[{"data":1,"prerenderedAt":8060},["ShallowReactive",2],{"nav:algorithms":3,"lesson:\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":374,"course-wordcounts":7928,"ref-card-index":7984},[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":302,"blurb":376,"body":377,"description":7890,"extension":7891,"meta":7892,"module":287,"navigation":2789,"path":303,"practice":7894,"rawbody":7909,"readingTime":7910,"seo":7915,"sources":7916,"status":7924,"stem":7925,"summary":305,"topics":7926,"__hash__":7927},"course\u002F01.algorithms\u002F09.backtracking\u002F03.branch-and-bound.md","",{"type":378,"value":379,"toc":7880},"minimark",[380,416,522,527,554,759,887,904,1146,1231,1236,1641,2254,2347,2730,2825,2988,3603,3995,3999,4053,4381,4421,4424,5026,5551,5611,6096,6446,6678,7101,7105,7111,7293,7368,7372,7837,7876],[381,382,383,384,390,391,395,396,399,400,403,404,407,408,411,412,415],"p",{},"The previous lessons built ",[385,386,387],"a",{"href":292},[388,389,288],"strong",{},":\na depth-first walk over a tree of partial solutions that ",[392,393,394],"em",{},"prunes"," a branch the\nmoment it becomes infeasible, such as a queen attacking another or a graph\ncoloring conflict. That kind of pruning asks a yes\u002Fno question: ",[392,397,398],{},"can this partial\nsolution still be completed at all?"," For ",[388,401,402],{},"optimization"," problems, maximize this\nor minimize that, we can ask a sharper question: ",[392,405,406],{},"even in the best case, can\ncompleting this partial solution beat the best answer I already have?"," If not, the\nentire subtree is dead, feasible or not. Pruning by ",[388,409,410],{},"value"," rather than mere\n",[388,413,414],{},"feasibility"," is the idea behind branch and bound, and it is often the\ndifference between minutes and millennia.",[381,417,418,419,482,483,499,500,517,518,521],{},"When even aggressive pruning is not enough, when the search tree is genuinely\n",[420,421,424],"span",{"className":422},[423],"katex",[420,425,429],{"className":426,"ariaHidden":428},[427],"katex-html","true",[420,430,433,438],{"className":431},[432],"base",[420,434],{"className":435,"style":437},[436],"strut","height:0.6644em;",[420,439,442,446],{"className":440},[441],"mord",[420,443,445],{"className":444},[441],"2",[420,447,450],{"className":448},[449],"msupsub",[420,451,454],{"className":452},[453],"vlist-t",[420,455,458],{"className":456},[457],"vlist-r",[420,459,462],{"className":460,"style":437},[461],"vlist",[420,463,465,470],{"style":464},"top:-3.063em;margin-right:0.05em;",[420,466],{"className":467,"style":469},[468],"pstrut","height:2.7em;",[420,471,477],{"className":472},[473,474,475,476],"sizing","reset-size6","size3","mtight",[420,478,481],{"className":479},[441,480,476],"mathnormal","n","-shaped and ",[420,484,486],{"className":485},[423],[420,487,489],{"className":488,"ariaHidden":428},[427],[420,490,492,496],{"className":491},[432],[420,493],{"className":494,"style":495},[436],"height:0.4306em;",[420,497,481],{"className":498},[441,480]," is, say, ",[420,501,503],{"className":502},[423],[420,504,506],{"className":505,"ariaHidden":428},[427],[420,507,509,513],{"className":508},[432],[420,510],{"className":511,"style":512},[436],"height:0.6444em;",[420,514,516],{"className":515},[441],"36",", a second technique buys a square root of the\nrunning time outright. ",[388,519,520],{},"Meet in the middle"," splits the instance, enumerates\neach half independently, and stitches the halves back together with sorting and\nbinary search. Both techniques attack the same enemy, the exponential, from\ndifferent sides.",[523,524,526],"h2",{"id":525},"branch-and-bound-pruning-by-value","Branch and bound: pruning by value",[381,528,529,530,533,534,537,538,541,542,545,546,549,550,553],{},"Branch and bound is backtracking with two extra pieces of bookkeeping. The first\nis the ",[388,531,532],{},"incumbent",": the value (and witness) of the best ",[392,535,536],{},"complete"," solution\nfound anywhere in the search so far. The second is a ",[388,539,540],{},"bound"," computed at every\nnode, an ",[392,543,544],{},"optimistic"," estimate of the best objective achievable by any\ncompletion of that partial solution. For a maximization problem the bound is an\n",[388,547,548],{},"upper bound"," (no completion can do better than this); for minimization it is a\n",[388,551,552],{},"lower bound",".",[555,556,558],"callout",{"type":557},"remark",[381,559,560,563,564,581,582,628,629,693,694,754,755,758],{},[388,561,562],{},"Remark (Pruning rule)."," At a node with bound ",[420,565,567],{"className":566},[423],[420,568,570],{"className":569,"ariaHidden":428},[427],[420,571,573,577],{"className":572},[432],[420,574],{"className":575,"style":576},[436],"height:0.6944em;",[420,578,580],{"className":579},[441,480],"b"," and incumbent value ",[420,583,585],{"className":584},[423],[420,586,588],{"className":587,"ariaHidden":428},[427],[420,589,591,595],{"className":590},[432],[420,592],{"className":593,"style":594},[436],"height:0.6887em;",[420,596,598,603],{"className":597},[441],[420,599,602],{"className":600,"style":601},[441,480],"margin-right:0.044em;","z",[420,604,606],{"className":605},[449],[420,607,609],{"className":608},[453],[420,610,612],{"className":611},[457],[420,613,615],{"className":614,"style":594},[461],[420,616,617,620],{"style":464},[420,618],{"className":619,"style":469},[468],[420,621,623],{"className":622},[473,474,475,476],[420,624,627],{"className":625},[626,476],"mbin","∗",":\nif ",[420,630,632],{"className":631},[423],[420,633,635,658],{"className":634,"ariaHidden":428},[427],[420,636,638,642,645,650,655],{"className":637},[432],[420,639],{"className":640,"style":641},[436],"height:0.8304em;vertical-align:-0.136em;",[420,643,580],{"className":644},[441,480],[420,646],{"className":647,"style":649},[648],"mspace","margin-right:0.2778em;",[420,651,654],{"className":652},[653],"mrel","≤",[420,656],{"className":657,"style":649},[648],[420,659,661,664],{"className":660},[432],[420,662],{"className":663,"style":594},[436],[420,665,667,670],{"className":666},[441],[420,668,602],{"className":669,"style":601},[441,480],[420,671,673],{"className":672},[449],[420,674,676],{"className":675},[453],[420,677,679],{"className":678},[457],[420,680,682],{"className":681,"style":594},[461],[420,683,684,687],{"style":464},[420,685],{"className":686,"style":469},[468],[420,688,690],{"className":689},[473,474,475,476],[420,691,627],{"className":692},[626,476]," (maximization) — or ",[420,695,697],{"className":696},[423],[420,698,700,719],{"className":699,"ariaHidden":428},[427],[420,701,703,706,709,712,716],{"className":702},[432],[420,704],{"className":705,"style":641},[436],[420,707,580],{"className":708},[441,480],[420,710],{"className":711,"style":649},[648],[420,713,715],{"className":714},[653],"≥",[420,717],{"className":718,"style":649},[648],[420,720,722,725],{"className":721},[432],[420,723],{"className":724,"style":594},[436],[420,726,728,731],{"className":727},[441],[420,729,602],{"className":730,"style":601},[441,480],[420,732,734],{"className":733},[449],[420,735,737],{"className":736},[453],[420,738,740],{"className":739},[457],[420,741,743],{"className":742,"style":594},[461],[420,744,745,748],{"style":464},[420,746],{"className":747,"style":469},[468],[420,749,751],{"className":750},[473,474,475,476],[420,752,627],{"className":753},[626,476]," (minimization) — then ",[392,756,757],{},"no","\ncompletion of this partial solution can improve on the incumbent, so prune the\nentire subtree. Otherwise branch on the next decision and recurse.",[381,760,761,762,821,822,832,833,836,837,840,841,844,845,886],{},"Correctness is immediate: the bound is optimistic, so ",[420,763,765],{"className":764},[423],[420,766,768,786],{"className":767,"ariaHidden":428},[427],[420,769,771,774,777,780,783],{"className":770},[432],[420,772],{"className":773,"style":641},[436],[420,775,580],{"className":776},[441,480],[420,778],{"className":779,"style":649},[648],[420,781,654],{"className":782},[653],[420,784],{"className":785,"style":649},[648],[420,787,789,792],{"className":788},[432],[420,790],{"className":791,"style":594},[436],[420,793,795,798],{"className":794},[441],[420,796,602],{"className":797,"style":601},[441,480],[420,799,801],{"className":800},[449],[420,802,804],{"className":803},[453],[420,805,807],{"className":806},[457],[420,808,810],{"className":809,"style":594},[461],[420,811,812,815],{"style":464},[420,813],{"className":814,"style":469},[468],[420,816,818],{"className":817},[473,474,475,476],[420,819,627],{"className":820},[626,476]," means even the\nbest descendant is no better than a solution we already hold; discarding it loses\nnothing.",[823,824,825],"sup",{},[385,826,831],{"href":827,"ariaDescribedBy":828,"dataFootnoteRef":376,"id":830},"#user-content-fn-erickson-bb",[829],"footnote-label","user-content-fnref-erickson-bb","1"," The ",[392,834,835],{},"power"," of the method lives entirely in two design\nchoices. A ",[388,838,839],{},"tighter bound"," prunes more nodes; a bound equal to the true\noptimum would prune everything but the answer. And a ",[388,842,843],{},"better search order",",\nfinding a strong incumbent early, raises ",[420,846,848],{"className":847},[423],[420,849,851],{"className":850,"ariaHidden":428},[427],[420,852,854,857],{"className":853},[432],[420,855],{"className":856,"style":594},[436],[420,858,860,863],{"className":859},[441],[420,861,602],{"className":862,"style":601},[441,480],[420,864,866],{"className":865},[449],[420,867,869],{"className":868},[453],[420,870,872],{"className":871},[457],[420,873,875],{"className":874,"style":594},[461],[420,876,877,880],{"style":464},[420,878],{"className":879,"style":469},[468],[420,881,883],{"className":882},[473,474,475,476],[420,884,627],{"className":885},[626,476]," sooner, which retroactively\nprunes more of the tree. The two interact: a good incumbent makes a mediocre\nbound effective.",[555,888,890],{"type":889},"note",[381,891,892,895,896,899,900,903],{},[388,893,894],{},"Intuition."," Backtracking explores until a branch is ",[392,897,898],{},"impossible",". Branch and\nbound explores until a branch is ",[392,901,902],{},"pointless",". The incumbent is a moving\nfloor; the bound is a ceiling on each subtree; whenever a subtree's ceiling\ndrops below the floor, the subtree is swept away.",[905,906,910,1140],"figure",{"className":907},[908,909],"tikz-figure","tikz-diagram-rendered",[911,912,917],"svg",{"xmlns":913,"width":914,"height":915,"viewBox":916},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","394.392","157.479","-75 -75 295.794 