[{"data":1,"prerenderedAt":9393},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fgreedy\u002Fmatroids":374,"course-wordcounts":9261,"ref-card-index":9317},[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":226,"blurb":376,"body":377,"description":9221,"extension":9222,"meta":9223,"module":206,"navigation":9225,"path":227,"practice":9226,"rawbody":9241,"readingTime":9242,"seo":9247,"sources":9248,"status":9257,"stem":9258,"summary":229,"topics":9259,"__hash__":9260},"course\u002F01.algorithms\u002F07.greedy\u002F04.matroids.md","",{"type":378,"value":379,"toc":9211},"minimark",[380,406,411,414,503,538,669,696,700,710,1177,1188,1260,1433,1753,1888,1892,2122,2167,2369,2372,5014,5635,6722,6841,6845,7159,7244,7357,7495,7723,8018,8136,8158,8165,8168,8549,8932,8938,8976,8979,8983,9170,9207],[381,382,383,384,388,389,392,393,397,398,402,403],"p",{},"Across this module we have watched greedy algorithms win (activity selection,\nHuffman codes, Kruskal's and Prim's spanning trees) and watched greedy lose, on\nthe 0\u002F1 knapsack. Each victory came with a ",[385,386,387],"em",{},"proof",": a guarantee that taking the\nlocally best option never forecloses the global optimum. This lesson is about the\nproofs themselves. First we distill the two arguments every greedy correctness\nproof is built from. Then we ask the deeper question: is there a structural\nproperty that ",[385,390,391],{},"predicts"," when greedy will work, before we attempt the proof? The\nanswer is yes, and its name is the ",[394,395,396],"strong",{},"matroid",". The matroid theory makes precise\nthe boundary we kept bumping into, why Kruskal is correct and 0\u002F1 knapsack is\nnot, and reduces ",[399,400,401],"q",{},"is greedy optimal here?"," to ",[399,404,405],{},"is this structure a matroid?",[407,408,410],"h2",{"id":409},"two-templates-for-proving-greedy-optimal","Two templates for proving greedy optimal",[381,412,413],{},"Every greedy correctness proof in this module is an instance of one of two\npatterns. They are interchangeable in power but feel different to wield.",[415,416,418],"callout",{"type":417},"remark",[381,419,420,423,424,450,451,466,467,470,471,486,487,502],{},[394,421,422],{},"Remark (Greedy-stays-ahead)."," Order both solutions by the steps of the algorithm.\nProve by induction on ",[425,426,429],"span",{"className":427},[428],"katex",[425,430,434],{"className":431,"ariaHidden":433},[432],"katex-html","true",[425,435,438,443],{"className":436},[437],"base",[425,439],{"className":440,"style":442},[441],"strut","height:0.6944em;",[425,444,449],{"className":445,"style":448},[446,447],"mord","mathnormal","margin-right:0.0315em;","k"," that after greedy's first ",[425,452,454],{"className":453},[428],[425,455,457],{"className":456,"ariaHidden":433},[432],[425,458,460,463],{"className":459},[437],[425,461],{"className":462,"style":442},[441],[425,464,449],{"className":465,"style":448},[446,447]," choices, greedy's\npartial solution is ",[385,468,469],{},"at least as good"," as the first ",[425,472,474],{"className":473},[428],[425,475,477],{"className":476,"ariaHidden":433},[432],[425,478,480,483],{"className":479},[437],[425,481],{"className":482,"style":442},[441],[425,484,449],{"className":485,"style":448},[446,447]," choices of any other\nsolution, by a measure that matters (finishes earliest, covers the most,\nspends the least). The base case is the first choice; the inductive step shows\ngreedy's ",[425,488,490],{"className":489},[428],[425,491,493],{"className":492,"ariaHidden":433},[432],[425,494,496,499],{"className":495},[437],[425,497],{"className":498,"style":442},[441],[425,500,449],{"className":501,"style":448},[446,447],"-th choice keeps it ahead. If greedy is never behind, it cannot\nlose at the end.",[381,504,505,506,521,522,537],{},"This is the activity-selection argument: greedy's ",[425,507,509],{"className":508},[428],[425,510,512],{"className":511,"ariaHidden":433},[432],[425,513,515,518],{"className":514},[437],[425,516],{"className":517,"style":442},[441],[425,519,449],{"className":520,"style":448},[446,447],"-th activity finishes no\nlater than the ",[425,523,525],{"className":524},[428],[425,526,528],{"className":527,"ariaHidden":433},[432],[425,529,531,534],{"className":530},[437],[425,532],{"className":533,"style":442},[441],[425,535,449],{"className":536,"style":448},[446,447],"-th activity of any valid schedule, so greedy always has at\nleast as much room left for future activities and ends up with at least as many.",[415,539,540],{"type":417},[381,541,542,545,546,549,550,568,569,585,586,601,602,617,618,633,634,652,653,668],{},[394,543,544],{},"Remark (Exchange argument)."," Take ",[385,547,548],{},"any"," optimal solution ",[425,551,553],{"className":552},[428],[425,554,556],{"className":555,"ariaHidden":433},[432],[425,557,559,563],{"className":558},[437],[425,560],{"className":561,"style":562},[441],"height:0.6833em;",[425,564,567],{"className":565,"style":566},[446,447],"margin-right:0.0278em;","O"," and transform it,\nstep by step, into the greedy solution ",[425,570,572],{"className":571},[428],[425,573,575],{"className":574,"ariaHidden":433},[432],[425,576,578,581],{"className":577},[437],[425,579],{"className":580,"style":562},[441],[425,582,584],{"className":583},[446,447],"G",", never decreasing its quality.\nEach step swaps one element of ",[425,587,589],{"className":588},[428],[425,590,592],{"className":591,"ariaHidden":433},[432],[425,593,595,598],{"className":594},[437],[425,596],{"className":597,"style":562},[441],[425,599,567],{"className":600,"style":566},[446,447]," for the element greedy would have chosen,\narguing the swap is feasible and no worse. After finitely many swaps ",[425,603,605],{"className":604},[428],[425,606,608],{"className":607,"ariaHidden":433},[432],[425,609,611,614],{"className":610},[437],[425,612],{"className":613,"style":562},[441],[425,615,567],{"className":616,"style":566},[446,447]," has\nbecome ",[425,619,621],{"className":620},[428],[425,622,624],{"className":623,"ariaHidden":433},[432],[425,625,627,630],{"className":626},[437],[425,628],{"className":629,"style":562},[441],[425,631,584],{"className":632},[446,447]," with value ",[425,635,637],{"className":636},[428],[425,638,640],{"className":639,"ariaHidden":433},[432],[425,641,643,647],{"className":642},[437],[425,644],{"className":645,"style":646},[441],"height:0.7719em;vertical-align:-0.136em;",[425,648,651],{"className":649},[650],"mrel","≥"," its original, so ",[425,654,656],{"className":655},[428],[425,657,659],{"className":658,"ariaHidden":433},[432],[425,660,662,665],{"className":661},[437],[425,663],{"className":664,"style":562},[441],[425,666,584],{"className":667},[446,447]," is optimal too.",[381,670,671,672,675,676,687,688,691,692,695],{},"This is the one you reach for: it proves the ",[385,673,674],{},"greedy-choice property"," by showing that\nwhatever an optimal solution does first, we may exchange it for greedy's first\nchoice without harm.",[677,678,679],"sup",{},[680,681,686],"a",{"href":682,"ariaDescribedBy":683,"dataFootnoteRef":376,"id":685},"#user-content-fn-clrs-greedy",[684],"footnote-label","user-content-fnref-clrs-greedy","1"," The two templates differ in direction:\nstays-ahead pushes greedy ",[385,689,690],{},"forward"," and shows it never falls behind; exchange\npulls an optimum ",[385,693,694],{},"toward"," greedy and shows the pull never hurts. The exchange\nargument is the one that generalizes into a theory, because its swap step is\nliterally an axiom of the structure we are about to define.",[407,697,699],{"id":698},"matroids","Matroids",[381,701,702,703,705,706,709],{},"A ",[394,704,396],{}," abstracts the notion of ",[399,707,708],{},"independence",": linear independence of\nvectors, acyclicity of edges, freedom to add one more element without breaking a\nrule.",[415,711,713,845],{"type":712},"definition",[381,714,715,718,719,721,722,788,789,804,805,808,809,824,825,840,841,844],{},[394,716,717],{},"Definition."," A ",[394,720,396],{}," is a pair ",[425,723,725],{"className":724},[428],[425,726,728,751],{"className":727,"ariaHidden":433},[432],[425,729,731,734,739,744,748],{"className":730},[437],[425,732],{"className":733,"style":562},[441],[425,735,738],{"className":736,"style":737},[446,447],"margin-right:0.109em;","M",[425,740],{"className":741,"style":743},[742],"mspace","margin-right:0.2778em;",[425,745,747],{"className":746},[650],"=",[425,749],{"className":750,"style":743},[742],[425,752,754,758,763,768,773,777,783],{"className":753},[437],[425,755],{"className":756,"style":757},[441],"height:1em;vertical-align:-0.25em;",[425,759,762],{"className":760},[761],"mopen","(",[425,764,767],{"className":765,"style":766},[446,447],"margin-right:0.0576em;","S",[425,769,772],{"className":770},[771],"mpunct",",",[425,774],{"className":775,"style":776},[742],"margin-right:0.1667em;",[425,778,782],{"className":779,"style":781},[446,780],"mathcal","margin-right:0.0738em;","I",[425,784,787],{"className":785},[786],"mclose",")"," where ",[425,790,792],{"className":791},[428],[425,793,795],{"className":794,"ariaHidden":433},[432],[425,796,798,801],{"className":797},[437],[425,799],{"className":800,"style":562},[441],[425,802,767],{"className":803,"style":766},[446,447]," is a\nfinite ",[394,806,807],{},"ground set"," and ",[425,810,812],{"className":811},[428],[425,813,815],{"className":814,"ariaHidden":433},[432],[425,816,818,821],{"className":817},[437],[425,819],{"className":820,"style":562},[441],[425,822,782],{"className":823,"style":781},[446,780]," is a nonempty family of subsets of ",[425,826,828],{"className":827},[428],[425,829,831],{"className":830,"ariaHidden":433},[432],[425,832,834,837],{"className":833},[437],[425,835],{"className":836,"style":562},[441],[425,838,767],{"className":839,"style":766},[446,447],",\nthe ",[394,842,843],{},"independent sets",", satisfying:",[846,847,848,963],"ol",{},[849,850,851,854,855,808,892,928,929,962],"li",{},[394,852,853],{},"(Hereditary \u002F downward-closed)"," If ",[425,856,858],{"className":857},[428],[425,859,861,883],{"className":860,"ariaHidden":433},[432],[425,862,864,868,873,876,880],{"className":863},[437],[425,865],{"className":866,"style":867},[441],"height:0.7224em;vertical-align:-0.0391em;",[425,869,872],{"className":870,"style":871},[446,447],"margin-right:0.0502em;","B",[425,874],{"className":875,"style":743},[742],[425,877,879],{"className":878},[650],"∈",[425,881],{"className":882,"style":743},[742],[425,884,886,889],{"className":885},[437],[425,887],{"className":888,"style":562},[441],[425,890,782],{"className":891,"style":781},[446,780],[425,893,895],{"className":894},[428],[425,896,898,919],{"className":897,"ariaHidden":433},[432],[425,899,901,905,909,912,916],{"className":900},[437],[425,902],{"className":903,"style":904},[441],"height:0.8193em;vertical-align:-0.136em;",[425,906,908],{"className":907},[446,447],"A",[425,910],{"className":911,"style":743},[742],[425,913,915],{"className":914},[650],"⊆",[425,917],{"className":918,"style":743},[742],[425,920,922,925],{"className":921},[437],[425,923],{"className":924,"style":562},[441],[425,926,872],{"className":927,"style":871},[446,447],",\nthen ",[425,930,932],{"className":931},[428],[425,933,935,953],{"className":934,"ariaHidden":433},[432],[425,936,938,941,944,947,950],{"className":937},[437],[425,939],{"className":940,"style":867},[441],[425,942,908],{"className":943},[446,447],[425,945],{"className":946,"style":743},[742],[425,948,879],{"className":949},[650],[425,951],{"className":952,"style":743},[742],[425,954,956,959],{"className":955},[437],[425,957],{"className":958,"style":562},[441],[425,960,782],{"className":961,"style":781},[446,780],". Every subset of an independent set is independent.",[849,964,965,854,968,808,1011,1058,1059,1115,1116,1176],{},[394,966,967],{},"(Exchange property)",[425,969,971],{"className":970},[428],[425,972,974,1002],{"className":973,"ariaHidden":433},[432],[425,975,977,981,984,987,990,993,996,999],{"className":976},[437],[425,978],{"className":979,"style":980},[441],"height:0.8778em;vertical-align:-0.1944em;",[425,982,908],{"className":983},[446,447],[425,985,772],{"className":986},[771],[425,988],{"className":989,"style":776},[742],[425,991,872],{"className":992,"style":871},[446,447],[425,994],{"className":995,"style":743},[742],[425,997,879],{"className":998},[650],[425,1000],{"className":1001,"style":743},[742],[425,1003,1005,1008],{"className":1004},[437],[425,1006],{"className":1007,"style":562},[441],[425,1009,782],{"className":1010,"style":781},[446,780],[425,1012,1014],{"className":1013},[428],[425,1015,1017,1043],{"className":1016,"ariaHidden":433},[432],[425,1018,1020,1023,1027,1030,1033,1036,1040],{"className":1019},[437],[425,1021],{"className":1022,"style":757},[441],[425,1024,1026],{"className":1025},[446],"∣",[425,1028,908],{"className":1029},[446,447],[425,1031,1026],{"className":1032},[446],[425,1034],{"className":1035,"style":743},[742],[425,1037,1039],{"className":1038},[650],"\u003C",[425,1041],{"className":1042,"style":743},[742],[425,1044,1046,1049,1052,1055],{"className":1045},[437],[425,1047],{"className":1048,"style":757},[441],[425,1050,1026],{"className":1051},[446],[425,1053,872],{"className":1054,"style":871},[446,447],[425,1056,1026],{"className":1057},[446],", then there\nexists some ",[425,1060,1062],{"className":1061},[428],[425,1063,1065,1085,1106],{"className":1064,"ariaHidden":433},[432],[425,1066,1068,1072,1076,1079,1082],{"className":1067},[437],[425,1069],{"className":1070,"style":1071},[441],"height:0.5782em;vertical-align:-0.0391em;",[425,1073,1075],{"className":1074},[446,447],"x",[425,1077],{"className":1078,"style":743},[742],[425,1080,879],{"className":1081},[650],[425,1083],{"className":1084,"style":743},[742],[425,1086,1088,1091,1094,1098,1103],{"className":1087},[437],[425,1089],{"className":1090,"style":757},[441],[425,1092,872],{"className":1093,"style":871},[446,447],[425,1095],{"className":1096,"style":1097},[742],"margin-right:0.2222em;",[425,1099,1102],{"className":1100},[1101],"mbin","∖",[425,1104],{"className":1105,"style":1097},[742],[425,1107,1109,1112],{"className":1108},[437],[425,1110],{"className":1111,"style":562},[441],[425,1113,908],{"className":1114},[446,447]," such that ",[425,1117,1119],{"className":1118},[428],[425,1120,1122,1141,1167],{"className":1121,"ariaHidden":433},[432],[425,1123,1125,1128,1131,1134,1138],{"className":1124},[437],[425,1126],{"className":1127,"style":562},[441],[425,1129,908],{"className":1130},[446,447],[425,1132],{"className":1133,"style":1097},[742],[425,1135,1137],{"className":1136},[1101],"∪",[425,1139],{"className":1140,"style":1097},[742],[425,1142,1144,1147,1151,1154,1158,1161,1164],{"className":1143},[437],[425,1145],{"className":1146,"style":757},[441],[425,1148,1150],{"className":1149},[761],"{",[425,1152,1075],{"className":1153},[446,447],[425,1155,1157],{"className":1156},[786],"}",[425,1159],{"className":1160,"style":743},[742],[425,1162,879],{"className":1163},[650],[425,1165],{"className":1166,"style":743},[742],[425,1168,1170,1173],{"className":1169},[437],[425,1171],{"className":1172,"style":562},[441],[425,1174,782],{"className":1175,"style":781},[446,780],".",[381,1178,1179,1180,1183,1184,1187],{},"The first axiom is mild; most natural notions of independence are closed under\ntaking subsets. The ",[394,1181,1182],{},"exchange property"," is the engine: a smaller independent\nset can always be grown by stealing ",[385,1185,1186],{},"some"," element from any larger one. It\nforbids the situation where you get stuck early with a maximal-but-small\nindependent set while a much larger one exists.",[415,1189,1190],{"type":712},[381,1191,1192,1194,1195,1210,1211,1213,1214,1229,1230,1233,1234,1259],{},[394,1193,717],{}," A maximal independent set, one that cannot be extended by any\nelement of ",[425,1196,1198],{"className":1197},[428],[425,1199,1201],{"className":1200,"ariaHidden":433},[432],[425,1202,1204,1207],{"className":1203},[437],[425,1205],{"className":1206,"style":562},[441],[425,1208,767],{"className":1209,"style":766},[446,447],", is a ",[394,1212,437],{}," of ",[425,1215,1217],{"className":1216},[428],[425,1218,1220],{"className":1219,"ariaHidden":433},[432],[425,1221,1223,1226],{"className":1222},[437],[425,1224],{"className":1225,"style":562},[441],[425,1227,738],{"className":1228,"style":737},[446,447],". The ",[394,1231,1232],{},"rank"," ",[425,1235,1237],{"className":1236},[428],[425,1238,1240],{"className":1239,"ariaHidden":433},[432],[425,1241,1243,1246,1250,1253,1256],{"className":1242},[437],[425,1244],{"className":1245,"style":757},[441],[425,1247,1249],{"className":1248,"style":566},[446,447],"r",[425,1251,762],{"className":1252},[761],[425,1254,738],{"className":1255,"style":737},[446,447],[425,1257,787],{"className":1258},[786]," is the size of a base.",[381,1261,1262,1263,1266,1267,1291,1292,1337,1338,402,1353,1368,1369,1384,1385,1387,1388,1412,1413,1428,1429,1432],{},"A first consequence of the exchange property is that ",[385,1264,1265],{},"all bases have the same\nsize",": if two bases ",[425,1268,1270],{"className":1269},[428],[425,1271,1273],{"className":1272,"ariaHidden":433},[432],[425,1274,1276,1279,1282,1285,1288],{"className":1275},[437],[425,1277],{"className":1278,"style":980},[441],[425,1280,908],{"className":1281},[446,447],[425,1283,772],{"className":1284},[771],[425,1286],{"className":1287,"style":776},[742],[425,1289,872],{"className":1290,"style":871},[446,447]," had ",[425,1293,1295],{"className":1294},[428],[425,1296,1298,1322],{"className":1297,"ariaHidden":433},[432],[425,1299,1301,1304,1307,1310,1313,1316,1319],{"className":1300},[437],[425,1302],{"className":1303,"style":757},[441],[425,1305,1026],{"className":1306},[446],[425,1308,908],{"className":1309},[446,447],[425,1311,1026],{"className":1312},[446],[425,1314],{"className":1315,"style":743},[742],[425,1317,1039],{"className":1318},[650],[425,1320],{"className":1321,"style":743},[742],[425,1323,1325,1328,1331,1334],{"className":1324},[437],[425,1326],{"className":1327,"style":757},[441],[425,1329,1026],{"className":1330},[446],[425,1332,872],{"className":1333,"style":871},[446,447],[425,1335,1026],{"className":1336},[446],", the exchange property would let us add\nan element of ",[425,1339,1341],{"className":1340},[428],[425,1342,1344],{"className":1343,"ariaHidden":433},[432],[425,1345,1347,1350],{"className":1346},[437],[425,1348],{"className":1349,"style":562},[441],[425,1351,872],{"className":1352,"style":871},[446,447],[425,1354,1356],{"className":1355},[428],[425,1357,1359],{"className":1358,"ariaHidden":433},[432],[425,1360,1362,1365],{"className":1361},[437],[425,1363],{"className":1364,"style":562},[441],[425,1366,908],{"className":1367},[446,447],", contradicting that ",[425,1370,1372],{"className":1371},[428],[425,1373,1375],{"className":1374,"ariaHidden":433},[432],[425,1376,1378,1381],{"className":1377},[437],[425,1379],{"className":1380,"style":562},[441],[425,1382,908],{"className":1383},[446,447]," is