],[385,13445,13447],{"className":13446},[462],[385,13448,13450],{"className":13449,"style":497},[466],[385,13451],{},[385,13453,733],{"className":13454},[732],[385,13456],{"className":13457,"style":737},[665],[385,13459,13461,13464],{"className":13460},[406],[385,13462,13287],{"className":13463},[406,407],[385,13465,13467],{"className":13466},[453],[385,13468,13470,13490],{"className":13469},[457,458],[385,13471,13473,13487],{"className":13472},[462],[385,13474,13476],{"className":13475,"style":705},[466],[385,13477,13478,13481],{"style":708},[385,13479],{"className":13480,"style":475},[474],[385,13482,13484],{"className":13483},[479,480,481,482],[385,13485,767],{"className":13486},[406,482],[385,13488,490],{"className":13489},[489],[385,13491,13493],{"className":13492},[462],[385,13494,13496],{"className":13495,"style":497},[466],[385,13497],{}," — polynomial in the ",[1013,13500,13501],{},"bit length",", not the value.\n",[558,13504,13505,13506,8168],{},"Most likely not, unless ",[385,13507,13509],{"className":13508},[388],[385,13510,13512,13531],{"className":13511,"ariaHidden":393},[392],[385,13513,13515,13518,13522,13525,13528],{"className":13514},[397],[385,13516],{"className":13517,"style":402},[401],[385,13519,13521],{"className":13520},[406,5340],"P",[385,13523],{"className":13524,"style":666},[665],[385,13526,671],{"className":13527},[670],[385,13529],{"className":13530,"style":666},[665],[385,13532,13534,13537],{"className":13533},[397],[385,13535],{"className":13536,"style":402},[401],[385,13538,13540],{"className":13539},[406],[385,13541,13543],{"className":13542},[406,5340],"NP",[385,13545,13547],{"className":13546},[388],[385,13548,13550],{"className":13549,"ariaHidden":393},[392],[385,13551,13553,13556],{"className":13552},[397],[385,13554],{"className":13555,"style":574},[401],[385,13557,13559],{"className":13558},[578,579],[385,13560,13562],{"className":13561},[406,583],[385,13563,629],{"className":13564},[406],"\n(and hence 0\u002F1 knapsack) is ",[385,13567,13569],{"className":13568},[388],[385,13570,13572],{"className":13571,"ariaHidden":393},[392],[385,13573,13575,13578],{"className":13574},[397],[385,13576],{"className":13577,"style":9944},[401],[385,13579,13581],{"className":13580},[578,579],[385,13582,13584],{"className":13583},[406,583],[385,13585,13587],{"className":13586},[406],"NP-complete",[381,13589,13590,13591,13618,13619,13634,13635,13651,13652,1427],{},"So the ",[385,13592,13594],{"className":13593},[388],[385,13595,13597],{"className":13596,"ariaHidden":393},[392],[385,13598,13600,13603,13606,13609,13612,13615],{"className":13599},[397],[385,13601],{"className":13602,"style":681},[401],[385,13604,4477],{"className":13605},[406],[385,13607,1049],{"className":13608},[685],[385,13610,829],{"className":13611},[406,407],[385,13613,863],{"className":13614},[406,407],[385,13616,1066],{"className":13617},[844]," DP is not a flaw in our analysis but the best we currently\nknow how to do. It is fast and practical precisely when the numbers are small\n(a target in the thousands, say), and uselessly slow when ",[385,13620,13622],{"className":13621},[388],[385,13623,13625],{"className":13624,"ariaHidden":393},[392],[385,13626,13628,13631],{"className":13627},[397],[385,13629],{"className":13630,"style":939},[401],[385,13632,863],{"className":13633},[406,407]," is a ",[385,13636,13638],{"className":13637},[388],[385,13639,13641],{"className":13640,"ariaHidden":393},[392],[385,13642,13644,13647],{"className":13643},[397],[385,13645],{"className":13646,"style":880},[401],[385,13648,13650],{"className":13649},[406],"200","-bit\ninteger, even though both instances have the same handful of elements. The\nlesson generalizes: any DP whose table is indexed by a numeric quantity (a\ntarget sum, a capacity) inherits this pseudo-polynomial character. The same\npattern recurs in\n",[598,13653,13654],{"href":257},"coin change and the unbounded knapsack",[632,13656,13658],{"id":13657},"subset-sum-as-a-special-case","Subset-sum as a special case",[381,13660,13661,13662,13665,13666,13782,13783,13798,13799,13961,13962,13995,13996,14011,14012,14027,14028,889,14031,14046,14047,14062],{},"We opened with subset-sum and built knapsack on top of it; the reduction also\nruns the other way, and making it precise shows the two problems are the ",[1013,13663,13664],{},"same\nmachine",". Take any subset-sum instance ",[385,13667,13669],{"className":13668},[388],[385,13670,13672],{"className":13671,"ariaHidden":393},[392],[385,13673,13675,13678,13681,13721,13724,13727,13730,13733,13736,13739,13779],{"className":13674},[397],[385,13676],{"className":13677,"style":681},[401],[385,13679,686],{"className":13680},[685],[385,13682,13684,13687],{"className":13683},[406],[385,13685,598],{"className":13686},[406,407],[385,13688,13690],{"className":13689},[453],[385,13691,13693,13713],{"className":13692},[457,458],[385,13694,13696,13710],{"className":13695},[462],[385,13697,13699],{"className":13698,"style":705},[466],[385,13700,13701,13704],{"style":708},[385,13702],{"className":13703,"style":475},[474],[385,13705,13707],{"className":13706},[479,480,481,482],[385,13708,604],{"className":13709},[406,482],[385,13711,490],{"className":13712},[489],[385,13714,13716],{"className":13715},[462],[385,13717,13719],{"className":13718,"style":497},[466],[385,13720],{},[385,13722,733],{"className":13723},[732],[385,13725],{"className":13726,"style":737},[665],[385,13728,789],{"className":13729},[788],[385,13731],{"className":13732,"style":737},[665],[385,13734,733],{"className":13735},[732],[385,13737],{"className":13738,"style":737},[665],[385,13740,13742,13745],{"className":13741},[406],[385,13743,598],{"className":13744},[406,407],[385,13746,13748],{"className":13747},[453],[385,13749,13751,13771],{"className":13750},[457,458],[385,13752,13754,13768],{"className":13753},[462],[385,13755,13757],{"className":13756,"style":817},[466],[385,13758,13759,13762],{"style":708},[385,13760],{"className":13761,"style":475},[474],[385,13763,13765],{"className":13764},[479,480,481,482],[385,13766,829],{"className":13767},[406,407,482],[385,13769,490],{"className":13770},[489],[385,13772,13774],{"className":13773},[462],[385,13775,13777],{"className":13776,"style":497},[466],[385,13778],{},[385,13780,845],{"className":13781},[844]," with\ntarget ",[385,13784,13786],{"className":13785},[388],[385,13787,13789],{"className":13788,"ariaHidden":393},[392],[385,13790,13792,13795],{"className":13791},[397],[385,13793],{"className":13794,"style":939},[401],[385,13796,863],{"className":13797},[406,407],", and feed it to 0\u002F1 knapsack with each item's value equal to its\nweight, ",[385,13800,13802],{"className":13801},[388],[385,13803,13805,13860,13915],{"className":13804,"ariaHidden":393},[392],[385,13806,13808,13811,13851,13854,13857],{"className":13807},[397],[385,13809],{"className":13810,"style":441},[401],[385,13812,13814,13817],{"className":13813},[406],[385,13815,520],{"className":13816,"style":519},[406,407],[385,13818,13820],{"className":13819},[453],[385,13821,13823,13843],{"className":13822},[457,458],[385,13824,13826,13840],{"className":13825},[462],[385,13827,13829],{"className":13828,"style":467},[466],[385,13830,13831,13834],{"style":535},[385,13832],{"className":13833,"style":475},[474],[385,13835,13837],{"className":13836},[479,480,481,482],[385,13838,427],{"className":13839},[406,407,482],[385,13841,490],{"className":13842},[489],[385,13844,13846],{"className":13845},[462],[385,13847,13849],{"className":13848,"style":497},[466],[385,13850],{},[385,13852],{"className":13853,"style":666},[665],[385,13855,671],{"className":13856},[670],[385,13858],{"className":13859,"style":666},[665],[385,13861,13863,13866,13906,13909,13912],{"className":13862},[397],[385,13864],{"className":13865,"style":441},[401],[385,13867,13869,13872],{"className":13868},[406],[385,13870,449],{"className":13871,"style":448},[406,407],[385,13873,13875],{"className":13874},[453],[385,13876,13878,13898],{"className":13877},[457,458],[385,13879,13881,13895],{"className":13880},[462],[385,13882,13884],{"className":13883,"style":467},[466],[385,13885,13886,13889],{"style":470},[385,13887],{"className":13888,"style":475},[474],[385,13890,13892],{"className":13891},[479,480,481,482],[385,13893,427],{"className":13894},[406,407,482],[385,13896,490],{"className":13897},[489],[385,13899,13901],{"className":13900},[462],[385,13902,13904],{"className":13903,"style":497},[466],[385,13905],{},[385,13907],{"className":13908,"style":666},[665],[385,13910,671],{"className":13911},[670],[385,13913],{"className":13914,"style":666},[665],[385,13916,13918,13921],{"className":13917},[397],[385,13919],{"className":13920,"style":441},[401],[385,13922,13924,13927],{"className":13923},[406],[385,13925,598],{"className":13926},[406,407],[385,13928,13930],{"className":13929},[453],[385,13931,13933,13953],{"className":13932},[457,458],[385,13934,13936,13950],{"className":13935},[462],[385,13937,13939],{"className":13938,"style":467},[466],[385,13940,13941,13944],{"style":708},[385,13942],{"className":13943,"style":475},[474],[385,13945,13947],{"className":13946},[479,480,481,482],[385,13948,427],{"className":13949},[406,407,482],[385,13951,490],{"className":13952},[489],[385,13954,13956],{"className":13955},[462],[385,13957,13959],{"className":13958,"style":497},[466],[385,13960],{},", and