118.109",[918,919,922,927,936,950,953,960,963,975,978,992,1017,1020,1038,1072,1087],"g",{"stroke":920,"style":921},"currentColor","stroke-miterlimit:10;stroke-width:.4",[923,924],"path",{"fill":925,"d":926},"none","M102.969-59.067c0-7.071-5.732-12.803-12.804-12.803-7.071 0-12.804 5.732-12.804 12.803s5.733 12.804 12.804 12.804 12.804-5.732 12.804-12.804Zm-12.804 0",[918,928,930],{"transform":929},"translate(-4.625 2.9)",[923,931],{"d":932,"fill":920,"stroke":920,"className":933,"style":935},"M92.863-60.544L90.424-60.544L90.424-60.860L93.250-65.008Q93.294-65.061 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149.073-60.227Q149.073-60.149 149.024-60.100Q148.975-60.051 148.905-60.044M148.905-61.731L143.592-61.731Q143.514-61.739 143.465-61.788Q143.417-61.837 143.417-61.915Q143.417-61.985 143.464-62.036Q143.510-62.087 143.592-62.098L148.905-62.098Q148.979-62.087 149.026-62.036Q149.073-61.985 149.073-61.915Q149.073-61.837 149.024-61.788Q148.975-61.739 148.905-61.731",[934],[918,1135,1136],{"transform":1093},[923,1137],{"d":1138,"fill":938,"stroke":938,"className":1139,"style":1031},"M152.709-59.700Q152.900-59.426 153.256-59.299Q153.611-59.172 153.994-59.172Q154.330-59.172 154.539-59.358Q154.748-59.544 154.844-59.837Q154.939-60.130 154.939-60.442Q154.939-60.766 154.842-61.061Q154.744-61.356 154.531-61.540Q154.318-61.723 153.986-61.723L153.420-61.723Q153.389-61.723 153.359-61.753Q153.330-61.782 153.330-61.809L153.330-61.891Q153.330-61.926 153.359-61.952Q153.389-61.977 153.420-61.977L153.900-62.012Q154.186-62.012 154.383-62.217Q154.580-62.422 154.676-62.717Q154.771-63.012 154.771-63.290Q154.771-63.669 154.572-63.907Q154.373-64.145 153.994-64.145Q153.674-64.145 153.385-64.038Q153.096-63.930 152.932-63.708Q153.111-63.708 153.234-63.581Q153.357-63.454 153.357-63.282Q153.357-63.110 153.232-62.985Q153.107-62.860 152.932-62.860Q152.760-62.860 152.635-62.985Q152.510-63.110 152.510-63.282Q152.510-63.649 152.734-63.897Q152.959-64.145 153.299-64.266Q153.639-64.387 153.994-64.387Q154.342-64.387 154.705-64.266Q155.068-64.145 155.316-63.895Q155.564-63.645 155.564-63.290Q155.564-62.805 155.246-62.422Q154.928-62.040 154.451-61.868Q155.002-61.758 155.402-61.372Q155.803-60.985 155.803-60.450Q155.803-59.993 155.539-59.637Q155.275-59.282 154.853-59.090Q154.432-58.899 153.994-58.899Q153.584-58.899 153.191-59.034Q152.799-59.169 152.533-59.454Q152.268-59.739 152.268-60.157Q152.268-60.352 152.400-60.481Q152.533-60.610 152.725-60.610Q152.850-60.610 152.953-60.551Q153.057-60.493 153.119-60.387Q153.182-60.282 153.182-60.157Q153.182-59.962 153.047-59.831Q152.912-59.700 152.709-59.700M156.514-60.290Q156.514-60.786 156.840-61.151Q157.166-61.516 157.689-61.762L157.420-61.922Q157.123-62.106 156.939-62.401Q156.756-62.696 156.756-63.036Q156.756-63.430 156.975-63.741Q157.193-64.051 157.547-64.219Q157.900-64.387 158.283-64.387Q158.557-64.387 158.830-64.309Q159.103-64.231 159.320-64.083Q159.537-63.934 159.674-63.708Q159.811-63.481 159.811-63.188Q159.811-62.782 159.541-62.475Q159.271-62.169 158.850-61.954L159.299-61.684Q159.518-61.547 159.686-61.354Q159.853-61.161 159.951-60.922Q160.049-60.684 160.049-60.426Q160.049-60.087 159.898-59.799Q159.748-59.512 159.502-59.315Q159.256-59.118 158.932-59.008Q158.607-58.899 158.283-58.899Q157.853-58.899 157.447-59.061Q157.041-59.223 156.777-59.542Q156.514-59.860 156.514-60.290M157.002-60.290Q157.002-59.805 157.393-59.489Q157.783-59.172 158.283-59.172Q158.576-59.172 158.875-59.284Q159.174-59.395 159.367-59.614Q159.561-59.833 159.561-60.145Q159.561-60.376 159.420-60.587Q159.279-60.797 159.072-60.915L157.963-61.594Q157.545-61.391 157.273-61.055Q157.002-60.719 157.002-60.290M157.588-62.723L158.576-62.122Q158.924-62.305 159.150-62.575Q159.377-62.844 159.377-63.188Q159.377-63.407 159.285-63.583Q159.193-63.758 159.039-63.881Q158.885-64.005 158.684-64.075Q158.482-64.145 158.283-64.145Q157.877-64.145 157.531-63.934Q157.186-63.723 157.186-63.340Q157.186-63.157 157.297-62.995Q157.408-62.833 157.588-62.723",[934],[1141,1142,1145],"figcaption",{"className":1143},[1144],"tikz-cap","Prune any node whose optimistic bound can't beat the incumbent",[381,1147,1148,1149,1230],{},"The right subtree carries bound ",[420,1150,1152],{"className":1151},[423],[420,1153,1155,1175,1220],{"className":1154,"ariaHidden":428},[427],[420,1156,1158,1162,1166,1169,1172],{"className":1157},[432],[420,1159],{"className":1160,"style":1161},[436],"height:0.7804em;vertical-align:-0.136em;",[420,1163,1165],{"className":1164},[441],"29",[420,1167],{"className":1168,"style":649},[648],[420,1170,654],{"className":1171},[653],[420,1173],{"className":1174,"style":649},[648],[420,1176,1178,1181,1210,1213,1217],{"className":1177},[432],[420,1179],{"className":1180,"style":594},[436],[420,1182,1184,1187],{"className":1183},[441],[420,1185,602],{"className":1186,"style":601},[441,480],[420,1188,1190],{"className":1189},[449],[420,1191,1193],{"className":1192},[453],[420,1194,1196],{"className":1195},[457],[420,1197,1199],{"className":1198,"style":594},[461],[420,1200,1201,1204],{"style":464},[420,1202],{"className":1203,"style":469},[468],[420,1205,1207],{"className":1206},[473,474,475,476],[420,1208,627],{"className":1209},[626,476],[420,1211],{"className":1212,"style":649},[648],[420,1214,1216],{"className":1215},[653],"=",[420,1218],{"className":1219,"style":649},[648],[420,1221,1223,1226],{"className":1222},[432],[420,1224],{"className":1225,"style":512},[436],[420,1227,1229],{"className":1228},[441],"38",", so it is pruned without ever\nbeing expanded; the blue path is the active best completion that established the\nincumbent.",[1232,1233,1235],"h3",{"id":1234},"worked-example-01-knapsack-by-branch-and-bound","Worked example: 0\u002F1 knapsack by branch and bound",[381,1237,1238,1239,1254,1255,1317,1318,1373,1374,1392,1393,1421,1422,1438,1439,1480,1481,1484,1485,1488,1489,1492,1589,1590,1607,1608,1624,1625,1640],{},"We have ",[420,1240,1242],{"className":1241},[423],[420,1243,1245],{"className":1244,"ariaHidden":428},[427],[420,1246,1248,1251],{"className":1247},[432],[420,1249],{"className":1250,"style":495},[436],[420,1252,481],{"className":1253},[441,480]," items with values ",[420,1256,1258],{"className":1257},[423],[420,1259,1261],{"className":1260,"ariaHidden":428},[427],[420,1262,1264,1268],{"className":1263},[432],[420,1265],{"className":1266,"style":1267},[436],"height:0.5806em;vertical-align:-0.15em;",[420,1269,1271,1276],{"className":1270},[441],[420,1272,1275],{"className":1273,"style":1274},[441,480],"margin-right:0.0359em;","v",[420,1277,1279],{"className":1278},[449],[420,1280,1283,1308],{"className":1281},[453,1282],"vlist-t2",[420,1284,1286,1303],{"className":1285},[457],[420,1287,1290],{"className":1288,"style":1289},[461],"height:0.3117em;",[420,1291,1293,1296],{"style":1292},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[420,1294],{"className":1295,"style":469},[468],[420,1297,1299],{"className":1298},[473,474,475,476],[420,1300,1302],{"className":1301},[441,480,476],"i",[420,1304,1307],{"className":1305},[1306],"vlist-s","​",[420,1309,1311],{"className":1310},[457],[420,1312,1315],{"className":1313,"style":1314},[461],"height:0.15em;",[420,1316],{}," and weights ",[420,1319,1321],{"className":1320},[423],[420,1322,1324],{"className":1323,"ariaHidden":428},[427],[420,1325,1327,1330],{"className":1326},[432],[420,1328],{"className":1329,"style":1267},[436],[420,1331,1333,1338],{"className":1332},[441],[420,1334,1337],{"className":1335,"style":1336},[441,480],"margin-right:0.0269em;","w",[420,1339,1341],{"className":1340},[449],[420,1342,1344,1365],{"className":1343},[453,1282],[420,1345,1347,1362],{"className":1346},[457],[420,1348,1350],{"className":1349,"style":1289},[461],[420,1351,1353,1356],{"style":1352},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[420,1354],{"className":1355,"style":469},[468],[420,1357,1359],{"className":1358},[473,474,475,476],[420,1360,1302],{"className":1361},[441,480,476],[420,1363,1307],{"className":1364},[1306],[420,1366,1368],{"className":1367},[457],[420,1369,1371],{"className":1370,"style":1314},[461],[420,1372],{}," and a capacity ",[420,1375,1377],{"className":1376},[423],[420,1378,1380],{"className":1379,"ariaHidden":428},[427],[420,1381,1383,1387],{"className":1382},[432],[420,1384],{"className":1385,"style":1386},[436],"height:0.6833em;",[420,1388,1391],{"className":1389,"style":1390},[441,480],"margin-right:0.1389em;","W",";\nchoose a subset of maximum total value with total weight ",[420,1394,1396],{"className":1395},[423],[420,1397,1399,1412],{"className":1398,"ariaHidden":428},[427],[420,1400,1402,1406,1409],{"className":1401},[432],[420,1403],{"className":1404,"style":1405},[436],"height:0.7719em;vertical-align:-0.136em;",[420,1407,654],{"className":1408},[653],[420,1410],{"className":1411,"style":649},[648],[420,1413,1415,1418],{"className":1414},[432],[420,1416],{"className":1417,"style":1386},[436],[420,1419,1391],{"className":1420,"style":1390},[441,480],". The decision\ntree is binary, take item ",[420,1423,1425],{"className":1424},[423],[420,1426,1428],{"className":1427,"ariaHidden":428},[427],[420,1429,1431,1435],{"className":1430},[432],[420,1432],{"className":1433,"style":1434},[436],"height:0.6595em;",[420,1436,1302],{"className":1437},[441,480]," or skip it, so it has ",[420,1440,1442],{"className":1441},[423],[420,1443,1445],{"className":1444,"ariaHidden":428},[427],[420,1446,1448,1451],{"className":1447},[432],[420,1449],{"className":1450,"style":437},[436],[420,1452,1454,1457],{"className":1453},[441],[420,1455,445],{"className":1456},[441],[420,1458,1460],{"className":1459},[449],[420,1461,1463],{"className":1462},[453],[420,1464,1466],{"className":1465},[457],[420,1467,1469],{"className":1468,"style":437},[461],[420,1470,1471,1474],{"style":464},[420,1472],{"className":1473,"style":469},[468],[420,1475,1477],{"className":1476},[473,474,475,476],[420,1478,481],{"className":1479},[441,480,476]," leaves. To bound a\nnode, we use the ",[388,1482,1483],{},"LP relaxation",": relax the integrality constraint and allow a\n",[392,1486,1487],{},"fraction"," of the next item. First order all items by ",[388,1490,1491],{},"value