maximal. So ",[399,1386,1232],{}," is\nwell-defined: every base has exactly ",[425,1389,1391],{"className":1390},[428],[425,1392,1394],{"className":1393,"ariaHidden":433},[432],[425,1395,1397,1400,1403,1406,1409],{"className":1396},[437],[425,1398],{"className":1399,"style":757},[441],[425,1401,1249],{"className":1402,"style":566},[446,447],[425,1404,762],{"className":1405},[761],[425,1407,738],{"className":1408,"style":737},[446,447],[425,1410,787],{"className":1411},[786]," elements, exactly as every basis of a\nvector space has the same dimension. (Indeed, the columns of a matrix, with\n",[425,1414,1416],{"className":1415},[428],[425,1417,1419],{"className":1418,"ariaHidden":433},[432],[425,1420,1422,1425],{"className":1421},[437],[425,1423],{"className":1424,"style":562},[441],[425,1426,782],{"className":1427,"style":781},[446,780]," the linearly independent subsets, form the ",[394,1430,1431],{},"linear matroid","; this\nis where the vocabulary comes from.)",[1434,1435,1439,1679],"figure",{"className":1436},[1437,1438],"tikz-figure","tikz-diagram-rendered",[1440,1441,1446],"svg",{"xmlns":1442,"width":1443,"height":1444,"viewBox":1445},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","273.941","210.970","-75 -75 205.456 158.227",[1447,1448,1451,1456,1465,1468,1475,1527,1530,1536,1539,1546,1549,1556,1607,1610,1623,1632],"g",{"stroke":1449,"style":1450},"currentColor","stroke-miterlimit:10;stroke-width:.4",[1452,1453],"path",{"fill":1454,"d":1455},"none","M-32.067-12.602a8.536 8.536 0 1 0-17.071 0 8.536 8.536 0 0 0 17.071 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4.346Q-6.117 4.346-5.734 4.167Q-5.351 3.987-5.100 3.667Q-4.848 3.346-4.848 2.963Q-4.848 2.596-5.094 2.372Q-5.340 2.147-5.711 2.147L-6.918 2.147Q-7.090 2.147-7.152 2.161Q-7.215 2.174-7.248 2.233Q-7.281 2.292-7.324 2.452",[1463],[1447,1668,1669],{"transform":1637},[1452,1670],{"d":1671,"fill":1612,"stroke":1612,"className":1672,"style":1642},"M1.695 9.210L-1.231 1.573Q-1.246 1.518-1.246 1.499Q-1.246 1.424-1.192 1.370Q-1.137 1.315-1.063 1.315Q-0.934 1.315-0.895 1.420L2.035 9.057Q2.051 9.112 2.051 9.131Q2.051 9.210 1.998 9.262Q1.945 9.315 1.867 9.315Q1.738 9.315 1.695 9.210",[1463],[1447,1674,1675],{"transform":1637},[1452,1676],{"d":1677,"fill":1612,"stroke":1612,"className":1678,"style":1642},"M6.389 7.315L4.826 7.315Q4.787 7.315 4.760 7.274Q4.732 7.233 4.732 7.186L4.756 7.085Q4.799 7.026 4.854 7.018Q5.178 7.018 5.420 6.874Q5.565 6.788 5.682 6.635Q5.799 6.483 5.846 6.444L8.795 1.713Q8.861 1.604 8.986 1.604L9.068 1.604Q9.182 1.604 9.205 1.713L9.846 6.858Q9.900 7.018 10.443 7.018Q10.541 7.049 10.541 7.139L10.518 7.245Q10.482 7.303 10.420 7.315L8.420 7.315Q8.385 7.315 8.354 7.274Q8.322 7.233 8.322 7.186L8.350 7.085Q8.381 7.030 8.443 7.018Q9.037 7.018 9.068 6.811L8.908 5.506L6.748 5.506L6.092 6.545Q6.084 6.592 6.053 6.665Q6.022 6.737 6.022 6.780Q6.022 6.913 6.141 6.965Q6.260 7.018 6.404 7.018Q6.498 7.049 6.498 7.139L6.475 7.245Q6.443 7.303 6.389 7.315M8.549 2.628L6.932 5.210L8.869 5.210",[1463],[1680,1681,1684,1685,1700,1701,1752],"figcaption",{"className":1682},[1683],"tikz-cap","The exchange property — any smaller independent set ",[425,1686,1688],{"className":1687},[428],[425,1689,1691],{"className":1690,"ariaHidden":433},[432],[425,1692,1694,1697],{"className":1693},[437],[425,1695],{"className":1696,"style":562},[441],[425,1698,908],{"className":1699},[446,447]," can absorb some ",[425,1702,1704],{"className":1703},[428],[425,1705,1707,1725,1743],{"className":1706,"ariaHidden":433},[432],[425,1708,1710,1713,1716,1719,1722],{"className":1709},[437],[425,1711],{"className":1712,"style":1071},[441],[425,1714,1075],{"className":1715},[446,447],[425,1717],{"className":1718,"style":743},[742],[425,1720,879],{"className":1721},[650],[425,1723],{"className":1724,"style":743},[742],[425,1726,1728,1731,1734,1737,1740],{"className":1727},[437],[425,1729],{"className":1730,"style":757},[441],[425,1732,872],{"className":1733,"style":871},[446,447],[425,1735],{"className":1736,"style":1097},[742],[425,1738,1102],{"className":1739},[1101],[425,1741],{"className":1742,"style":1097},[742],[425,1744,1746,1749],{"className":1745},[437],[425,1747],{"className":1748,"style":562},[441],[425,1750,908],{"className":1751},[446,447]," from a larger one",[381,1754,1755,1756,1801,1802,1853,1854,1871,1872,1887],{},"Because ",[425,1757,1759],{"className":1758},[428],[425,1760,1762,1786],{"className":1761,"ariaHidden":433},[432],[425,1763,1765,1768,1771,1774,1777,1780,1783],{"className":1764},[437],[425,1766],{"className":1767,"style":757},[441],[425,1769,1026],{"className":1770},[446],[425,1772,908],{"className":1773},[446,447],[425,1775,1026],{"className":1776},[446],[425,1778],{"className":1779,"style":743},[742],[425,1781,1039],{"className":1782},[650],[425,1784],{"className":1785,"style":743},[742],[425,1787,1789,1792,1795,1798],{"className":1788},[437],[425,1790],{"className":1791,"style":757},[441],[425,1793,1026],{"className":1794},[446],[425,1796,872],{"className":1797,"style":871},[446,447],[425,1799,1026],{"className":1800},[446],", some ",[425,1803,1805],{"className":1804},[428],[425,1806,1808,1826,1844],{"className":1807,"ariaHidden":433},[432],[425,1809,1811,1814,1817,1820,1823],{"className":1810},[437],[425,1812],{"className":1813,"style":1071},[441],[425,1815,1075],{"className":1816},[446,447],[425,1818],{"className":1819,"style":743},[742],[425,1821,879],{"className":1822},[650],[425,1824],{"className":1825,"style":743},[742],[425,1827,1829,1832,1835,1838,1841],{"className":1828},[437],[425,1830],{"className":1831,"style":757},[441],[425,1833,872],{"className":1834,"style":871},[446,447],[425,1836],{"className":1837,"style":1097},[742],[425,1839,1102],{"className":1840},[1101],[425,1842],{"className":1843,"style":1097},[742],[425,1845,1847,1850],{"className":1846},[437],[425,1848],{"className":1849,"style":562},[441],[425,1851,908],{"className":1852},[446,447]," (here ",[425,1855,1857],{"className":1856},[428],[425,1858,1860],{"className":1859,"ariaHidden":433},[432],[425,1861,1863,1867],{"className":1862},[437],[425,1864],{"className":1865,"style":1866},[441],"height:0.4306em;",[425,1868,1870],{"className":1869},[446,447],"c",") joins ",[425,1873,1875],{"className":1874},[428],[425,1876,1878],{"className":1877,"ariaHidden":433},[432],[425,1879,1881,1884],{"className":1880},[437],[425,1882],{"className":1883,"style":562},[441],[425,1885,908],{"className":1886},[446,447]," keeping it\nindependent, so no maximal independent set is left stranded below the rank.",[407,1889,1891],{"id":1890},"the-matroidgreedy-theorem","The matroid–greedy theorem",[381,1893,702,1894,1897,1898,1945,1946,1979,1980,2105,2106,2121],{},[394,1895,1896],{},"weighted matroid"," attaches a positive weight ",[425,1899,1901],{"className":1900},[428],[425,1902,1904,1934],{"className":1903,"ariaHidden":433},[432],[425,1905,1907,1910,1915,1918,1921,1924,1927,1931],{"className":1906},[437],[425,1908],{"className":1909,"style":757},[441],[425,1911,1914],{"className":1912,"style":1913},[446,447],"margin-right:0.0269em;","w",[425,1916,762],{"className":1917},[761],[425,1919,1075],{"className":1920},[446,447],[425,1922,787],{"className":1923},[786],[425,1925],{"className":1926,"style":743},[742],[425,1928,1930],{"className":1929},[650],">",[425,1932],{"className":1933,"style":743},[742],[425,1935,1937,1941],{"className":1936},[437],[425,1938],{"className":1939,"style":1940},[441],"height:0.6444em;",[425,1942,1944],{"className":1943},[446],"0"," to each element\n",[425,1947,1949],{"className":1948},[428],[425,1950,1952,1970],{"className":1951,"ariaHidden":433},[432],[425,1953,1955,1958,1961,1964,1967],{"className":1954},[437],[425,1956],{"className":1957,"style":1071},[441],[425,1959,1075],{"className":1960},[446,447],[425,1962],{"className":1963,"style":743},[742],[425,1965,879],{"className":1966},[650],[425,1968],{"className":1969,"style":743},[742],[425,1971,1973,1976],{"className":1972},[437],[425,1974],{"className":1975,"style":562},[441],[425,1977,767],{"className":1978,"style":766},[446,447],", with ",[425,1981,1983],{"className":1982},[428],[425,1984,1986,2013],{"className":1985,"ariaHidden":433},[432],[425,1987,1989,1992,1995,1998,2001,2004,2007,2010],{"className":1988},[437],[425,1990],{"className":1991,"style":757},[441],[425,1993,1914],{"className":1994,"style":1913},[446,447],[425,1996,762],{"className":1997},[761],[425,1999,908],{"className":2000},[446,447],[425,2002,787],{"className":2003},[786],[425,2005],{"className":2006,"style":743},[742],[425,2008,747],{"className":2009},[650],[425,2011],{"className":2012,"style":743},[742],[425,2014,2016,2020,2090,2093,2096,2099,2102],{"className":2015},[437],[425,2017],{"className":2018,"style":2019},[441],"height:1.0771em;vertical-align:-0.3271em;",[425,2021,2024,2031],{"className":2022},[2023],"mop",[425,2025,2030],{"className":2026,"style":2029},[2023,2027,2028],"op-symbol","small-op","position:relative;top:0em;","∑",[425,2032,2035],{"className":2033},[2034],"msupsub",[425,2036,2040,2081],{"className":2037},[2038,2039],"vlist-t","vlist-t2",[425,2041,2044,2076],{"className":2042},[2043],"vlist-r",[425,2045,2049],{"className":2046,"style":2048},[2047],"vlist","height:0.1786em;",[425,2050,2052,2057],{"style":2051},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[425,2053],{"className":2054,"style":2056},[2055],"pstrut","height:2.7em;",[425,2058,2064],{"className":2059},[2060,2061,2062,2063],"sizing","reset-size6","size3","mtight",[425,2065,2067,2070,2073],{"className":2066},[446,2063],[425,2068,1075],{"className":2069},[446,447,2063],[425,2071,879],{"className":2072},[650,2063],[425,2074,908],{"className":2075},[446,447,2063],[425,2077,2080],{"className":2078},[2079],"vlist-s","​",[425,2082,2084],{"className":2083},[2043],[425,2085,2088],{"className":2086,"style":2087},[2047],"height:0.3271em;",[425,2089],{},[425,2091],{"className":2092,"style":776},[742],[425,2094,1914],{"className":2095,"style":1913},[446,447],[425,2097,762],{"className":2098},[761],[425,2100,1075],{"className":2101},[446,447],[425,2103,787],{"className":2104},[786],". The natural optimization problem:\nfind a maximum-weight independent set. Since weights are positive and\n",[425,2107,2109],{"className":2108},[428],[425,2110,2112],{"className":2111,"ariaHidden":433},[432],[425,2113,2115,2118],{"className":2114},[437],[425,2116],{"className":2117,"style":562},[441],[425,2119,782],{"className":2120,"style":781},[446,780]," is hereditary, a maximum-weight independent set is always a base.\nThe greedy algorithm is the obvious one: sort by weight descending, add each\nelement if it keeps the set independent.",[2123,2124,2128],"pre",{"className":2125,"code":2126,"language":2127,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{Greedy}(M=(S,\\mathcal I),\\, w)$ — maximum-weight independent set\n$A \\gets \\varnothing$\nsort $S$ into nonincreasing order by weight $w$\nfor each $x \\in S$ in sorted order do\n  if $A \\cup \\{x\\} \\in \\mathcal{I}$ then   \u002F\u002F independence test\n    $A \\gets A \\cup \\{x\\}$\nreturn $A$\n","algorithm",[2129,2130,2131,2137,2142,2147,2152,2157,2162],"code",{"__ignoreMap":376},[425,2132,2134],{"class":2133,"line":6},"line",[425,2135,2136],{},"caption: $\\textsc{Greedy}(M=(S,\\mathcal I),\\, w)$ — maximum-weight independent set\n",[425,2138,2139],{"class":2133,"line":18},[425,2140,2141],{},"$A \\gets \\varnothing$\n",[425,2143,2144],{"class":2133,"line":24},[425,2145,2146],{},"sort $S$ into nonincreasing order by weight $w$\n",[425,2148,2149],{"class":2133,"line":73},[425,2150,2151],{},"for each $x \\in S$ in sorted order do\n",[425,2153,2154],{"class":2133,"line":102},[425,2155,2156],{},"  if $A \\cup \\{x\\} \\in \\mathcal{I}$ then   \u002F\u002F independence test\n",[425,2158,2159],{"class":2133,"line":108},[425,2160,2161],{},"    $A \\gets A \\cup \\{x\\}$\n",[425,2163,2164],{"class":2133,"line":116},[425,2165,2166],{},"return $A$\n",[415,2168,2170],{"type":2169},"theorem",[381,2171,2172,854,2175,2223,2224,2239,2240,2282,2283,2298,2299,2329,2330,2333,2334,2356,2357,2360,2361],{},[394,2173,2174],{},"Theorem (Rado–Edmonds).",[425,2176,2178],{"className":2177},[428],[425,2179,2181,2199],{"className":2180,"ariaHidden":433},[432],[425,2182,2184,2187,2190,2193,2196],{"className":2183},[437],[425,2185],{"className":2186,"style":562},[441],[425,2188,738],{"className":2189,"style":737},[446,447],[425,2191],{"className":2192,"style":743},[742],[425,2194,747],{"className":2195},[650],[425,2197],{"className":2198,"style":743},[742],[425,2200,2202,2205,2208,2211,2214,2217,2220],{"className":2201},[437],[425,2203],{"className":2204,"style":757},[441],[425,2206,762],{"className":2207},[761],[425,2209,767],{"className":2210,"style":766},[446,447],[425,2212,772],{"className":2213},[771],[425,2215],{"className":2216,"style":776},[742],[425,2218,782],{"className":2219,"style":781},[446,780],[425,2221,787],{"className":2222},[786]," is a matroid with weight\nfunction ",[425,2225,2227],{"className":2226},[428],[425,2228,2230],{"className":2229,"ariaHidden":433},[432],[425,2231,2233,2236],{"className":2232},[437],[425,2234],{"className":2235,"style":1866},[441],[425,2237,1914],{"className":2238,"style":1913},[446,447],", then ",[425,2241,2243],{"className":2242},[428],[425,2244,2246],{"className":2245,"ariaHidden":433},[432],[425,2247,2249,2252,2264,2267,2270,2273,2276,2279],{"className":2248},[437],[425,2250],{"className":2251,"style":757},[441],[425,2253,2257],{"className":2254},[2255,2256],"enclosing","textsc",[425,2258,2261],{"className":2259},[446,2260],"text",[425,2262,218],{"className":2263},[446],[425,2265,762],{"className":2266},[761],[425,2268,738],{"className":2269,"style":737},[446,447],[425,2271,772],{"className":2272},[771],[425,2274],{"className":2275,"style":776},[742],[425,2277,1914],{"className":2278,"style":1913},[446,447],[425,2280,787],{"className":2281},[786]," returns a maximum-weight base of ",[425,2284,2286],{"className":2285},[428],[425,2287,2289],{"className":2288,"ariaHidden":433},[432],[425,2290,2292,2295],{"className":2291},[437],[425,2293],{"className":2294,"style":562},[441],[425,2296,738],{"className":2297,"style":737},[446,447],".\nConversely, if ",[425,2300,2302],{"className":2301},[428],[425,2303,2305],{"className":2304,"ariaHidden":433},[432],[425,2306,2308,2311,2314,2317,2320,2323,2326],{"className":2307},[437],[425,2309],{"className":2310,"style":757},[441],[425,2312,762],{"className":2313},[761],[425,2315,767],{"className":2316,"style":766},[446,447],[425,2318,772],{"className":2319},[771],[425,2321],{"className":2322,"style":776},[742],[425,2324,782],{"className":2325,"style":781},[446,780],[425,2327,787],{"className":2328},[786]," is hereditary but ",[385,2331,2332],{},"not"," a matroid, then there\nis a weight function on which ",[425,2335,2337],{"className":2336},[428],[425,2338,2340],{"className":2339,"ariaHidden":433},[432],[425,2341,2343,2347],{"className":2342},[437],[425,2344],{"className":2345,"style":2346},[441],"height:0.8889em;vertical-align:-0.1944em;",[425,2348,2350],{"className":2349},[2255,2256],[425,2351,2353],{"className":2352},[446,2260],[425,2354,218],{"className":2355},[446]," fails. So greedy is optimal on a\nhereditary structure ",[394,2358,2359],{},"if and only if"," that structure is a matroid.",[677,2362,2363],{},[680,2364,2368],{"href":2365,"ariaDescribedBy":2366,"dataFootnoteRef":376,"id":2367},"#user-content-fn-clrs-matroid",[684],"user-content-fnref-clrs-matroid","2",[381,2370,2371],{},"The forward direction is a clean exchange argument; it is worth seeing in full,\nbecause it is the abstract skeleton of every concrete greedy proof in this module.",[415,2373,2374,3484,4448],{"type":387},[381,2375,2376,2379,2380,2544,2545,2753,2754,2888,2889,2904,2905,2956,2957,2972,2973,3085,3086,3128,3129,3144,3145,3271,3272,3287,3288,3415,3416,3432,3433,1176],{},[394,2377,2378],{},"Proof (greedy is optimal on a matroid)."," Let ",[425,2381,2383],{"className":2382},[428],[425,2384,2386],{"className":2385,"ariaHidden":433},[432],[425,2387,2389,2393,2437,2440,2443,2483,2486,2489,2494,2497,2500,2503],{"className":2388},[437],[425,2390],{"className":2391,"style":2392},[441],"height:0.625em;vertical-align:-0.1944em;",[425,2394,2396,2400],{"className":2395},[446],[425,2397,1447],{"className":2398,"style":2399},[446,447],"margin-right:0.0359em;",[425,2401,2403],{"className":2402},[2034],[425,2404,2406,2428],{"className":2405},[2038,2039],[425,2407,2409,2425],{"className":2408},[2043],[425,2410,2413],{"className":2411,"style":2412},[2047],"height:0.3011em;",[425,2414,2416,2419],{"style":2415},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[425,2417],{"className":2418,"style":2056},[2055],[425,2420,2422],{"className":2421},[2060,2061,2062,2063],[425,2423,686],{"className":2424},[446,2063],[425,2426,2080],{"className":2427},[2079],[425,2429,2431],{"className":2430},[2043],[425,2432,2435],{"className":2433,"style":2434},[2047],"height:0.15em;",[425,2436],{},[425,2438,772],{"className":2439},[771],[425,2441],{"className":2442,"style":776},[742],[425,2444,2446,2449],{"className":2445},[446],[425,2447,1447],{"className":2448,"style":2399},[446,447],[425,2450,2452],{"className":2451},[2034],[425,2453,2455,2475],{"className":2454},[2038,2039],[425,2456,2458,2472],{"className":2457},[2043],[425,2459,2461],{"className":2460,"style":2412},[2047],[425,2462,2463,2466],{"style":2415},[425,2464],{"className":2465,"style":2056},[2055],[425,2467,2469],{"className":2468},[2060,2061,2062,2063],[425,2470,2368],{"className":2471},[446,2063],[425,2473,2080],{"className":2474},[2079],[425,2476,2478],{"className":2477},[2043],[425,2479,2481],{"className":2480,"style":2434},[2047],[425,2482],{},[425,2484,772],{"className":2485},[771],[425,2487],{"className":2488,"style":776},[742],[425,2490,2493],{"className":2491},[2492],"minner","…",[425,2495],{"className":2496,"style":776},[742],[425,2498,772],{"className":2499},[771],[425,2501],{"className":2502,"style":776},[742],[425,2504,2506,2509],{"className":2505},[446],[425,2507,1447],{"className":2508,"style":2399},[446,447],[425,2510,2512],{"className":2511},[2034],[425,2513,2515,2536],{"className":2514},[2038,2039],[425,2516,2518,2533],{"className":2517},[2043],[425,2519,2522],{"className":2520,"style":2521},[2047],"height:0.1514em;",[425,2523,2524,2527],{"style":2415},[425,2525],{"className":2526,"style":2056},[2055],[425,2528,2530],{"className":2529},[2060,2061,2062,2063],[425,2531,1249],{"className":2532,"style":566},[446,447,2063],[425,2534,2080],{"className":2535},[2079],[425,2537,2539],{"className":2538},[2043],[425,2540,2542],{"className":2541,"style":2434},[2047],[425,2543],{}," be greedy's\npicks in the order chosen, so ",[425,2546,2548],{"className":2547},[428],[425,2549,2551,2615,2679,2698],{"className":2550,"ariaHidden":433},[432],[425,2552,2554,2557,2560,2563,2603,2606,2609,2612],{"className":2553},[437],[425,2555],{"className":2556,"style":757},[441],[425,2558,1914],{"className":2559,"style":1913},[446,447],[425,2561,762],{"className":2562},[761],[425,2564,2566,2569],{"className":2565},[446],[425,2567,1447],{"className":2568,"style":2399},[446,447],[425,2570,2572],{"className":2571},[2034],[425,2573,2575,2595],{"className":2574},[2038,2039],[425,2576,2578,2592],{"className":2577},[2043],[425,2579,2581],{"className":2580,"style":2412},[2047],[425,2582,2583,2586],{"style":2415},[425,2584],{"className":2585,"style":2056},[2055],[425,2587,2589],{"className":2588},[2060,2061,2062,2063],[425,2590,686],{"className":2591},[446,2063],[425,2593,2080],{"className":2594},[2079],[425,2596,2598],{"className":2597},[2043],[425,2599,2601],{"className":2600,"style":2434},[2047],[425,2602],{},[425,2604,787],{"className":2605},[786],[425,2607],{"className":2608,"style":743},[742],[425,2610,651],{"className":2611},[650],[425,2613],{"className":2614,"style":743},[742],[425,2616,2618,2621,2624,2627,2667,2670,2673,2676],{"className":2617},[437],[425,2619],{"className":2620,"style":757},[441],[425,2622,1914],{"className":2623,"style":1913},[446,447],[425,2625,762],{"className":2626},[761],[425,2628,2630,2633],{"className":2629},[446],[425,2631,1447],{"className":2632,"style":2399},[446,447],[425,2634,2636],{"className":2635},[2034],[425,2637,2639,2659],{"className":2638},[2038,2039],[425,2640,2642,2656],{"className":2641},[2043],[425,2643,2645],{"className":2644,"style":2412},[2047],[425,2646,2647,2650],{"style":2415},[425,2648],{"className":2649,"style":2056},[2055],[425,2651,2653],{"className":2652},[2060,2061,2062,2063],[425,2654,2368],{"className":2655},[446,2063],[425,2657,2080],{"className":2658},[2079],[425,2660,2662],{"className":2661},[2043],[425,2663,2665],{"className":2664,"style":2434},[2047],[425,2666],{},[425,2668,787],{"className":2669},[786],[425,2671],{"className":2672,"style":743},[742],[425,2674,651],{"className":2675},[650],[425,2677],{"className":2678,"style":743},[742],[425,2680,2682,2685,2689,2692,2695],{"className":2681},[437],[425,2683],{"className":2684,"style":646},[441],[425,2686,2688],{"className":2687},[2492],"⋯",[425,2690],{"className":2691,"style":743},[742],[425,2693,651],{"className":2694},[650],[425,2696],{"className":2697,"style":743},[742],[425,2699,2701,2704,2707,2710,2750],{"className":2700},[437],[425,2702],{"className":2703,"style":757},[441],[425,2705,1914],{"className":2706,"style":1913},[446,447],[425,2708,762],{"className":2709},[761],[425,2711,2713,2716],{"className":2712},[446],[425,2714,1447],{"className":2715,"style":2399},[446,447],[425,2717,2719],{"className":2718},[2034],[425,2720,2722,2742],{"className":2721},[2038,2039],[425,2723,2725,2739],{"className":2724},[2043],[425,2726,2728],{"className":2727,"style":2521},[2047],[425,2729,2730,2733],{"style":2415},[425,2731],{"className":2732,"style":2056},[2055],[425,2734,2736],{"className":2735},[2060,2061,2062,2063],[425,2737,1249],{"className":2738,"style":566},[446,447,2063],[425,2740,2080],{"className":2741},[2079],[425,2743,2745],{"className":2744},[2043],[425,2746,2748],{"className":2747,"style":2434},[2047],[425,2749],{},[425,2751,787],{"className":2752},[786],", and let\n",[425,2755,2757],{"className":2756},[428],[425,2758,2760,2778],{"className":2759,"ariaHidden":433},[432],[425,2761,2763,2766,2769,2772,2775],{"className":2762},[437],[425,2764],{"className":2765,"style":562},[441],[425,2767,584],{"className":2768},[446,447],[425,2770],{"className":2771,"style":743},[742],[425,2773,747],{"className":2774},[650],[425,2776],{"className":2777,"style":743},[742],[425,2779,2781,2784,2787,2827,2830,2833,2836,2839,2842,2845,2885],{"className":2780},[437],[425,2782],{"className":2783,"style":757},[441],[425,2785,1150],{"className":2786},[761],[425,2788,2790,2793],{"className":2789},[446],[425,2791,1447],{"className":2792,"style":2399},[446,447],[425,2794,2796],{"className":2795},[2034],[425,2797,2799,2819],{"className":2798},[2038,2039],[425,2800,2802,2816],{"className":2801},[2043],[425,2803,2805],{"className":2804,"style":2412},[2047],[425,2806,2807,2810],{"style":2415},[425,2808],{"className":2809,"style":2056},[2055],[425,2811,2813],{"className":2812},[2060,2061,2062,2063],[425,2814,686],{"className":2815},[446,2063],[425,2817,2080],{"className":2818},[2079],[425,2820,2822],{"className":2821},[2043],[425,2823,2825],{"className":2824,"style":2434},[2047],[425,2826],{},[425,2828,772],{"className":2829},[771],[425,2831],{"className":2832,"style":776},[742],[425,2834,2493],{"className":2835},[2492],[425,2837],{"className":2838,"style":776},[742],[425,2840,772],{"className":2841},[771],[425,2843],{"className":2844,"style":776},[742],[425,2846,2848,2851],{"className":2847},[446],[425,2849,1447],{"className":2850,"style":2399},[446,447],[425,2852,2854],{"className":2853},[2034],[425,2855,2857,2877],{"className":2856},[2038,2039],[425,2858,2860,2874],{"className":2859},[2043],[425,2861,2863],{"className":2862,"style":2521},[2047],[425,2864,2865,2868],{"style":2415},[425,2866],{"className":2867,"style":2056},[2055],[425,2869,2871],{"className":2870},[2060,2061,2062,2063],[425,2872,1249],{"className":2873,"style":566},[446,447,2063],[425,2875,2080],{"className":2876},[2079],[425,2878,2880],{"className":2879},[2043],[425,2881,2883],{"className":2882,"style":2434},[2047],[425,2884],{},[425,2886,1157],{"className":2887},[786],". Suppose for contradiction some base ",[425,2890,2892],{"className":2891},[428],[425,2893,2895],{"className":2894,"ariaHidden":433},[432],[425,2896,2898,2901],{"className":2897},[437],[425,2899],{"className":2900,"style":562},[441],[425,2902,567],{"className":2903,"style":566},[446,447]," has\n",[425,2906,2908],{"className":2907},[428],[425,2909,2911,2938],{"className":2910,"ariaHidden":433},[432],[425,2912,2914,2917,2920,2923,2926,2929,2932,2935],{"className":2913},[437],[425,2915],{"className":2916,"style":757},[441],[425,2918,1914],{"className":2919,"style":1913},[446,447],[425,2921,762],{"className":2922},[761],[425,2924,567],{"className":2925,"style":566},[446,447],[425,2927,787],{"className":2928},[786],[425,2930],{"className":2931,"style":743},[742],[425,2933,1930],{"className":2934},[650],[425,2936],{"className":2937,"style":743},[742],[425,2939,2941,2944,2947,2950,2953],{"className":2940},[437],[425,2942],{"className":2943,"style":757},[441],[425,2945,1914],{"className":2946,"style":1913},[446,447],[425,2948,762],{"className":2949},[761],[425,2951,584],{"className":2952},[446,447],[425,2954,787],{"className":2955},[786],". List ",[425,2958,2960],{"className":2959},[428],[425,2961,2963],{"className":2962,"ariaHidden":433},[432],[425,2964,2966,2969],{"className":2965},[437],[425,2967],{"className":2968,"style":562},[441],[425,2970,567],{"className":2971,"style":566},[446,447],"'s elements in nonincreasing weight order as\n",[425,2974,2976],{"className":2975},[428],[425,2977,2979],{"className":2978,"ariaHidden":433},[432],[425,2980,2982,2985,3027,3030,3033,3036,3039,3042,3045],{"className":2981},[437],[425,2983],{"className":2984,"style":2392},[441],[425,2986,2988,2992],{"className":2987},[446],[425,2989,2991],{"className":2990},[446,447],"o",[425,2993,2995],{"className":2994},[2034],[425,2996,2998,3019],{"className":2997},[2038,2039],[425,2999,3001,3016],{"className":3000},[2043],[425,3002,3004],{"className":3003,"style":2412},[2047],[425,3005,3007,3010],{"style":3006},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[425,3008],{"className":3009,"style":2056},[2055],[425,3011,3013],{"className":3012},[2060,2061,2062,2063],[425,3014,686],{"className":3015},[446,2063],[425,3017,2080],{"className":3018},[2079],[425,3020,3022],{"className":3021},[2043],[425,3023,3025],{"className":3024,"style":2434},[2047],[425,3026],{},[425,3028,772],{"className":3029},[771],[425,3031],{"className":3032,"style":776},[742],[425,3034,2493],{"className":3035},[2492],[425,3037],{"className":3038,"style":776},[742],[425,3040,772],{"className":3041},[771],[425,3043],{"className":3044,"style":776},[742],[425,3046,3048,3051],{"className":3047},[446],[425,3049,2991],{"className":3050},[446,447],[425,3052,3054],{"className":3053},[2034],[425,3055,3057,3077],{"className":3056},[2038,2039],[425,3058,3060,3074],{"className":3059},[2043],[425,3061,3063],{"className":3062,"style":2521},[2047],[425,3064,3065,3068],{"style":3006},[425,3066],{"className":3067,"style":2056},[2055],[425,3069,3071],{"className":3070},[2060,2061,2062,2063],[425,3072,1249],{"className":3073,"style":566},[446,447,2063],[425,3075,2080],{"className":3076},[2079],[425,3078,3080],{"className":3079},[2043],[425,3081,3083],{"className":3082,"style":2434},[2047],[425,3084],{}," (both bases have size ",[425,3087,3089],{"className":3088},[428],[425,3090,3092,3110],{"className":3091,"ariaHidden":433},[432],[425,3093,3095,3098,3101,3104,3107],{"className":3094},[437],[425,3096],{"className":3097,"style":1866},[441],[425,3099,1249],{"className":3100,"style":566},[446,447],[425,3102],{"className":3103,"style":743},[742],[425,3105,747],{"className":3106},[650],[425,3108],{"className":3109,"style":743},[742],[425,3111,3113,3116,3119,3122,3125],{"className":3112},[437],[425,3114],{"className":3115,"style":757},[441],[425,3117,1249],{"className":3118,"style":566},[446,447],[425,3120,762],{"className":3121},[761],[425,3123,738],{"className":3124,"style":737},[446,447],[425,3126,787],{"className":3127},[786],"). Let ",[425,3130,3132],{"className":3131},[428],[425,3133,3135],{"className":3134,"ariaHidden":433},[432],[425,3136,3138,3141],{"className":3137},[437],[425,3139],{"className":3140,"style":442},[441],[425,3142,449],{"className":3143,"style":448},[446,447]," be the first index\nwhere ",[425,3146,3148],{"className":3147},[428],[425,3149,3151,3216],{"className":3150,"ariaHidden":433},[432],[425,3152,3154,3157,3160,3163,3204,3207,3210,3213],{"className":3153},[437],[425,3155],{"className":3156,"style":757},[441],[425,3158,1914],{"className":3159,"style":1913},[446,447],[425,3161,762],{"className":3162},[761],[425,3164,3166,3169],{"className":3165},[446],[425,3167,1447],{"className":3168,"style":2399},[446,447],[425,3170,3172],{"className":3171},[2034],[425,3173,3175,3196],{"className":3174},[2038,2039],[425,3176,3178,3193],{"className":3177},[2043],[425,3179,3182],{"className":3180,"style":3181},[2047],"height:0.3361em;",[425,3183,3184,3187],{"style":2415},[425,3185],{"className":3186,"style":2056},[2055],[425,3188,3190],{"className":3189},[2060,2061,2062,2063],[425,3191,449],{"className":3192,"style":448},[446,447,2063],[425,3194,2080],{"className":3195},[2079],[425,3197,3199],{"className":3198},[2043],[425,3200,3202],{"className":3201,"style":2434},[2047],[425,3203],{},[425,3205,787],{"className":3206},[786],[425,3208],{"className":3209,"style":743},[742],[425,3211,1039],{"className":3212},[650],[425,3214],{"className":3215,"style":743},[742],[425,3217,3219,3222,3225,3228,3268],{"className":3218},[437],[425,3220],{"className":3221,"style":757},[441],[425,3223,1914],{"className":3224,"style":1913},[446,447],[425,3226,762],{"className":3227},[761],[425,3229,3231,3234],{"className":3230},[446],[425,3232,2991],{"className":3233},[446,447],[425,3235,3237],{"className":3236},[2034],[425,3238,3240,3260],{"className":3239},[2038,2039],[425,3241,3243,3257],{"className":3242},[2043],[425,3244,3246],{"className":3245,"style":3181},[2047],[425,3247,3248,3251],{"style":3006},[425,3249],{"className":3250,"style":2056},[2055],[425,3252,3254],{"className":3253},[2060,2061,2062,2063],[425,3255,449],{"className":3256,"style":448},[446,447,2063],[425,3258,2080],{"className":3259},[2079],[425,3261,3263],{"className":3262},[2043],[425,3264,3266],{"className":3265,"style":2434},[2047],[425,3267],{},[425,3269,787],{"className":3270},[786],"; such a ",[425,3273,3275],{"className":3274},[428],[425,3276,3278],{"className":3277,"ariaHidden":433},[432],[425,3279,3281,3284],{"className":3280},[437],[425,3282],{"className":3283,"style":442},[441],[425,3285,449],{"className":3286,"style":448},[446,447]," exists, else 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Each\n",[425,4062,4064],{"className":4063},[428],[425,4065,4067,4123],{"className":4066,"ariaHidden":433},[432],[425,4068,4070,4074,4114,4117,4120],{"className":4069},[437],[425,4071],{"className":4072,"style":4073},[441],"height:0.6891em;vertical-align:-0.15em;",[425,4075,4077,4080],{"className":4076},[446],[425,4078,2991],{"className":4079},[446,447],[425,4081,4083],{"className":4082},[2034],[425,4084,4086,4106],{"className":4085},[2038,2039],[425,4087,4089,4103],{"className":4088},[2043],[425,4090,4092],{"className":4091,"style":3324},[2047],[425,4093,4094,4097],{"style":3006},[425,4095],{"className":4096,"style":2056},[2055],[425,4098,4100],{"className":4099},[2060,2061,2062,2063],[425,4101,3336],{"className":4102},[446,447,2063],[425,4104,2080],{"className":4105},[2079],[425,4107,4109],{"className":4108},[2043],[425,4110,4112],{"className":4111,"style":2434},[2047],[425,4113],{},[425,4115],{"className":4116,"style":743},[742],[425,4118,879],{"className":4119},[650],[425,4121],{"className":4122,"style":743},[742],[425,4124,4126,4129],{"className":4125},[437],[425,4127],{"className":4128,"style":562},[441],[425,4130,872],{"className":4131,"style":871},[446,447]," has ",[425,4134,4136],{"className":4135},[428],[425,4137,4139,4203,4267],{"className":4138,"ariaHidden":433},[432],[425,4140,4142,4145,4148,4151,4191,4194,4197,4200],{"className":4141},[437],[425,4143],{"className":4144,"style":757},[441],[425,4146,1914],{"className":4147,"style":1913},[446,447],[425,4149,762],{"className":4150},[761],[425,4152,4154,4157],{"className":4153},[446],[425,4155,2991],{"className":4156},[446,447],[425,4158,4160],{"className":4159},[2034],[425,4161,4163,4183],{"className":4162},[2038,2039],[425,4164,4166,4180],{"className":4165},[2043],[425,4167,4169],{"className":4168,"style":3324},[2047],[425,4170,4171,4174],{"style":3006},[425,4172],{"className":4173,"style":2056},[2055],[425,4175,4177],{"className":4176},[2060,2061,2062,2063],[425,4178,3336],{"className":4179},[446,447,2063],[425,4181,2080],{"className":4182},[2079],[425,4184,4186],{"className":4185},[2043],[425,4187,4189],{"className":4188,"style":2434},[2047],[425,4190],{},[425,4192,787],{"className":4193},[786],[425,4195],{"className":4196,"style":743},[742],[425,4198,651],{"className":4199},[650],[425,4201],{"className":4202,"style":743},[742],[425,4204,4206,4209,4212,4215,4255,4258,4261,4264],{"className":4205},[437],[425,4207],{"className":4208,"style":757},[441],[425,4210,1914],{"className":4211,"style":1913},[446,447],[425,4213,762],{"className":4214},[761],[425,4216,4218,4221],{"className":4217},[446],[425,4219,2991],{"className":4220},[446,447],[425,4222,4224],{"className":4223},[2034],[425,4225,4227,4247],{"className":4226},[2038,2039],[425,4228,4230,4244],{"className":4229},[2043],[425,4231,4233],{"className":4232,"style":3181},[2047],[425,4234,4235,4238],{"style":3006},[425,4236],{"className":4237,"style":2056},[2055],[425,4239,4241],{"className":4240},[2060,2061,2062,2063],[425,4242,449],{"className":4243,"style":448},[446,447,2063],[425,4245,2080],{"className":4246},[2079],[425,4248,4250],{"className":4249},[2043],[425,4251,4253],{"className":4252,"style":2434},[2047],[425,4254],{},[425,4256,787],{"className":4257},[786],[425,4259],{"className":4260,"style":743},[742],[425,4262,1930],{"className":4263},[650],[425,4265],{"className":4266,"style":743},[742],[425,4268,4270,4273,4276,4279,4319],{"className":4269},[437],[425,4271],{"className":4272,"style":757},[441],[425,4274,1914],{"className":4275,"style":1913},[446,447],[425,4277,762],{"className":4278},[761],[425,4280,4282,4285],{"className":4281},[446],[425,4283,1447],{"className":4284,"style":2399},[446,447],[425,4286,4288],{"className":4287},[2034],[425,4289,4291,4311],{"className":4290},[2038,2039],[425,4292,4294,4308],{"className":4293},[2043],[425,4295,4297],{"className":4296,"style":3181},[2047],[425,4298,4299,4302],{"style":2415},[425,4300],{"className":4301,"style":2056},[2055],[425,4303,4305],{"className":4304},[2060,2061,2062,2063],[425,4306,449],{"className":4307,"style":448},[446,447,2063],[425,4309,2080],{"className":4310},[2079],[425,4312,4314],{"className":4313},[2043],[425,4315,4317],{"className":4316,"style":2434},[2047],[425,4318],{},[425,4320,787],{"className":4321},[786],", so ",[425,4324,4326],{"className":4325},[428],[425,4327,4329,4393],{"className":4328,"ariaHidden":433},[432],[425,4330,4332,4335,4338,4341,4381,4384,4387,4390],{"className":4331},[437],[425,4333],{"className":4334,"style":3995},[441],[425,4336,1914],{"className":4337,"style":1913},[446,447],[425,4339,762],{"className":4340},[761],[425,4342,4344,4347],{"className":4343},[446],[425,4345,2991],{"className":4346},[446,447],[425,4348,4350],{"className":4349},[2034],[425,4351,4353,4373],{"className":4352},[2038,2039],[425,4354,4356,4370],{"className":4355},[2043],[425,4357,4359],{"className":4358,"style":3324},[2047],[425,4360,4361,4364],{"style":3006},[425,4362],{"className":4363,"style":2056},[2055],[425,4365,4367],{"className":4366},[2060,2061,2062,2063],[425,4368,3915],{"className":4369,"style":3914},[446,447,2063],[425,4371,2080],{"className":4372},[2079],[425,4374,4376],{"className":4375},[2043],[425,4377,4379],{"className":4378,"style":3925},[2047],[425,4380],{},[425,4382,787],{"className":4383},[786],[425,4385],{"className":4386,"style":743},[742],[425,4388,1930],{"className":4389},[650],[425,4391],{"className":4392,"style":743},[742],[425,4394,4396,4399,4402,4405,4445],{"className":4395},[437],[425,4397],{"className":4398,"style":757},[441],[425,4400,1914],{"className":4401,"style":1913},[446,447],[425,4403,762],{"className":4404},[761],[425,4406,4408,4411],{"className":4407},[446],[425,4409,1447],{"className":4410,"style":2399},[446,447],[425,4412,4414],{"className":4413},[2034],[425,4415,4417,4437],{"className":4416},[2038,2039],[425,4418,4420,4434],{"className":4419},[2043],[425,4421,4423],{"className":4422,"style":3181},[2047],[425,4424,4425,4428],{"style":2415},[425,4426],{"className":4427,"style":2056},[2055],[425,4429,4431],{"className":4430},[2060,2061,2062,2063],[425,4432,449],{"className":4433,"style":448},[446,447,2063],[425,4435,2080],{"className":4436},[2079],[425,4438,4440],{"className":4439},[2043],[425,4441,4443],{"className":4442,"style":2434},[2047],[425,4444],{},[425,4446,787],{"className":4447},[786],[381,4449,4450,4451,4503,4504,4647,4648,4724,4725,4850,4851,1233,4904,4907,4959,4960,4975,4976,4991,4992],{},"But when greedy considered the elements in weight order, at the moment it