capacity ",[385,13963,13965],{"className":13964},[388],[385,13966,13968,13986],{"className":13967,"ariaHidden":393},[392],[385,13969,13971,13974,13977,13980,13983],{"className":13970},[397],[385,13972],{"className":13973,"style":402},[401],[385,13975,409],{"className":13976,"style":408},[406,407],[385,13978],{"className":13979,"style":666},[665],[385,13981,671],{"className":13982},[670],[385,13984],{"className":13985,"style":666},[665],[385,13987,13989,13992],{"className":13988},[397],[385,13990],{"className":13991,"style":939},[401],[385,13993,863],{"className":13994},[406,407],". The most value you can pack into\na capacity-",[385,13997,13999],{"className":13998},[388],[385,14000,14002],{"className":14001,"ariaHidden":393},[392],[385,14003,14005,14008],{"className":14004},[397],[385,14006],{"className":14007,"style":939},[401],[385,14009,863],{"className":14010},[406,407]," knapsack is at most ",[385,14013,14015],{"className":14014},[388],[385,14016,14018],{"className":14017,"ariaHidden":393},[392],[385,14019,14021,14024],{"className":14020},[397],[385,14022],{"className":14023,"style":939},[401],[385,14025,863],{"className":14026},[406,407],", and it ",[1013,14029,14030],{},"equals",[385,14032,14034],{"className":14033},[388],[385,14035,14037],{"className":14036,"ariaHidden":393},[392],[385,14038,14040,14043],{"className":14039},[397],[385,14041],{"className":14042,"style":939},[401],[385,14044,863],{"className":14045},[406,407]," exactly when some\nsubset of weights sums to ",[385,14048,14050],{"className":14049},[388],[385,14051,14053],{"className":14052,"ariaHidden":393},[392],[385,14054,14056,14059],{"className":14055},[397],[385,14057],{"className":14058,"style":939},[401],[385,14060,863],{"className":14061},[406,407]," without waste. So",[385,14064,14066],{"className":14065},[1833],[385,14067,14069],{"className":14068},[388],[385,14070,14072,14108,14133],{"className":14071,"ariaHidden":393},[392],[385,14073,14075,14078,14081,14084,14087,14090,14093,14096,14099,14102,14105],{"className":14074},[397],[385,14076],{"className":14077,"style":681},[401],[385,14079,5522],{"className":14080,"style":5521},[406,407],[385,14082,1049],{"className":14083},[685],[385,14085,829],{"className":14086},[406,407],[385,14088,733],{"className":14089},[732],[385,14091],{"className":14092,"style":737},[665],[385,14094,863],{"className":14095},[406,407],[385,14097,1066],{"className":14098},[844],[385,14100],{"className":14101,"style":666},[665],[385,14103,671],{"className":14104},[670],[385,14106],{"className":14107,"style":666},[665],[385,14109,14111,14114,14117,14120,14123,14127,14130],{"className":14110},[397],[385,14112],{"className":14113,"style":939},[401],[385,14115,863],{"className":14116},[406,407],[385,14118],{"className":14119,"style":666},[665],[385,14121],{"className":14122,"style":666},[665],[385,14124,14126],{"className":14125},[670],"↔",[385,14128],{"className":14129,"style":666},[665],[385,14131],{"className":14132,"style":666},[665],[385,14134,14136,14139,14146,14149,14156,14159],{"className":14135},[397],[385,14137],{"className":14138,"style":9944},[401],[385,14140,14142],{"className":14141},[406,583],[385,14143,14145],{"className":14144},[406],"some sublist of ",[385,14147,661],{"className":14148},[406,407],[385,14150,14152],{"className":14151},[406,583],[385,14153,14155],{"className":14154},[406]," sums to ",[385,14157,863],{"className":14158},[406,407],[385,14160,733],{"className":14161},[732],[381,14163,14164,14165,14186,14187,14202,14203,14221,14222,14249,14250,14271],{},"and one call to ",[385,14166,14168],{"className":14167},[388],[385,14169,14171],{"className":14170,"ariaHidden":393},[392],[385,14172,14174,14177],{"className":14173},[397],[385,14175],{"className":14176,"style":9944},[401],[385,14178,14180],{"className":14179},[578,579],[385,14181,14183],{"className":14182},[406,583],[385,14184,11233],{"className":14185},[406]," answers subset-sum. The two recurrences are\nthe same shape with the ",[385,14188,14190],{"className":14189},[388],[385,14191,14193],{"className":14192,"ariaHidden":393},[392],[385,14194,14196,14199],{"className":14195},[397],[385,14197],{"className":14198,"style":1808},[401],[385,14200,1812],{"className":14201},[406]," of the boolean table promoted to a ",[385,14204,14206],{"className":14205},[388],[385,14207,14209],{"className":14208,"ariaHidden":393},[392],[385,14210,14212,14215],{"className":14211},[397],[385,14213],{"className":14214,"style":956},[401],[385,14216,14218],{"className":14217},[4948],[385,14219,5341],{"className":14220},[406,5340]," over\nvalues, which is why everything transfers: ",[385,14223,14225],{"className":14224},[388],[385,14226,14228],{"className":14227,"ariaHidden":393},[392],[385,14229,14231,14234,14237,14240,14243,14246],{"className":14230},[397],[385,14232],{"className":14233,"style":681},[401],[385,14235,4477],{"className":14236},[406],[385,14238,1049],{"className":14239},[685],[385,14241,829],{"className":14242},[406,407],[385,14244,863],{"className":14245},[406,407],[385,14247,1066],{"className":14248},[844]," time,\npseudo-polynomial, and ",[385,14251,14253],{"className":14252},[388],[385,14254,14256],{"className":14255,"ariaHidden":393},[392],[385,14257,14259,14262],{"className":14258},[397],[385,14260],{"className":14261,"style":9944},[401],[385,14263,14265],{"className":14264},[578,579],[385,14266,14268],{"className":14267},[406,583],[385,14269,13587],{"className":14270},[406]," \u002F NP-hard in general.",[632,14273,14275,14276,14279],{"id":14274},"where-greed-does-work-fractional-knapsack","Where greed ",[1013,14277,14278],{},"does"," work: fractional knapsack",[381,14281,14282,14283,14372,14373,14388,14389,14392,14393,14396,14397,14492],{},"Change one rule, letting items be split so that we may take any fraction ",[385,14284,14286],{"className":14285},[388],[385,14287,14289,14307,14363],{"className":14288,"ariaHidden":393},[392],[385,14290,14292,14295,14298,14301,14304],{"className":14291},[397],[385,14293],{"className":14294,"style":9187},[401],[385,14296,884],{"className":14297},[406],[385,14299],{"className":14300,"style":666},[665],[385,14302,5052],{"className":14303},[670],[385,14305],{"className":14306,"style":666},[665],[385,14308,14310,14313,14354,14357,14360],{"className":14309},[397],[385,14311],{"className":14312,"style":6865},[401],[385,14314,14316,14320],{"className":14315},[406],[385,14317,14319],{"className":14318},[406,407],"x",[385,14321,14323],{"className":14322},[453],[385,14324,14326,14346],{"className":14325},[457,458],[385,14327,14329,14343],{"className":14328},[462],[385,14330,14332],{"className":14331,"style":467},[466],[385,14333,14334,14337],{"style":708},[385,14335],{"className":14336,"style":475},[474],[385,14338,14340],{"className":14339},[479,480,481,482],[385,14341,427],{"className":14342},[406,407,482],[385,14344,490],{"className":14345},[489],[385,14347,14349],{"className":14348},[462],[385,14350,14352],{"className":14351,"style":497},[466],[385,14353],{},[385,14355],{"className":14356,"style":666},[665],[385,14358,5052],{"className":14359},[670],[385,14361],{"className":14362,"style":666},[665],[385,14364,14366,14369],{"className":14365},[397],[385,14367],{"className":14368,"style":880},[401],[385,14370,604],{"className":14371},[406]," of item ",[385,14374,14376],{"className":14375},[388],[385,14377,14379],{"className":14378,"ariaHidden":393},[392],[385,14380,14382,14385],{"className":14381},[397],[385,14383],{"className":14384,"style":423},[401],[385,14386,427],{"className":14387},[406,407],", and