density",[420,1493,1495],{"className":1494},[423],[420,1496,1498],{"className":1497,"ariaHidden":428},[427],[420,1499,1501,1505,1545,1549],{"className":1500},[432],[420,1502],{"className":1503,"style":1504},[436],"height:1em;vertical-align:-0.25em;",[420,1506,1508,1511],{"className":1507},[441],[420,1509,1275],{"className":1510,"style":1274},[441,480],[420,1512,1514],{"className":1513},[449],[420,1515,1517,1537],{"className":1516},[453,1282],[420,1518,1520,1534],{"className":1519},[457],[420,1521,1523],{"className":1522,"style":1289},[461],[420,1524,1525,1528],{"style":1292},[420,1526],{"className":1527,"style":469},[468],[420,1529,1531],{"className":1530},[473,474,475,476],[420,1532,1302],{"className":1533},[441,480,476],[420,1535,1307],{"className":1536},[1306],[420,1538,1540],{"className":1539},[457],[420,1541,1543],{"className":1542,"style":1314},[461],[420,1544],{},[420,1546,1548],{"className":1547},[441],"\u002F",[420,1550,1552,1555],{"className":1551},[441],[420,1553,1337],{"className":1554,"style":1336},[441,480],[420,1556,1558],{"className":1557},[449],[420,1559,1561,1581],{"className":1560},[453,1282],[420,1562,1564,1578],{"className":1563},[457],[420,1565,1567],{"className":1566,"style":1289},[461],[420,1568,1569,1572],{"style":1352},[420,1570],{"className":1571,"style":469},[468],[420,1573,1575],{"className":1574},[473,474,475,476],[420,1576,1302],{"className":1577},[441,480,476],[420,1579,1307],{"className":1580},[1306],[420,1582,1584],{"className":1583},[457],[420,1585,1587],{"className":1586,"style":1314},[461],[420,1588],{},", descending. At a node that has fixed a prefix of decisions, with\naccumulated value ",[420,1591,1593],{"className":1592},[423],[420,1594,1596],{"className":1595,"ariaHidden":428},[427],[420,1597,1599,1602],{"className":1598},[432],[420,1600],{"className":1601,"style":1386},[436],[420,1603,1606],{"className":1604,"style":1605},[441,480],"margin-right:0.2222em;","V"," and remaining capacity ",[420,1609,1611],{"className":1610},[423],[420,1612,1614],{"className":1613,"ariaHidden":428},[427],[420,1615,1617,1620],{"className":1616},[432],[420,1618],{"className":1619,"style":495},[436],[420,1621,1623],{"className":1622},[441,480],"c",", greedily fill ",[420,1626,1628],{"className":1627},[423],[420,1629,1631],{"className":1630,"ariaHidden":428},[427],[420,1632,1634,1637],{"className":1633},[432],[420,1635],{"className":1636,"style":495},[436],[420,1638,1623],{"className":1639},[441,480]," with the\nnot-yet-decided items in density order, taking the last one fractionally:",[420,1642,1645],{"className":1643},[1644],"katex-display",[420,1646,1648],{"className":1647},[423],[420,1649,1651,1679,1705,1886,1923,2093],{"className":1650,"ariaHidden":428},[427],[420,1652,1654,1657,1664,1667,1670,1673,1676],{"className":1653},[432],[420,1655],{"className":1656,"style":576},[436],[420,1658,1661],{"className":1659},[441,1660],"text",[420,1662,540],{"className":1663},[441],[420,1665],{"className":1666,"style":649},[648],[420,1668],{"className":1669,"style":649},[648],[420,1671,1216],{"className":1672},[653],[420,1674],{"className":1675,"style":649},[648],[420,1677],{"className":1678,"style":649},[648],[420,1680,1682,1686,1689,1692,1695,1699,1702],{"className":1681},[432],[420,1683],{"className":1684,"style":1685},[436],"height:0.7667em;vertical-align:-0.0833em;",[420,1687,1606],{"className":1688,"style":1605},[441,480],[420,1690],{"className":1691,"style":649},[648],[420,1693],{"className":1694,"style":1605},[648],[420,1696,1698],{"className":1697},[626],"+",[420,1700],{"className":1701,"style":649},[648],[420,1703],{"className":1704,"style":1605},[648],[420,1706,1708,1712,1827,1831,1871,1874,1877,1880,1883],{"className":1707},[432],[420,1709],{"className":1710,"style":1711},[436],"height:2.4502em;vertical-align:-1.4002em;",[420,1713,1717],{"className":1714},[1715,1716],"mop","op-limits",[420,1718,1720,1818],{"className":1719},[453,1282],[420,1721,1723,1815],{"className":1722},[457],[420,1724,1727,1801],{"className":1725,"style":1726},[461],"height:1.05em;",[420,1728,1730,1734],{"style":1729},"top:-1.8557em;margin-left:0em;",[420,1731],{"className":1732,"style":1733},[468],"height:3.05em;",[420,1735,1737],{"className":1736},[473,474,475,476],[420,1738,1740,1743,1747],{"className":1739},[441,476],[420,1741,1302],{"className":1742},[441,480,476],[420,1744,1746],{"className":1745},[653,476],"∈",[420,1748,1750,1754],{"className":1749},[441,476],[420,1751,1753],{"className":1752,"style":1390},[441,480,476],"F",[420,1755,1757],{"className":1756},[449],[420,1758,1760,1792],{"className":1759},[453,1282],[420,1761,1763,1789],{"className":1762},[457],[420,1764,1767],{"className":1765,"style":1766},[461],"height:0.3448em;",[420,1768,1770,1774],{"style":1769},"top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;",[420,1771],{"className":1772,"style":1773},[468],"height:2.5em;",[420,1775,1779],{"className":1776},[473,1777,1778,476],"reset-size3","size1",[420,1780,1782],{"className":1781},[441,476],[420,1783,1785],{"className":1784},[441,1660,476],[420,1786,1788],{"className":1787},[441,476],"full",[420,1790,1307],{"className":1791},[1306],[420,1793,1795],{"className":1794},[457],[420,1796,1799],{"className":1797,"style":1798},[461],"height:0.1512em;",[420,1800],{},[420,1802,1804,1807],{"style":1803},"top:-3.05em;",[420,1805],{"className":1806,"style":1733},[468],[420,1808,1809],{},[420,1810,1814],{"className":1811},[1715,1812,1813],"op-symbol","large-op","∑",[420,1816,1307],{"className":1817},[1306],[420,1819,1821],{"className":1820},[457],[420,1822,1825],{"className":1823,"style":1824},[461],"height:1.4002em;",[420,1826],{},[420,1828],{"className":1829,"style":1830},[648],"margin-right:0.1667em;",[420,1832,1834,1837],{"className":1833},[441],[420,1835,1275],{"className":1836,"style":1274},[441,480],[420,1838,1840],{"className":1839},[449],[420,1841,1843,1863],{"className":1842},[453,1282],[420,1844,1846,1860],{"className":1845},[457],[420,1847,1849],{"className":1848,"style":1289},[461],[420,1850,1851,1854],{"style":1292},[420,1852],{"className":1853,"style":469},[468],[420,1855,1857],{"className":1856},[473,474,475,476],[420,1858,1302],{"className":1859},[441,480,476],[420,1861,1307],{"className":1862},[1306],[420,1864,1866],{"className":1865},[457],[420,1867,1869],{"className":1868,"style":1314},[461],[420,1870],{},[420,1872],{"className":1873,"style":649},[648],[420,1875],{"className":1876,"style":1605},[648],[420,1878,1698],{"className":1879},[626],[420,1881],{"className":1882,"style":649},[648],[420,1884],{"className":1885,"style":1605},[648],[420,1887,1889,1893,1903,1906,1909,1913,1917,1920],{"className":1888},[432],[420,1890],{"className":1891,"style":1892},[436],"height:1.8em;vertical-align:-0.65em;",[420,1894,1897],{"className":1895},[1896],"mopen",[420,1898,1902],{"className":1899},[1900,1901],"delimsizing","size2","(",[420,1904,1623],{"className":1905},[441,480],[420,1907],{"className":1908,"style":1605},[648],[420,1910,1912],{"className":1911},[626],"−",[420,1914],{"className":1915,"style":1916},[648],"margin-right:-0.1667em;",[420,1918],{"className":1919,"style":1916},[648],[420,1921],{"className":1922,"style":1605},[648],[420,1924,1926,1930,2026,2029,2032,2035,2075,2083,2086,2090],{"className":1925},[432],[420,1927],{"className":1928,"style":1929},[436],"height:2.5502em;vertical-align:-1.4002em;",[420,1931,1933],{"className":1932},[1715,1716],[420,1934,1936,2018],{"className":1935},[453,1282],[420,1937,1939,2015],{"className":1938},[457],[420,1940,1942,2005],{"className":1941,"style":1726},[461],[420,1943,1944,1947],{"style":1729},[420,1945],{"className":1946,"style":1733},[468],[420,1948,1950],{"className":1949},[473,474,475,476],[420,1951,1953,1956,1959],{"className":1952},[441,476],[420,1954,1302],{"className":1955},[441,480,476],[420,1957,1746],{"className":1958},[653,476],[420,1960,1962,1965],{"className":1961},[441,476],[420,1963,1753],{"className":1964,"style":1390},[441,480,476],[420,1966,1968],{"className":1967},[449],[420,1969,1971,1997],{"className":1970},[453,1282],[420,1972,1974,1994],{"className":1973},[457],[420,1975,1977],{"className":1976,"style":1766},[461],[420,1978,1979,1982],{"style":1769},[420,1980],{"className":1981,"style":1773},[468],[420,1983,1985],{"className":1984},[473,1777,1778,476],[420,1986,1988],{"className":1987},[441,476],[420,1989,1991],{"className":1990},[441,1660,476],[420,1992,1788],{"className":1993},[441,476],[420,1995,1307],{"className":1996},[1306],[420,1998,2000],{"className":1999},[457],[420,2001,2003],{"className":2002,"style":1798},[461],[420,2004],{},[420,2006,2007,2010],{"style":1803},[420,2008],{"className":2009,"style":1733},[468],[420,2011,2012],{},[420,2013,1814],{"className":2014},[1715,1812,1813],[420,2016,1307],{"className":2017},[1306],[420,2019,2021],{"className":2020},[457],[420,2022,2024],{"className":2023,"style":1824},[461],[420,2025],{},[420,2027],{"className":2028,"style":1916},[648],[420,2030],{"className":2031,"style":1916},[648],[420,2033],{"className":2034,"style":1830},[648],[420,2036,2038,2041],{"className":2037},[441],[420,2039,1337],{"className":2040,"style":1336},[441,480],[420,2042,2044],{"className":2043},[449],[420,2045,2047,2067],{"className":2046},[453,1282],[420,2048,2050,2064],{"className":2049},[457],[420,2051,2053],{"className":2052,"style":1289},[461],[420,2054,2055,2058],{"style":1352},[420,2056],{"className":2057,"style":469},[468],[420,2059,2061],{"className":2060},[473,474,475,476],[420,2062,1302],{"className":2063},[441,480,476],[420,2065,1307],{"className":2066},[1306],[420,2068,2070],{"className":2069},[457],[420,2071,2073],{"className":2072,"style":1314},[461],[420,2074],{},[420,2076,2079],{"className":2077},[2078],"mclose",[420,2080,2082],{"className":2081},[1900,1901],")",[420,2084],{"className":2085,"style":1605},[648],[420,2087,2089],{"className":2088},[626],"⋅",[420,2091],{"className":2092,"style":1605},[648],[420,2094,2096,2100,2249],{"className":2095},[432],[420,2097],{"className":2098,"style":2099},[436],"height:2.0797em;vertical-align:-0.9721em;",[420,2101,2103,2107,2246],{"className":2102},[441],[420,2104],{"className":2105},[1896,2106],"nulldelimiter",[420,2108,2111],{"className":2109},[2110],"mfrac",[420,2112,2114,2237],{"className":2113},[453,1282],[420,2115,2117,2234],{"className":2116},[457],[420,2118,2121,2174,2185],{"className":2119,"style":2120},[461],"height:1.1076em;",[420,2122,2124,2128],{"style":2123},"top:-2.314em;",[420,2125],{"className":2126,"style":2127},[468],"height:3em;",[420,2129,2131],{"className":2130},[441],[420,2132,2134,2137],{"className":2133},[441],[420,2135,1337],{"className":2136,"style":1336},[441,480],[420,2138,2140],{"className":2139},[449],[420,2141,2143,2165],{"className":2142},[453,1282],[420,2144,2146,2162],{"className":2145},[457],[420,2147,2149],{"className":2148,"style":1289},[461],[420,2150,2151,2154],{"style":1352},[420,2152],{"className":2153,"style":469},[468],[420,2155,2157],{"className":2156},[473,474,475,476],[420,2158,2161],{"className":2159,"style":2160},[441,480,476],"margin-right:0.0572em;","j",[420,2163,1307],{"className":2164},[1306],[420,2166,2168],{"className":2167},[457],[420,2169,2172],{"className":2170,"style":2171},[461],"height:0.2861em;",[420,2173],{},[420,2175,2177,2180],{"style":2176},"top:-3.23em;",[420,2178],{"className":2179,"style":2127},[468],[420,2181],{"className":2182,"style":2184},[2183],"frac-line","border-bottom-width:0.04em;",[420,2186,2188,2191],{"style":2187},"top:-3.677em;",[420,2189],{"className":2190,"style":2127},[468],[420,2192,2194],{"className":2193},[441],[420,2195,2197,2200],{"className":2196},[441],[420,2198,1275],{"className":2199,"style":1274},[441,480],[420,2201,2203],{"className":2202},[449],[420,2204,2206,2226],{"className":2205},[453,1282],[420,2207,2209,2223],{"className":2208},[457],[420,2210,2212],{"className":2211,"style":1289},[461],[420,2213,2214,2217],{"style":1292},[420,2215],{"className":2216,"style":469},[468],[420,2218,2220],{"className":2219},[473,474,475,476],[420,2221,2161],{"className":2222,"style":2160},[441,480,476],[420,2224,1307],{"className":2225},[1306],[420,2227,2229],{"className":2228},[457],[420,2230,2232],{"className":2231,"style":2171},[461],[420,2233],{},[420,2235,1307],{"className":2236},[1306],[420,2238,2240],{"className":2239},[457],[420,2241,2244],{"className":2242,"style":2243},[461],"height:0.9721em;",[420,2245],{},[420,2247],{"className":2248},[2078,2106],[420,2250,2253],{"className":2251},[2252],"mpunct",",",[381,2255,2256,2257,2318,2319,2335,2336,2339,2340],{},"where ",[420,2258,2260],{"className":2259},[423],[420,2261,2263],{"className":2262,"ariaHidden":428},[427],[420,2264,2266,2270],{"className":2265},[432],[420,2267],{"className":2268,"style":2269},[436],"height:0.8333em;vertical-align:-0.15em;",[420,2271,2273,2276],{"className":2272},[441],[420,2274,1753],{"className":2275,"style":1390},[441,480],[420,2277,2279],{"className":2278},[449],[420,2280,2282,2310],{"className":2281},[453,1282],[420,2283,2285,2307],{"className":2284},[457],[420,2286,2289],{"className":2287,"style":2288},[461],"height:0.3361em;",[420,2290,2292,2295],{"style":2291},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[420,2293],{"className":2294,"style":469},[468],[420,2296,2298],{"className":2297},[473,474,475,476],[420,2299,2301],{"className":2300},[441,476],[420,2302,2304],{"className":2303},[441,1660,476],[420,2305,1788],{"className":2306},[441,476],[420,2308,1307],{"className":2309},[1306],[420,2311,2313],{"className":2312},[457],[420,2314,2316],{"className":2315,"style":1314},[461],[420,2317],{}," are the remaining items that fit wholly and ",[420,2320,2322],{"className":2321},[423],[420,2323,2325],{"className":2324,"ariaHidden":428},[427],[420,2326,2328,2332],{"className":2327},[432],[420,2329],{"className":2330,"style":2331},[436],"height:0.854em;vertical-align:-0.1944em;",[420,2333,2161],{"className":2334,"style":2160},[441,480]," is the\nfirst item that overflows (filled fractionally). 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",[420,2715,2717],{"className":2716},[423],[420,2718,2720],{"className":2719,"ariaHidden":428},[427],[420,2721,2723,2726],{"className":2722},[432],[420,2724],{"className":2725,"style":495},[436],[420,2727,1623],{"className":2728},[441,480]," is the relaxation's over-estimate)",[2731,2732,2736],"pre",{"className":2733,"code":2734,"language":2735,"meta":376,"style":376},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{KnapsackBnB}$ — maximize value within capacity $W$ (items sorted by $v_i\u002Fw_i$)\n$z^* \\gets 0$ \u002F\u002F incumbent value\n$\\textsc{Expand}(i = 0,\\ V = 0,\\ \\text{weight} = 0)$:\n  if $\\text{weight} > W$ then return \u002F\u002F infeasible\n  if $V > z^*$ then $z^* \\gets V$ \u002F\u002F new incumbent\n  if $i = n$ then return\n  if $\\textsc{Bound}(i, V, \\text{weight}) \\le z^*$ then return \u002F\u002F prune by value\n  $\\textsc{Expand}(i+1,\\ V + v_i,\\ \\text{weight} + w_i)$ \u002F\u002F take item $i$\n  $\\textsc{Expand}(i+1,\\ V,\\ \\text{weight})$ \u002F\u002F skip item $i$\n\n$\\textsc{Bound}(i, V, \\text{weight})$:\n  $b \\gets V;\\quad c \\gets W - \\text{weight}$\n  for $j \\gets i$ to $n-1$ do\n    if $w_j \\le c$ then $b \\gets b + v_j;\\ c \\gets c - w_j$\n    else return $b + c \\cdot v_j \u002F w_j$ \u002F\u002F fractional fill\n  return $b$\n","algorithm",[2737,2738,2739,2745,2750,2755,2760,2765,2770,2775,2780,2785,2791,2796,2801,2807,2813,2819],"code",{"__ignoreMap":376},[420,2740,2742],{"class":2741,"line":6},"line",[420,2743,2744],{},"caption: $\\textsc{KnapsackBnB}$ — maximize value within capacity $W$ (items sorted by $v_i\u002Fw_i$)\n",[420,2746,2747],{"class":2741,"line":18},[420,2748,2749],{},"$z^* \\gets 0$ \u002F\u002F incumbent value\n",[420,2751,2752],{"class":2741,"line":24},[420,2753,2754],{},"$\\textsc{Expand}(i = 0,\\ V = 0,\\ \\text{weight} = 0)$:\n",[420,2756,2757],{"class":2741,"line":73},[420,2758,2759],{},"  if $\\text{weight} > W$ then return \u002F\u002F infeasible\n",[420,2761,2762],{"class":2741,"line":102},[420,2763,2764],{},"  if $V > z^*$ then $z^* \\gets V$ \u002F\u002F new incumbent\n",[420,2766,2767],{"class":2741,"line":108},[420,2768,2769],{},"  if $i = n$ then return\n",[420,2771,2772],{"class":2741,"line":116},[420,2773,2774],{},"  if $\\textsc{Bound}(i, V, \\text{weight}) \\le z^*$ then return \u002F\u002F prune by value\n",[420,2776,2777],{"class":2741,"line":196},[420,2778,2779],{},"  $\\textsc{Expand}(i+1,\\ V + v_i,\\ \\text{weight} + w_i)$ \u002F\u002F take item $i$\n",[420,2781,2782],{"class":2741,"line":202},[420,2783,2784],{},"  $\\textsc{Expand}(i+1,\\ V,\\ \\text{weight})$ \u002F\u002F skip item $i$\n",[420,2786,2787],{"class":2741,"line":283},[420,2788,2790],{"emptyLinePlaceholder":2789},true,"\n",[420,2792,2793],{"class":2741,"line":333},[420,2794,2795],{},"$\\textsc{Bound}(i, V, \\text{weight})$:\n",[420,2797,2798],{"class":2741,"line":354},[420,2799,2800],{},"  $b \\gets V;\\quad c \\gets W - \\text{weight}$\n",[420,2802,2804],{"class":2741,"line":2803},13,[420,2805,2806],{},"  for $j \\gets i$ to $n-1$ do\n",[420,2808,2810],{"class":2741,"line":2809},14,[420,2811,2812],{},"    if $w_j \\le c$ then $b \\gets b + v_j;\\ c \\gets c - w_j$\n",[420,2814,2816],{"class":2741,"line":2815},15,[420,2817,2818],{},"    else return $b + c \\cdot v_j \u002F w_j$ \u002F\u002F fractional fill\n",[420,2820,2822],{"class":2741,"line":2821},16,[420,2823,2824],{},"  return $b$\n",[381,2826,2827,2828,2831,2832,2835,2836,2840,2841,2867,2868,2871,2872,2887,2888,2903,2904,2920,2921,2936,2937],{},"Branching on ",[392,2829,2830],{},"take"," before ",[392,2833,2834],{},"skip"," tends to find a heavy, valuable incumbent\nearly, which makes the bound bite sooner. Note what this buys us against the\n",[385,2837,2838],{"href":252},[388,2839,232],{}," solution. The\nDP runs in ",[420,2842,2844],{"className":2843},[423],[420,2845,2847],{"className":2846,"ariaHidden":428},[427],[420,2848,2850,2853,2857,2860,2864],{"className":2849},[432],[420,2851],{"className":2852,"style":1504},[436],[420,2854,2856],{"className":2855},[441],"Θ",[420,2858,1902],{"className":2859},[1896],[420,2861,2863],{"className":2862,"style":1390},[441,480],"nW",[420,2865,2082],{"className":2866},[2078]," time, ",[392,2869,2870],{},"pseudo","-polynomial, because ",[420,2873,2875],{"className":2874},[423],[420,2876,2878],{"className":2877,"ariaHidden":428},[427],[420,2879,2881,2884],{"className":2880},[432],[420,2882],{"className":2883,"style":1386},[436],[420,2885,1391],{"className":2886,"style":1390},[441,480]," enters as a\nmagnitude, not a bit-length. When ",[420,2889,2891],{"className":2890},[423],[420,2892,2894],{"className":2893,"ariaHidden":428},[427],[420,2895,2897,2900],{"className":2896},[432],[420,2898],{"className":2899,"style":1386},[436],[420,2901,1391],{"className":2902,"style":1390},[441,480]," is enormous (say weights are ",[420,2905,2907],{"className":2906},[423],[420,2908,2910],{"className":2909,"ariaHidden":428},[427],[420,2911,2913,2916],{"className":2912},[432],[420,2914],{"className":2915,"style":512},[436],[420,2917,2919],{"className":2918},[441],"9","-digit\nnumbers) the DP table is hopeless, yet if ",[420,2922,2924],{"className":2923},[423],[420,2925,2927],{"className":2926,"ariaHidden":428},[427],[420,2928,2930,2933],{"className":2929},[432],[420,2931],{"className":2932,"style":495},[436],[420,2934,481],{"className":2935},[441,480]," is moderate the branch-and-bound\ntree, ferociously pruned, finishes quickly. The two methods are complementary:\n",[388,2938,2939,2940,2955,2956,2971,2972,2987],{},"DP wins when ",[420,2941,2943],{"className":2942},[423],[420,2944,2946],{"className":2945,"ariaHidden":428},[427],[420,2947,2949,2952],{"className":2948},[432],[420,2950],{"className":2951,"style":1386},[436],[420,2953,1391],{"className":2954,"style":1390},[441,480]," is small; branch and bound wins when ",[420,2957,2959],{"className":2958},[423],[420,2960,2962],{"className":2961,"ariaHidden":428},[427],[420,2963,2965,2968],{"className":2964},[432],[420,2966],{"className":2967,"style":1386},[436],[420,2969,1391],{"className":2970,"style":1390},[441,480]," is huge but ",[420,2973,2975],{"className":2974},[423],[420,2976,2978],{"className":2977,"ariaHidden":428},[427],[420,2979,2981,2984],{"className":2980},[432],[420,2982],{"className":2983,"style":495},[436],[420,2985,481],{"className":2986},[441,480]," is\nmoderate.",[905,2989,2991,3368],{"className":2990},[908,909],[911,2992,2996],{"xmlns":913,"width":2993,"height":2994,"viewBox":2995},"411.994","163.142","-75 -75 308.996 122.356",[918,2997,2998,3001,3040,3084,3252,3301,3323],{"stroke":920,"style":921},[923,2999],{"fill":925,"d":3000},"M59.235-72.07H40.827a4 4 0 0 0-4 4v15.15a4 4 0 0 0 4 4h18.408a4 4 0 0 0 4-4v-15.15a4 4 0 0 0-4-4ZM36.827-48.92",[918,3002,3003,3010,3016,3022,3028,3034],{"stroke":925,"fontSize":2919},[918,3004,3006],{"transform":3005},"translate(-10.204 8.575)",[923,3007],{"d":3008,"fill":920,"stroke":920,"className":3009,"style":935},"M52.679-71.407L51.870-77.146Q51.804-77.327 51.220-77.327Q51.114-77.327 51.114-77.445Q51.114-77.643 51.277-77.643L53.461-77.643Q53.505-77.643 53.533-77.608Q53.562-77.573 53.562-77.533Q53.562-77.454 53.531-77.390Q53.500-77.327 53.434-77.327Q52.714-77.327 52.714-77.085L53.382-72.370L56.216-76.795Q56.225-76.839 56.265-76.922Q56.304-77.006 56.304-77.058Q56.304-77.208 