chose\n",[425,4452,4454],{"className":4453},[428],[425,4455,4457],{"className":4456,"ariaHidden":433},[432],[425,4458,4460,4463],{"className":4459},[437],[425,4461],{"className":4462,"style":2392},[441],[425,4464,4466,4469],{"className":4465},[446],[425,4467,1447],{"className":4468,"style":2399},[446,447],[425,4470,4472],{"className":4471},[2034],[425,4473,4475,4495],{"className":4474},[2038,2039],[425,4476,4478,4492],{"className":4477},[2043],[425,4479,4481],{"className":4480,"style":3181},[2047],[425,4482,4483,4486],{"style":2415},[425,4484],{"className":4485,"style":2056},[2055],[425,4487,4489],{"className":4488},[2060,2061,2062,2063],[425,4490,449],{"className":4491,"style":448},[446,447,2063],[425,4493,2080],{"className":4494},[2079],[425,4496,4498],{"className":4497},[2043],[425,4499,4501],{"className":4500,"style":2434},[2047],[425,4502],{}," the set ",[425,4505,4507],{"className":4506},[428],[425,4508,4510,4528],{"className":4509,"ariaHidden":433},[432],[425,4511,4513,4516,4519,4522,4525],{"className":4512},[437],[425,4514],{"className":4515,"style":562},[441],[425,4517,908],{"className":4518},[446,447],[425,4520],{"className":4521,"style":743},[742],[425,4523,747],{"className":4524},[650],[425,4526],{"className":4527,"style":743},[742],[425,4529,4531,4534,4537,4577,4580,4583,4586,4589,4592,4595,4644],{"className":4530},[437],[425,4532],{"className":4533,"style":757},[441],[425,4535,1150],{"className":4536},[761],[425,4538,4540,4543],{"className":4539},[446],[425,4541,1447],{"className":4542,"style":2399},[446,447],[425,4544,4546],{"className":4545},[2034],[425,4547,4549,4569],{"className":4548},[2038,2039],[425,4550,4552,4566],{"className":4551},[2043],[425,4553,4555],{"className":4554,"style":2412},[2047],[425,4556,4557,4560],{"style":2415},[425,4558],{"className":4559,"style":2056},[2055],[425,4561,4563],{"className":4562},[2060,2061,2062,2063],[425,4564,686],{"className":4565},[446,2063],[425,4567,2080],{"className":4568},[2079],[425,4570,4572],{"className":4571},[2043],[425,4573,4575],{"className":4574,"style":2434},[2047],[425,4576],{},[425,4578,772],{"className":4579},[771],[425,4581],{"className":4582,"style":776},[742],[425,4584,2493],{"className":4585},[2492],[425,4587],{"className":4588,"style":776},[742],[425,4590,772],{"className":4591},[771],[425,4593],{"className":4594,"style":776},[742],[425,4596,4598,4601],{"className":4597},[446],[425,4599,1447],{"className":4600,"style":2399},[446,447],[425,4602,4604],{"className":4603},[2034],[425,4605,4607,4636],{"className":4606},[2038,2039],[425,4608,4610,4633],{"className":4609},[2043],[425,4611,4613],{"className":4612,"style":3181},[2047],[425,4614,4615,4618],{"style":2415},[425,4616],{"className":4617,"style":2056},[2055],[425,4619,4621],{"className":4620},[2060,2061,2062,2063],[425,4622,4624,4627,4630],{"className":4623},[446,2063],[425,4625,449],{"className":4626,"style":448},[446,447,2063],[425,4628,3613],{"className":4629},[1101,2063],[425,4631,686],{"className":4632},[446,2063],[425,4634,2080],{"className":4635},[2079],[425,4637,4639],{"className":4638},[2043],[425,4640,4642],{"className":4641,"style":3626},[2047],[425,4643],{},[425,4645,1157],{"className":4646},[786]," was already in hand, and\n",[425,4649,4651],{"className":4650},[428],[425,4652,4654,4672],{"className":4653,"ariaHidden":433},[432],[425,4655,4657,4660,4663,4666,4669],{"className":4656},[437],[425,4658],{"className":4659,"style":562},[441],[425,4661,908],{"className":4662},[446,447],[425,4664],{"className":4665,"style":1097},[742],[425,4667,1137],{"className":4668},[1101],[425,4670],{"className":4671,"style":1097},[742],[425,4673,4675,4678,4681,4721],{"className":4674},[437],[425,4676],{"className":4677,"style":3995},[441],[425,4679,1150],{"className":4680},[761],[425,4682,4684,4687],{"className":4683},[446],[425,4685,2991],{"className":4686},[446,447],[425,4688,4690],{"className":4689},[2034],[425,4691,4693,4713],{"className":4692},[2038,2039],[425,4694,4696,4710],{"className":4695},[2043],[425,4697,4699],{"className":4698,"style":3324},[2047],[425,4700,4701,4704],{"style":3006},[425,4702],{"className":4703,"style":2056},[2055],[425,4705,4707],{"className":4706},[2060,2061,2062,2063],[425,4708,3915],{"className":4709,"style":3914},[446,447,2063],[425,4711,2080],{"className":4712},[2079],[425,4714,4716],{"className":4715},[2043],[425,4717,4719],{"className":4718,"style":3925},[2047],[425,4720],{},[425,4722,1157],{"className":4723},[786]," is independent with ",[425,4726,4728],{"className":4727},[428],[425,4729,4731,4795],{"className":4730,"ariaHidden":433},[432],[425,4732,4734,4737,4740,4743,4783,4786,4789,4792],{"className":4733},[437],[425,4735],{"className":4736,"style":3995},[441],[425,4738,1914],{"className":4739,"style":1913},[446,447],[425,4741,762],{"className":4742},[761],[425,4744,4746,4749],{"className":4745},[446],[425,4747,2991],{"className":4748},[446,447],[425,4750,4752],{"className":4751},[2034],[425,4753,4755,4775],{"className":4754},[2038,2039],[425,4756,4758,4772],{"className":4757},[2043],[425,4759,4761],{"className":4760,"style":3324},[2047],[425,4762,4763,4766],{"style":3006},[425,4764],{"className":4765,"style":2056},[2055],[425,4767,4769],{"className":4768},[2060,2061,2062,2063],[425,4770,3915],{"className":4771,"style":3914},[446,447,2063],[425,4773,2080],{"className":4774},[2079],[425,4776,4778],{"className":4777},[2043],[425,4779,4781],{"className":4780,"style":3925},[2047],[425,4782],{},[425,4784,787],{"className":4785},[786],[425,4787],{"className":4788,"style":743},[742],[425,4790,1930],{"className":4791},[650],[425,4793],{"className":4794,"style":743},[742],[425,4796,4798,4801,4804,4807,4847],{"className":4797},[437],[425,4799],{"className":4800,"style":757},[441],[425,4802,1914],{"className":4803,"style":1913},[446,447],[425,4805,762],{"className":4806},[761],[425,4808,4810,4813],{"className":4809},[446],[425,4811,1447],{"className":4812,"style":2399},[446,447],[425,4814,4816],{"className":4815},[2034],[425,4817,4819,4839],{"className":4818},[2038,2039],[425,4820,4822,4836],{"className":4821},[2043],[425,4823,4825],{"className":4824,"style":3181},[2047],[425,4826,4827,4830],{"style":2415},[425,4828],{"className":4829,"style":2056},[2055],[425,4831,4833],{"className":4832},[2060,2061,2062,2063],[425,4834,449],{"className":4835,"style":448},[446,447,2063],[425,4837,2080],{"className":4838},[2079],[425,4840,4842],{"className":4841},[2043],[425,4843,4845],{"className":4844,"style":2434},[2047],[425,4846],{},[425,4848,787],{"className":4849},[786],". Greedy scans in\nnonincreasing weight, so it would have reached and accepted ",[425,4852,4854],{"className":4853},[428],[425,4855,4857],{"className":4856,"ariaHidden":433},[432],[425,4858,4860,4864],{"className":4859},[437],[425,4861],{"className":4862,"style":4863},[441],"height:0.7167em;vertical-align:-0.2861em;",[425,4865,4867,4870],{"className":4866},[446],[425,4868,2991],{"className":4869},[446,447],[425,4871,4873],{"className":4872},[2034],[425,4874,4876,4896],{"className":4875},[2038,2039],[425,4877,4879,4893],{"className":4878},[2043],[425,4880,4882],{"className":4881,"style":3324},[2047],[425,4883,4884,4887],{"style":3006},[425,4885],{"className":4886,"style":2056},[2055],[425,4888,4890],{"className":4889},[2060,2061,2062,2063],[425,4891,3915],{"className":4892,"style":3914},[446,447,2063],[425,4894,2080],{"className":4895},[2079],[425,4897,4899],{"className":4898},[2043],[425,4900,4902],{"className":4901,"style":3925},[2047],[425,4903],{},[394,4905,4906],{},"before",[425,4908,4910],{"className":4909},[428],[425,4911,4913],{"className":4912,"ariaHidden":433},[432],[425,4914,4916,4919],{"className":4915},[437],[425,4917],{"className":4918,"style":2392},[441],[425,4920,4922,4925],{"className":4921},[446],[425,4923,1447],{"className":4924,"style":2399},[446,447],[425,4926,4928],{"className":4927},[2034],[425,4929,4931,4951],{"className":4930},[2038,2039],[425,4932,4934,4948],{"className":4933},[2043],[425,4935,4937],{"className":4936,"style":3181},[2047],[425,4938,4939,4942],{"style":2415},[425,4940],{"className":4941,"style":2056},[2055],[425,4943,4945],{"className":4944},[2060,2061,2062,2063],[425,4946,449],{"className":4947,"style":448},[446,447,2063],[425,4949,2080],{"className":4950},[2079],[425,4952,4954],{"className":4953},[2043],[425,4955,4957],{"className":4956,"style":2434},[2047],[425,4958],{},", a contradiction. Hence no such ",[425,4961,4963],{"className":4962},[428],[425,4964,4966],{"className":4965,"ariaHidden":433},[432],[425,4967,4969,4972],{"className":4968},[437],[425,4970],{"className":4971,"style":562},[441],[425,4973,567],{"className":4974,"style":566},[446,447]," exists and ",[425,4977,4979],{"className":4978},[428],[425,4980,4982],{"className":4981,"ariaHidden":433},[432],[425,4983,4985,4988],{"className":4984},[437],[425,4986],{"className":4987,"style":562},[441],[425,4989,584],{"className":4990},[446,447]," is maximum-weight.\n",[425,4993,4995],{"className":4994},[428],[425,4996,4998],{"className":4997,"ariaHidden":433},[432],[425,4999,5001,5005],{"className":5000},[437],[425,5002],{"className":5003,"style":5004},[441],"height:0.675em;",[425,5006,5009],{"className":5007},[2255,5008],"qed",[425,5010,5013],{"className":5011},[446,5012],"amsrm","□",[381,5015,5016,5017,5032,5033,5158,5159,5302,5303,5437,5438,5508,5509,5524,5525,5528,5529,5581,5582,5634],{},"Laid out side by side, greedy's picks and the optimum's, both sorted by weight,\nmake the contradiction visible. They agree on weight up to the first index ",[425,5018,5020],{"className":5019},[428],[425,5021,5023],{"className":5022,"ariaHidden":433},[432],[425,5024,5026,5029],{"className":5025},[437],[425,5027],{"className":5028,"style":442},[441],[425,5030,449],{"className":5031,"style":448},[446,447],"\nwhere greedy falls behind (",[425,5034,5036],{"className":5035},[428],[425,5037,5039,5103],{"className":5038,"ariaHidden":433},[432],[425,5040,5042,5045,5048,5051,5091,5094,5097,5100],{"className":5041},[437],[425,5043],{"className":5044,"style":757},[441],[425,5046,1914],{"className":5047,"style":1913},[446,447],[425,5049,762],{"className":5050},[761],[425,5052,5054,5057],{"className":5053},[446],[425,5055,1447],{"className":5056,"style":2399},[446,447],[425,5058,5060],{"className":5059},[2034],[425,5061,5063,5083],{"className":5062},[2038,2039],[425,5064,5066,5080],{"className":5065},[2043],[425,5067,5069],{"className":5068,"style":3181},[2047],[425,5070,5071,5074],{"style":2415},[425,5072],{"className":5073,"style":2056},[2055],[425,5075,5077],{"className":5076},[2060,2061,2062,2063],[425,5078,449],{"className":5079,"style":448},[446,447,2063],[425,5081,2080],{"className":5082},[2079],[425,5084,5086],{"className":5085},[2043],[425,5087,5089],{"className":5088,"style":2434},[2047],[425,5090],{},[425,5092,787],{"className":5093},[786],[425,5095],{"className":5096,"style":743},[742],[425,5098,1039],{"className":5099},[650],[425,5101],{"className":5102,"style":743},[742],[425,5104,5106,5109,5112,5115,5155],{"className":5105},[437],[425,5107],{"className":5108,"style":757},[441],[425,5110,1914],{"className":5111,"style":1913},[446,447],[425,5113,762],{"className":5114},[761],[425,5116,5118,5121],{"className":5117},[446],[425,5119,2991],{"className":5120},[446,447],[425,5122,5124],{"className":5123},[2034],[425,5125,5127,5147],{"className":5126},[2038,2039],[425,5128,5130,5144],{"className":5129},[2043],[425,5131,5133],{"className":5132,"style":3181},[2047],[425,5134,5135,5138],{"style":3006},[425,5136],{"className":5137,"style":2056},[2055],[425,5139,5141],{"className":5140},[2060,2061,2062,2063],[425,5142,449],{"className":5143,"style":448},[446,447,2063],[425,5145,2080],{"className":5146},[2079],[425,5148,5150],{"className":5149},[2043],[425,5151,5153],{"className":5152,"style":2434},[2047],[425,5154],{},[425,5156,787],{"className":5157},[786],"). The prefix ",[425,5160,5162],{"className":5161},[428],[425,5163,5165,5183],{"className":5164,"ariaHidden":433},[432],[425,5166,5168,5171,5174,5177,5180],{"className":5167},[437],[425,5169],{"className":5170,"style":562},[441],[425,5172,908],{"className":5173},[446,447],[425,5175],{"className":5176,"style":743},[742],[425,5178,747],{"className":5179},[650],[425,5181],{"className":5182,"style":743},[742],[425,5184,5186,5189,5192,5232,5235,5238,5241,5244,5247,5250,5299],{"className":5185},[437],[425,5187],{"className":5188,"style":757},[441],[425,5190,1150],{"className":5191},[761],[425,5193,5195,5198],{"className":5194},[446],[425,5196,1447],{"className":5197,"style":2399},[446,447],[425,5199,5201],{"className":5200},[2034],[425,5202,5204,5224],{"className":5203},[2038,2039],[425,5205,5207,5221],{"className":5206},[2043],[425,5208,5210],{"className":5209,"style":2412},[2047],[425,5211,5212,5215],{"style":2415},[425,5213],{"className":5214,"style":2056},[2055],[425,5216,5218],{"className":5217},[2060,2061,2062,2063],[425,5219,686],{"className":5220},[446,2063],[425,5222,2080],{"className":5223},[2079],[425,5225,5227],{"className":5226},[2043],[425,5228,5230],{"className":5229,"style":2434},[2047],[425,5231],{},[425,5233,772],{"className":5234},[771],[425,5236],{"className":5237,"style":776},[742],[425,5239,2493],{"className":5240},[2492],[425,5242],{"className":5243,"style":776},[742],[425,5245,772],{"className":5246},[771],[425,5248],{"className":5249,"style":776},[742],[425,5251,5253,5256],{"className":5252},[446],[425,5254,1447],{"className":5255,"style":2399},[446,447],[425,5257,5259],{"className":5258},[2034],[425,5260,5262,5291],{"className":5261},[2038,2039],[425,5263,5265,5288],{"className":5264},[2043],[425,5266,5268],{"className":5267,"style":3181},[2047],[425,5269,5270,5273],{"style":2415},[425,5271],{"className":5272,"style":2056},[2055],[425,5274,5276],{"className":5275},[2060,2061,2062,2063],[425,5277,5279,5282,5285],{"className":5278},[446,2063],[425,5280,449],{"className":5281,"style":448},[446,447,2063],[425,5283,3613],{"className":5284},[1101,2063],[425,5286,686],{"className":5287},[446,2063],[425,5289,2080],{"className":5290},[2079],[425,5292,5294],{"className":5293},[2043],[425,5295,5297],{"className":5296,"style":3626},[2047],[425,5298],{},[425,5300,1157],{"className":5301},[786],"\nis smaller than ",[425,5304,5306],{"className":5305},[428],[425,5307,5309,5327],{"className":5308,"ariaHidden":433},[432],[425,5310,5312,5315,5318,5321,5324],{"className":5311},[437],[425,5313],{"className":5314,"style":562},[441],[425,5316,872],{"className":5317,"style":871},[446,447],[425,5319],{"className":5320,"style":743},[742],[425,5322,747],{"className":5323},[650],[425,5325],{"className":5326,"style":743},[742],[425,5328,5330,5333,5336,5376,5379,5382,5385,5388,5391,5394,5434],{"className":5329},[437],[425,5331],{"className":5332,"style":757},[441],[425,5334,1150],{"className":5335},[761],[425,5337,5339,5342],{"className":5338},[446],[425,5340,2991],{"className":5341},[446,447],[425,5343,5345],{"className":5344},[2034],[425,5346,5348,5368],{"className":5347},[2038,2039],[425,5349,5351,5365],{"className":5350},[2043],[425,5352,5354],{"className":5353,"style":2412},[2047],[425,5355,5356,5359],{"style":3006},[425,5357],{"className":5358,"style":2056},[2055],[425,5360,5362],{"className":5361},[2060,2061,2062,2063],[425,5363,686],{"className":5364},[446,2063],[425,5366,2080],{"className":5367},[2079],[425,5369,5371],{"className":5370},[2043],[425,5372,5374],{"className":5373,"style":2434},[2047],[425,5375],{},[425,5377,772],{"className":5378},[771],[425,5380],{"className":5381,"style":776},[742],[425,5383,2493],{"className":5384},[2492],[425,5386],{"className":5387,"style":776},[742],[425,5389,772],{"className":5390},[771],[425,5392],{"className":5393,"style":776},[742],[425,5395,5397,5400],{"className":5396},[446],[425,5398,2991],{"className":5399},[446,447],[425,5401,5403],{"className":5402},[2034],[425,5404,5406,5426],{"className":5405},[2038,2039],[425,5407,5409,5423],{"className":5408},[2043],[425,5410,5412],{"className":5411,"style":3181},[2047],[425,5413,5414,5417],{"style":3006},[425,5415],{"className":5416,"style":2056},[2055],[425,5418,5420],{"className":5419},[2060,2061,2062,2063],[425,5421,449],{"className":5422,"style":448},[446,447,2063],[425,5424,2080],{"className":5425},[2079],[425,5427,5429],{"className":5428},[2043],[425,5430,5432],{"className":5431,"style":2434},[2047],[425,5433],{},[425,5435,1157],{"className":5436},[786],", so the exchange property hands us an element\n",[425,5439,5441],{"className":5440},[428],[425,5442,5444,5499],{"className":5443,"ariaHidden":433},[432],[425,5445,5447,5450,5490,5493,5496],{"className":5446},[437],[425,5448],{"className":5449,"style":3884},[441],[425,5451,5453,5456],{"className":5452},[446],[425,5454,2991],{"className":5455},[446,447],[425,5457,5459],{"className":5458},[2034],[425,5460,5462,5482],{"className":5461},[2038,2039],[425,5463,5465,5479],{"className":5464},[2043],[425,5466,5468],{"className":5467,"style":3324},[2047],[425,5469,5470,5473],{"style":3006},[425,5471],{"className":5472,"style":2056},[2055],[425,5474,5476],{"className":5475},[2060,2061,2062,2063],[425,5477,3915],{"className":5478,"style":3914},[446,447,2063],[425,5480,2080],{"className":5481},[2079],[425,5483,5485],{"className":5484},[2043],[425,5486,5488],{"className":5487,"style":3925},[2047],[425,5489],{},[425,5491],{"className":5492,"style":743},[742],[425,5494,879],{"className":5495},[650],[425,5497],{"className":5498,"style":743},[742],[425,5500,5502,5505],{"className":5501},[437],[425,5503],{"className":5504,"style":562},[441],[425,5506,872],{"className":5507,"style":871},[446,447]," that keeps ",[425,5510,5512],{"className":5511},[428],[425,5513,5515],{"className":5514,"ariaHidden":433},[432],[425,5516,5518,5521],{"className":5517},[437],[425,5519],{"className":5520,"style":562},[441],[425,5522,908],{"className":5523},[446,447]," independent and is ",[385,5526,5527],{},"heavier"," than ",[425,5530,5532],{"className":5531},[428],[425,5533,5535],{"className":5534,"ariaHidden":433},[432],[425,5536,5538,5541],{"className":5537},[437],[425,5539],{"className":5540,"style":2392},[441],[425,5542,5544,5547],{"className":5543},[446],[425,5545,1447],{"className":5546,"style":2399},[446,447],[425,5548,5550],{"className":5549},[2034],[425,5551,5553,5573],{"className":5552},[2038,2039],[425,5554,5556,5570],{"className":5555},[2043],[425,5557,5559],{"className":5558,"style":3181},[2047],[425,5560,5561,5564],{"style":2415},[425,5562],{"className":5563,"style":2056},[2055],[425,5565,5567],{"className":5566},[2060,2061,2062,2063],[425,5568,449],{"className":5569,"style":448},[446,447,2063],[425,5571,2080],{"className":5572},[2079],[425,5574,5576],{"className":5575},[2043],[425,5577,5579],{"className":5578,"style":2434},[2047],[425,5580],{}," — which greedy,\nscanning in weight order, must have reached and accepted before ",[425,5583,5585],{"className":5584},[428],[425,5586,5588],{"className":5587,"ariaHidden":433},[432],[425,5589,5591,5594],{"className":5590},[437],[425,5592],{"className":5593,"style":2392},[441],[425,5595,5597,5600],{"className":5596},[446],[425,5598,1447],{"className":5599,"style":2399},[446,447],[425,5601,5603],{"className":5602},[2034],[425,5604,5606,5626],{"className":5605},[2038,2039],[425,5607,5609,5623],{"className":5608},[2043],[425,5610,5612],{"className":5611,"style":3181},[2047],[425,5613,5614,5617],{"style":2415},[425,5615],{"className":5616,"style":2056},[2055],[425,5618,5620],{"className":5619},[2060,2061,2062,2063],[425,5621,449],{"className":5622,"style":448},[446,447,2063],[425,5624,2080],{"className":5625},[2079],[425,5627,5629],{"className":5628},[2043],[425,5630,5632],{"className":5631,"style":2434},[2047],[425,5633],{},". That