the problem ",[558,14390,14391],{},"fractional knapsack"," becomes ",[1013,14394,14395],{},"easy",", solvable\nby the very greedy strategy that failed the 0\u002F1 version. Sort items by\nvalue-per-weight ratio ",[385,14398,14400],{"className":14399},[388],[385,14401,14403],{"className":14402,"ariaHidden":393},[392],[385,14404,14406,14409,14449,14452],{"className":14405},[397],[385,14407],{"className":14408,"style":681},[401],[385,14410,14412,14415],{"className":14411},[406],[385,14413,520],{"className":14414,"style":519},[406,407],[385,14416,14418],{"className":14417},[453],[385,14419,14421,14441],{"className":14420},[457,458],[385,14422,14424,14438],{"className":14423},[462],[385,14425,14427],{"className":14426,"style":467},[466],[385,14428,14429,14432],{"style":535},[385,14430],{"className":14431,"style":475},[474],[385,14433,14435],{"className":14434},[479,480,481,482],[385,14436,427],{"className":14437},[406,407,482],[385,14439,490],{"className":14440},[489],[385,14442,14444],{"className":14443},[462],[385,14445,14447],{"className":14446,"style":497},[466],[385,14448],{},[385,14450,5623],{"className":14451},[406],[385,14453,14455,14458],{"className":14454},[406],[385,14456,449],{"className":14457,"style":448},[406,407],[385,14459,14461],{"className":14460},[453],[385,14462,14464,14484],{"className":14463},[457,458],[385,14465,14467,14481],{"className":14466},[462],[385,14468,14470],{"className":14469,"style":467},[466],[385,14471,14472,14475],{"style":470},[385,14473],{"className":14474,"style":475},[474],[385,14476,14478],{"className":14477},[479,480,481,482],[385,14479,427],{"className":14480},[406,407,482],[385,14482,490],{"className":14483},[489],[385,14485,14487],{"className":14486},[462],[385,14488,14490],{"className":14489,"style":497},[466],[385,14491],{}," and take them greedily, highest ratio first,\nslicing the last item to fill the knapsack exactly.",[4327,14494,14496],{"className":4329,"code":14495,"language":4331,"meta":376,"style":376},"caption: $\\textsc{Fractional-Knapsack}(w, v, n, W)$ — greedy, fractions allowed\nnumber: 4\nsort items so that $v[1]\u002Fw[1] \\ge v[2]\u002Fw[2] \\ge \\cdots \\ge v[n]\u002Fw[n]$\n$value \\gets 0$;  $b \\gets W$ \u002F\u002F remaining capacity\nfor $i \\gets 1$ to $n$ do\n  if $w[i] \\le b$ then\n    $value \\gets value + v[i]$ \u002F\u002F take all of item $i$\n    $b \\gets b - w[i]$\n  else\n    $value \\gets value + v[i] \\cdot (b \u002F w[i])$ \u002F\u002F fraction fills exactly\n    return $value$\nreturn $value$\n",[4333,14497,14498,14503,14508,14513,14518,14522,14527,14532,14536,14541,14546,14551],{"__ignoreMap":376},[385,14499,14500],{"class":4337,"line":6},[385,14501,14502],{},"caption: $\\textsc{Fractional-Knapsack}(w, v, n, W)$ — greedy, fractions allowed\n",[385,14504,14505],{"class":4337,"line":18},[385,14506,14507],{},"number: 4\n",[385,14509,14510],{"class":4337,"line":24},[385,14511,14512],{},"sort items so that $v[1]\u002Fw[1] \\ge v[2]\u002Fw[2] \\ge \\cdots \\ge v[n]\u002Fw[n]$\n",[385,14514,14515],{"class":4337,"line":73},[385,14516,14517],{},"$value \\gets 0$;  $b \\gets W$ \u002F\u002F remaining capacity\n",[385,14519,14520],{"class":4337,"line":102},[385,14521,4360],{},[385,14523,14524],{"class":4337,"line":108},[385,14525,14526],{},"  if $w[i] \\le b$ then\n",[385,14528,14529],{"class":4337,"line":116},[385,14530,14531],{},"    $value \\gets value + v[i]$ \u002F\u002F take all of item $i$\n",[385,14533,14534],{"class":4337,"line":196},[385,14535,10163],{},[385,14537,14538],{"class":4337,"line":202},[385,14539,14540],{},"  else\n",[385,14542,14543],{"class":4337,"line":283},[385,14544,14545],{},"    $value \\gets value + v[i] \\cdot (b \u002F w[i])$ \u002F\u002F fraction fills exactly\n",[385,14547,14548],{"class":4337,"line":333},[385,14549,14550],{},"    return $value$\n",[385,14552,14553],{"class":4337,"line":354},[385,14554,14555],{},"return $value$\n",[381,14557,14558,14559,14598],{},"This runs in ",[385,14560,14562],{"className":14561},[388],[385,14563,14565],{"className":14564,"ariaHidden":393},[392],[385,14566,14568,14571,14574,14577,14580,14583,14589,14592,14595],{"className":14567},[397],[385,14569],{"className":14570,"style":681},[401],[385,14572,4477],{"className":14573},[406],[385,14575,1049],{"className":14576},[685],[385,14578,829],{"className":14579},[406,407],[385,14581],{"className":14582,"style":737},[665],[385,14584,14586],{"className":14585},[4948],[385,14587,13118],{"className":14588,"style":13117},[406,5340],[385,14590],{"className":14591,"style":737},[665],[385,14593,829],{"className":14594},[406,407],[385,14596,1066],{"className":14597},[844],", since the sort dominates, and is provably optimal by\nan exchange argument: any optimal solution can be rearranged, without losing\nvalue, to match the greedy choice. The contrast is the moral of the lesson.",[637,14600,14601],{"type":12390},[381,14602,14603,14606,14607,14640,14641,5972,14643,14646],{},[558,14604,14605],{},"Remark (Greedy vs. DP)."," The fractional relaxation has a greedy-choice property; the 0\u002F1 problem does\nnot. When we may take fractions, the most valuable per-unit material should\nalways come first, and filling greedily can never be improved. When items are\nindivisible, that local argument breaks, since a high-ratio item can crowd out a\nbetter combination, and we are forced to consider subsets, which is exactly\nwhat the ",[385,14608,14610],{"className":14609},[388],[385,14611,14613],{"className":14612,"ariaHidden":393},[392],[385,14614,14616,14619,14622,14625,14628,14631,14634,14637],{"className":14615},[397],[385,14617],{"className":14618,"style":681},[401],[385,14620,5522],{"className":14621,"style":5521},[406,407],[385,14623,1049],{"className":14624},[685],[385,14626,427],{"className":14627},[406,407],[385,14629,733],{"className":14630},[732],[385,14632],{"className":14633,"style":737},[665],[385,14635,449],{"className":14636,"style":448},[406,407],[385,14638,1066],{"className":14639},[844]," dynamic program does. One small change to the rules moves a\nproblem across the line between ",[1013,14642,207],{},[1013,14644,14645],{},"dynamic programming",", and across\nthe line between polynomial and NP-hard.",[632,14648,14650],{"id":14649},"takeaways","Takeaways",[1367,14652,14653,14759,14913,15103,15226],{},[1370,14654,14655,5972,14676,14679,14680,14683,14684,14717,14718,14751,14752,5972,14755,14758],{},[385,14656,14658],{"className":14657},[388],[385,14659,14661],{"className":14660,"ariaHidden":393},[392],[385,14662,14664,14667],{"className":14663},[397],[385,14665],{"className":14666,"style":574},[401],[385,14668,14670],{"className":14669},[578,579],[385,14671,14673],{"className":14672},[406,583],[385,14674,629],{"className":14675},[406],[558,14677,14678],{},"0\u002F1 knapsack"," share one engine: a ",[1013,14681,14682],{},"two-dimensional","\nsubproblem (",[385,14685,14687],{"className":14686},[388],[385,14688,14690],{"className":14689,"ariaHidden":393},[392],[385,14691,14693,14696,14699,14702,14705,14708,14711,14714],{"className":14692},[397],[385,14694],{"className":14695,"style":681},[401],[385,14697,1045],{"className":14698},[406,407],[385,14700,1049],{"className":14701},[685],[385,14703,427],{"className":14704},[406,407],[385,14706,733],{"className":14707},[732],[385,14709],{"className":14710,"style":737},[665],[385,14712,1062],{"className":14713},[406,407],[385,14715,1066],{"className":14716},[844]," \u002F ",[385,14719,14721],{"className":14720},[388],[385,14722,14724],{"className":14723,"ariaHidden":393},[392],[385,14725,14727,14730,14733,14736,14739,14742,14745,14748],{"className":14726},[397],[385,14728],{"className":14729,"style":681},[401],[385,14731,5522],{"className":14732,"style":5521},[406,407],[385,14734,1049],{"className":14735},[685],[385,14737,427],{"className":14738},[406,407],[385,14740,733],{"className":14741},[732],[385,14743],{"className":14744,"style":737},[665],[385,14746,449],{"className":14747,"style":448},[406,407],[385,14749,1066],{"className":14750},[844],") indexed by ",[1013,14753,14754],{},"items available",[1013,14756,14757],{},"budget\nremaining",", because how much budget is left is part of the state.",[1370,14760,14761,14762,14764,14765,14780,14781,14799,14800,14803,14804,5623,14856,14908,14909,14912],{},"The