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",[420,3399,3401],{"className":3400},[423],[420,3402,3404],{"className":3403,"ariaHidden":428},[427],[420,3405,3407,3410,3414,3421,3425,3429,3432,3435,3438,3442,3445,3448,3453,3459,3462,3466,3469,3472,3476,3479,3482,3485,3490,3496,3499,3502,3505,3508,3512],{"className":3406},[432],[420,3408],{"className":3409,"style":1504},[436],[420,3411,3413],{"className":3412},[441,480],"A",[420,3415,3417],{"className":3416},[441],[420,3418,3420],{"className":3419},[653],":",[420,3422,3424],{"className":3423},[1896],"⟨",[420,3426,3428],{"className":3427},[441],"10",[420,3430,2253],{"className":3431},[2252],[420,3433],{"className":3434,"style":1830},[648],[420,3436,445],{"className":3437},[441],[420,3439,3441],{"className":3440},[2078],"⟩",[420,3443,2253],{"className":3444},[2252],[420,3446],{"className":3447,"style":1830},[648],[420,3449,3452],{"className":3450,"style":3451},[441,480],"margin-right:0.0502em;","B",[420,3454,3456],{"className":3455},[441],[420,3457,3420],{"className":3458},[653],[420,3460,3424],{"className":3461},[1896],[420,3463,3465],{"className":3464},[441],"12",[420,3467,2253],{"className":3468},[2252],[420,3470],{"className":3471,"style":1830},[648],[420,3473,3475],{"className":3474},[441],"4",[420,3477,3441],{"className":3478},[2078],[420,3480,2253],{"className":3481},[2252],[420,3483],{"className":3484,"style":1830},[648],[420,3486,3489],{"className":3487,"style":3488},[441,480],"margin-right:0.0715em;","C",[420,3491,3493],{"className":3492},[441],[420,3494,3420],{"className":3495},[653],[420,3497,3424],{"className":3498},[1896],[420,3500,3396],{"className":3501},[441],[420,3503,2253],{"className":3504},[2252],[420,3506],{"className":3507,"style":1830},[648],[420,3509,3511],{"className":3510},[441],"3",[420,3513,3441],{"className":3514},[2078]," by density). The skip-",[420,3517,3519],{"className":3518},[423],[420,3520,3522],{"className":3521,"ariaHidden":428},[427],[420,3523,3525,3528],{"className":3524},[432],[420,3526],{"className":3527,"style":1386},[436],[420,3529,3413],{"className":3530},[441,480]," node's LP bound ",[420,3533,3535],{"className":3534},[423],[420,3536,3538,3557],{"className":3537,"ariaHidden":428},[427],[420,3539,3541,3544,3548,3551,3554],{"className":3540},[432],[420,3542],{"className":3543,"style":1161},[436],[420,3545,3547],{"className":3546},[441],"16",[420,3549],{"className":3550,"style":649},[648],[420,3552,654],{"className":3553},[653],[420,3555],{"className":3556,"style":649},[648],[420,3558,3560,3563,3592,3598],{"className":3559},[432],[420,3561],{"className":3562,"style":594},[436],[420,3564,3566,3569],{"className":3565},[441],[420,3567,602],{"className":3568,"style":601},[441,480],[420,3570,3572],{"className":3571},[449],[420,3573,3575],{"className":3574},[453],[420,3576,3578],{"className":3577},[457],[420,3579,3581],{"className":3580,"style":594},[461],[420,3582,3583,3586],{"style":464},[420,3584],{"className":3585,"style":469},[468],[420,3587,3589],{"className":3588},[473,474,475,476],[420,3590,627],{"className":3591},[626,476],[420,3593,3595],{"className":3594},[441],[420,3596,1216],{"className":3597},[653],[420,3599,3601],{"className":3600},[441],"22",", so that subtree is pruned (red)",[381,3604,3605,3606,3608,3609,3641,3642,3692,3693,3708,3709,3724,3725,3740,3741,3826,3827,3842,3843,3978,3979,3994],{},"Branching ",[392,3607,2830],{},"-first dives straight to the incumbent ",[420,3610,3612],{"className":3611},[423],[420,3613,3615],{"className":3614,"ariaHidden":428},[427],[420,3616,3618,3621,3625,3628,3631,3634,3637],{"className":3617},[432],[420,3619],{"className":3620,"style":1504},[436],[420,3622,3624],{"className":3623},[1896],"{",[420,3626,3413],{"className":3627},[441,480],[420,3629,2253],{"className":3630},[2252],[420,3632],{"className":3633,"style":1830},[648],[420,3635,3452],{"className":3636,"style":3451},[441,480],[420,3638,3640],{"className":3639},[2078],"}"," with value\n",[420,3643,3645],{"className":3644},[423],[420,3646,3648],{"className":3647,"ariaHidden":428},[427],[420,3649,3651,3654,3683,3689],{"className":3650},[432],[420,3652],{"className":3653,"style":594},[436],[420,3655,3657,3660],{"className":3656},[441],[420,3658,602],{"className":3659,"style":601},[441,480],[420,3661,3663],{"className":3662},[449],[420,3664,3666],{"className":3665},[453],[420,3667,3669],{"className":3668},[457],[420,3670,3672],{"className":3671,"style":594},[461],[420,3673,3674,3677],{"style":464},[420,3675],{"className":3676,"style":469},[468],[420,3678,3680],{"className":3679},[473,474,475,476],[420,3681,627],{"className":3682},[626,476],[420,3684,3686],{"className":3685},[441],[420,3687,1216],{"className":3688},[653],[420,3690,3601],{"className":3691},[441],". The skip-",[420,3694,3696],{"className":3695},[423],[420,3697,3699],{"className":3698,"ariaHidden":428},[427],[420,3700,3702,3705],{"className":3701},[432],[420,3703],{"className":3704,"style":1386},[436],[420,3706,3413],{"className":3707},[441,480]," subtree's optimistic LP bound is only ",[420,3710,3712],{"className":3711},[423],[420,3713,3715],{"className":3714,"ariaHidden":428},[427],[420,3716,3718,3721],{"className":3717},[432],[420,3719],{"className":3720,"style":512},[436],[420,3722,3547],{"className":3723},[441]," (take ",[420,3726,3728],{"className":3727},[423],[420,3729,3731],{"className":3730,"ariaHidden":428},[427],[420,3732,3734,3737],{"className":3733},[432],[420,3735],{"className":3736,"style":1386},[436],[420,3738,3452],{"className":3739,"style":3451},[441,480],"\nwhole, then ",[420,3742,3744],{"className":3743},[423],[420,3745,3747],{"className":3746,"ariaHidden":428},[427],[420,3748,3750,3754],{"className":3749},[432],[420,3751],{"className":3752,"style":3753},[436],"height:1.1901em;vertical-align:-0.345em;",[420,3755,3757,3760,3823],{"className":3756},[441],[420,3758],{"className":3759},[1896,2106],[420,3761,3763],{"className":3762},[2110],[420,3764,3766,3814],{"className":3765},[453,1282],[420,3767,3769,3811],{"className":3768},[457],[420,3770,3773,3788,3796],{"className":3771,"style":3772},[461],"height:0.8451em;",[420,3774,3776,3779],{"style":3775},"top:-2.655em;",[420,3777],{"className":3778,"style":2127},[468],[420,3780,3782],{"className":3781},[473,474,475,476],[420,3783,3785],{"className":3784},[441,476],[420,3786,3511],{"className":3787},[441,476],[420,3789,3790,3793],{"style":2176},[420,3791],{"className":3792,"style":2127},[468],[420,3794],{"className":3795,"style":2184},[2183],[420,3797,3799,3802],{"style":3798},"top:-3.394em;",[420,3800],{"className":3801,"style":2127},[468],[420,3803,3805],{"className":3804},[473,474,475,476],[420,3806,3808],{"className":3807},[441,476],[420,3809,445],{"className":3810},[441,476],[420,3812,1307],{"className":3813},[1306],[420,3815,3817],{"className":3816},[457],[420,3818,3821],{"className":3819,"style":3820},[461],"height:0.345em;",[420,3822],{},[420,3824],{"className":3825},[2078,2106]," of ",[420,3828,3830],{"className":3829},[423],[420,3831,3833],{"className":3832,"ariaHidden":428},[427],[420,3834,3836,3839],{"className":3835},[432],[420,3837],{"className":3838,"style":1386},[436],[420,3840,3489],{"className":3841,"style":3488},[441,480],": ",[420,3844,3846],{"className":3845},[423],[420,3847,3849,3868,3886,3969],{"className":3848,"ariaHidden":428},[427],[420,3850,3852,3856,3859,3862,3865],{"className":3851},[432],[420,3853],{"className":3854,"style":3855},[436],"height:0.7278em;vertical-align:-0.0833em;",[420,3857,3465],{"className":3858},[441],[420,3860],{"className":3861,"style":1605},[648],[420,3863,1698],{"className":3864},[626],[420,3866],{"className":3867,"style":1605},[648],[420,3869,3871,3874,3877,3880,3883],{"className":3870},[432],[420,3872],{"className":3873,"style":512},[436],[420,3875,445],{"className":3876},[441],[420,3878],{"className":3879,"style":1605},[648],[420,3881,2089],{"className":3882},[626],[420,3884],{"className":3885,"style":1605},[648],[420,3887,3889,3892,3960,3963,3966],{"className":3888},[432],[420,3890],{"className":3891,"style":3753},[436],[420,3893,3895,3898,3957],{"className":3894},[441],[420,3896],{"className":3897},[1896,2106],[420,3899,3901],{"className":3900},[2110],[420,3902,3904,3949],{"className":3903},[453,1282],[420,3905,3907,3946],{"className":3906},[457],[420,3908,3910,3924,3932],{"className":3909,"style":3772},[461],[420,3911,3912,3915],{"style":3775},[420,3913],{"className":3914,"style":2127},[468],[420,3916,3918],{"className":3917},[473,474,475,476],[420,3919,3921],{"className":3920},[441,476],[420,3922,3511],{"className":3923},[441,476],[420,3925,3926,3929],{"style":2176},[420,3927],{"className":3928,"style":2127},[468],[420,3930],{"className":3931,"style":2184},[2183],[420,3933,3934,3937],{"style":3798},[420,3935],{"className":3936,"style":2127},[468],[420,3938,3940],{"className":3939},[473,474,475,476],[420,3941,3943],{"className":3942},[441,476],[420,3944,3396],{"className":3945},[441,476],[420,3947,1307],{"className":3948},[1306],[420,3950,3952],{"className":3951},[457],[420,3953,3955],{"className":3954,"style":3820},[461],[420,3956],{},[420,3958],{"className":3959},[2078,2106],[420,3961],{"className":3962,"style":649},[648],[420,3964,1216],{"className":3965},[653],[420,3967],{"className":3968,"style":649},[648],[420,3970,3972,3975],{"className":3971},[432],[420,3973],{"className":3974,"style":512},[436],[420,3976,3547],{"className":3977},[441],"), which cannot beat\n",[420,3980,3982],{"className":3981},[423],[420,3983,3985],{"className":3984,"ariaHidden":428},[427],[420,3986,3988,3991],{"className":3987},[432],[420,3989],{"className":3990,"style":512},[436],[420,3992,3601],{"className":3993},[441],", so the entire right half is discarded before a single completion is built.",[1232,3996,3998],{"id":3997},"search-order-depth-first-vs-best-first","Search order: depth-first vs best-first",[381,4000,4001,4002,4005,4006,4009,4010,4036,4037,4040,4041,4044,4045,4048,4049,4052],{},"The skeleton above is ",[388,4003,4004],{},"depth-first"," branch and bound: it recurses to a leaf\nfast, so it finds ",[392,4007,4008],{},"some"," complete solution, an incumbent, almost immediately,\nand it uses only ",[420,4011,4013],{"className":4012},[423],[420,4014,4016],{"className":4015,"ariaHidden":428},[427],[420,4017,4019,4022,4027,4030,4033],{"className":4018},[432],[420,4020],{"className":4021,"style":1504},[436],[420,4023,4026],{"className":4024,"style":4025},[441,480],"margin-right:0.0278em;","O",[420,4028,1902],{"className":4029},[1896],[420,4031,481],{"className":4032},[441,480],[420,4034,2082],{"className":4035},[2078]," stack. The cost is that the first incumbent may be poor,\nweakening early pruning. The alternative is ",[388,4038,4039],{},"best-first"," search: keep a\n",[385,4042,4043],{"href":56},"priority queue"," of live nodes keyed by\ntheir bound, and always expand the node with the most promising bound. Best-first\ntends to drive toward the optimum with the fewest ",[392,4046,4047],{},"expansions"," and, for many\nproblems, expands the optimal node first, but it can hold an exponential frontier\nof live nodes in the queue, so its ",[388,4050,4051],{},"memory"," is the liability. The practical\ncompromise is to seed the incumbent with a quick greedy solution, then