is the\ncontradiction.",[1434,5636,5638,6216],{"className":5637},[1437,1438],[1440,5639,5643],{"xmlns":1442,"width":5640,"height":5641,"viewBox":5642},"460.752","186.304","-75 -75 345.564 139.728",[1447,5644,5645,5656,5665,5674,5698,5713,5727,5739,5752,5764,5785,5796,5807,5819,5831,5840,6062,6065,6118,6121],{"stroke":1449,"style":1450},[1447,5646,5648],{"fill":5647,"stroke":5647},"var(--tk-line)",[1447,5649,5651],{"transform":5650},"translate(-1.701 -76.634)",[1452,5652],{"d":5653,"fill":5647,"stroke":5647,"className":5654,"style":5655},"M8.310 18.779L6.269 18.779L6.269 18.540Q7.050 18.540 7.050 18.420L7.050 15.874Q6.872 15.949 6.668 15.979Q6.464 16.008 6.235 16.008L6.235 15.769Q6.571 15.769 6.855 15.700Q7.138 15.632 7.348 15.449L7.453 15.449Q7.480 15.449 7.504 15.473Q7.529 15.498 7.529 15.525L7.529 18.420Q7.529 18.540 8.310 18.540",[1463],"stroke-width:0.150",[1447,5657,5658],{"fill":5647,"stroke":5647},[1447,5659,5661],{"transform":5660},"translate(32.442 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20.779",[1463],[1680,6217,6219,6220,6272,6273,6326,6327,788,6342,6467,6468,808,6599,6721],{"className":6218},[1683],"The Rado–Edmonds exchange step. Greedy's picks ",[425,6221,6223],{"className":6222},[428],[425,6224,6226],{"className":6225,"ariaHidden":433},[432],[425,6227,6229,6232],{"className":6228},[437],[425,6230],{"className":6231,"style":2392},[441],[425,6233,6235,6238],{"className":6234},[446],[425,6236,1447],{"className":6237,"style":2399},[446,447],[425,6239,6241],{"className":6240},[2034],[425,6242,6244,6264],{"className":6243},[2038,2039],[425,6245,6247,6261],{"className":6246},[2043],[425,6248,6250],{"className":6249,"style":3324},[2047],[425,6251,6252,6255],{"style":2415},[425,6253],{"className":6254,"style":2056},[2055],[425,6256,6258],{"className":6257},[2060,2061,2062,2063],[425,6259,3336],{"className":6260},[446,447,2063],[425,6262,2080],{"className":6263},[2079],[425,6265,6267],{"className":6266},[2043],[425,6268,6270],{"className":6269,"style":2434},[2047],[425,6271],{}," and an allegedly better base's picks ",[425,6274,6276],{"className":6275},[428],[425,6277,6279],{"className":6278,"ariaHidden":433},[432],[425,6280,6282,6286],{"className":6281},[437],[425,6283],{"className":6284,"style":6285},[441],"height:0.5806em;vertical-align:-0.15em;",[425,6287,6289,6292],{"className":6288},[446],[425,6290,2991],{"className":6291},[446,447],[425,6293,6295],{"className":6294},[2034],[425,6296,6298,6318],{"className":6297},[2038,2039],[425,6299,6301,6315],{"className":6300},[2043],[425,6302,6304],{"className":6303,"style":3324},[2047],[425,6305,6306,6309],{"style":3006},[425,6307],{"className":6308,"style":2056},[2055],[425,6310,6312],{"className":6311},[2060,2061,2062,2063],[425,6313,3336],{"className":6314},[446,447,2063],[425,6316,2080],{"className":6317},[2079],[425,6319,6321],{"className":6320},[2043],[425,6322,6324],{"className":6323,"style":2434},[2047],[425,6325],{},", both in nonincreasing weight, agree until index ",[425,6328,6330],{"className":6329},[428],[425,6331,6333],{"className":6332,"ariaHidden":433},[432],[425,6334,6336,6339],{"className":6335},[437],[425,6337],{"className":6338,"style":442},[441],[425,6340,449],{"className":6341,"style":448},[446,447],[425,6343,6345],{"className":6344},[428],[425,6346,6348,6412],{"className":6347,"ariaHidden":433},[432],[425,6349,6351,6354,6357,6360,6400,6403,6406,6409],{"className":6350},[437],[425,6352],{"className":6353,"style":757},[441],[425,6355,1914],{"className":6356,"style":1913},[446,447],[425,6358,762],{"className":6359},[761],[425,6361,6363,6366],{"className":6362},[446],[425,6364,1447],{"className":6365,"style":2399},[446,447],[425,6367,6369],{"className":6368},[2034],[425,6370,6372,6392],{"className":6371},[2038,2039],[425,6373,6375,6389],{"className":6374},[2043],[425,6376,6378],{"className":6377,"style":3181},[2047],[425,6379,6380,6383],{"style":2415},[425,6381],{"className":6382,"style":2056},[2055],[425,6384,6386],{"className":6385},[2060,2061,2062,2063],[425,6387,449],{"className":6388,"style":448},[446,447,2063],[425,6390,2080],{"className":6391},[2079],[425,6393,6395],{"className":6394},[2043],[425,6396,6398],{"className":6397,"style":2434},[2047],[425,6399],{},[425,6401,787],{"className":6402},[786],[425,6404],{"className":6405,"style":743},[742],[425,6407,1039],{"className":6408},[650],[425,6410],{"className":6411,"style":743},[742],[425,6413,6415,6418,6421,6424,6464],{"className":6414},[437],[425,6416],{"className":6417,"style":757},[441],[425,6419,1914],{"className":6420,"style":1913},[446,447],[425,6422,762],{"className":6423},[761],[425,6425,6427,6430],{"className":6426},[446],[425,6428,2991],{"className":6429},[446,447],[425,6431,6433],{"className":6432},[2034],[425,6434,6436,6456],{"className":6435},[2038,2039],[425,6437,6439,6453],{"className":6438},[2043],[425,6440,6442],{"className":6441,"style":3181},[2047],[425,6443,6444,6447],{"style":3006},[425,6445],{"className":6446,"style":2056},[2055],[425,6448,6450],{"className":6449},[2060,2061,2062,2063],[425,6451,449],{"className":6452,"style":448},[446,447,2063],[425,6454,2080],{"className":6455},[2079],[425,6457,6459],{"className":6458},[2043],[425,6460,6462],{"className":6461,"style":2434},[2047],[425,6463],{},[425,6465,787],{"className":6466},[786]," (red). The exchange property on ",[425,6469,6471],{"className":6470},[428],[425,6472,6474,6492],{"className":6473,"ariaHidden":433},[432],[425,6475,6477,6480,6483,6486,6489],{"className":6476},[437],[425,6478],{"className":6479,"style":562},[441],[425,6481,908],{"className":6482},[446,447],[425,6484],{"className":6485,"style":743},[742],[425,6487,747],{"className":6488},[650],[425,6490],{"className":6491,"style":743},[742],[425,6493,6495,6498,6501,6541,6547,6596],{"className":6494},[437],[425,6496],{"className":6497,"style":757},[441],[425,6499,1150],{"className":6500},[761],[425,6502,6504,6507],{"className":6503},[446],[425,6505,1447],{"className":6506,"style":2399},[446,447],[425,6508,6510],{"className":6509},[2034],[425,6511,6513,6533],{"className":6512},[2038,2039],[425,6514,6516,6530],{"className":6515},[2043],[425,6517,6519],{"className":6518,"style":2412},[2047],[425,6520,6521,6524],{"style":2415},[425,6522],{"className":6523,"style":2056},[2055],[425,6525,6527],{"className":6526},[2060,2061,2062,2063],[425,6528,686],{"className":6529},[446,2063],[425,6531,2080],{"className":6532},[2079],[425,6534,6536],{"className":6535},[2043],[425,6537,6539],{"className":6538,"style":2434},[2047],[425,6540],{},[425,6542,6544],{"className":6543},[446],[425,6545,2493],{"className":6546},[2492],[425,6548,6550,6553],{"className":6549},[446],[425,6551,1447],{"className":6552,"style":2399},[446,447],[425,6554,6556],{"className":6555},[2034],[425,6557,6559,6588],{"className":6558},[2038,2039],[425,6560,6562,6585],{"className":6561},[2043],[425,6563,6565],{"className":6564,"style":3181},[2047],[425,6566,6567,6570],{"style":2415},[425,6568],{"className":6569,"style":2056},[2055],[425,6571,6573],{"className":6572},[2060,2061,2062,2063],[425,6574,6576,6579,6582],{"className":6575},[446,2063],[425,6577,449],{"className":6578,"style":448},[446,447,2063],[425,6580,3613],{"className":6581},[1101,2063],[425,6583,686],{"className":6584},[446,2063],[425,6586,2080],{"className":6587},[2079],[425,6589,6591],{"className":6590},[2043],[425,6592,6594],{"className":6593,"style":3626},[2047],[425,6595],{},[425,6597,1157],{"className":6598},[786],[425,6600,6602],{"className":6601},[428],[425,6603,6605,6623],{"className":6604,"ariaHidden":433},[432],[425,6606,6608,6611,6614,6617,6620],{"className":6607},[437],[425,6609],{"className":6610,"style":562},[441],[425,6612,872],{"className":6613,"style":871},[446,447],[425,6615],{"className":6616,"style":743},[742],[425,6618,747],{"className":6619},[650],[425,6621],{"className":6622,"style":743},[742],[425,6624,6626,6629,6632,6672,6678,6718],{"className":6625},[437],[425,6627],{"className":6628,"style":757},[441],[425,6630,1150],{"className":6631},[761],[425,6633,6635,6638],{"className":6634},[446],[425,6636,2991],{"className":6637},[446,447],[425,6639,6641],{"className":6640},[2034],[425,6642,6644,6664],{"className":6643},[2038,2039],[425,6645,6647,6661],{"className":6646},[2043],[425,6648,6650],{"className":6649,"style":2412},[2047],[425,6651,6652,6655],{"style":3006},[425,6653],{"className":6654,"style":2056},[2055],[425,6656,6658],{"className":6657},[2060,2061,2062,2063],[425,6659,686],{"className":6660},[446,2063],[425,6662,2080],{"className":6663},[2079],[425,6665,6667],{"className":6666},[2043],[425,6668,6670],{"className":6669,"style":2434},[2047],[425,6671],{},[425,6673,6675],{"className":6674},[446],[425,6676,2493],{"className":6677},[2492],[425,6679,6681,6684],{"className":6680},[446],[425,6682,2991],{"className":6683},[446,447],[425,6685,6687],{"className":6686},[2034],[425,6688,6690,6710],{"className":6689},[2038,2039],[425,6691,6693,6707],{"className":6692},[2043],[425,6694,6696],{"className":6695,"style":3181},[2047],[425,6697,6698,6701],{"style":3006},[425,6699],{"className":6700,"style":2056},[2055],[425,6702,6704],{"className":6703},[2060,2061,2062,2063],[425,6705,449],{"className":6706,"style":448},[446,447,2063],[425,6708,2080],{"className":6709},[2079],[425,6711,6713],{"className":6712},[2043],[425,6714,6716],{"className":6715,"style":2434},[2047],[425,6717],{},[425,6719,1157],{"className":6720},[786]," yields a heavier independent extension greedy would have taken first.",[381,6723,6724,6725,6780,6781,6784,6785,6809,6810,6825,6826,1176],{},"The single step that does all the work is ",[399,6726,6727,6728,1176],{},"the exchange property gives an element ",[425,6729,6731],{"className":6730},[428],[425,6732,6734],{"className":6733,"ariaHidden":433},[432],[425,6735,6737,6740],{"className":6736},[437],[425,6738],{"className":6739,"style":4863},[441],[425,6741,6743,6746],{"className":6742},[446],[425,6744,2991],{"className":6745},[446,447],[425,6747,6749],{"className":6748},[2034],[425,6750,6752,6772],{"className":6751},[2038,2039],[425,6753,6755,6769],{"className":6754},[2043],[425,6756,6758],{"className":6757,"style":3324},[2047],[425,6759,6760,6763],{"style":3006},[425,6761],{"className":6762,"style":2056},[2055],[425,6764,6766],{"className":6765},[2060,2061,2062,2063],[425,6767,3915],{"className":6768,"style":3914},[446,447,2063],[425,6770,2080],{"className":6771},[2079],[425,6773,6775],{"className":6774},[2043],[425,6776,6778],{"className":6777,"style":3925},[2047],[425,6779],{}," That is the matroid axiom standing in for the ad-hoc swap we constructed\nby hand for activity selection. The converse, that a non-matroid hereditary\nstructure has a weight function defeating greedy, is what makes the matroid the\n",[385,6782,6783],{},"exact"," characterization, not merely a sufficient condition: if the exchange\nproperty fails for some ",[425,6786,6788],{"className":6787},[428],[425,6789,6791],{"className":6790,"ariaHidden":433},[432],[425,6792,6794,6797,6800,6803,6806],{"className":6793},[437],[425,6795],{"className":6796,"style":980},[441],[425,6798,908],{"className":6799},[446,447],[425,6801,772],{"className":6802},[771],[425,6804],{"className":6805,"style":776},[742],[425,6807,872],{"className":6808,"style":871},[446,447],", one can place weights that lure greedy into the\nstranded maximal set ",[425,6811,6813],{"className":6812},[428],[425,6814,6816],{"className":6815,"ariaHidden":433},[432],[425,6817,6819,6822],{"className":6818},[437],[425,6820],{"className":6821,"style":562},[441],[425,6823,908],{"className":6824},[446,447]," and away from the heavier ",[425,6827,6829],{"className":6828},[428],[425,6830,6832],{"className":6831,"ariaHidden":433},[432],[425,6833,6835,6838],{"className":6834},[437],[425,6836],{"className":6837,"style":562},[441],[425,6839,872],{"className":6840,"style":871},[446,447],[407,6842,6844],{"id":6843},"examples-that-are-matroids","Examples that are matroids",[381,6846,6847,2379,6850,6900,6901,6934,6935,6950,6951,6954,6955,6970,6971,6995,6996,2239,7041,7056,7057,7072,7073,7088,7089,7092,7093,1176],{},[394,6848,6849],{},"The graphic matroid.",[425,6851,6853],{"className":6852},[428],[425,6854,6856,6874],{"className":6855,"ariaHidden":433},[432],[425,6857,6859,6862,6865,6868,6871],{"className":6858},[437],[425,6860],{"className":6861,"style":562},[441],[425,6863,584],{"className":6864},[446,447],[425,6866],{"className":6867,"style":743},[742],[425,6869,747],{"className":6870},[650],[425,6872],{"className":6873,"style":743},[742],[425,6875,6877,6880,6883,6887,6890,6893,6897],{"className":6876},[437],[425,6878],{"className":6879,"style":757},[441],[425,6881,762],{"className":6882},[761],[425,6884,6886],{"className":6885,"style":1097},[446,447],"V",[425,6888,772],{"className":6889},[771],[425,6891],{"className":6892,"style":776},[742],[425,6894,6896],{"className":6895,"style":766},[446,447],"E",[425,6898,787],{"className":6899},[786]," be a graph. Take ",[425,6902,6904],{"className":6903},[428],[425,6905,6907,6925],{"className":6906,"ariaHidden":433},[432],[425,6908,6910,6913,6916,6919,6922],{"className":6909},[437],[425,6911],{"className":6912,"style":562},[441],[425,6914,767],{"className":6915,"style":766},[446,447],[425,6917],{"className":6918,"style":743},[742],[425,6920,747],{"className":6921},[650],[425,6923],{"className":6924,"style":743},[742],[425,6926,6928,6931],{"className":6927},[437],[425,6929],{"className":6930,"style":562},[441],[425,6932,6896],{"className":6933,"style":766},[446,447]," and let\n",[425,6936,6938],{"className":6937},[428],[425,6939,6941],{"className":6940,"ariaHidden":433},[432],[425,6942,6944,6947],{"className":6943},[437],[425,6945],{"className":6946,"style":562},[441],[425,6948,782],{"className":6949,"style":781},[446,780]," be the ",[394,6952,6953],{},"acyclic"," edge sets, the forests of ",[425,6956,6958],{"className":6957},[428],[425,6959,6961],{"className":6960,"ariaHidden":433},[432],[425,6962,6964,6967],{"className":6963},[437],[425,6965],{"className":6966,"style":562},[441],[425,6968,584],{"className":6969},[446,447],". This is a\nmatroid: a subset of a forest is a forest (hereditary), and if forests ",[425,6972,6974],{"className":6973},[428],[425,6975,6977],{"className":6976,"ariaHidden":433},[432],[425,6978,6980,6983,6986,6989,6992],{"className":6979},[437],[425,6981],{"className":6982,"style":980},[441],[425,6984,908],{"className":6985},[446,447],[425,6987,772],{"className":6988},[771],[425,6990],{"className":6991,"style":776},[742],[425,6993,872],{"className":6994,"style":871},[446,447],"\nhave ",[425,6997,6999],{"className":6998},[428],[425,7000,7002,7026],{"className":7001,"ariaHidden":433},[432],[425,7003,7005,7008,7011,7014,7017,7020,7023],{"className":7004},[437],[425,7006],{"className":7007,"style":757},[441],[425,7009,1026],{"className":7010},[446],[425,7012,908],{"className":7013},[446,447],[425,7015,1026],{"className":7016},[446],[425,7018],{"className":7019,"style":743},[742],[425,7021,1039],{"className":7022},[650],[425,7024],{"className":7025,"style":743},[742],[425,7027,7029,7032,7035,7038],{"className":7028},[437],[425,7030],{"className":7031,"style":757},[441],[425,7033,1026],{"className":7034},[446],[425,7036,872],{"className":7037,"style":871},[446,447],[425,7039,1026],{"className":7040},[446],[425,7042,7044],{"className":7043},[428],[425,7045,7047],{"className":7046,"ariaHidden":433},[432],[425,7048,7050,7053],{"className":7049},[437],[425,7051],{"className":7052,"style":562},[441],[425,7054,872],{"className":7055,"style":871},[446,447]," touches more components, so some edge of ",[425,7058,7060],{"className":7059},[428],[425,7061,7063],{"className":7062,"ariaHidden":433},[432],[425,7064,7066,7069],{"className":7065},[437],[425,7067],{"className":7068,"style":562},[441],[425,7070,872],{"className":7071,"style":871},[446,447]," joins two\ntrees of ",[425,7074,7076],{"className":7075},[428],[425,7077,7079],{"className":7078,"ariaHidden":433},[432],[425,7080,7082,7085],{"className":7081},[437],[425,7083],{"className":7084,"style":562},[441],[425,7086,908],{"className":7087},[446,447]," without creating a cycle (exchange property). The bases are the\n",[394,7090,7091],{},"spanning forests","; for a connected graph, the spanning trees, all of size\n",[425,7094,7096],{"className":7095},[428],[425,7097,7099,7123,7141],{"className":7098,"ariaHidden":433},[432],[425,7100,7102,7105,7108,7111,7114,7117,7120],{"className":7101},[437],[425,7103],{"className":7104,"style":757},[441],[425,7106,1026],{"className":7107},[446],[425,7109,6886],{"className":7110,"style":1097},[446,447],[425,7112,1026],{"className":7113},[446],[425,7115],{"className":7116,"style":1097},[742],[425,7118,3613],{"className":7119},[1101],[425,7121],{"className":7122,"style":1097},[742],[425,7124,7126,7129,7132,7135,7138],{"className":7125},[437],[425,7127],{"className":7128,"style":1940},[441],[425,7130,686],{"className":7131},[446],[425,7133],{"className":7134,"style":743},[742],[425,7136,747],{"className":7137},[650],[425,7139],{"className":7140,"style":743},[742],[425,7142,7144,7147,7150,7153,7156],{"className":7143},[437],[425,7145],{"className":7146,"style":757},[441],[425,7148,1249],{"className":7149,"style":566},[446,447],[425,7151,762],{"className":7152},[761],[425,7154,738],{"className":7155,"style":737},[446,447],[425,7157,787],{"className":7158},[786],[381,7160,7161,7162,7183,7184,7187,7188,7191,7192,7235,7236,7239,7240,7243],{},"Now run ",[425,7163,7165],{"className":7164},[428],[425,7166,7168],{"className":7167,"ariaHidden":433},[432],[425,7169,7171,7174],{"className":7170},[437],[425,7172],{"className":7173,"style":2346},[441],[425,7175,7177],{"className":7176},[2255,2256],[425,7178,7180],{"className":7179},[446,2260],[425,7181,218],{"className":7182},[446]," on the graphic matroid with ",[394,7185,7186],{},"negated"," edge weights\n(maximum-weight independent set becomes minimum-weight spanning tree): sort edges,\nadd each edge that does not form a cycle. That is ",[394,7189,7190],{},"Kruskal's algorithm",",\nexactly: the independence test ",[399,7193,7194,7234],{},[425,7195,7197],{"className":7196},[428],[425,7198,7200,7218],{"className":7199,"ariaHidden":433},[432],[425,7201,7203,7206,7209,7212,7215],{"className":7202},[437],[425,7204],{"className":7205,"style":562},[441],[425,7207,908],{"className":7208},[446,447],[425,7210],{"className":7211,"style":1097},[742],[425,7213,1137],{"className":7214},[1101],[425,7216],{"className":7217,"style":1097},[742],[425,7219,7221,7224,7227,7231],{"className":7220},[437],[425,7222],{"className":7223,"style":757},[441],[425,7225,1150],{"className":7226},[761],[425,7228,7230],{"className":7229},[446,447],"e",[425,7232,1157],{"className":7233},[786]," stays