recurrence is the ",[558,14763,1364],{}," choice on the last element: a ",[385,14766,14768],{"className":14767},[388],[385,14769,14771],{"className":14770,"ariaHidden":393},[392],[385,14772,14774,14777],{"className":14773},[397],[385,14775],{"className":14776,"style":1808},[401],[385,14778,1812],{"className":14779},[406],"\nof two previous-row cells for subset-sum, a ",[385,14782,14784],{"className":14783},[388],[385,14785,14787],{"className":14786,"ariaHidden":393},[392],[385,14788,14790,14793],{"className":14789},[397],[385,14791],{"className":14792,"style":956},[401],[385,14794,14796],{"className":14795},[4948],[385,14797,5341],{"className":14798},[406,5340]," for knapsack. Each cell\ndepends on the one ",[1013,14801,14802],{},"directly above"," (exclude) and the one ",[385,14805,14807],{"className":14806},[388],[385,14808,14810],{"className":14809,"ariaHidden":393},[392],[385,14811,14813,14816],{"className":14812},[397],[385,14814],{"className":14815,"style":441},[401],[385,14817,14819,14822],{"className":14818},[406],[385,14820,598],{"className":14821},[406,407],[385,14823,14825],{"className":14824},[453],[385,14826,14828,14848],{"className":14827},[457,458],[385,14829,14831,14845],{"className":14830},[462],[385,14832,14834],{"className":14833,"style":467},[466],[385,14835,14836,14839],{"style":708},[385,14837],{"className":14838,"style":475},[474],[385,14840,14842],{"className":14841},[479,480,481,482],[385,14843,427],{"className":14844},[406,407,482],[385,14846,490],{"className":14847},[489],[385,14849,14851],{"className":14850},[462],[385,14852,14854],{"className":14853,"style":497},[466],[385,14855],{},[385,14857,14859],{"className":14858},[388],[385,14860,14862],{"className":14861,"ariaHidden":393},[392],[385,14863,14865,14868],{"className":14864},[397],[385,14866],{"className":14867,"style":441},[401],[385,14869,14871,14874],{"className":14870},[406],[385,14872,449],{"className":14873,"style":448},[406,407],[385,14875,14877],{"className":14876},[453],[385,14878,14880,14900],{"className":14879},[457,458],[385,14881,14883,14897],{"className":14882},[462],[385,14884,14886],{"className":14885,"style":467},[466],[385,14887,14888,14891],{"style":470},[385,14889],{"className":14890,"style":475},[474],[385,14892,14894],{"className":14893},[479,480,481,482],[385,14895,427],{"className":14896},[406,407,482],[385,14898,490],{"className":14899},[489],[385,14901,14903],{"className":14902},[462],[385,14904,14906],{"className":14905,"style":497},[466],[385,14907],{}," columns\n",[1013,14910,14911],{},"to the left"," (include); the optimal subset is recovered by walking backward.",[1370,14914,14915,14717,14942,2747,14966,14968,14969,14971,14972,15014,15015,5770,15084,15086,15087,15102],{},[385,14916,14918],{"className":14917},[388],[385,14919,14921],{"className":14920,"ariaHidden":393},[392],[385,14922,14924,14927,14930,14933,14936,14939],{"className":14923},[397],[385,14925],{"className":14926,"style":681},[401],[385,14928,4477],{"className":14929},[406],[385,14931,1049],{"className":14932},[685],[385,14934,829],{"className":14935},[406,407],[385,14937,863],{"className":14938},[406,407],[385,14940,1066],{"className":14941},[844],[385,14943,14945],{"className":14944},[388],[385,14946,14948],{"className":14947,"ariaHidden":393},[392],[385,14949,14951,14954,14957,14960,14963],{"className":14950},[397],[385,14952],{"className":14953,"style":681},[401],[385,14955,4477],{"className":14956},[406],[385,14958,1049],{"className":14959},[685],[385,14961,11259],{"className":14962,"style":408},[406,407],[385,14964,1066],{"className":14965},[844],[558,14967,13048],{},": polynomial in the\nnumeric ",[1013,14970,5175],{}," of the target, but with an honest size of ",[385,14973,14975],{"className":14974},[388],[385,14976,14978,14999],{"className":14977,"ariaHidden":393},[392],[385,14979,14981,14984,14987,14990,14993,14996],{"className":14980},[397],[385,14982],{"className":14983,"style":681},[401],[385,14985,1049],{"className":14986},[685],[385,14988,829],{"className":14989},[406,407],[385,14991],{"className":14992,"style":1448},[665],[385,14994,4418],{"className":14995},[1452],[385,14997],{"className":14998,"style":1448},[665],[385,15000,15002,15005,15008,15011],{"className":15001},[397],[385,15003],{"className":15004,"style":681},[401],[385,15006,604],{"className":15007},[406],[385,15009,1066],{"className":15010},[844],[385,15012,9916],{"className":15013},[406,407]," bits it is\n",[385,15016,15018],{"className":15017},[388],[385,15019,15021,15045],{"className":15020,"ariaHidden":393},[392],[385,15022,15024,15027,15030,15033,15036,15039,15042],{"className":15023},[397],[385,15025],{"className":15026,"style":681},[401],[385,15028,4477],{"className":15029},[406],[385,15031,1049],{"className":15032},[685],[385,15034,829],{"className":15035},[406,407],[385,15037],{"className":15038,"style":1448},[665],[385,15040,12945],{"className":15041},[1452],[385,15043],{"className":15044,"style":1448},[665],[385,15046,15048,15052,15081],{"className":15047},[397],[385,15049],{"className":15050,"style":15051},[401],"height:1.0991em;vertical-align:-0.25em;",[385,15053,15055,15058],{"className":15054},[406],[385,15056,767],{"className":15057},[406],[385,15059,15061],{"className":15060},[453],[385,15062,15064],{"className":15063},[457],[385,15065,15067],{"className":15066},[462],[385,15068,15070],{"className":15069,"style":12570},[466],[385,15071,15072,15075],{"style":1001},[385,15073],{"className":15074,"style":475},[474],[385,15076,15078],{"className":15077},[479,480,481,482],[385,15079,9916],{"className":15080},[406,407,482],[385,15082,1066],{"className":15083},[844],[1013,15085,13004],{}," in the bit length ",[385,15088,15090],{"className":15089},[388],[385,15091,15093],{"className":15092,"ariaHidden":393},[392],[385,15094,15096,15099],{"className":15095},[397],[385,15097],{"className":15098,"style":574},[401],[385,15100,9916],{"className":15101},[406,407]," of one integer.",[1370,15104,15105,15106,15109,15110,15146,15147,2747,15168,15189,15190,1427],{},"A ",[1013,15107,15108],{},"truly"," polynomial ",[385,15111,15113],{"className":15112},[388],[385,15114,15116],{"className":15115,"ariaHidden":393},[392],[385,15117,15119,15122,15128,15131,15134,15137,15140,15143],{"className":15118},[397],[385,15120],{"className":15121,"style":681},[401],[385,15123,15125],{"className":15124},[4948],[385,15126,13212],{"className":15127,"style":13117},[406,5340],[385,15129,1049],{"className":15130},[685],[385,15132,829],{"className":15133},[406,407],[385,15135,733],{"className":15136},[732],[385,15138],{"className":15139,"style":737},[665],[385,15141,9916],{"className":15142},[406,407],[385,15144,1066],{"className":15145},[844]," algorithm is unlikely:\n",[385,15148,15150],{"className":15149},[388],[385,15151,15153],{"className":15152,"ariaHidden":393},[392],[385,15154,15156,15159],{"className":15155},[397],[385,15157],{"className":15158,"style":574},[401],[385,15160,15162],{"className":15161},[578,579],[385,15163,15165],{"className":15164},[406,583],[385,15166,629],{"className":15167},[406],[385,15169,15171],{"className":15170},[388],[385,15172,15174],{"className":15173,"ariaHidden":393},[392],[385,15175,15177,15180],{"className":15176},[397],[385,15178],{"className":15179,"style":9944},[401],[385,15181,15183],{"className":15182},[578,579],[385,15184,15186],{"className":15185},[406,583],[385,15187,13587],{"className":15188},[406],", so one exists only if\n",[385,15191,15193],{"className":15192},[388],[385,15194,15196,15214],{"className":15195,"ariaHidden":393},[392],[385,15197,15199,15202,15205,15208,15211],{"className":15198},[397],[385,15200],{"className":15201,"style":402},[401],[385,15203,13521],{"className":15204},[406,5340],[385,15206],{"className":15207,"style":666},[665],[385,15209,671],{"className":15210},[670],[385,15212],{"className":15213,"style":666},[665],[385,15215,15217,15220],{"className":15216},[397],[385,15218],{"className":15219,"style":402},[401],[385,15221,15223],{"className":15222},[406],[385,15224,13543],{"className":15225},[406,5340],[1370,15227,15228,15231,15232,889,15271,15273],{},[558,15229,15230],{},"Fractional knapsack"," flips to a ",[385,15233,15235],{"className":15234},[388],[385,15236,15238],{"className":15237,"ariaHidden":393},[392],[385,15239,15241,15244,15247,15250,15253,15256,15262,15265,15268],{"className":15240},[397],[385,15242],{"className":15243,"style":681},[401],[385,15245,4477],{"className":15246},[406],[385,15248,1049],{"className":15249},[685],[385,15251,829],{"className":15252},[406,407],[385,15254],{"className":15255,"style":737},[665],[385,15257,15259],{"className":15258},[4948],[385,15260,13118],{"className":15261,"style":13117},[406,5340],[385,15263],{"className":15264,"style":737},[665],[385,15266,829],{"className":15267},[406,407],[385,15269,1066],{"className":15270},[844],[1013,15272,207],{}," algorithm:\nallowing fractions restores the greedy-choice property that indivisibility\ndestroys, so one rule change crosses the line from NP-hard to easy.",[15275,15276,15279,15284],"section",{"className":15277,"dataFootnotes":376},[15278],"footnotes",[632,15280,15283],{"className":15281,"id":602},[15282],"sr-only","Footnotes",[15285,15286,15287,15301,15347],"ol",{},[1370,15288,15290,15293,15294],{"id":15289},"user-content-fn-skiena-knap",[558,15291,15292],{},"Skiena",", §10 — Dynamic Programming: 0\u002F1 knapsack as a canonical NP-hard resource-allocation problem solved by DP. ",[598,15295,15300],{"href":15296,"ariaLabel":15297,"className":15298,"dataFootnoteBackref":376},"#user-content-fnref-skiena-knap","Back to reference 1",[15299],"data-footnote-backref","↩",[1370,15302,15304,15307,15308,15341,15342],{"id":15303},"user-content-fn-clrs-knap",[558,15305,15306],{},"CLRS",", Ch. 15 — Dynamic Programming: the 0\u002F1 knapsack value recurrence ",[385,15309,15311],{"className":15310},[388],[385,15312,15314],{"className":15313,"ariaHidden":393},[392],[385,15315,15317,15320,15323,15326,15329,15332,15335,15338],{"className":15316},[397],[385,15318],{"className":15319,"style":681},[401],[385,15321,5522],{"className":15322,"style":5521},[406,407],[385,15324,1049],{"className":15325},[685],[385,15327,427],{"className":15328},[406,407],[385,15330,733],{"className":15331},[732],[385,15333],{"className":15334,"style":737},[665],[385,15336,449],{"className":15337,"style":448},[406,407],[385,15339,1066],{"className":15340},[844]," taking the max of exclude and include. ",[598,15343,15300],{"href":15344,"ariaLabel":15345,"className":15346,"dataFootnoteBackref":376},"#user-content-fnref-clrs-knap","Back to reference 2",[15299],[1370,15348,15350,15353,15354,15381,15382],{"id":15349},"user-content-fn-erickson-knap",[558,15351,15352],{},"Erickson",", Ch. 3 — Dynamic Programming: the pseudo-polynomial ",[385,15355,15357],{"className":15356},[388],[385,15358,15360],{"className":15359,"ariaHidden":393},[392],[385,15361,15363,15366,15369,15372,15375,15378],{"className":15362},[397],[385,15364],{"className":15365,"style":681},[401],[385,15367,4477],{"className":15368},[406],[385,15370,1049],{"className":15371},[685],[385,15373,829],{"className":15374},[406,407],[385,15376,863],{"className":15377},[406,407],[385,15379,1066],{"className":15380},[844]," running time, exponential in the input's bit length. ",[598,15383,15300],{"href":15384,"ariaLabel":15385,"className":15386,"dataFootnoteBackref":376},"#user-content-fnref-erickson-knap","Back to reference 3",[15299],[15388,15389,15390],"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":15392},[15393,15397,15398,15399,15400,15401,15402,15403,15405,15406],{"id":634,"depth":18,"text":635,"children":15394},[15395,15396],{"id":2589,"depth":24,"text":2590},{"id":4324,"depth":24,"text":4325},{"id":4599,"depth":18,"text":4600},{"id":6495,"depth":18,"text":6496},{"id":8108,"depth":18,"text":8109},{"id":9765,"depth":18,"text":9766},{"id":11115,"depth":18,"text":11116},{"id":13657,"depth":18,"text":13658},{"id":14274,"depth":18,"text":15404},"Where greed does work: fractional knapsack",{"id":14649,"depth":18,"text":14650},{"id":602,"depth":18,"text":15283},"A thief breaks into a warehouse carrying a knapsack that holds at most W units\nof weight. Item i weighs wi​ and is worth vi​. Each item is taken whole or\nleft behind, with no fractions and no duplicates. Which subset maximizes the value\ncarried out without exceeding the capacity? This is the 0\u002F1 knapsack problem,\nand despite its toy framing it is one of the most important problems in\ncomputing: it is the kernel of resource allocation, budgeting, and cutting-stock\nproblems, and a canonical NP-hard optimization problem whose dynamic program\nteaches a subtle lesson about what polynomial really means.1","md",{"moduleNumber":116,"lessonNumber":24,"order":15410},703,true,[15413,15417,15420,15423,15426],{"title":15414,"slug":15415,"difficulty":15416},"Partition Equal Subset Sum","partition-equal-subset-sum","Medium",{"title":15418,"slug":15419,"difficulty":15416},"Coin Change","coin-change",{"title":15421,"slug":15422,"difficulty":15416},"Coin Change II","coin-change-ii",{"title":15424,"slug":15425,"difficulty":15416},"Target Sum","target-sum",{"title":15427,"slug":15428,"difficulty":15416},"Ones and Zeroes","ones-and-zeroes","---\ntitle: Knapsack & Subset Problems\nmodule: Dynamic Programming\nmoduleNumber: 7\nlessonNumber: 3\norder: 703\nsummary: >-\n  We start from $\\textsc{Subset-sum}$ — does some sublist hit a target $t$? — and its\n  include\u002Fexclude recurrence over a boolean table $A(i, u)$, then bolt on values\n  to get 0\u002F1 knapsack as the same machine with $\\lor$ promoted to $\\max$. We fill\n  both tables, recover the chosen items, and confront the surprise that the\n  $\\Theta(nt)$ running time is only _pseudo-polynomial_ — exponential in the bit\n  length $b$, and unimprovable unless $\\mathrm{P}=\\mathrm{NP}$ since subset-sum is\n  $\\textsc{NP-complete}$. The fractional variant reveals the sharp line between greedy\n  and dynamic programming.\ntopics: [Dynamic Programming]\nsources:\n  - book: CLRS\n    ref: \"Ch. 15 — Dynamic Programming\"\n  - book: Skiena\n    ref: \"§10 — Dynamic Programming\"\n  - book: Erickson\n    ref: \"Ch. 3 — Dynamic Programming\"\npractice:\n  - title: 'Partition Equal Subset Sum'\n    slug: partition-equal-subset-sum\n    difficulty: Medium\n  - title: 'Coin Change'\n    slug: coin-change\n    difficulty: Medium\n  - title: 'Coin Change II'\n    slug: coin-change-ii\n    difficulty: Medium\n  - title: 'Target Sum'\n    slug: target-sum\n    difficulty: Medium\n  - title: 'Ones and Zeroes'\n    slug: ones-and-zeroes\n    difficulty: Medium\n---\n\nA thief breaks into a warehouse carrying a knapsack that holds at most $W$ units\nof weight. Item $i$ weighs $w_i$ and is worth $v_i$. Each item is taken whole or\nleft behind, with no fractions and no duplicates. Which subset maximizes the value\ncarried out without exceeding the capacity? This is the **0\u002F1 knapsack problem**,\nand despite its toy framing it is one of the most important problems in\ncomputing: it is the kernel of resource allocation, budgeting, and cutting-stock\nproblems, and a canonical $\\textsc{NP-hard}$ optimization problem whose dynamic program\nteaches a subtle lesson about what \"polynomial\" really means.[^skiena-knap]\n\nWe will arrive at knapsack through its simpler decision cousin, $\\textsc{Subset-sum}$,\nwhich strips away the values and asks a plain yes\u002Fno question. The include\u002Fexclude\nrecurrence is identical, the running-time subtlety is identical, and subset-sum is\nthe cleanest place to see both.\n\n## Subset-sum: the decision core\n\n> **Input:** a list $L = \\langle a_1, a_2, \\dots, a_n\\rangle$ of positive integers\n> and a target integer $t > 0$.\n> **Output (simplified):** $\\text{yes}$ if some sublist of $L$ sums to exactly\n> $t$, and $\\text{no}$ otherwise.\n\nThe brute-force space is the $2^n$ sublists. To get a recurrence we shrink the\ninstance one element at a time, and the trick that supplies the _second_\ndimension is to make the _target itself_ a parameter of the subproblem.\n\n> **Definition (Subset-sum subproblem).** Let $A(i, u)$ be $\\text{true}$ exactly when some sublist of the prefix\n> $L[1..i] = \\langle a_1, \\dots, a_i\\rangle$ sums to $u$, and $\\text{false}$\n> otherwise.\n\nThe answer we want is $A(n, t)$. Now look at $a_i$, the last element we are\nallowed to use, and make the **include\u002Fexclude** decision that defines every\n0\u002F1 dynamic program:\n\n- **Exclude $a_i$.** Then some sublist of the first $i-1$ elements must already\n  hit $u$ on its own: $A(i-1, u)$.\n- **Include $a_i$.** Then it contributes $a_i$ toward the target, and the first\n  $i-1$ elements must cover the shortfall: $A(i-1, u - a_i)$.