run\ndepth-first with strong bounds: cheap memory, and a floor high enough that the\nbound prunes hard from the start.",[905,4054,4056,4352],{"className":4055},[908,909],[911,4057,4061],{"xmlns":913,"width":4058,"height":4059,"viewBox":4060},"387.252","207.595","-75 -75 290.439 155.697",[918,4062,4063,4068,4073,4093,4096,4099,4102,4105,4108,4111,4144,4209,4212,4224,4244,4247,4254,4257,4260,4263,4266,4269,4272,4311],{"stroke":920,"style":921},[918,4064,4065],{"stroke":938,"style":939},[923,4066],{"fill":925,"d":4067},"M1.015-63.134a8.536 8.536 0 1 0-17.072 0 8.536 8.536 0 0 0 17.072 0Zm-8.536 0",[918,4069,4070],{"stroke":938,"style":939},[923,4071],{"fill":925,"d":4072},"M-21.747-31.836a8.536 8.536 0 1 0-17.072 0 8.536 8.536 0 0 0 17.072 0Zm-8.536 0",[918,4074,4075,4078,4081,4084,4087,4090],{"stroke":938,"style":939},[923,4076],{"fill":925,"d":4077},"m-12.894-55.746-10.134 13.934",[923,4079],{"fill":938,"stroke":925,"d":4080},"m-24.91-39.225 5.082-2.635-3.2.048-.941-3.059",[923,4082],{"fill":925,"d":4083},"M-33.129-.538a8.536 8.536 0 1 0-17.071 0 8.536 8.536 0 0 0 17.071 0ZM-33.405-23.25l-4.044 11.119",[923,4085],{"fill":938,"stroke":925,"d":4086},"m-38.542-9.124 4.155-3.937-3.062.93-1.75-2.68",[923,4088],{"fill":925,"d":4089},"M-10.366-.538a8.536 8.536 0 1 0-17.072 0 8.536 8.536 0 0 0 17.072 0ZM-27.161-23.25l4.59 12.622",[923,4091],{"stroke":925,"d":4092},"m-21.887-8.748.41-3.554-1.094 1.674-1.914-.58",[923,4094],{"fill":925,"d":4095},"M23.777-31.836a8.536 8.536 0 1 0-17.072 0 8.536 8.536 0 0 0 17.072 0ZM-2.148-55.746 8.927-40.519",[923,4097],{"stroke":925,"d":4098},"m10.103-38.901-.588-3.529-.588 1.911-2-.029",[923,4100],{"fill":925,"d":4101},"M12.396-.538a8.536 8.536 0 1 0-17.072 0 8.536 8.536 0 0 0 17.072 0ZM12.256-23.626 7.529-10.628",[923,4103],{"stroke":925,"d":4104},"m6.845-8.748 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Depth-first dives to a leaf for an early incumbent with ",[420,4357,4359],{"className":4358},[423],[420,4360,4362],{"className":4361,"ariaHidden":428},[427],[420,4363,4365,4368,4371,4374,4377],{"className":4364},[432],[420,4366],{"className":4367,"style":1504},[436],[420,4369,4026],{"className":4370,"style":4025},[441,480],[420,4372,1902],{"className":4373},[1896],[420,4375,481],{"className":4376},[441,480],[420,4378,2082],{"className":4379},[2078]," stack; best-first expands the highest-bound live node — fewer expansions, but an exponential frontier",[381,4382,4383,4384,4420],{},"Depth-first (left) follows one accented path to a leaf, banking an incumbent fast\nwhile holding only the current root-to-node stack. Best-first (right) instead pops\nthe live node of highest bound (",[420,4385,4387],{"className":4386},[423],[420,4388,4390,4411],{"className":4389,"ariaHidden":428},[427],[420,4391,4393,4397,4401,4404,4408],{"className":4392},[432],[420,4394],{"className":4395,"style":4396},[436],"height:0.6835em;vertical-align:-0.0391em;",[420,4398,4400],{"className":4399},[441],"37",[420,4402],{"className":4403,"style":649},[648],[420,4405,4407],{"className":4406},[653],">",[420,4409],{"className":4410,"style":649},[648],[420,4412,4414,4417],{"className":4413},[432],[420,4415],{"className":4416,"style":512},[436],[420,4418,1165],{"className":4419},[441],") from a priority queue, steering toward\nthe optimum in fewer expansions at the cost of keeping the whole frontier in\nmemory.",[523,4422,520],{"id":4423},"meet-in-the-middle",[381,4425,4426,4427,4442,4443,4477,4478,4481,4482,4497,4498,4513,4514,4547,4548,4551,4552,4600,4601,4616,4617,4673,4674,4689,4690,4742,4743,4758,4759,4774,4775,4883,4884,5025],{},"Some problems resist pruning entirely: the bound is weak, the structure\nsymmetric, every branch genuinely live. If the instance is a subset problem over\n",[420,4428,4430],{"className":4429},[423],[420,4431,4433],{"className":4432,"ariaHidden":428},[427],[420,4434,4436,4439],{"className":4435},[432],[420,4437],{"className":4438,"style":495},[436],[420,4440,481],{"className":4441},[441,480]," items and ",[420,4444,4446],{"className":4445},[423],[420,4447,4449,4467],{"className":4448,"ariaHidden":428},[427],[420,4450,4452,4455,4458,4461,4464],{"className":4451},[432],[420,4453],{"className":4454,"style":1405},[436],[420,4456,481],{"className":4457},[441,480],[420,4459],{"className":4460,"style":649},[648],[420,4462,654],{"className":4463},[653],[420,4465],{"className":4466,"style":649},[648],[420,4468,4470,4473],{"className":4469},[432],[420,4471],{"className":4472,"style":512},[436],[420,4474,4476],{"className":4475},[441],"40",", ",[388,4479,4480],{},"meet in the middle"," sidesteps pruning and attacks the\nexponent directly. Split the items into two halves ",[420,4483,4485],{"className":4484},[423],[420,4486,4488],{"className":4487,"ariaHidden":428},[427],[420,4489,4491,4494],{"className":4490},[432],[420,4492],{"className":4493,"style":1386},[436],[420,4495,3413],{"className":4496},[441,480]," and ",[420,4499,4501],{"className":4500},[423],[420,4502,4504],{"className":4503,"ariaHidden":428},[427],[420,4505,4507,4510],{"className":4506},[432],[420,4508],{"className":4509,"style":1386},[436],[420,4511,3452],{"className":4512,"style":3451},[441,480]," of size ",[420,4515,4517],{"className":4516},[423],[420,4518,4520,4534],{"className":4519,"ariaHidden":428},[427],[420,4521,4523,4527,4531],{"className":4522},[432],[420,4524],{"className":4525,"style":4526},[436],"height:0.4831em;",[420,4528,4530],{"className":4529},[653],"≈",[420,4532],{"className":4533,"style":649},[648],[420,4535,4537,4540,4543],{"className":4536},[432],[420,4538],{"className":4539,"style":1504},[436],[420,4541,481],{"className":4542},[441,480],[420,4544,4546],{"className":4545},[441],"\u002F2",". Enumerate ",[392,4549,4550],{},"all"," ",[420,4553,4555],{"className":4554},[423],[420,4556,4558],{"className":4557,"ariaHidden":428},[427],[420,4559,4561,4565],{"className":4560},[432],[420,4562],{"className":4563,"style":4564},[436],"height:0.888em;",[420,4566,4568,4571],{"className":4567},[441],[420,4569,445],{"className":4570},[441],[420,4572,4574],{"className":4573},[449],[420,4575,4577],{"className":4576},[453],[420,4578,4580],{"className":4579},[457],[420,4581,4583],{"className":4582,"style":4564},[461],[420,4584,4585,4588],{"style":464},[420,4586],{"className":4587,"style":469},[468],[420,4589,4591],{"className":4590},[473,474,475,476],[420,4592,4594,4597],{"className":4593},[441,476],[420,4595,481],{"className":4596},[441,480,476],[420,4598,4546],{"className":4599},[441,476]," subset sums of ",[420,4602,4604],{"className":4603},[423],[420,4605,4607],{"className":4606,"ariaHidden":428},[427],[420,4608,4610,4613],{"className":4609},[432],[420,4611],{"className":4612,"style":1386},[436],[420,4614,3413],{"className":4615},[441,480]," into a list ",[420,4618,4620],{"className":4619},[423],[420,4621,4623],{"className":4622,"ariaHidden":428},[427],[420,4624,4626,4629],{"className":4625},[432],[420,4627],{"className":4628,"style":2269},[436],[420,4630,4632,4637],{"className":4631},[441],[420,4633,4636],{"className":4634,"style":4635},[441,480],"margin-right:0.0576em;","S",[420,4638,4640],{"className":4639},[449],[420,4641,4643,4665],{"className":4642},[453,1282],[420,4644,4646,4662],{"className":4645},[457],[420,4647,4650],{"className":4648,"style":4649},[461],"height:0.3283em;",[420,4651,4653,4656],{"style":4652},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[420,4654],{"className":4655,"style":469},[468],[420,4657,4659],{"className":4658},[473,474,475,476],[420,4660,3413],{"className":4661},[441,480,476],[420,4663,1307],{"className":4664},[1306],[420,4666,4668],{"className":4667},[457],[420,4669,4671],{"className":4670,"style":1314},[461],[420,4672],{},", and\nlikewise all subset sums of ",[420,4675,4677],{"className":4676},[423],[420,4678,4680],{"className":4679,"ariaHidden":428},[427],[420,4681,4683,4686],{"className":4682},[432],[420,4684],{"className":4685,"style":1386},[436],[420,4687,3452],{"className":4688,"style":3451},[441,480]," into ",[420,4691,4693],{"className":4692},[423],[420,4694,4696],{"className":4695,"ariaHidden":428},[427],[420,4697,4699,4702],{"className":4698},[432],[420,4700],{"className":4701,"style":2269},[436],[420,4703,4705,4708],{"className":4704},[441],[420,4706,4636],{"className":4707,"style":4635},[441,480],[420,4709,4711],{"className":4710},[449],[420,4712,4714,4734],{"className":4713},[453,1282],[420,4715,4717,4731],{"className":4716},[457],[420,4718,4720],{"className":4719,"style":4649},[461],[420,4721,4722,4725],{"style":4652},[420,4723],{"className":4724,"style":469},[468],[420,4726,4728],{"className":4727},[473,474,475,476],[420,4729,3452],{"className":4730,"style":3451},[441,480,476],[420,4732,1307],{"className":4733},[1306],[420,4735,4737],{"className":4736},[457],[420,4738,4740],{"className":4739,"style":1314},[461],[420,4741],{},". Every subset of the whole is one\nchoice from ",[420,4744,4746],{"className":4745},[423],[420,4747,4749],{"className":4748,"ariaHidden":428},[427],[420,4750,4752,4755],{"className":4751},[432],[420,4753],{"className":4754,"style":1386},[436],[420,4756,3413],{"className":4757},[441,480]," paired with one from ",[420,4760,4762],{"className":4761},[423],[420,4763,4765],{"className":4764,"ariaHidden":428},[427],[420,4766,4768,4771],{"className":4767},[432],[420,4769],{"className":4770,"style":1386},[436],[420,4772,3452],{"className":4773,"style":3451},[441,480],", so the full answer is recovered by\n",[388,4776,4777,4778,4830,4831],{},"combining one element of ",[420,4779,4781],{"className":4780},[423],[420,4782,4784],{"className":4783,"ariaHidden":428},[427],[420,4785,4787,4790],{"className":4786},[432],[420,4788],{"className":4789,"style":2269},[436],[420,4791,4793,4796],{"className":4792},[441],[420,4794,4636],{"className":4795,"style":4635},[441,480],[420,4797,4799],{"className":4798},[449],[420,4800,4802,4822],{"className":4801},[453,1282],[420,4803,4805,4819],{"className":4804},[457],[420,4806,4808],{"className":4807,"style":4649},[461],[420,4809,4810,4813],{"style":4652},[420,4811],{"className":4812,"style":469},[468],[420,4814,4816],{"className":4815},[473,474,475,476],[420,4817,3413],{"className":4818},[441,480,476],[420,4820,1307],{"className":4821},[1306],[420,4823,4825],{"className":4824},[457],[420,4826,4828],{"className":4827,"style":1314},[461],[420,4829],{}," with one of ",[420,4832,4834],{"className":4833},[423],[420,4835,4837],{"className":4836,"ariaHidden":428},[427],[420,4838,4840,4843],{"className":4839},[432],[420,4841],{"className":4842,"style":2269},[436],[420,4844,4846,4849],{"className":4845},[441],[420,4847,4636],{"className":4848,"style":4635},[441,480],[420,4850,4852],{"className":4851},[449],[420,4853,4855,4875],{"className":4854},[453,1282],[420,4856,4858,4872],{"className":4857},[457],[420,4859,4861],{"className":4860,"style":4649},[461],[420,4862,4863,4866],{"style":4652},[420,4864],{"className":4865,"style":469},[468],[420,4867,4869],{"className":4868},[473,474,475,476],[420,4870,3452],{"className":4871,"style":3451},[441,480,476],[420,4873,1307],{"className":4874},[1306],[420,4876,4878],{"className":4877},[457],[420,4879,4881],{"className":4880,"style":1314},[461],[420,4882],{},", but we do that\ncombination cleverly, not by trying all ",[420,4885,4887],{"className":4886},[423],[420,4888,4890,4940,4990],{"className":4889,"ariaHidden":428},[427],[420,4891,4893,4896,4931,4934,4937],{"className":4892},[432],[420,4894],{"className":4895,"style":4564},[436],[420,4897,4899,4902],{"className":4898},[441],[420,4900,445],{"className":4901},[441],[420,4903,4905],{"className":4904},[449],[420,4906,4908],{"className":4907},[453],[420,4909,4911],{"className":4910},[457],[420,4912,4914],{"className":4913,"style":4564},[461],[420,4915,4916,4919],{"style":464},[420,4917],{"className":4918,"style":469},[468],[420,4920,4922],{"className":4921},[473,474,475,476],[420,4923,4925,4928],{"className":4924},[441,476],[420,4926,481],{"className":4927},[441,480,476],[420,4929,4546],{"className":4930},[441,476],[420,4932],{"className":4933,"style":1605},[648],[420,4935,2089],{"className":4936},[626],[420,4938],{"className":4939,"style":1605},[648],[420,4941,4943,4946,4981,4984,4987],{"className":4942},[432],[420,4944],{"className":4945,"style":4564},[436],[420,4947,4949,4952],{"className":4948},[441],[420,4950,445],{"className":4951},[441],[420,4953,4955],{"className":4954},[449],[420,4956,4958],{"className":4957},[453],[420,4959,4961],{"className":4960},[457],[420,4962,4964],{"className":4963,"style":4564},[461],[420,4965,4966,4969],{"style":464},[420,4967],{"className":4968,"style":469},[468],[420,4970,4972],{"className":4971},[473,474,475,476],[420,4973,4975,4978],{"className":4974},[441,476],[420,4976,481],{"className":4977},[441,480,476],[420,4979,4546],{"className":4980},[441,476],[420,4982],{"className":4983,"style":649},[648],[420,4985,1216],{"className":4986},[653],[420,4988],{"className":4989,"style":649},[648],[420,4991,4993,4996],{"className":4992},[432],[420,4994],{"className":4995,"style":437},[436],[420,4997,4999,5002],{"className":4998},[441],[420,5000,445],{"className":5001},[441],[420,5003,5005],{"className":5004},[449],[420,5006,5008],{"className":5007},[453],[420,5009,5011],{"className":5010},[457],[420,5012,5014],{"className":5013,"style":437},[461],[420,5015,5016,5019],{"style":464},[420,5017],{"className":5018,"style":469},[468],[420,5020,5022],{"className":5021},[473,474,475,476],[420,5023,481],{"className":5024},[441,480,476]," pairs.",[381,5027,5028,5029,5048,5049,5179,5180,5232,5233,5304,5305,5357,5358,5391,5392,5488,5489,553],{},"For the canonical task, find a subset whose sum is ",[388,5030,5031,5032],{},"closest to a target ",[420,5033,5035],{"className":5034},[423],[420,5036,5038],{"className":5037,"ariaHidden":428},[427],[420,5039,5041,5044],{"className":5040},[432],[420,5042],{"className":5043,"style":1386},[436],[420,5045,5047],{"className":5046,"style":1390},[441,480],"T","\n(this is the minimum-partition-difference problem with ",[420,5050,5052],{"className":5051},[423],[420,5053,5055,5073],{"className":5054,"ariaHidden":428},[427],[420,5056,5058,5061,5064,5067,5070],{"className":5057},[432],[420,5059],{"className":5060,"style":1386},[436],[420,5062,5047],{"className":5063,"style":1390},[441,480],[420,5065],{"className":5066,"style":649},[648],[420,5068,1216],{"className":5069},[653],[420,5071],{"className":5072,"style":649},[648],[420,5074,5076,5080,5083,5128,5131,5173,5176],{"className":5075},[432],[420,5077],{"className":5078,"style":5079},[436],"height:1.0497em;vertical-align:-0.2997em;",[420,5081,1902],{"className":5082},[1896],[420,5084,5086,5091],{"className":5085},[1715],[420,5087,1814],{"className":5088,"style":5090},[1715,1812,5089],"small-op","position:relative;top:0em;",[420,5092,5094],{"className":5093},[449],[420,5095,5097,5119],{"className":5096},[453,1282],[420,5098,5100,5116],{"className":5099},[457],[420,5101,5104],{"className":5102,"style":5103},[461],"height:0.162em;",[420,5105,5107,5110],{"style":5106},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[420,5108],{"className":5109,"style":469},[468],[420,5111,5113],{"className":5112},[473,474,475,476],[420,5114,1302],{"className":5115},[441,480,476],[420,5117,1307],{"className":5118},[1306],[420,5120,5122],{"className":5121},[457],[420,5123,5126],{"className":5124,"style":5125},[461],"height:0.2997em;",[420,5127],{},[420,5129],{"className":5130,"style":1830},[648],[420,5132,5134,5138],{"className":5133},[441],[420,5135,5137],{"className":5136},[441,480],"x",[420,5139,5141],{"className":5140},[449],[420,5142,5144,5165],{"className":5143},[453,1282],[420,5145,5147,5162],{"className":5146},[457],[420,5148,5150],{"className":5149,"style":1289},[461],[420,5151,5153,5156],{"style":5152},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[420,5154],{"className":5155,"style":469},[468],[420,5157,5159],{"className":5158},[473,474,475,476],[420,5160,1302],{"className":5161},[441,480,476],[420,5163,1307],{"className":5164},[1306],[420,5166,5168],{"className":5167},[457],[420,5169,5171],{"className":5170,"style":1314},[461],[420,5172],{},[420,5174,2082],{"className":5175},[2078],[420,5177,4546],{"className":5178},[441],"), sort\n",[420,5181,5183],{"className":5182},[423],[420,5184,5186],{"className":5185,"ariaHidden":428},[427],[420,5187,5189,5192],{"className":5188},[432],[420,5190],{"className":5191,"style":2269},[436],[420,5193,5195,5198],{"className":5194},[441],[420,5196,4636],{"className":5197,"style":4635},[441,480],[420,5199,5201],{"className":5200},[449],[420,5202,5204,5224],{"className":5203},[453,1282],[420,5205,5207,5221],{"className":5206},[457],[420,5208,5210],{"className":5209,"style":4649},[461],[420,5211,5212,5215],{"style":4652},[420,5213],{"className":5214,"style":469},[468],[420,5216,5218],{"className":5217},[473,474,475,476],[420,5219,3452],{"className":5220,"style":3451},[441,480,476],[420,5222,1307],{"className":5223},[1306],[420,5225,5227],{"className":5226},[457],[420,5228,5230],{"className":5229,"style":1314},[461],[420,5231],{},", then for each ",[420,5234,5236],{"className":5235},[423],[420,5237,5239,5258],{"className":5238,"ariaHidden":428},[427],[420,5240,5242,5246,5249,5252,5255],{"className":5241},[432],[420,5243],{"className":5244,"style":5245},[436],"height:0.5782em;vertical-align:-0.0391em;",[420,5247,385],{"className":5248},[441,480],[420,5250],{"className":5251,"style":649},[648],[420,5253,1746],{"className":5254},[653],[420,5256],{"className":5257,"style":649},[648],[420,5259,5261,5264],{"className":5260},[432],[420,5262],{"className":5263,"style":2269},[436],[420,5265,5267,5270],{"className":5266},[441],[420,5268,4636],{"className":5269,"style":4635},[441,480],[420,5271,5273],{"className":5272},[449],[420,5274,5276,5296],{"className":5275},[453,1282],[420,5277,5279,5293],{"className":5278},[457],[420,5280,5282],{"className":5281,"style":4649},[461],[420,5283,5284,5287],{"style":4652},[420,5285],{"className":5286,"style":469},[468],[420,5288,5290],{"className":5289},[473,474,475,476],[420,5291,3413],{"className":5292},[441,480,476],[420,5294,1307],{"className":5295},[1306],[420,5297,5299],{"className":5298},[457],[420,5300,5302],{"className":5301,"style":1314},[461],[420,5303],{}," binary-search ",[420,5306,5308],{"className":5307},[423],[420,5309,5311],{"className":5310,"ariaHidden":428},[427],[420,5312,5314,5317],{"className":5313},[432],[420,5315],{"className":5316,"style":2269},[436],[420,5318,5320,5323],{"className":5319},[441],[420,5321,4636],{"className":5322,"style":4635},[441,480],[420,5324,5326],{"className":5325},[449],[420,5327,5329,5349],{"className":5328},[453,1282],[420,5330,5332,5346],{"className":5331},[457],[420,5333,5335],{"className":5334,"style":4649},[461],[420,5336,5337,5340],{"style":4652},[420,5338],{"className":5339,"style":469},[468],[420,5341,5343],{"className":5342},[473,474,475,476],[420,5344,3452],{"className":5345,"style":3451},[441,480,476],[420,5347,1307],{"className":5348},[1306],[420,5350,5352],{"className":5351},[457],[420,5353,5355],{"className":5354,"style":1314},[461],[420,5356],{}," for the value nearest\n",[420,5359,5361],{"className":5360},[423],[420,5362,5364,5382],{"className":5363,"ariaHidden":428},[427],[420,5365,5367,5370,5373,5376,5379],{"className":5366},[432],[420,5368],{"className":5369,"style":1685},[436],[420,5371,5047],{"className":5372,"style":1390},[441,480],[420,5374],{"className":5375,"style":1605},[648],[420,5377,1912],{"className":5378},[626],[420,5380],{"className":5381,"style":1605},[648],[420,5383,5385,5388],{"className":5384},[432],[420,5386],{"className":5387,"style":495},[436],[420,5389,385],{"className":5390},[441,480],". Each query is ",[420,5393,5395],{"className":5394},[423],[420,5396,5398,5470],{"className":5397,"ariaHidden":428},[427],[420,5399,5401,5405,5408,5411,5420,5423,5458,5461,5464,5467],{"className":5400},[432],[420,5402],{"className":5403,"style":5404},[436],"height:1.138em;vertical-align:-0.25em;",[420,5406,4026],{"className":5407,"style":4025},[441,480],[420,5409,1902],{"className":5410},[1896],[420,5412,5414],{"className":5413},[1715],[420,5415,5419],{"className":5416,"style":5418},[441,5417],"mathrm","margin-right:0.0139em;","log",[420,5421],{"className":5422,"style":1830},[648],[420,5424,5426,5429],{"className":5425},[441],[420,5427,445],{"className":5428},[441],[420,5430,5432],{"className":5431},[449],[420,5433,5435],{"className":5434},[453],[420,5436,5438],{"className":5437},[457],[420,5439,5441],{"className":5440,"style":4564},[461],[420,5442,5443,5446],{"style":464},[420,5444],{"className":5445,"style":469},[468],[420,5447,5449],{"className":5448},[473,474,475,476],[420,5450,5452,5455],{"className":5451},[441,476],[420,5453,481],{"className":5454},[441,480,476],[420,5456,4546],{"className":5457},[441,476],[420,5459,2082],{"className":5460},[2078],[420,5462],{"className":5463,"style":649},[648],[420,5465,1216],{"className":5466},[653],[420,5468],{"className":5469,"style":649},[648],[420,5471,5473,5476,5479,5482,5485],{"className":5472},[432],[420,5474],{"className":5475,"style":1504},[436],[420,5477,4026],{"className":5478,"style":4025},[441,480],[420,5480,1902],{"className":5481},[1896],[420,5483,481],{"className":5484},[441,480],[420,5486,2082],{"className":5487},[2078],", so the whole combine