acyclic"," is the\n",[680,7237,7238],{"href":107},"union-find","\ncycle check from the ",[680,7241,7242],{"href":170},"minimum spanning trees","\nlesson. So Kruskal's correctness is not a special miracle; it is the\nRado–Edmonds theorem instantiated on the graphic matroid.",[1434,7245,7247,7353],{"className":7246},[1437,1438],[1440,7248,7252],{"xmlns":1442,"width":7249,"height":7250,"viewBox":7251},"185.412","184.344","-75 -75 139.059 138.258",[1447,7253,7254,7257,7264,7267,7274,7277,7284,7287,7294,7306,7318,7330],{"stroke":1449,"style":1450},[1452,7255],{"fill":1454,"d":7256},"M-35.128-51.753c0-5.5-4.458-9.959-9.958-9.959s-9.959 4.459-9.959 9.959 4.459 9.958 9.959 9.958 9.958-4.458 9.958-9.958Zm-9.958 0",[1447,7258,7260],{"transform":7259},"translate(-2.322 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edge sets form a matroid — Kruskal is matroid-greedy, taking edges by weight and rejecting any that close a cycle",[381,7358,7359,7360,7395,7396,7412,7413,808,7428,7444,7445,7478,7479,7494],{},"Edges ",[425,7361,7363],{"className":7362},[428],[425,7364,7366],{"className":7365,"ariaHidden":433},[432],[425,7367,7369,7373,7376,7379,7382,7385,7388,7391],{"className":7368},[437],[425,7370],{"className":7371,"style":7372},[441],"height:0.8389em;vertical-align:-0.1944em;",[425,7374,686],{"className":7375},[446],[425,7377,772],{"className":7378},[771],[425,7380],{"className":7381,"style":776},[742],[425,7383,2368],{"className":7384},[446],[425,7386,772],{"className":7387},[771],[425,7389],{"className":7390,"style":776},[742],[425,7392,7394],{"className":7393},[446],"3"," are accepted in weight order (blue); edge ",[425,7397,7399],{"className":7398},[428],[425,7400,7402],{"className":7401,"ariaHidden":433},[432],[425,7403,7405,7408],{"className":7404},[437],[425,7406],{"className":7407,"style":1940},[441],[425,7409,7411],{"className":7410},[446],"4"," is rejected because\n",[425,7414,7416],{"className":7415},[428],[425,7417,7419],{"className":7418,"ariaHidden":433},[432],[425,7420,7422,7425],{"className":7421},[437],[425,7423],{"className":7424,"style":1866},[441],[425,7426,1249],{"className":7427,"style":566},[446,447],[425,7429,7431],{"className":7430},[428],[425,7432,7434],{"className":7433,"ariaHidden":433},[432],[425,7435,7437,7440],{"className":7436},[437],[425,7438],{"className":7439,"style":1866},[441],[425,7441,7443],{"className":7442},[446,447],"s"," are already connected, so adding it would close the cycle ",[425,7446,7448],{"className":7447},[428],[425,7449,7451],{"className":7450,"ariaHidden":433},[432],[425,7452,7454,7457,7460,7463,7466,7469,7472,7475],{"className":7453},[437],[425,7455],{"className":7456,"style":2392},[441],[425,7458,381],{"className":7459},[446,447],[425,7461],{"className":7462,"style":776},[742],[425,7464,399],{"className":7465,"style":2399},[446,447],[425,7467],{"className":7468,"style":776},[742],[425,7470,7443],{"className":7471},[446,447],[425,7473],{"className":7474,"style":776},[742],[425,7476,1249],{"className":7477,"style":566},[446,447],",\nleaving ",[425,7480,7482],{"className":7481},[428],[425,7483,7485],{"className":7484,"ariaHidden":433},[432],[425,7486,7488,7491],{"className":7487},[437],[425,7489],{"className":7490,"style":562},[441],[425,7492,782],{"className":7493,"style":781},[446,780],". The accepted set is a base: a spanning tree.",[381,7496,7497,7500,7501,6934,7516,7617,7618,7633,7634,1233,7637,7702,7703,7722],{},[394,7498,7499],{},"The uniform \u002F partition matroid."," Even simpler: fix ",[425,7502,7504],{"className":7503},[428],[425,7505,7507],{"className":7506,"ariaHidden":433},[432],[425,7508,7510,7513],{"className":7509},[437],[425,7511],{"className":7512,"style":442},[441],[425,7514,449],{"className":7515,"style":448},[446,447],[425,7517,7519],{"className":7518},[428],[425,7520,7522,7540,7561,7580,7605],{"className":7521,"ariaHidden":433},[432],[425,7523,7525,7528,7531,7534,7537],{"className":7524},[437],[425,7526],{"className":7527,"style":562},[441],[425,7529,782],{"className":7530,"style":781},[446,780],[425,7532],{"className":7533,"style":743},[742],[425,7535,747],{"className":7536},[650],[425,7538],{"className":7539,"style":743},[742],[425,7541,7543,7546,7549,7552,7555,7558],{"className":7542},[437],[425,7544],{"className":7545,"style":757},[441],[425,7547,1150],{"className":7548},[761],[425,7550,908],{"className":7551},[446,447],[425,7553],{"className":7554,"style":743},[742],[425,7556,915],{"className":7557},[650],[425,7559],{"className":7560,"style":743},[742],[425,7562,7564,7567,7570,7573,7577],{"className":7563},[437],[425,7565],{"className":7566,"style":562},[441],[425,7568,767],{"className":7569,"style":766},[446,447],[425,7571],{"className":7572,"style":743},[742],[425,7574,7576],{"className":7575},[650],":",[425,7578],{"className":7579,"style":743},[742],[425,7581,7583,7586,7589,7592,7595,7598,7602],{"className":7582},[437],[425,7584],{"className":7585,"style":757},[441],[425,7587,1026],{"className":7588},[446],[425,7590,908],{"className":7591},[446,447],[425,7593,1026],{"className":7594},[446],[425,7596],{"className":7597,"style":743},[742],[425,7599,7601],{"className":7600},[650],"≤",[425,7603],{"className":7604,"style":743},[742],[425,7606,7608,7611,7614],{"className":7607},[437],[425,7609],{"className":7610,"style":757},[441],[425,7612,449],{"className":7613,"style":448},[446,447],[425,7615,1157],{"className":7616},[786],", so every set of size at most ",[425,7619,7621],{"className":7620},[428],[425,7622,7624],{"className":7623,"ariaHidden":433},[432],[425,7625,7627,7630],{"className":7626},[437],[425,7628],{"className":7629,"style":442},[441],[425,7631,449],{"className":7632,"style":448},[446,447]," is\nindependent. This ",[394,7635,7636],{},"uniform matroid",[425,7638,7640],{"className":7639},[428],[425,7641,7643],{"className":7642,"ariaHidden":433},[432],[425,7644,7646,7650],{"className":7645},[437],[425,7647],{"className":7648,"style":7649},[441],"height:0.9694em;vertical-align:-0.2861em;",[425,7651,7653,7657],{"className":7652},[446],[425,7654,7656],{"className":7655,"style":737},[446,447],"U",[425,7658,7660],{"className":7659},[2034],[425,7661,7663,7694],{"className":7662},[2038,2039],[425,7664,7666,7691],{"className":7665},[2043],[425,7667,7669],{"className":7668,"style":3181},[2047],[425,7670,7672,7675],{"style":7671},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[425,7673],{"className":7674,"style":2056},[2055],[425,7676,7678],{"className":7677},[2060,2061,2062,2063],[425,7679,7681,7684,7687],{"className":7680},[446,2063],[425,7682,449],{"className":7683,"style":448},[446,447,2063],[425,7685,772],{"className":7686},[771,2063],[425,7688,7690],{"className":7689},[446,447,2063],"n",[425,7692,2080],{"className":7693},[2079],[425,7695,7697],{"className":7696},[2043],[425,7698,7700],{"className":7699,"style":3925},[2047],[425,7701],{}," is plainly hereditary, and the\nexchange property is trivial (any larger set has a spare element). Greedy reduces\nto ",[399,7704,7705,7706,7721],{},"pick the ",[425,7707,7709],{"className":7708},[428],[425,7710,7712],{"className":7711,"ariaHidden":433},[432],[425,7713,7715,7718],{"className":7714},[437],[425,7716],{"className":7717,"style":442},[441],[425,7719,449],{"className":7720,"style":448},[446,447]," heaviest elements,"," which is obviously optimal; the matroid\nmachinery confirms the trivial.",[1434,7724,7726,7877],{"className":7725},[1437,1438],[1440,7727,7731],{"xmlns":1442,"width":7728,"height":7729,"viewBox":7730},"353.821","162.910","-75 -75 265.366 122.182",[1447,7732,7733,7737,7744,7747,7754,7757,7764,7767,7774,7777,7784,7787,7794,7797,7832,7841],{"stroke":1449,"style":1450},[1452,7734],{"fill":7735,"stroke":1449,"d":7736,"style":5716},"var(--tk-soft-accent)","M-59.968 31.333v-71.132h23.558v71.132Zm23.558-71.132",[1447,7738,7740],{"transform":7739},"translate(9.654 -77.09)",[1452,7741],{"d":7742,"fill":1449,"stroke":1449,"className":7743,"style":1642},"M-58.976 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28.202Q-18.457 28.020-18.774 28.020Q-19.074 28.020-19.305 28.208Q-19.535 28.395-19.658 28.687Q-19.781 28.978-19.781 29.270M-14.750 31.333L-16.731 31.333L-16.731 31.036Q-16.461 31.036-16.293 30.991Q-16.125 30.946-16.125 30.774L-16.125 28.638Q-16.125 28.423-16.188 28.327Q-16.250 28.231-16.367 28.210Q-16.485 28.188-16.731 28.188L-16.731 27.892L-15.563 27.806L-15.563 28.591Q-15.485 28.380-15.332 28.194Q-15.180 28.009-14.981 27.907Q-14.781 27.806-14.555 27.806Q-14.309 27.806-14.117 27.950Q-13.926 28.095-13.926 28.325Q-13.926 28.481-14.031 28.591Q-14.137 28.700-14.293 28.700Q-14.449 28.700-14.559 28.591Q-14.668 28.481-14.668 28.325Q-14.668 28.165-14.563 28.060Q-14.887 28.060-15.102 28.288Q-15.317 28.517-15.412 28.856Q-15.508 29.196-15.508 29.501L-15.508 30.774Q-15.508 30.942-15.281 30.989Q-15.055 31.036-14.750 31.036",[1463],[1680,7878,7880,7881,7943,7944,7983,7984,8017],{"className":7879},[1683],"The uniform matroid ",[425,7882,7884],{"className":7883},[428],[425,7885,7887],{"className":7886,"ariaHidden":433},[432],[425,7888,7890,7893],{"className":7889},[437],[425,7891],{"className":7892,"style":7649},[441],[425,7894,7896,7899],{"className":7895},[446],[425,7897,7656],{"className":7898,"style":737},[446,447],[425,7900,7902],{"className":7901},[2034],[425,7903,7905,7935],{"className":7904},[2038,2039],[425,7906,7908,7932],{"className":7907},[2043],[425,7909,7911],{"className":7910,"style":2412},[2047],[425,7912,7913,7916],{"style":7671},[425,7914],{"className":7915,"style":2056},[2055],[425,7917,7919],{"className":7918},[2060,2061,2062,2063],[425,7920,7922,7925,7928],{"className":7921},[446,2063],[425,7923,7394],{"className":7924},[446,2063],[425,7926,772],{"className":7927},[771,2063],[425,7929,7931],{"className":7930},[446,2063],"6",[425,7933,2080],{"className":7934},[2079],[425,7936,7938],{"className":7937},[2043],[425,7939,7941],{"className":7940,"style":3925},[2047],[425,7942],{},": independent means ",[425,7945,7947],{"className":7946},[428],[425,7948,7950,7974],{"className":7949,"ariaHidden":433},[432],[425,7951,7953,7956,7959,7962,7965,7968,7971],{"className":7952},[437],[425,7954],{"className":7955,"style":757},[441],[425,7957,1026],{"className":7958},[446],[425,7960,908],{"className":7961},[446,447],[425,7963,1026],{"className":7964},[446],[425,7966],{"className":7967,"style":743},[742],[425,7969,7601],{"className":7970},[650],[425,7972],{"className":7973,"style":743},[742],[425,7975,7977,7980],{"className":7976},[437],[425,7978],{"className":7979,"style":1940},[441],[425,7981,7394],{"className":7982},[446],". Greedy sorts by weight descending and takes the first ",[425,7985,7987],{"className":7986},[428],[425,7988,7990,8008],{"className":7989,"ariaHidden":433},[432],[425,7991,7993,7996,7999,8002,8005],{"className":7992},[437],[425,7994],{"className":7995,"style":442},[441],[425,7997,449],{"className":7998,"style":448},[446,447],[425,8000],{"className":8001,"style":743},[742],[425,8003,747],{"className":8004},[650],[425,8006],{"className":8007,"style":743},[742],[425,8009,8011,8014],{"className":8010},[437],[425,8012],{"className":8013,"style":1940},[441],[425,8015,7394],{"className":8016},[446]," (blue), which is trivially the maximum-weight base; the exchange property holds because any larger set always has a spare element.",[381,8019,8020,8021,8024,8025,8040,8041,8095,8096,8111,8112,8115,8116,8135],{},"The ",[394,8022,8023],{},"partition matroid"," generalizes it: partition\n",[425,8026,8028],{"className":8027},[428],[425,8029,8031],{"className":8030,"ariaHidden":433},[432],[425,8032,8034,8037],{"className":8033},[437],[425,8035],{"className":8036,"style":562},[441],[425,8038,767],{"className":8039,"style":766},[446,447]," into groups and allow at most ",[425,8042,8044],{"className":8043},[428],[425,8045,8047],{"className":8046,"ariaHidden":433},[432],[425,8048,8050,8054],{"className":8049},[437],[425,8051],{"className":8052,"style":8053},[441],"height:0.8444em;vertical-align:-0.15em;",[425,8055,8057,8060],{"className":8056},[446],[425,8058,449],{"className":8059,"style":448},[446,447],[425,8061,8063],{"className":8062},[2034],[425,8064,8066,8087],{"className":8065},[2038,2039],[425,8067,8069,8084],{"className":8068},[2043],[425,8070,8072],{"className":8071,"style":3324},[2047],[425,8073,8075,8078],{"style":8074},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[425,8076],{"className":8077,"style":2056},[2055],[425,8079,8081],{"className":8080},[2060,2061,2062,2063],[425,8082,3336],{"className":8083},[446,447,2063],[425,8085,2080],{"className":8086},[2079],[425,8088,8090],{"className":8089},[2043],[425,8091,8093],{"className":8092,"style":2434},[2047],[425,8094],{}," elements from group ",[425,8097,8099],{"className":8098},[428],[425,8100,8102],{"className":8101,"ariaHidden":433},[432],[425,8103,8105,8108],{"className":8104},[437],[425,8106],{"className":8107,"style":3428},[441],[425,8109,3336],{"className":8110},[446,447],"; independence is\n",[399,8113,8114],{},"within every group's cap."," Scheduling and assignment constraints of the form ",[399,8117,8118,8119,8134],{},"no more than ",[425,8120,8122],{"className":8121},[428],[425,8123,8125],{"className":8124,"ariaHidden":433},[432],[425,8126,8128,8131],{"className":8127},[437],[425,8129],{"className":8130,"style":1866},[441],[425,8132,1870],{"className":8133},[446,447]," of this kind"," are partition-matroid constraints, which is why greedy\nsolves so many of them.",[381,8137,8138,8141,8142,8145,8146,8149,8150,8153,8154,8157],{},[394,8139,8140],{},"Beyond greedy: matroid intersection."," A single matroid yields to greedy; the\ncommon independent sets of ",[385,8143,8144],{},"two"," matroids over the same ground set do not, in\ngeneral. Still, ",[394,8147,8148],{},"matroid intersection"," solves the maximum-weight common\nindependent set in polynomial time by augmenting-path methods (bipartite matching\nis a special case). The intersection of ",[385,8151,8152],{},"three"," matroids is already NP-hard. So\nthe matroid is the precise frontier of ",[399,8155,8156],{},"greedy works","; one step beyond it you need\nheavier machinery.",[407,8159,8161,8162,8164],{"id":8160},"why-01-knapsack-and-tsp-are-not-matroids","Why 0\u002F1 knapsack and TSP are ",[385,8163,2332],{}," matroids",[381,8166,8167],{},"The theorem cuts both ways, and it explains the failures we have already seen.",[381,8169,8170,8173,8174,8177,8178,8181,8182,8197,8198,8201,8202,8205,8206,8223,8224,8227,8228,8261,8262,8284,8285,8315,8316,8418,8419,8422,8423,8453,8454,8475,8476,8548],{},[394,8171,8172],{},"0\u002F1 knapsack."," Recall from the ",[680,8175,8176],{"href":211},"greedy method","\nlesson that greedy fails on the 0\u002F1 knapsack, and from the\n",[680,8179,8180],{"href":252},"knapsack DP"," lesson that it\nneeds dynamic programming. The reason, in matroid language: let ",[425,8183,8185],{"className":8184},[428],[425,8186,8188],{"className":8187,"ariaHidden":433},[432],[425,8189,8191,8194],{"className":8190},[437],[425,8192],{"className":8193,"style":562},[441],[425,8195,767],{"className":8196,"style":766},[446,447]," be the items\nand call a set ",[399,8199,8200],{},"independent"," if its total ",[385,8203,8204],{},"weight"," fits in capacity ",[425,8207,8209],{"className":8208},[428],[425,8210,8212],{"className":8211,"ariaHidden":433},[432],[425,8213,8215,8218],{"className":8214},[437],[425,8216],{"className":8217,"style":562},[441],[425,8219,8222],{"className":8220,"style":8221},[446,447],"margin-right:0.1389em;","W",". This is\nhereditary: drop items and it still fits. But the ",[394,8225,8226],{},"exchange property fails",".\nTake capacity ",[425,8229,8231],{"className":8230},[428],[425,8232,8234,8252],{"className":8233,"ariaHidden":433},[432],[425,8235,8237,8240,8243,8246,8249],{"className":8236},[437],[425,8238],{"className":8239,"style":562},[441],[425,8241,8222],{"className":8242,"style":8221},[446,447],[425,8244],{"className":8245,"style":743},[742],[425,8247,747],{"className":8248},[650],[425,8250],{"className":8251,"style":743},[742],[425,8253,8255,8258],{"className":8254},[437],[425,8256],{"className":8257,"style":1940},[441],[425,8259,1634],{"className":8260},[446]," and item sizes giving the independent sets ",[425,8263,8265],{"className":8264},[428],[425,8266,8268],{"className":8267,"ariaHidden":433},[432],[425,8269,8271,8274,8277,8281],{"className":8270},[437],[425,8272],{"className":8273,"style":757},[441],[425,8275,1150],{"className":8276},[761],[425,8278,8280],{"className":8279},[446],"7",[425,8282,1157],{"className":8283},[786]," and\n",[425,8286,8288],{"className":8287},[428],[425,8289,8291],{"className":8290,"ariaHidden":433},[432],[425,8292,8294,8297,8300,8303,8306,8309,8312],{"className":8293},[437],[425,8295],{"className":8296,"style":757},[441],[425,8298,1150],{"className":8299},[761],[425,8301,7411],{"className":8302},[446],[425,8304,772],{"className":8305},[771],[425,8307],{"className":8308,"style":776},[742],[425,8310,7411],{"className":8311},[446],[425,8313,1157],{"className":8314},[786],". Both fit, and ",[425,8317,8319],{"className":8318},[428],[425,8320,8322,8361,8379,8397],{"className":8321,"ariaHidden":433},[432],[425,8323,8325,8328,8331,8334,8337,8340,8343,8346,8349,8352,8355,8358],{"className":8324},[437],[425,8326],{"className":8327,"style":757},[441],[425,8329,1026],{"className":8330},[446],[425,8332,1150],{"className":8333},[761],[425,8335,7411],{"className":8336},[446],[425,8338,772],{"className":8339},[771],[425,8341],{"className":8342,"style":776},[742],[425,8344,7411],{"className":8345},[446],[425,8347,1157],{"className":8348},[786],[425,8350,1026],{"className":8351},[446],[425,8353],{"className":8354,"style":743},[742],[425,8356,747],{"className":8357},[650],[425,8359],{"className":8360,"style":743},[742],[425,8362,8364,8367,8370,8373,8376],{"className":8363},[437],[425,8365],{"className":8366,"style":3822},[441],[425,8368,2368],{"className":8369},[446],[425,8371],{"className":8372,"style":743},[742],[425,8374,1930],{"className":8375},[650],[425,8377],{"className":8378,"style":743},[742],[425,8380,8382,8385,8388,8391,8394],{"className":8381},[437],[425,8383],{"className":8384,"style":1940},[441],[425,8386,686],{"className":8387},[446],[425,8389],{"className":8390,"style":743},[742],[425,8392,747],{"className":8393},[650],[425,8395],{"className":8396,"style":743},[742],[425,8398,8400,8403,8406,8409,8412,8415],{"className":8399},[437],[425,8401],{"className":8402,"style":757},[441],[425,8404,1026],{"className":8405},[446],[425,8407,1150],{"className":8408},[761],[425,8410,8280],{"className":8411},[446],[425,8413,1157],{"className":8414},[786],[425,8416,1026],{"className":8417},[446],", yet ",[385,8420,8421],{},"no"," element of\n",[425,8424,8426],{"className":8425},[428],[425,8427,8429],{"className":8428,"ariaHidden":433},[432],[425,8430,8432,8435,8438,8441,8444,8447,8450],{"className":8431},[437],[425,8433],{"className":8434,"style":757},[441],[425,8436,1150],{"className":8437},[761],[425,8439,7411],{"className":8440},[446],[425,8442,772],{"className":8443},[771],[425,8445],{"className":8446,"style":776},[742],[425,8448,7411],{"className":8449},[446],[425,8451,1157],{"className":8452},[786]," can join ",[425,8455,8457],{"className":8456},[428],[425,8458,8460],{"className":8459,"ariaHidden":433},[432],[425,8461,8463,8466,8469,8472],{"className":8462},[437],[425,8464],{"className":8465,"style":757},[441],[425,8467,1150],{"className":8468},[761],[425,8470,8280],{"className":8471},[446],[425,8473,1157],{"className":8474},[786],", since ",[425,8477,8479],{"className":8478},[428],[425,8480,8482,8502,8520,8539],{"className":8481,"ariaHidden":433},[432],[425,8483,8485,8489,8492,8495,8499],{"className":8484},[437],[425,8486],{"className":8487,"style":8488},[441],"height:0.7278em;vertical-align:-0.0833em;",[425,8490,8280],{"className":8491},[446],[425,8493],{"className":8494,"style":1097},[742],[425,8496,8498],{"className":8497},[1101],"+",[425,8500],{"className":8501,"style":1097},[742],[425,8503,8505,8508,8511,8514,8517],{"className":8504},[437],[425,8506],{"className":8507,"style":1940},[441],[425,8509,7411],{"className":8510},[446],[425,8512],{"className":8513,"style":743},[742],[425,8515,747],{"className":8516},[650],[425,8518],{"className":8519,"style":743},[742],[425,8521,8523,8526,8530,8533,8536],{"className":8522},[437],[425,8524],{"className":8525,"style":3822},[441],[425,8527,8529],{"className":8528},[446],"11",[425,8531],{"className":8532,"style":743},[742],[425,8534,1930],{"className":8535},[650],[425,8537],{"className":8538,"style":743},[742],[425,8540,8542,8545],{"className":8541},[437],[425,8543],{"className":8544,"style":1940},[441],[425,8546,1634],{"className":8547},[446],". 