\n\nThe element can be used in _either_ way, so the subproblem is true if _either_\nbranch succeeds, a logical $\\lor$. The base cases pin down the boundary: a\nbudget of $0$ is always reachable by the empty set; a positive budget is\nunreachable with no elements; a negative budget is impossible:\n$$\nA(i,u) =\n\\begin{cases}\n\\text{true} & \\text{if } u = 0, \\\\[3pt]\n\\text{false} & \\text{if } u \u003C 0, \\\\[3pt]\n\\text{false} & \\text{if } i = 0 \\text{ and } u > 0, \\\\[3pt]\nA(i-1, u)\\ \\lor\\ A(i-1,\\, u - a_i) & \\text{if } i > 0,\\ u > 0.\n\\end{cases}\n$$\nEvery entry depends only on two entries of the _previous row_ ($i-1$): the one\ndirectly above (exclude) and the one above and $a_i$ columns to the left\n(include). So we fill the table **row by row** in increasing $i$, sweeping the\nbudget $u = 0, \\dots, t$ within each row.\n\n### The subset-sum table, filled\n\nTake $L = \\langle 1, 3, 4, 2\\rangle$ and target $t = 6$. The table holds the\nboolean $A(i, u)$: row $0$ is $\\text{true}$ only in column $0$ (the empty set\nsums to $0$), and each later row ORs the _exclude_ cell directly above with the\n_include_ cell $a_i$ columns to its left. The shaded cell $A(4,6)$ is the answer;\nthe two arrows show its include\u002Fexclude dependency.\n\n$$\n% caption: Boolean subset-sum table with the answer cell and its include and exclude\n%          predecessors.\n\\begin{tikzpicture}[\n  >=Stealth,\n  T\u002F.style={draw, minimum size=8mm, inner sep=1pt, font=\\small, fill=acc!15},\n  F\u002F.style={draw, minimum size=8mm, inner sep=1pt, font=\\small},\n  lbl\u002F.style={draw=none, font=\\small}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % header: budgets u = 0..6\n  \\node[lbl] at (0,0) {$A(i,u)$};\n  \\foreach \\u in {0,...,6} { \\node[lbl] at (\\u+1,0) {$u{=}\\u$}; }\n  % row labels (with a_i)\n  \\node[lbl] at (0,-1) {$i{=}0$};\n  \\node[lbl] at (0,-2) {$1\\ (a{=}1)$};\n  \\node[lbl] at (0,-3) {$2\\ (a{=}3)$};\n  \\node[lbl] at (0,-4) {$3\\ (a{=}4)$};\n  \\node[lbl] at (0,-5) {$4\\ (a{=}2)$};\n  % row 0: only u=0 true\n  \\node[T] at (1,-1) {T};\n  \\foreach \\c in {2,...,7} { \\node[F] at (\\c,-1) {F}; }\n  % row 1: a=1 -> sums {0,1}\n  \\node[T] at (1,-2) {T}; \\node[T] at (2,-2) {T};\n  \\foreach \\c in {3,...,7} { \\node[F] at (\\c,-2) {F}; }\n  % row 2: a=3 -> sums {0,1,3,4}\n  \\node[T] at (1,-3) {T}; \\node[T] at (2,-3) {T}; \\node[F] at (3,-3) {F};\n  \\node[T] at (4,-3) {T}; \\node[T] at (5,-3) {T}; \\node[F] at (6,-3) {F}; \\node[F] at (7,-3) {F};\n  % row 3: a=4 -> sums {0,1,3,4,5,7,8}∩[0,6] = {0,1,3,4,5}\n  \\node[T] at (1,-4) {T}; \\node[T] at (2,-4) {T}; \\node[F] at (3,-4) {F};\n  \\node[T] at (4,-4) {T}; \\node[T] at (5,-4) {T}; \\node[T] at (6,-4) {T}; \\node[F] at (7,-4) {F};\n  % row 4: a=2 -> add 2 to {0,1,3,4,5} -> {2,3,5,6,7}; union {0,1,3,4,5} = {0,1,2,3,4,5,6}\n  \\node[T] at (1,-5) {T}; \\node[T] at (2,-5) {T}; \\node[T] at (3,-5) {T};\n  \\node[T] at (4,-5) {T}; \\node[T] at (5,-5) {T}; \\node[T] at (6,-5) {T};\n  \\node[T, fill=acc!45] (ans) at (7,-5) {T};\n  % --- the answer's two predecessors, dashed accent borders (the T\u002FF letter stays visible) ---\n  \\node[draw=acc, very thick, dashed, minimum size=8mm, inner sep=1pt] (exc) at (7,-4) {};\n  \\node[draw=acc, very thick, dashed, minimum size=8mm, inner sep=1pt] (inc) at (5,-4) {};\n  % EXCLUDE arrow: predecessor sits directly above, so a short bow down the right edge.\n  \\draw[->, very thick, red!75!black] (exc.east) to[out=0, in=0, looseness=2.3] (ans.east);\n  \\node[lbl, anchor=west, align=left, text=red!75!black] at (8.15,-4)\n    {\\itshape exclude $a_4$:\\\\skip it, copy the\\\\cell above, $A(3,6)$};\n  % INCLUDE arrow: predecessor is two columns left (u-a_4). Route it cleanly around the\n  % outside — down the column-4\u002F5 gap, along below the table, up into the answer — so it\n  % never cuts through a cell.\n  \\draw[->, very thick, red!75!black]\n    (inc.south) -- (4.5,-4.5) -- (4.5,-6.1) -- (7,-6.1) -- (ans.south);\n  \\node[lbl, align=center, text=red!75!black] at (5.75,-6.55)\n    {\\itshape include $a_4{=}2$: jump back $2$ columns, $A(3,4)$};\n\\end{tikzpicture}\n$$\n\nThe answer is $A(4,6) = \\text{true}$, witnessed by $\\set{4, 2}$ or\n$\\set{1,3,2}$. Reading its two dashed predecessors: _exclude_ $a_4$ asks\n$A(3,6) = \\text{false}$ (no subset of $\\langle1,3,4\\rangle$ hits $6$), while\n_include_ $a_4{=}2$ asks $A(3, 6-2) = A(3,4) = \\text{true}$; the $\\lor$ makes\nthe cell true through the include branch.\n\n### Subset-sum in pseudocode\n\n```algorithm\ncaption: $\\textsc{Subset-Sum}(L[1..n], t)$ — does some sublist sum to $t$?\nnumber: 1\n$A[0..n][0..t] \\gets \\text{false}$\n$A[0..n][0] \\gets \\text{true}$ \u002F\u002F empty set reaches $0$\nfor $i \\gets 1$ to $n$ do\n  for $u \\gets 1$ to $t$ do\n    if $a_i > u$ then\n      $A[i][u] \\gets A[i-1][u]$ \u002F\u002F $a_i$ too big: exclude\n    else\n      $A[i][u] \\gets A[i-1][u] \\lor A[i-1][u - a_i]$ \u002F\u002F exclude or include\nreturn $A[n][t]$\n```\n\nThis fills $(n+1)(t+1)$ boolean cells in $\\Theta(1)$ apiece, for $\\Theta(nt)$\ntime and $\\Theta(nt)$ space, and the space drops to $\\Theta(t)$ because each row\nreads only the one above it. We will return to that innocent-looking $\\Theta(nt)$\nbelow, since it hides a trap. First, let us see that 0\u002F1 knapsack is the same machine\nwith values bolted on.\n\n## The problem and why greed fails\n\n> **Input:** $n$ items with positive integer weights $w_1, \\dots, w_n$ and values\n> $v_1, \\dots, v_n$, and an integer capacity $W$.\n> **Output:** a subset $S \\subseteq \\set{1, \\dots, n}$ with $\\sum_{i \\in S} w_i\n> \\le W$ maximizing $\\sum_{i \\in S} v_i$.\n\nKnapsack is subset-sum with two upgrades: each item carries a separate **value**\n$v_i$ as well as a weight $w_i$, and the capacity $W$ is an _upper bound_ rather\nthan an exact target, so we **maximize** value instead of answering yes\u002Fno. The\ninclude\u002Fexclude decision is unchanged: the $\\lor$ of subset-sum becomes a $\\max$,\nand the boolean cell becomes a value. (Set $v_i = w_i = a_i$ and the answer\n$K(n,t) = t$ recovers subset-sum exactly, a reduction we make precise\n[below](#subset-sum-as-a-special-case).)\n\nThe tempting greedy idea, taking items in decreasing order of value-per-weight\nratio $v_i \u002F w_i$, is _wrong_ for the 0\u002F1 problem. Suppose $W = 10$ and the\nitems are $A = (w{=}6, v{=}10)$, $B = (w{=}5, v{=}9)$, $C = (w{=}5, v{=}9)$.\nItem $A$ has the best ratio ($1.67$ per unit) so greed grabs it first, then\nnothing else fits, for a total value of $10$. But taking $B$ and $C$ together fills the\nknapsack exactly for value $18$. Greed is fooled because a locally efficient item can\nblock a globally better combination; the 0\u002F1 problem lacks the structure that lets\ngreedy choices succeed on [matroids](\u002Falgorithms\u002Fgreedy\u002Fmatroids). We need to consider\nsubsets, and that calls for dynamic programming.\n\n$$\n% caption: Why greedy fails for 0\u002F1 knapsack ($W=10$). Best ratio first grabs $A$\n%          ($v\u002Fw = 1.67$) and then nothing fits: value $10$. Taking $B$ and $C$ instead\n%          fills capacity exactly for value $18$.\n\\begin{tikzpicture}[\n  >=Stealth,\n  it\u002F.style={draw, font=\\footnotesize, inner sep=2pt},\n  lbl\u002F.style={draw=none, font=\\footnotesize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[lbl] at (0,1.5) {\\itshape greedy: ratio first};\n  \\node[it, minimum width=15.6mm, fill=acc!15] (gA) at (-0.26,0) {$A\\ (w6,v10)$};\n  \\draw[thick] (-1.3,-0.55) rectangle (1.3,0.55);\n  \\node[lbl] at (-1.7,-0.95) {$0$};\n  \\node[lbl] at (1.3,-0.95) {$W{=}10$};\n  \\node[lbl, acc] at (0,-1.4) {value $= 10$};\n  \\node[lbl] at (5,1.5) {\\itshape optimal: $B+C$};\n  \\node[it, minimum width=10.4mm, fill=acc!15] (oB) at (4.06,0) {$B$};\n  \\node[it, minimum width=10.4mm, fill=acc!15] (oC) at (5.94,0) {$C$};\n  \\draw[thick] (3.7,-0.55) rectangle (6.3,0.55);\n  \\node[lbl] at (3.3,-0.95) {$0$};\n  \\node[lbl] at (6.3,-0.95) {$W{=}10$};\n  \\node[lbl, acc] at (5,-1.4) {value $= 18$};\n\\end{tikzpicture}\n$$\n\nThe high-ratio item $A$ is a trap: it is locally efficient yet blocks the\n$B + C$ pair that packs the knapsack with no waste.\n\n## The subproblem and recurrence\n\nThe brute-force space is the $2^n$ subsets. To get a recurrence we need a\nsubproblem definition that shrinks the instance one decision at a time. The key\ninsight, the source of the _second_ dimension, is that we must track not only\n_which items_ remain available but _how much capacity_ is left.\n\n> **Definition (Knapsack DP state).** Let $K(i, w)$ be the maximum value achievable using only the first $i$ items\n> $\\set{1, \\dots, i}$ within a weight budget of $w$.\n\nThe answer is $K(n, W)$. Now consider item $i$, the last one we are allowed to\nuse, and make the same include\u002Fexclude decision as in subset-sum; this is the\n\"0\u002F1\" in the name.\n\n**Exclude item $i$.** Then the best we can do is whatever the first $i-1$ items\nachieve within the same budget: $K(i-1, w)$.\n\n**Include item $i$.** This is only possible if it fits, $w_i \\le w$. We collect\nits value $v_i$ and spend $w_i$ of the budget, leaving $w - w_i$ for the first\n$i-1$ items: $v_i + K(i-1, w - w_i)$.\n\nWe take the better of the two, and if item $i$ does not fit, only the first\noption is available:[^clrs-knap]\n$$\nK(i,w) =\n\\begin{cases}\n0 & \\text{if } i = 0 \\text{ or } w = 0, \\\\[3pt]\nK(i-1, w) & \\text{if } w_i > w, \\\\[3pt]\n\\max\\bigl(K(i-1, w),\\ v_i + K(i-1, w - w_i)\\bigr) & \\text{if } w_i \\le w.\n\\end{cases}\n$$\nThe base case says: with no items, or no capacity, the value is $0$. Each entry\ndepends only on entries in the _previous row_ ($i-1$), so we fill the table **row\nby row** in increasing $i$, and within each row over all budgets $w = 0, \\dots,\nW$.\n\n## The DP table, filled\n\nTake capacity $W = 5$ and four items: $1{:}(w{=}1, v{=}1)$, $2{:}(w{=}2,\nv{=}6)$, $3{:}(w{=}3, v{=}10)$, $4{:}(w{=}5, v{=}16)$. The table holds $K(i, w)$;\nrow $0$ is all zeros (no items), and each later row applies the recurrence across\nbudgets $0$ through $5$.\n\n$$\n% caption: Filled 0\u002F1 knapsack value table with include and exclude arrows into the\n%          answer.\n\\begin{tikzpicture}[\n  >=Stealth,\n  every node\u002F.style={draw, minimum size=8mm, inner sep=1pt, font=\\small},\n  lbl\u002F.style={draw=none}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % header: budgets\n  \\node[lbl] at (0,0) {$K(i,w)$};\n  \\node[lbl] at (1,0) {$w{=}0$};\n  \\node[lbl] at (2,0) {$1$};\n  \\node[lbl] at (3,0) {$2$};\n  \\node[lbl] at (4,0) {$3$};\n  \\node[lbl] at (5,0) {$4$};\n  \\node[lbl] at (6,0) {$5$};\n  % row labels\n  \\node[lbl] at (0,-1) {$i{=}0$};\n  \\node[lbl] at (0,-2) {$1\\ (w{=}1)$};\n  \\node[lbl] at (0,-3) {$2\\ (w{=}2)$};\n  \\node[lbl] at (0,-4) {$3\\ (w{=}3)$};\n  \\node[lbl] at (0,-5) {$4\\ (w{=}5)$};\n  % row 0\n  \\node at (1,-1) {0}; \\node at (2,-1) {0}; \\node at (3,-1) {0}; \\node at (4,-1) {0}; \\node at (5,-1) {0}; \\node at (6,-1) {0};\n  % row 1: (1,1)\n  \\node at (1,-2) {0}; \\node at (2,-2) {1}; \\node at (3,-2) {1}; \\node at (4,-2) {1}; \\node at (5,-2) {1}; \\node at (6,-2) {1};\n  % row 2: (2,6) -- include\u002Fexclude sources for K(3,5)\n  \\node at (1,-3) {0}; \\node at (2,-3) {1}; \\node[draw=acc, dashed] (kinc) at (3,-3) {6}; \\node at (4,-3) {7}; \\node at (5,-3) {7}; \\node[draw=acc, dashed] (kexc) at (6,-3) {7};\n  % row 3: (3,10)\n  \\node at (1,-4) {0}; \\node at (2,-4) {1}; \\node at (3,-4) {6}; \\node at (4,-4) {10}; \\node at (5,-4) {11}; \\node[fill=acc!15, draw=acc, very thick] (k35) at (6,-4) {16};\n  % row 4: (5,16)\n  \\node at (1,-5) {0}; \\node at (2,-5) {1}; \\node at (3,-5) {6}; \\node at (4,-5) {10}; \\node at (5,-5) {11}; \\node[fill=acc!15] at (6,-5) {16};\n  % dependency arrows into K(3,5)=16\n  \\draw[->, thick, red!75!black] (kexc.south) -- (k35.north);\n  \\draw[->, thick, red!75!black] (kinc.south east) to[bend left=12] (k35.north west);\n  \\node[lbl, anchor=west] at (6.55,-3.0) {\\itshape exclude $3$};\n  \\node[lbl] (inclbl3) at (1.7,-6) {\\itshape include $3$ ($w{-}3$)};\n  \\draw[->, thin, red!75!black] (inclbl3.west) to[out=180, in=270] (kinc.south west);\n\\end{tikzpicture}\n$$\n\nThe shaded answer is $K(4, 5) = 16$. The arrows trace the highlighted entry\n$K(3,5)$: item $3$ weighs $3 \\le 5$, so we compare _excluding it_ ($K(2,5) = 7$,\nthe cell directly above) against _including it_ ($v_3 + K(2, 5-3) = 10 + K(2,2) =\n10 + 6 = 16$, reaching $w_3 = 3$ columns to the left); the include branch wins at\n$16$. One row down, $K(4,5)$ compares _excluding item $4$_ ($K(3,5) = 16$)\nagainst _including it_ ($16 + K(3, 0) = 16$), a tie at $16$.\n\n## The algorithm\n\n```algorithm\ncaption: $\\textsc{Knapsack-01}(w, v, n, W)$ — maximum value within capacity $W$\nnumber: 2\nfor $b \\gets 0$ to $W$ do\n  $K[0][b] \\gets 0$ \u002F\u002F no items: value $0$\nfor $i \\gets 1$ to $n$ do\n  for $b \\gets 0$ to $W$ do\n    $K[i][b] \\gets K[i-1][b]$ \u002F\u002F skip item $i$\n    if $w[i] \\le b$ then\n      $take \\gets v[i] + K[i-1][b - w[i]]$\n      $K[i][b] \\gets \\max(K[i][b],\\ take)$ \u002F\u002F take item $i$\nreturn $K[n][W]$\n```\n\n**Recovering the chosen items.** As with every DP, the table holds the value; the\n_subset_ is recovered by walking backward from $K[n][W]$. At row $i$, if $K[i][b]\n\\neq K[i-1][b]$ then item $i$ was taken: record it and drop the budget to\n$b - w_i$; otherwise it was skipped. Continue down to row $0$.\n\n```algorithm\ncaption: $\\textsc{Knapsack-Items}(K, w, n, W)$ — recover the optimal subset\nnumber: 3\n$S \\gets \\emptyset$\n$b \\gets W$\nfor $i \\gets n$ downto $1$ do\n  if $K[i][b] \\neq K[i-1][b]$ then\n    $S \\gets S \\cup \\set{i}$ \u002F\u002F item $i$ taken\n    $b \\gets b - w[i]$\nreturn $S$\n```\n\nOn the filled table above ($W = 5$), the walk starts at $K[4][5]$ and reads off\none decision per row, dropping the budget by $w_i$ whenever an item is taken:\n\n$$\n% caption: Backward reconstruction from $K[4][5]=16$. At each row, $K[i][b]=K[i-1][b]$\n%          means item $i$ was skipped (budget unchanged); a jump means it was taken\n%          (budget drops by $w_i$). Items $3$ and $2$ are recovered, weight $3+2=5$, value\n%          $10+6=16$.\n\\begin{tikzpicture}[\n  >=Stealth,\n  row\u002F.style={draw, minimum width=34mm, minimum height=8mm, inner sep=2pt, font=\\small},\n  take\u002F.style={row, fill=acc!15, draw=acc, very thick},\n  skip\u002F.style={row},\n  bud\u002F.style={draw=none, font=\\footnotesize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[skip] (r4) at (0,0)    {$i{=}4$: $K[4][5]{=}K[3][5]$};\n  \\node[take] (r3) at (0,-1.1) {$i{=}3$: $K[3][5]{\\neq}K[2][5]$};\n  \\node[take] (r2) at (0,-2.2) {$i{=}2$: $K[2][2]{\\neq}K[1][2]$};\n  \\node[skip] (r1) at (0,-3.3) {$i{=}1$: $K[1][0]{=}K[0][0]$};\n  % decision + budget column on the right, dropping as items are taken\n  \\node[bud, anchor=west] at (2.05,0)    {skip, $b{=}5$};\n  \\node[bud, anchor=west, acc] at (2.05,-1.1) {take $3$, $b{=}5{\\to}2$};\n  \\node[bud, anchor=west, acc] at (2.05,-2.2) {take $2$, $b{=}2{\\to}0$};\n  \\node[bud, anchor=west] at (2.05,-3.3) {skip, $b{=}0$};\n  \\draw[->, red!75!black, thick] (r4.south) -- (r3.north);\n  \\draw[->, red!75!black, thick] (r3.south) -- (r2.north);\n  \\draw[->, red!75!black, thick] (r2.south) -- (r1.north);\n  \\node[bud, acc, align=center] at (0,-4.25) {$S=\\{2,3\\}$, \\ value $16$};\n\\end{tikzpicture}\n$$\n\n## Running time and the pseudo-polynomial trap\n\nThe table has $(n + 1)(W + 1)$ entries, each filled in $\\Theta(1)$, so\n$\\textsc{Knapsack-01}$ runs in\n$$\n\\Theta(nW)\n$$\ntime and space. (Space drops to $\\Theta(W)$ if only the value is needed, since\neach row reads only the previous one, so scan $b$ from high to low to reuse a\nsingle array.)\n\nThe descending sweep is not a detail but the whole correctness argument for the\n1-D version: when we apply an item, $K[b - w_i]$ must still hold the value from\n_before_ this item was available, so we must reach $b$ before we overwrite\n$b - w_i$ — that is, go high to low.\n\n$$\n% caption: 0\u002F1 knapsack on one rolling array, applying item $2$ ($w{=}2,v{=}6$) with the\n%          budget swept $b = 5 \\to 2$ (descending). Each update\n%          $K[b] \\gets \\max(K[b],\\,6 + K[b{-}2])$ reads $K[b{-}2]$ before it is\n%          overwritten, so item $2$ is used at most once.