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86.205Q-7.621 86.039-7.890 85.985L-8.511 85.871Q-8.918 85.793-9.218 85.537Q-9.519 85.281-9.519 84.906Q-9.519 84.539-9.318 84.317Q-9.117 84.094-8.793 83.996Q-8.468 83.899-8.129 83.899Q-7.664 83.899-7.367 84.106L-7.144 83.922Q-7.121 83.899-7.089 83.899L-7.039 83.899Q-7.007 83.899-6.980 83.926Q-6.953 83.953-6.953 83.985L-6.953 84.969Q-6.953 85-6.978 85.029Q-7.004 85.059-7.039 85.059L-7.144 85.059Q-7.179 85.059-7.207 85.031Q-7.234 85.004-7.234 84.969Q-7.234 84.570-7.486 84.350Q-7.738 84.129-8.136 84.129Q-8.492 84.129-8.775 84.252Q-9.058 84.375-9.058 84.680Q-9.058 84.899-8.857 85.031Q-8.656 85.164-8.410 85.207L-7.785 85.320Q-7.355 85.410-7.047 85.707Q-6.738 86.004-6.738 86.418Q-6.738 86.988-7.136 87.266Q-7.535 87.543-8.129 87.543Q-8.679 87.543-9.031 87.207L-9.328 87.520Q-9.351 87.543-9.386 87.543L-9.433 87.543Q-9.457 87.543-9.488 87.512Q-9.519 87.481-9.519 87.457",[1463],[1680,8771,8773,8774,8807,8808,8829,8830,8860,8861,402,8876,8897,8898,8931],{"className":8772},[1683],"0\u002F1 knapsack is not a matroid (",[425,8775,8777],{"className":8776},[428],[425,8778,8780,8798],{"className":8779,"ariaHidden":433},[432],[425,8781,8783,8786,8789,8792,8795],{"className":8782},[437],[425,8784],{"className":8785,"style":562},[441],[425,8787,8222],{"className":8788,"style":8221},[446,447],[425,8790],{"className":8791,"style":743},[742],[425,8793,747],{"className":8794},[650],[425,8796],{"className":8797,"style":743},[742],[425,8799,8801,8804],{"className":8800},[437],[425,8802],{"className":8803,"style":1940},[441],[425,8805,1634],{"className":8806},[446],"). The independent set ",[425,8809,8811],{"className":8810},[428],[425,8812,8814],{"className":8813,"ariaHidden":433},[432],[425,8815,8817,8820,8823,8826],{"className":8816},[437],[425,8818],{"className":8819,"style":757},[441],[425,8821,1150],{"className":8822},[761],[425,8824,8280],{"className":8825},[446],[425,8827,1157],{"className":8828},[786]," and the larger ",[425,8831,8833],{"className":8832},[428],[425,8834,8836],{"className":8835,"ariaHidden":433},[432],[425,8837,8839,8842,8845,8848,8851,8854,8857],{"className":8838},[437],[425,8840],{"className":8841,"style":757},[441],[425,8843,1150],{"className":8844},[761],[425,8846,7411],{"className":8847},[446],[425,8849,772],{"className":8850},[771],[425,8852],{"className":8853,"style":776},[742],[425,8855,7411],{"className":8856},[446],[425,8858,1157],{"className":8859},[786]," both fit, yet adding either ",[425,8862,8864],{"className":8863},[428],[425,8865,8867],{"className":8866,"ariaHidden":433},[432],[425,8868,8870,8873],{"className":8869},[437],[425,8871],{"className":8872,"style":1940},[441],[425,8874,7411],{"className":8875},[446],[425,8877,8879],{"className":8878},[428],[425,8880,8882],{"className":8881,"ariaHidden":433},[432],[425,8883,8885,8888,8891,8894],{"className":8884},[437],[425,8886],{"className":8887,"style":757},[441],[425,8889,1150],{"className":8890},[761],[425,8892,8280],{"className":8893},[446],[425,8895,1157],{"className":8896},[786]," gives ",[425,8899,8901],{"className":8900},[428],[425,8902,8904,8922],{"className":8903,"ariaHidden":433},[432],[425,8905,8907,8910,8913,8916,8919],{"className":8906},[437],[425,8908],{"className":8909,"style":3822},[441],[425,8911,8529],{"className":8912},[446],[425,8914],{"className":8915,"style":743},[742],[425,8917,1930],{"className":8918},[650],[425,8920],{"className":8921,"style":743},[742],[425,8923,8925,8928],{"className":8924},[437],[425,8926],{"className":8927,"style":1940},[441],[425,8929,1634],{"className":8930},[446],". No element of the larger set can grow the smaller, so the exchange property fails.",[381,8933,8934,8935,8937],{},"The\nexchange property is violated; the structure is not a matroid; and by Rado–Edmonds\n",[385,8936,1186],{}," weighting must defeat greedy, which is exactly the value\u002Fdensity\ncounterexample from the greedy-method lesson.",[381,8939,8940,8943,8944,8959,8960,8975],{},[394,8941,8942],{},"Traveling salesman."," For the TSP, let ",[425,8945,8947],{"className":8946},[428],[425,8948,8950],{"className":8949,"ariaHidden":433},[432],[425,8951,8953,8956],{"className":8952},[437],[425,8954],{"className":8955,"style":562},[441],[425,8957,782],{"className":8958,"style":781},[446,780]," be the edge sets that\nextend to a Hamiltonian tour (or the partial-tour fragments greedy builds). This\nis not even reliably hereditary, and the exchange property collapses badly: a\ncheap partial path can be a dead end that no edge of a longer, valid fragment can\nrescue without revisiting a vertex or exceeding degree ",[425,8961,8963],{"className":8962},[428],[425,8964,8966],{"className":8965,"ariaHidden":433},[432],[425,8967,8969,8972],{"className":8968},[437],[425,8970],{"className":8971,"style":1940},[441],[425,8973,2368],{"className":8974},[446],". There is no matroid,\nso no weighting guarantee, and indeed nearest-neighbor greedy can be made\narbitrarily bad. TSP's hardness is not an accident of greed; the independence\nstructure simply isn't a matroid (and TSP is NP-hard regardless).",[381,8977,8978],{},"The pattern is uniform. When greedy is provably optimal, you can almost always\nexhibit a matroid behind it. When greedy fails, the exchange property is the axiom\nthat breaks.",[407,8980,8982],{"id":8981},"takeaways","Takeaways",[8984,8985,8986,9013,9106,9120,9158],"ul",{},[849,8987,8988,8989,8992,8993,9008,9009,9012],{},"Every greedy correctness proof is a ",[394,8990,8991],{},"greedy-stays-ahead"," induction (greedy's\n",[425,8994,8996],{"className":8995},[428],[425,8997,8999],{"className":8998,"ariaHidden":433},[432],[425,9000,9002,9005],{"className":9001},[437],[425,9003],{"className":9004,"style":442},[441],[425,9006,449],{"className":9007,"style":448},[446,447],"-th partial solution is never behind any rival's) or an ",[394,9010,9011],{},"exchange argument","\n(transform any optimum into the greedy solution without loss). They are\nequivalent in power; exchange is the one that abstracts into a theory.",[849,9014,702,9015,1233,9017,9065,9066,9069,9070,9072,9073,9075,9076,9079,9080,3868,9104,1176],{},[394,9016,396],{},[425,9018,9020],{"className":9019},[428],[425,9021,9023,9041],{"className":9022,"ariaHidden":433},[432],[425,9024,9026,9029,9032,9035,9038],{"className":9025},[437],[425,9027],{"className":9028,"style":562},[441],[425,9030,738],{"className":9031,"style":737},[446,447],[425,9033],{"className":9034,"style":743},[742],[425,9036,747],{"className":9037},[650],[425,9039],{"className":9040,"style":743},[742],[425,9042,9044,9047,9050,9053,9056,9059,9062],{"className":9043},[437],[425,9045],{"className":9046,"style":757},[441],[425,9048,762],{"className":9049},[761],[425,9051,767],{"className":9052,"style":766},[446,447],[425,9054,772],{"className":9055},[771],[425,9057],{"className":9058,"style":776},[742],[425,9060,782],{"className":9061,"style":781},[446,780],[425,9063,787],{"className":9064},[786]," is a ground set plus a ",[394,9067,9068],{},"hereditary"," family\nof ",[394,9071,8200],{}," sets satisfying the ",[394,9074,1182],{},": any smaller\nindependent set can absorb some element of any larger one. Maximal independent\nsets are ",[394,9077,9078],{},"bases","; they all share size ",[425,9081,9083],{"className":9082},[428],[425,9084,9086],{"className":9085,"ariaHidden":433},[432],[425,9087,9089,9092,9095,9098,9101],{"className":9088},[437],[425,9090],{"className":9091,"style":757},[441],[425,9093,1249],{"className":9094,"style":566},[446,447],[425,9096,762],{"className":9097},[761],[425,9099,738],{"className":9100,"style":737},[446,447],[425,9102,787],{"className":9103},[786],[394,9105,1232],{},[849,9107,9108,9111,9112,9115,9116,9119],{},[394,9109,9110],{},"Rado–Edmonds theorem:"," on a weighted matroid, sorting by weight and greedily\nkeeping independence yields a ",[394,9113,9114],{},"maximum-weight base",", and greedy is optimal on\na hereditary structure ",[385,9117,9118],{},"iff"," it is a matroid. The proof's one nontrivial step is\nprecisely the exchange axiom.",[849,9121,8020,9122,9125,9126,9129,9130,9133,9134,9153,9154,9157],{},[394,9123,9124],{},"graphic matroid"," (acyclic edge sets) makes ",[394,9127,9128],{},"Kruskal's MST"," an exact\ninstance of matroid-greedy; the ",[394,9131,9132],{},"uniform \u002F partition matroid"," captures\n",[399,9135,9136,9137,9152],{},"at most ",[425,9138,9140],{"className":9139},[428],[425,9141,9143],{"className":9142,"ariaHidden":433},[432],[425,9144,9146,9149],{"className":9145},[437],[425,9147],{"className":9148,"style":442},[441],[425,9150,449],{"className":9151,"style":448},[446,447]," of each kind"," constraints. ",[394,9155,9156],{},"Matroid intersection"," is the\npolynomial-time frontier just beyond single-matroid greedy.",[849,9159,9160,808,9163,9166,9167,9169],{},[394,9161,9162],{},"0\u002F1 knapsack",[394,9164,9165],{},"TSP"," are ",[385,9168,2332],{}," matroids (the exchange property fails),\nwhich is the structural reason greedy fails on them and dynamic programming or\nexact search is required instead.",[9171,9172,9175,9180],"section",{"className":9173,"dataFootnotes":376},[9174],"footnotes",[407,9176,9179],{"className":9177,"id":684},[9178],"sr-only","Footnotes",[846,9181,9182,9196],{},[849,9183,9185,9188,9189],{"id":9184},"user-content-fn-clrs-greedy",[394,9186,9187],{},"CLRS",", Ch. 16 — Greedy Algorithms (§16.2): the greedy-choice property and optimal substructure, proved by exchanging an optimal solution's first choice for greedy's. ",[680,9190,9195],{"href":9191,"ariaLabel":9192,"className":9193,"dataFootnoteBackref":376},"#user-content-fnref-clrs-greedy","Back to reference 1",[9194],"data-footnote-backref","↩",[849,9197,9199,9201,9202],{"id":9198},"user-content-fn-clrs-matroid",[394,9200,9187],{},", Ch. 16 — Greedy Algorithms (§16.4): the Rado–Edmonds theorem — greedy returns a maximum-weight independent set exactly when the structure is a matroid; the proof rests on the exchange property. ",[680,9203,9195],{"href":9204,"ariaLabel":9205,"className":9206,"dataFootnoteBackref":376},"#user-content-fnref-clrs-matroid","Back to reference 2",[9194],[9208,9209,9210],"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":9212},[9213,9214,9215,9216,9217,9219,9220],{"id":409,"depth":18,"text":410},{"id":698,"depth":18,"text":699},{"id":1890,"depth":18,"text":1891},{"id":6843,"depth":18,"text":6844},{"id":8160,"depth":18,"text":9218},"Why 0\u002F1 knapsack and TSP are not matroids",{"id":8981,"depth":18,"text":8982},{"id":684,"depth":18,"text":9179},"Across this module we have watched greedy algorithms win (activity selection,\nHuffman codes, Kruskal's and Prim's spanning trees) and watched greedy lose, on\nthe 0\u002F1 knapsack. Each victory came with a proof: a guarantee that taking the\nlocally best option never forecloses the global optimum. This lesson is about the\nproofs themselves. First we distill the two arguments every greedy correctness\nproof is built from. Then we ask the deeper question: is there a structural\nproperty that predicts when greedy will work, before we attempt the proof? The\nanswer is yes, and its name is the matroid. The matroid theory makes precise\nthe boundary we kept bumping into, why Kruskal is correct and 0\u002F1 knapsack is\nnot, and reduces is greedy optimal here? to is this structure a matroid?","md",{"moduleNumber":116,"lessonNumber":73,"order":9224},704,true,[9227,9231,9235,9238],{"title":9228,"slug":9229,"difficulty":9230},"Maximum Number of Non-Overlapping Substrings","maximum-number-of-non-overlapping-substrings","Hard",{"title":9232,"slug":9233,"difficulty":9234},"Min Cost to Connect All Points","min-cost-to-connect-all-points","Medium",{"title":9236,"slug":9237,"difficulty":9230},"Maximum Performance of a Team","maximum-performance-of-a-team",{"title":9239,"slug":9240,"difficulty":9230},"Course Schedule III","course-schedule-iii","---\ntitle: Matroids & Exchange Arguments\nmodule: Greedy Algorithms\nmoduleNumber: 7\nlessonNumber: 4\norder: 704\nsummary: >-\n  The capstone of the greedy module: _why_ and _when_ a greedy algorithm is\n  provably optimal. We recap the two correctness templates — **greedy-stays-ahead**\n  and the **exchange argument** — then meet the **matroid** $M=(S,\\mathcal{I})$, an\n  abstraction whose **exchange property** is exactly the structure greedy needs.\n  The matroid–greedy theorem says sorting by weight and taking what stays\n  independent yields a maximum-weight basis _if and only if_ the structure is a\n  matroid. Kruskal's MST is the canonical instance; 0\u002F1 knapsack and TSP are the\n  canonical failures.\ntopics: [Greedy]\nsources:\n  - book: CLRS\n    ref: \"Ch. 16 — Greedy Algorithms (§16.4 Matroids)\"\n  - book: Erickson\n    ref: \"Ch. — Matroids\"\n  - book: Skiena\n    ref: \"§ — Greedy\"\npractice:\n  - title: 'Maximum Number of Non-Overlapping Substrings'\n    slug: maximum-number-of-non-overlapping-substrings\n    difficulty: Hard\n  - title: 'Min Cost to Connect All Points'\n    slug: min-cost-to-connect-all-points\n    difficulty: Medium\n  - title: 'Maximum Performance of a Team'\n    slug: maximum-performance-of-a-team\n    difficulty: Hard\n  - title: 'Course Schedule III'\n    slug: course-schedule-iii\n    difficulty: Hard\n---\n\nAcross this module we have watched greedy algorithms win (activity selection,\nHuffman codes, Kruskal's and Prim's spanning trees) and watched greedy lose, on\nthe 0\u002F1 knapsack. Each victory came with a _proof_: a guarantee that taking the\nlocally best option never forecloses the global optimum. This lesson is about the\nproofs themselves. First we distill the two arguments every greedy correctness\nproof is built from. Then we ask the deeper question: is there a structural\nproperty that _predicts_ when greedy will work, before we attempt the proof? The\nanswer is yes, and its name is the **matroid**. The matroid theory makes precise\nthe boundary we kept bumping into, why Kruskal is correct and 0\u002F1 knapsack is\nnot, and reduces \"is greedy optimal here?\" to \"is this structure a matroid?\"\n\n## Two templates for proving greedy optimal\n\nEvery greedy correctness proof in this module is an instance of one of two\npatterns. They are interchangeable in power but feel different to wield.\n\n> **Remark (Greedy-stays-ahead).** Order both solutions by the steps of the algorithm.\n> Prove by induction on $k$ that after greedy's first $k$ choices, greedy's\n> partial solution is _at least as good_ as the first $k$ choices of any other\n> solution, by a measure that matters (finishes earliest, covers the most,\n> spends the least). The base case is the first choice; the inductive step shows\n> greedy's $k$-th choice keeps it ahead. If greedy is never behind, it cannot\n> lose at the end.\n\nThis is the activity-selection argument: greedy's $k$-th activity finishes no\nlater than the $k$-th activity of any valid schedule, so greedy always has at\nleast as much room left for future activities and ends up with at least as many.\n\n> **Remark (Exchange argument).** Take _any_ optimal solution $O$ and transform it,\n> step by step, into the greedy solution $G$, never decreasing its quality.\n> Each step swaps one element of $O$ for the element greedy would have chosen,\n> arguing the swap is feasible and no worse. After finitely many swaps $O$ has\n> become $G$ with value $\\ge$ its original, so $G$ is optimal too.\n\nThis is the one you reach for: it proves the _greedy-choice property_ by showing that\nwhatever an optimal solution does first, we may exchange it for greedy's first\nchoice without harm.[^clrs-greedy] The two templates differ in direction:\nstays-ahead pushes greedy _forward_ and shows it never falls behind; exchange\npulls an optimum _toward_ greedy and shows the pull never hurts. The exchange\nargument is the one that generalizes into a theory, because its swap step is\nliterally an axiom of the structure we are about to define.\n\n## Matroids\n\nA **matroid** abstracts the notion of \"independence\": linear independence of\nvectors, acyclicity of edges, freedom to add one more element without breaking a\nrule.\n\n> **Definition.** A **matroid** is a pair $M = (S, \\mathcal{I})$ where $S$ is a\n> finite **ground set** and $\\mathcal{I}$ is a nonempty family of subsets of $S$,\n> the **independent sets**, satisfying:\n>\n> 1. **(Hereditary \u002F downward-closed)** If $B \\in \\mathcal{I}$ and $A \\subseteq B$,\n>    then $A \\in \\mathcal{I}$. Every subset of an independent set is independent.\n> 2. **(Exchange property)** If $A, B \\in \\mathcal{I}$ and $|A| \u003C |B|$, then there\n>    exists some $x \\in B \\setminus A$ such that $A \\cup \\{x\\} \\in \\mathcal{I}$.\n\nThe first axiom is mild; most natural notions of independence are closed under\ntaking subsets. The **exchange property** is the engine: a smaller independent\nset can always be grown by stealing _some_ element from any larger one. It\nforbids the situation where you get stuck early with a maximal-but-small\nindependent set while a much larger one exists.\n\n> **Definition.