\n\\begin{tikzpicture}[\n  >=Stealth,\n  cell\u002F.style={draw, minimum size=9mm, inner sep=1pt, font=\\small},\n  upd\u002F.style={cell, fill=acc!18},\n  lbl\u002F.style={draw=none, font=\\footnotesize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[lbl] at (-1.85,0) {before};\n  \\foreach \\v\u002F\\x in {0\u002F0,1\u002F1,1\u002F2,1\u002F3,1\u002F4,1\u002F5} { \\node[cell] at (\\x,0) {$\\v$}; }\n  \\foreach \\x in {0,...,5} { \\node[lbl] at (\\x,0.72) {$b{=}\\x$}; }\n  \\node[lbl] at (-1.85,-2.2) {after};\n  \\foreach \\v\u002F\\x\u002F\\s in {0\u002F0\u002Fcell,1\u002F1\u002Fcell,6\u002F2\u002Fupd,7\u002F3\u002Fupd,7\u002F4\u002Fupd,7\u002F5\u002Fupd} { \\node[\\s] at (\\x,-2.2) {$\\v$}; }\n  \\draw[->, thick, red!75!black] (5.55,-1.1) -- (1.55,-1.1) node[midway, above, draw=none, font=\\footnotesize, red!75!black] {sweep $b$ high $\\to$ low};\n\\end{tikzpicture}\n$$\n\nSweeping the other way, low to high, would let $K[b - w_i]$ already include item\n$2$, silently packing it twice — exactly the reuse that the\n[unbounded knapsack](\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded)\n_wants_, and that the ascending sweep there deliberately permits.\n\nThat $\\Theta(nW)$ looks polynomial, and here lies one of the most\nimportant subtleties in all of algorithms.\n\n**Is this *really* \"polynomial time\"?** It is tempting to call $\\Theta(nt)$ (or\n$\\Theta(nW)$) polynomial: the input is a list of $n+1$ numbers, so surely its\nsize is $\\approx \\Theta(n)$, and $nt$ is polynomial in $n$. That accounting is\nwrong, and seeing why is what matters here. We have to count the input's size _in\nbits_, not in numbers. This is the same bit-length accounting that underlies\n[asymptotic analysis](\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis).\n\n> **Remark (An honest size parameter).** Suppose each $a_i$ fits in $b$ bits, so\n> $0 \\le a_i \\le 2^b - 1$, and likewise $0 \\le t \\le 2^b - 1$. Then the _entire_\n> input — the $n$ integers plus the target — is written in about $(n+1)b$ bits.\n> That, not $n$ alone, is the honest size parameter.\n\nNow rewrite the running time against $b$. Since $t$ can be as large as $2^b - 1$,\n$$\n\\Theta(nt) \\;=\\; \\Theta\\!\\bigl(n \\cdot 2^{b}\\bigr),\n$$\nwhich is **exponential** in $b$, the number of bits of a single input integer.\nDoubling the bits used to write the target squares $t$ and thus squares the\nrunning time. An algorithm whose time is polynomial in the _numeric value_ of an\ninput but exponential in its _encoded length_ is called **pseudo-polynomial**: it\ncounts as \"polynomial\" only if we dishonestly pretend an integer $s$ has size\n$|s|$ rather than its true size $\\Theta(\\log |s|)$.[^erickson-knap] Such running\ntimes are central to\n[coping with hardness](\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness).\n\n> **Remark (Pseudo-polynomial complexity).** Can we get a *real* polynomial-time algorithm? That would mean running time\n> $T(n, b) = \\operatorname{poly}(n, b) = \\Theta(n^{c_1} b^{c_2})$ for some\n> constants $c_1, c_2$ — polynomial in the _bit length_, not the value.\n> **Most likely not, unless $\\mathrm{P} = \\mathrm{NP}$:** $\\textsc{Subset-sum}$\n> (and hence 0\u002F1 knapsack) is $\\textsc{NP-complete}$.\n\nSo the $\\Theta(nt)$ DP is not a flaw in our analysis but the best we currently\nknow how to do. It is fast and practical precisely when the numbers are small\n(a target in the thousands, say), and uselessly slow when $t$ is a $200$-bit\ninteger, even though both instances have the same handful of elements. The\nlesson generalizes: any DP whose table is indexed by a numeric quantity (a\ntarget sum, a capacity) inherits this pseudo-polynomial character. The same\npattern recurs in\n[coin change and the unbounded knapsack](\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded).\n\n## Subset-sum as a special case\n\nWe opened with subset-sum and built knapsack on top of it; the reduction also\nruns the other way, and making it precise shows the two problems are the _same\nmachine_. Take any subset-sum instance $\\langle a_1, \\dots, a_n\\rangle$ with\ntarget $t$, and feed it to 0\u002F1 knapsack with each item's value equal to its\nweight, $v_i = w_i = a_i$, and capacity $W = t$. The most value you can pack into\na capacity-$t$ knapsack is at most $t$, and it _equals_ $t$ exactly when some\nsubset of weights sums to $t$ without waste. So\n$$\nK(n, t) = t \\iff \\text{some sublist of } L \\text{ sums to } t,\n$$\nand one call to $\\textsc{Knapsack-01}$ answers subset-sum. The two recurrences are\nthe same shape with the $\\lor$ of the boolean table promoted to a $\\max$ over\nvalues, which is why everything transfers: $\\Theta(nt)$ time,\npseudo-polynomial, and $\\textsc{NP-complete}$ \u002F NP-hard in general.\n\n## Where greed _does_ work: fractional knapsack\n\nChange one rule, letting items be split so that we may take any fraction $0 \\le x_i \\le\n1$ of item $i$, and the problem **fractional knapsack** becomes _easy_, solvable\nby the very greedy strategy that failed the 0\u002F1 version. Sort items by\nvalue-per-weight ratio $v_i \u002F w_i$ and take them greedily, highest ratio first,\nslicing the last item to fill the knapsack exactly.\n\n```algorithm\ncaption: $\\textsc{Fractional-Knapsack}(w, v, n, W)$ — greedy, fractions allowed\nnumber: 4\nsort items so that $v[1]\u002Fw[1] \\ge v[2]\u002Fw[2] \\ge \\cdots \\ge v[n]\u002Fw[n]$\n$value \\gets 0$;  $b \\gets W$ \u002F\u002F remaining capacity\nfor $i \\gets 1$ to $n$ do\n  if $w[i] \\le b$ then\n    $value \\gets value + v[i]$ \u002F\u002F take all of item $i$\n    $b \\gets b - w[i]$\n  else\n    $value \\gets value + v[i] \\cdot (b \u002F w[i])$ \u002F\u002F fraction fills exactly\n    return $value$\nreturn $value$\n```\n\nThis runs in $\\Theta(n \\log n)$, since the sort dominates, and is provably optimal by\nan exchange argument: any optimal solution can be rearranged, without losing\nvalue, to match the greedy choice. The contrast is the moral of the lesson.\n\n> **Remark (Greedy vs. DP).** The fractional relaxation has a greedy-choice property; the 0\u002F1 problem does\n> not. When we may take fractions, the most valuable per-unit material should\n> always come first, and filling greedily can never be improved. When items are\n> indivisible, that local argument breaks, since a high-ratio item can crowd out a\n> better combination, and we are forced to consider subsets, which is exactly\n> what the $K(i, w)$ dynamic program does. One small change to the rules moves a\n> problem across the line between _greedy_ and _dynamic programming_, and across\n> the line between polynomial and NP-hard.\n\n## Takeaways\n\n- $\\textsc{Subset-sum}$ and **0\u002F1 knapsack** share one engine: a _two-dimensional_\n  subproblem ($A(i,u)$ \u002F $K(i,w)$) indexed by _items available_ and _budget\n  remaining_, because how much budget is left is part of the state.\n- The recurrence is the **include\u002Fexclude** choice on the last element: a $\\lor$\n  of two previous-row cells for subset-sum, a $\\max$ for knapsack. Each cell\n  depends on the one _directly above_ (exclude) and the one $a_i$\u002F$w_i$ columns\n  _to the left_ (include); the optimal subset is recovered by walking backward.\n- $\\Theta(nt)$ \u002F $\\Theta(nW)$ is **pseudo-polynomial**: polynomial in the\n  numeric _value_ of the target, but with an honest size of $(n+1)b$ bits it is\n  $\\Theta(n \\cdot 2^b)$, _exponential_ in the bit length $b$ of one integer.\n- A _truly_ polynomial $\\operatorname{poly}(n, b)$ algorithm is unlikely:\n  $\\textsc{Subset-sum}$ is $\\textsc{NP-complete}$, so one exists only if\n  $\\mathrm{P} = \\mathrm{NP}$.\n- **Fractional knapsack** flips to a $\\Theta(n\\log n)$ _greedy_ algorithm:\n  allowing fractions restores the greedy-choice property that indivisibility\n  destroys, so one rule change crosses the line from NP-hard to easy.\n\n[^skiena-knap]: **Skiena**, §10 — Dynamic Programming: 0\u002F1 knapsack as a canonical NP-hard resource-allocation problem solved by DP.\n[^clrs-knap]: **CLRS**, Ch. 15 — Dynamic Programming: the 0\u002F1 knapsack value recurrence $K(i,w)$ taking the max of exclude and include.\n[^erickson-knap]: **Erickson**, Ch. 3 — Dynamic Programming: the pseudo-polynomial $\\Theta(nt)$ running time, exponential in the input's bit length.\n",{"text":15431,"minutes":15432,"time":15433,"words":15434},"12 min read",11.44,686400,2288,{"title":251,"description":15407},[15437,15439,15441],{"book":15306,"ref":15438},"Ch. 15 — Dynamic Programming",{"book":15292,"ref":15440},"§10 — Dynamic Programming",{"book":15352,"ref":15442},"Ch. 3 — Dynamic 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