** A maximal independent set, one that cannot be extended by any\n> element of $S$, is a **base** of $M$. The **rank** $r(M)$ is the size of a base.\n\nA first consequence of the exchange property is that _all bases have the same\nsize_: if two bases $A, B$ had $|A| \u003C |B|$, the exchange property would let us add\nan element of $B$ to $A$, contradicting that $A$ is maximal. So \"rank\" is\nwell-defined: every base has exactly $r(M)$ elements, exactly as every basis of a\nvector space has the same dimension. (Indeed, the columns of a matrix, with\n$\\mathcal{I}$ the linearly independent subsets, form the **linear matroid**; this\nis where the vocabulary comes from.)\n\n$$\n% caption: The exchange property — any smaller independent set $A$ can absorb some\n%          $x \\in B \\setminus A$ from a larger one\n\\begin{tikzpicture}[\n  every node\u002F.style={circle, draw, minimum size=6mm, inner sep=1pt, font=\\small},\n  >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  % set A (smaller): 2 elements\n  \\node (a1) at (0,0.7) {$a$};\n  \\node (a2) at (0,-0.7) {$b$};\n  \\node[draw=none, font=\\small\\itshape] at (0,1.7) {$A$ ($|A|=2$)};\n  \\draw[rounded corners] (-0.7,-1.4) rectangle (0.7,1.4);\n  % set B (larger): 3 elements\n  \\node (b1) at (5,1.2) {$a$};\n  \\node (b2) at (5,0) {$c$};\n  \\node (b3) at (5,-1.2) {$d$};\n  \\node[draw=none, font=\\small\\itshape] at (5,1.9) {$B$ ($|B|=3$)};\n  \\draw[rounded corners] (4.3,-1.7) rectangle (5.7,1.6);\n  % the new element joining A\n  \\node[draw=acc, very thick, text=acc] (anew) at (1.9,-0.7) {$c$};\n  \\draw[->, red!75!black, very thick] (b2) .. controls (3.4,-0.4) .. (anew);\n  \\node[draw=none, text=acc, font=\\footnotesize] at (2.5,-1.6) {$x = c \\in B \\setminus A$};\n\\end{tikzpicture}\n$$\n\nBecause $|A| \u003C |B|$, some $x \\in B \\setminus A$ (here $c$) joins $A$ keeping it\nindependent, so no maximal independent set is left stranded below the rank.\n\n## The matroid–greedy theorem\n\nA **weighted matroid** attaches a positive weight $w(x) > 0$ to each element\n$x \\in S$, with $w(A) = \\sum_{x \\in A} w(x)$. The natural optimization problem:\nfind a maximum-weight independent set. Since weights are positive and\n$\\mathcal{I}$ is hereditary, a maximum-weight independent set is always a base.\nThe greedy algorithm is the obvious one: sort by weight descending, add each\nelement if it keeps the set independent.\n\n```algorithm\ncaption: $\\textsc{Greedy}(M=(S,\\mathcal I),\\, w)$ — maximum-weight independent set\n$A \\gets \\varnothing$\nsort $S$ into nonincreasing order by weight $w$\nfor each $x \\in S$ in sorted order do\n  if $A \\cup \\{x\\} \\in \\mathcal{I}$ then   \u002F\u002F independence test\n    $A \\gets A \\cup \\{x\\}$\nreturn $A$\n```\n\n> **Theorem (Rado–Edmonds).** If $M = (S, \\mathcal{I})$ is a matroid with weight\n> function $w$, then $\\textsc{Greedy}(M, w)$ returns a maximum-weight base of $M$.\n> Conversely, if $(S, \\mathcal{I})$ is hereditary but _not_ a matroid, then there\n> is a weight function on which $\\textsc{Greedy}$ fails. So greedy is optimal on a\n> hereditary structure **if and only if** that structure is a matroid.[^clrs-matroid]\n\nThe forward direction is a clean exchange argument; it is worth seeing in full,\nbecause it is the abstract skeleton of every concrete greedy proof in this module.\n\n> **Proof (greedy is optimal on a matroid).** Let $g_1, g_2, \\dots, g_r$ be greedy's\n> picks in the order chosen, so $w(g_1) \\ge w(g_2) \\ge \\cdots \\ge w(g_r)$, and let\n> $G = \\{g_1, \\dots, g_r\\}$. Suppose for contradiction some base $O$ has\n> $w(O) > w(G)$. List $O$'s elements in nonincreasing weight order as\n> $o_1, \\dots, o_r$ (both bases have size $r = r(M)$). Let $k$ be the first index\n> where $w(g_k) \u003C w(o_k)$; such a $k$ exists, else $w(g_i) \\ge w(o_i)$ for all $i$\n> and $w(G) \\ge w(O)$.\n>\n> Consider $A = \\{g_1, \\dots, g_{k-1}\\}$ and $B = \\{o_1, \\dots, o_k\\}$, both\n> independent (subsets of independent sets, by the hereditary axiom). Since\n> $|A| = k-1 \u003C k = |B|$, the **exchange property** gives an element\n> $o_j \\in B \\setminus A$ with $A \\cup \\{o_j\\} \\in \\mathcal{I}$. Each\n> $o_i \\in B$ has $w(o_i) \\ge w(o_k) > w(g_k)$, so $w(o_j) > w(g_k)$.\n>\n> But when greedy considered the elements in weight order, at the moment it chose\n> $g_k$ the set $A = \\{g_1,\\dots,g_{k-1}\\}$ was already in hand, and\n> $A \\cup \\{o_j\\}$ is independent with $w(o_j) > w(g_k)$. Greedy scans in\n> nonincreasing weight, so it would have reached and accepted $o_j$ **before**\n> $g_k$, a contradiction. Hence no such $O$ exists and $G$ is maximum-weight.\n> $\\qed$\n\nLaid out side by side, greedy's picks and the optimum's, both sorted by weight,\nmake the contradiction visible. They agree on weight up to the first index $k$\nwhere greedy falls behind ($w(g_k) \u003C w(o_k)$). The prefix $A=\\{g_1,\\dots,g_{k-1}\\}$\nis smaller than $B=\\{o_1,\\dots,o_k\\}$, so the exchange property hands us an element\n$o_j\\in B$ that keeps $A$ independent and is _heavier_ than $g_k$ — which greedy,\nscanning in weight order, must have reached and accepted before $g_k$. That is the\ncontradiction.\n\n$$\n% caption: The Rado–Edmonds exchange step. Greedy's picks $g_i$ and an allegedly better\n%          base's picks $o_i$, both in nonincreasing weight, agree until index $k$ where\n%          $w(g_k)\u003Cw(o_k)$ (red). The exchange property on $A=\\{g_1{\\dots}g_{k-1}\\}$ and\n%          $B=\\{o_1{\\dots}o_k\\}$ yields a heavier independent extension greedy would have\n%          taken first.\n\\begin{tikzpicture}[font=\\small,\n  cell\u002F.style={draw, thick, minimum width=9mm, minimum height=7mm, fill=black!6},\n  hit\u002F.style={draw=red!75!black, very thick, minimum width=9mm, minimum height=7mm, fill=red!18}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-2.5,-1.5) rectangle (9.2,3.2);\n  % column index headers (rank position of each pick); k is the first mismatch\n  \\foreach \\i\u002F\\x in {1\u002F0,2\u002F1.2,4\u002F3.6} \\node[font=\\tiny, black!55] at (\\x,2.75) {$\\i$};\n  \\node[font=\\tiny, red!75!black] at (2.4,2.75) {$k{=}3$};\n  % greedy row (weights 9 7 4 2): falls behind at column k\n  \\node[font=\\footnotesize, anchor=east] at (-0.7,2) {greedy $G$};\n  \\foreach \\w\u002F\\x [count=\\i] in {9\u002F0,7\u002F1.2,4\u002F2.4,2\u002F3.6} {\n    \\ifnum\\i=3 \\node[hit] at (\\x,2) {$\\w$};\\else \\node[cell] at (\\x,2) {$\\w$};\\fi }\n  % the allegedly better set O (weights 9 7 6 3)\n  \\node[font=\\footnotesize, anchor=east] at (-0.7,0) {rival $O$};\n  \\foreach \\w\u002F\\x [count=\\i] in {9\u002F0,7\u002F1.2,6\u002F2.4,3\u002F3.6} {\n    \\ifnum\\i=3 \\node[hit] at (\\x,0) {$\\w$};\\else \\node[cell] at (\\x,0) {$\\w$};\\fi }\n  % exchange: O's k-th weight (6) beats greedy's (4); arrow from the 6 up to the 4\n  \\draw[->, very thick, red!75!black] (2.85,-0.05) to[bend right=22] (2.85,2.05);\n  \\node[red!75!black, font=\\footnotesize, align=left, anchor=west] at (4.2,1.0)\n    {$o_k{=}6 > 4{=}g_k$, yet the\\\\exchange property lets greedy\\\\add $6$ at step $k$ — so its\\\\choice was not actually greedy.};\n  % bracket A: greedy's first k-1 picks (cols 1-2); label sits in the gap, clear of row O\n  \\draw[acc, thick] (-0.5,1.5) -- (-0.5,1.4) -- (1.7,1.4) -- (1.7,1.5);\n  \\node[acc, font=\\footnotesize] at (0.6,1.08) {$A$: first $k{-}1$ picks};\n  % bracket B: O's first k picks (cols 1-3) — one more element than A\n  \\draw[acc, thick] (-0.5,-0.5) -- (-0.5,-0.6) -- (2.9,-0.6) -- (2.9,-0.5);\n  \\node[acc, font=\\footnotesize] at (1.2,-1.02) {$B$: first $k$ picks \\;($|B|>|A|$)};\n\\end{tikzpicture}\n$$\n\nThe single step that does all the work is \"the exchange property gives an element\n$o_j$.\" That is the matroid axiom standing in for the ad-hoc swap we constructed\nby hand for activity selection. The converse, that a non-matroid hereditary\nstructure has a weight function defeating greedy, is what makes the matroid the\n_exact_ characterization, not merely a sufficient condition: if the exchange\nproperty fails for some $A, B$, one can place weights that lure greedy into the\nstranded maximal set $A$ and away from the heavier $B$.\n\n## Examples that are matroids\n\n**The graphic matroid.** Let $G = (V, E)$ be a graph. Take $S = E$ and let\n$\\mathcal{I}$ be the **acyclic** edge sets, the forests of $G$. This is a\nmatroid: a subset of a forest is a forest (hereditary), and if forests $A, B$\nhave $|A| \u003C |B|$, then $B$ touches more components, so some edge of $B$ joins two\ntrees of $A$ without creating a cycle (exchange property). The bases are the\n**spanning forests**; for a connected graph, the spanning trees, all of size\n$|V| - 1 = r(M)$.\n\nNow run $\\textsc{Greedy}$ on the graphic matroid with **negated** edge weights\n(maximum-weight independent set becomes minimum-weight spanning tree): sort edges,\nadd each edge that does not form a cycle. That is **Kruskal's algorithm**,\nexactly: the independence test \"$A \\cup \\{e\\}$ stays acyclic\" is the\n[union-find](\u002Falgorithms\u002Fdata-structures\u002Funion-find)\ncycle check from the [minimum spanning trees](\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees)\nlesson. So Kruskal's correctness is not a special miracle; it is the\nRado–Edmonds theorem instantiated on the graphic matroid.\n\n$$\n% caption: Acyclic edge sets form a matroid — Kruskal is matroid-greedy, taking edges by\n%          weight and rejecting any that close a cycle\n\\begin{tikzpicture}[\n  every node\u002F.style={circle, draw, minimum size=7mm, inner sep=1pt, font=\\small},\n  >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node (p) at (0,1.6) {$p$};\n  \\node (q) at (3,1.6) {$q$};\n  \\node (r) at (0,-1) {$r$};\n  \\node (s) at (3,-1) {$s$};\n  % accepted edges (in weight order 1,2,3) -> spanning tree\n  \\draw[acc, very thick] (p) -- node[draw=none, above, font=\\footnotesize] {$1$} (q);\n  \\draw[acc, very thick] (p) -- node[draw=none, left, font=\\footnotesize] {$2$} (r);\n  \\draw[acc, very thick] (q) -- node[draw=none, right, font=\\footnotesize] {$3$} (s);\n  % rejected edge: weight 4 would close cycle p-q-s-r-p\n  \\draw[dashed, gray] (r) -- node[draw=none, below, font=\\footnotesize] {$4$ (reject)} (s);\n\\end{tikzpicture}\n$$\n\nEdges $1, 2, 3$ are accepted in weight order (blue); edge $4$ is rejected because\n$r$ and $s$ are already connected, so adding it would close the cycle $p\\,q\\,s\\,r$,\nleaving $\\mathcal{I}$. The accepted set is a base: a spanning tree.\n\n**The uniform \u002F partition matroid.** Even simpler: fix $k$ and let\n$\\mathcal{I} = \\{A \\subseteq S : |A| \\le k\\}$, so every set of size at most $k$ is\nindependent. This **uniform matroid** $U_{k,n}$ is plainly hereditary, and the\nexchange property is trivial (any larger set has a spare element). Greedy reduces\nto \"pick the $k$ heaviest elements,\" which is obviously optimal; the matroid\nmachinery confirms the trivial.\n\n$$\n% caption: The uniform matroid $U_{3,6}$: independent means $|A|\\le 3$. Greedy sorts by\n%          weight descending and takes the first $k=3$ (blue), which is trivially the\n%          maximum-weight base; the exchange property holds because any larger set always\n%          has a spare element.\n\\begin{tikzpicture}[font=\\small, xscale=0.92, yscale=0.5]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\foreach \\i\u002F\\w\u002F\\lab\u002F\\sel in {0\u002F5.0\u002F9\u002F1, 1\u002F4.2\u002F8\u002F1, 2\u002F3.4\u002F6\u002F1, 3\u002F2.4\u002F4\u002F0, 4\u002F1.6\u002F3\u002F0, 5\u002F0.9\u002F1\u002F0} {\n    \\ifnum\\sel=1\n      \\fill[acc!28, draw=black, thick] (\\i*1.2,0) rectangle (\\i*1.2+0.9,\\w);\n    \\else\n      \\fill[black!10, draw=black, thick] (\\i*1.2,0) rectangle (\\i*1.2+0.9,\\w);\n    \\fi\n    \\node[font=\\footnotesize] at (\\i*1.2+0.45,\\w+0.6) {\\lab};\n  }\n  \\draw[acc, dashed] (3.5,-0.4) -- (3.5,6.2);\n  \\node[acc, font=\\footnotesize, anchor=south] at (1.8,6.3) {keep $k=3$};\n  \\node[font=\\footnotesize, anchor=south, black!55] at (5.2,6.3) {reject};\n  \\draw[->, thick] (-0.2,-0.4) -- (7.4,-0.4) node[right,font=\\footnotesize]{weight order};\n\\end{tikzpicture}\n$$\n\nThe **partition matroid** generalizes it: partition\n$S$ into groups and allow at most $k_i$ elements from group $i$; independence is\n\"within every group's cap.\" Scheduling and assignment constraints of the form \"no\nmore than $c$ of this kind\" are partition-matroid constraints, which is why greedy\nsolves so many of them.\n\n**Beyond greedy: matroid intersection.** A single matroid yields to greedy; the\ncommon independent sets of _two_ matroids over the same ground set do not, in\ngeneral. Still, **matroid intersection** solves the maximum-weight common\nindependent set in polynomial time by augmenting-path methods (bipartite matching\nis a special case). The intersection of _three_ matroids is already NP-hard. So\nthe matroid is the precise frontier of \"greedy works\"; one step beyond it you need\nheavier machinery.\n\n## Why 0\u002F1 knapsack and TSP are _not_ matroids\n\nThe theorem cuts both ways, and it explains the failures we have already seen.\n\n**0\u002F1 knapsack.** Recall from the [greedy method](\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method)\nlesson that greedy fails on the 0\u002F1 knapsack, and from the\n[knapsack DP](\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack) lesson that it\nneeds dynamic programming. The reason, in matroid language: let $S$ be the items\nand call a set \"independent\" if its total _weight_ fits in capacity $W$. This is\nhereditary: drop items and it still fits. But the **exchange property fails**.\nTake capacity $W = 8$ and item sizes giving the independent sets $\\{7\\}$ and\n$\\{4, 4\\}$. Both fit, and $|\\{4,4\\}| = 2 > 1 = |\\{7\\}|$, yet _no_ element of\n$\\{4,4\\}$ can join $\\{7\\}$, since $7 + 4 = 11 > 8$. The smaller independent set is\nstranded: it cannot grow toward the larger one, which is the exact pattern the\nexchange property forbids.\n\n$$\n% caption: 0\u002F1 knapsack is not a matroid ($W=8$). The independent set $\\{7\\}$ and the\n%          larger $\\{4,4\\}$ both fit, yet adding either $4$ to $\\{7\\}$ gives $11>8$. No\n%          element of the larger set can grow the smaller, so the exchange property fails.\n\\begin{tikzpicture}[font=\\small, yscale=0.5, xscale=0.9]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\draw[dashed, black!45] (-0.3,8) -- (8.6,8);\n  \\node[font=\\footnotesize, anchor=west] at (8.7,8) {$W=8$};\n  \\draw[thick] (0,0) rectangle (1.6,8);\n  \\fill[acc!22, draw=black] (0,0) rectangle (1.6,7);\n  \\node at (0.8,3.5) {$7$};\n  \\node[below, font=\\footnotesize] at (0.8,0) {$\\{7\\}$ fits};\n  \\draw[thick] (3,0) rectangle (4.6,8);\n  \\fill[acc!22, draw=black] (3,0) rectangle (4.6,4);\n  \\fill[acc!12, draw=black] (3,4) rectangle (4.6,8);\n  \\node at (3.8,2) {$4$}; \\node at (3.8,6) {$4$};\n  \\node[below, font=\\footnotesize] at (3.8,0) {$\\{4,4\\}$ fits};\n  \\draw[thick] (6.2,0) rectangle (7.8,11.2);\n  \\fill[acc!22, draw=black] (6.2,0) rectangle (7.8,7);\n  \\fill[red!25, draw=black] (6.2,7) rectangle (7.8,11);\n  \\node at (7,3.5) {$7$}; \\node at (7,9) {$4$};\n  \\node[red, below, font=\\footnotesize, align=center] at (7,0) {$7{+}4{=}11>8$\\\\ exchange fails};\n\\end{tikzpicture}\n$$\n\nThe\nexchange property is violated; the structure is not a matroid; and by Rado–Edmonds\n_some_ weighting must defeat greedy, which is exactly the value\u002Fdensity\ncounterexample from the greedy-method lesson.\n\n**Traveling salesman.** For the TSP, let $\\mathcal{I}$ be the edge sets that\nextend to a Hamiltonian tour (or the partial-tour fragments greedy builds). This\nis not even reliably hereditary, and the exchange property collapses badly: a\ncheap partial path can be a dead end that no edge of a longer, valid fragment can\nrescue without revisiting a vertex or exceeding degree $2$. There is no matroid,\nso no weighting guarantee, and indeed nearest-neighbor greedy can be made\narbitrarily bad. TSP's hardness is not an accident of greed; the independence\nstructure simply isn't a matroid (and TSP is NP-hard regardless).\n\nThe pattern is uniform. When greedy is provably optimal, you can almost always\nexhibit a matroid behind it. When greedy fails, the exchange property is the axiom\nthat breaks.\n\n## Takeaways\n\n- Every greedy correctness proof is a **greedy-stays-ahead** induction (greedy's\n  $k$-th partial solution is never behind any rival's) or an **exchange argument**\n  (transform any optimum into the greedy solution without loss). They are\n  equivalent in power; exchange is the one that abstracts into a theory.\n- A **matroid** $M = (S, \\mathcal{I})$ is a ground set plus a **hereditary** family\n  of **independent** sets satisfying the **exchange property**: any smaller\n  independent set can absorb some element of any larger one. Maximal independent\n  sets are **bases**; they all share size $r(M)$, the **rank**.\n- **Rado–Edmonds theorem:** on a weighted matroid, sorting by weight and greedily\n  keeping independence yields a **maximum-weight base**, and greedy is optimal on\n  a hereditary structure _iff_ it is a matroid. The proof's one nontrivial step is\n  precisely the exchange axiom.\n- The **graphic matroid** (acyclic edge sets) makes **Kruskal's MST** an exact\n  instance of matroid-greedy; the **uniform \u002F partition matroid** captures\n  \"at most $k$ of each kind\" constraints. **Matroid intersection** is the\n  polynomial-time frontier just beyond single-matroid greedy.\n- **0\u002F1 knapsack** and **TSP** are _not_ matroids (the exchange property fails),\n  which is the structural reason greedy fails on them and dynamic programming or\n  exact search is required instead.\n\n[^clrs-greedy]: **CLRS**, Ch. 16 — Greedy Algorithms (§16.2): the greedy-choice property and optimal substructure, proved by exchanging an optimal solution's first choice for greedy's.\n[^clrs-matroid]: **CLRS**, Ch. 16 — Greedy Algorithms (§16.4): the Rado–Edmonds theorem — greedy returns a maximum-weight independent set exactly when the structure is a matroid; the proof rests on the exchange property.\n[^erickson-matroid]: **Erickson**, Ch. — Matroids: matroids as the abstraction of independence (linear, graphic, uniform); bases, rank, and the equivalence of greedy optimality with the matroid axioms.\n[^skiena-greedy]: **Skiena**, § — Greedy: greedy heuristics, when local choices are globally optimal, and matroid structure as the underlying explanation for Kruskal-style algorithms.\n",{"text":9243,"minutes":9244,"time":9245,"words":9246},"11 min read",10.19,611400,2038,{"title":226,"description":9221},[9249,9251,9254],{"book":9187,"ref":9250},"Ch. 16 — Greedy Algorithms (§16.4 Matroids)",{"book":9252,"ref":9253},"Erickson","Ch. — Matroids",{"book":9255,"ref":9256},"Skiena","§ — 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