is\n",[420,5490,5492],{"className":5491},[423],[420,5493,5495],{"className":5494,"ariaHidden":428},[427],[420,5496,5498,5501,5504,5507,5542,5545,5548],{"className":5497},[432],[420,5499],{"className":5500,"style":5404},[436],[420,5502,4026],{"className":5503,"style":4025},[441,480],[420,5505,1902],{"className":5506},[1896],[420,5508,5510,5513],{"className":5509},[441],[420,5511,445],{"className":5512},[441],[420,5514,5516],{"className":5515},[449],[420,5517,5519],{"className":5518},[453],[420,5520,5522],{"className":5521},[457],[420,5523,5525],{"className":5524,"style":4564},[461],[420,5526,5527,5530],{"style":464},[420,5528],{"className":5529,"style":469},[468],[420,5531,5533],{"className":5532},[473,474,475,476],[420,5534,5536,5539],{"className":5535},[441,476],[420,5537,481],{"className":5538},[441,480,476],[420,5540,4546],{"className":5541},[441,476],[420,5543],{"className":5544,"style":1830},[648],[420,5546,481],{"className":5547},[441,480],[420,5549,2082],{"className":5550},[2078],[2731,5552,5554],{"className":2733,"code":5553,"language":2735,"meta":376,"style":376},"caption: $\\textsc{MeetInTheMiddle}$ — subset sum closest to target $T$\nsplit items into halves $A$ (size $\\lceil n\u002F2\\rceil$) and $B$\n$S_A \\gets$ all $2^{|A|}$ subset sums of $A$\n$S_B \\gets$ all $2^{|B|}$ subset sums of $B$\nsort $S_B$\n$best \\gets \\infty$\nfor each $a$ in $S_A$ do\n  $r \\gets T - a$ \u002F\u002F complement from $B$\n  $s \\gets$ value in $S_B$ nearest $r$ (binary search: floor and ceiling of $r$)\n  $best \\gets \\min(best,\\ |\\,a + s - T\\,|)$\nreturn $best$\n",[2737,5555,5556,5561,5566,5571,5576,5581,5586,5591,5596,5601,5606],{"__ignoreMap":376},[420,5557,5558],{"class":2741,"line":6},[420,5559,5560],{},"caption: $\\textsc{MeetInTheMiddle}$ — subset sum closest to target $T$\n",[420,5562,5563],{"class":2741,"line":18},[420,5564,5565],{},"split items into halves $A$ (size $\\lceil n\u002F2\\rceil$) and $B$\n",[420,5567,5568],{"class":2741,"line":24},[420,5569,5570],{},"$S_A \\gets$ all $2^{|A|}$ subset sums of $A$\n",[420,5572,5573],{"class":2741,"line":73},[420,5574,5575],{},"$S_B \\gets$ all $2^{|B|}$ subset sums of $B$\n",[420,5577,5578],{"class":2741,"line":102},[420,5579,5580],{},"sort $S_B$\n",[420,5582,5583],{"class":2741,"line":108},[420,5584,5585],{},"$best \\gets \\infty$\n",[420,5587,5588],{"class":2741,"line":116},[420,5589,5590],{},"for each $a$ in $S_A$ do\n",[420,5592,5593],{"class":2741,"line":196},[420,5594,5595],{},"  $r \\gets T - a$ \u002F\u002F complement from $B$\n",[420,5597,5598],{"class":2741,"line":202},[420,5599,5600],{},"  $s \\gets$ value in $S_B$ nearest $r$ (binary search: floor and ceiling of $r$)\n",[420,5602,5603],{"class":2741,"line":283},[420,5604,5605],{},"  $best \\gets \\min(best,\\ |\\,a + s - T\\,|)$\n",[420,5607,5608],{"class":2741,"line":333},[420,5609,5610],{},"return $best$\n",[381,5612,5613,5614,5670,5671,5733,5734,5796,5797,5859,5860,5910,5911,5956,5957,6005,6006,5956,6051,6095],{},"The enumeration is ",[420,5615,5617],{"className":5616},[423],[420,5618,5620],{"className":5619,"ariaHidden":428},[427],[420,5621,5623,5626,5629,5632,5667],{"className":5622},[432],[420,5624],{"className":5625,"style":5404},[436],[420,5627,4026],{"className":5628,"style":4025},[441,480],[420,5630,1902],{"className":5631},[1896],[420,5633,5635,5638],{"className":5634},[441],[420,5636,445],{"className":5637},[441],[420,5639,5641],{"className":5640},[449],[420,5642,5644],{"className":5643},[453],[420,5645,5647],{"className":5646},[457],[420,5648,5650],{"className":5649,"style":4564},[461],[420,5651,5652,5655],{"style":464},[420,5653],{"className":5654,"style":469},[468],[420,5656,5658],{"className":5657},[473,474,475,476],[420,5659,5661,5664],{"className":5660},[441,476],[420,5662,481],{"className":5663},[441,480,476],[420,5665,4546],{"className":5666},[441,476],[420,5668,2082],{"className":5669},[2078]," per half, the sort is ",[420,5672,5674],{"className":5673},[423],[420,5675,5677],{"className":5676,"ariaHidden":428},[427],[420,5678,5680,5683,5686,5689,5724,5727,5730],{"className":5679},[432],[420,5681],{"className":5682,"style":5404},[436],[420,5684,4026],{"className":5685,"style":4025},[441,480],[420,5687,1902],{"className":5688},[1896],[420,5690,5692,5695],{"className":5691},[441],[420,5693,445],{"className":5694},[441],[420,5696,5698],{"className":5697},[449],[420,5699,5701],{"className":5700},[453],[420,5702,5704],{"className":5703},[457],[420,5705,5707],{"className":5706,"style":4564},[461],[420,5708,5709,5712],{"style":464},[420,5710],{"className":5711,"style":469},[468],[420,5713,5715],{"className":5714},[473,474,475,476],[420,5716,5718,5721],{"className":5717},[441,476],[420,5719,481],{"className":5720},[441,480,476],[420,5722,4546],{"className":5723},[441,476],[420,5725],{"className":5726,"style":1830},[648],[420,5728,481],{"className":5729},[441,480],[420,5731,2082],{"className":5732},[2078],", and the\ncombine is ",[420,5735,5737],{"className":5736},[423],[420,5738,5740],{"className":5739,"ariaHidden":428},[427],[420,5741,5743,5746,5749,5752,5787,5790,5793],{"className":5742},[432],[420,5744],{"className":5745,"style":5404},[436],[420,5747,4026],{"className":5748,"style":4025},[441,480],[420,5750,1902],{"className":5751},[1896],[420,5753,5755,5758],{"className":5754},[441],[420,5756,445],{"className":5757},[441],[420,5759,5761],{"className":5760},[449],[420,5762,5764],{"className":5763},[453],[420,5765,5767],{"className":5766},[457],[420,5768,5770],{"className":5769,"style":4564},[461],[420,5771,5772,5775],{"style":464},[420,5773],{"className":5774,"style":469},[468],[420,5776,5778],{"className":5777},[473,474,475,476],[420,5779,5781,5784],{"className":5780},[441,476],[420,5782,481],{"className":5783},[441,480,476],[420,5785,4546],{"className":5786},[441,476],[420,5788],{"className":5789,"style":1830},[648],[420,5791,481],{"className":5792},[441,480],[420,5794,2082],{"className":5795},[2078],", so the whole algorithm is ",[420,5798,5800],{"className":5799},[423],[420,5801,5803],{"className":5802,"ariaHidden":428},[427],[420,5804,5806,5809,5812,5815,5850,5853,5856],{"className":5805},[432],[420,5807],{"className":5808,"style":5404},[436],[420,5810,4026],{"className":5811,"style":4025},[441,480],[420,5813,1902],{"className":5814},[1896],[420,5816,5818,5821],{"className":5817},[441],[420,5819,445],{"className":5820},[441],[420,5822,5824],{"className":5823},[449],[420,5825,5827],{"className":5826},[453],[420,5828,5830],{"className":5829},[457],[420,5831,5833],{"className":5832,"style":4564},[461],[420,5834,5835,5838],{"style":464},[420,5836],{"className":5837,"style":469},[468],[420,5839,5841],{"className":5840},[473,474,475,476],[420,5842,5844,5847],{"className":5843},[441,476],[420,5845,481],{"className":5846},[441,480,476],[420,5848,4546],{"className":5849},[441,476],[420,5851],{"className":5852,"style":1830},[648],[420,5854,481],{"className":5855},[441,480],[420,5857,2082],{"className":5858},[2078],", a quadratic\nimprovement over the ",[420,5861,5863],{"className":5862},[423],[420,5864,5866],{"className":5865,"ariaHidden":428},[427],[420,5867,5869,5872,5875,5878,5907],{"className":5868},[432],[420,5870],{"className":5871,"style":1504},[436],[420,5873,4026],{"className":5874,"style":4025},[441,480],[420,5876,1902],{"className":5877},[1896],[420,5879,5881,5884],{"className":5880},[441],[420,5882,445],{"className":5883},[441],[420,5885,5887],{"className":5886},[449],[420,5888,5890],{"className":5889},[453],[420,5891,5893],{"className":5892},[457],[420,5894,5896],{"className":5895,"style":437},[461],[420,5897,5898,5901],{"style":464},[420,5899],{"className":5900,"style":469},[468],[420,5902,5904],{"className":5903},[473,474,475,476],[420,5905,481],{"className":5906},[441,480,476],[420,5908,2082],{"className":5909},[2078]," brute force. Concretely, ",[420,5912,5914],{"className":5913},[423],[420,5915,5917],{"className":5916,"ariaHidden":428},[427],[420,5918,5920,5924],{"className":5919},[432],[420,5921],{"className":5922,"style":5923},[436],"height:0.8141em;",[420,5925,5927,5930],{"className":5926},[441],[420,5928,445],{"className":5929},[441],[420,5931,5933],{"className":5932},[449],[420,5934,5936],{"className":5935},[453],[420,5937,5939],{"className":5938},[457],[420,5940,5942],{"className":5941,"style":5923},[461],[420,5943,5944,5947],{"style":464},[420,5945],{"className":5946,"style":469},[468],[420,5948,5950],{"className":5949},[473,474,475,476],[420,5951,5953],{"className":5952},[441,476],[420,5954,4476],{"className":5955},[441,476]," is about ",[420,5958,5960],{"className":5959},[423],[420,5961,5963],{"className":5962,"ariaHidden":428},[427],[420,5964,5966,5969,5972],{"className":5965},[432],[420,5967],{"className":5968,"style":5923},[436],[420,5970,831],{"className":5971},[441],[420,5973,5975,5979],{"className":5974},[441],[420,5976,5978],{"className":5977},[441],"0",[420,5980,5982],{"className":5981},[449],[420,5983,5985],{"className":5984},[453],[420,5986,5988],{"className":5987},[457],[420,5989,5991],{"className":5990,"style":5923},[461],[420,5992,5993,5996],{"style":464},[420,5994],{"className":5995,"style":469},[468],[420,5997,5999],{"className":5998},[473,474,475,476],[420,6000,6002],{"className":6001},[441,476],[420,6003,3465],{"className":6004},[441,476],"\n(out of reach) while ",[420,6007,6009],{"className":6008},[423],[420,6010,6012],{"className":6011,"ariaHidden":428},[427],[420,6013,6015,6018],{"className":6014},[432],[420,6016],{"className":6017,"style":5923},[436],[420,6019,6021,6024],{"className":6020},[441],[420,6022,445],{"className":6023},[441],[420,6025,6027],{"className":6026},[449],[420,6028,6030],{"className":6029},[453],[420,6031,6033],{"className":6032},[457],[420,6034,6036],{"className":6035,"style":5923},[461],[420,6037,6038,6041],{"style":464},[420,6039],{"className":6040,"style":469},[468],[420,6042,6044],{"className":6043},[473,474,475,476],[420,6045,6047],{"className":6046},[441,476],[420,6048,6050],{"className":6049},[441,476],"20",[420,6052,6054],{"className":6053},[423],[420,6055,6057],{"className":6056,"ariaHidden":428},[427],[420,6058,6060,6063,6066],{"className":6059},[432],[420,6061],{"className":6062,"style":5923},[436],[420,6064,831],{"className":6065},[441],[420,6067,6069,6072],{"className":6068},[441],[420,6070,5978],{"className":6071},[441],[420,6073,6075],{"className":6074},[449],[420,6076,6078],{"className":6077},[453],[420,6079,6081],{"className":6080},[457],[420,6082,6084],{"className":6083,"style":5923},[461],[420,6085,6086,6089],{"style":464},[420,6087],{"className":6088,"style":469},[468],[420,6090,6092],{"className":6091},[473,474,475,476],[420,6093,3396],{"className":6094},[441,476]," (instant). The technique is exact,\nno approximation and no pruning luck, and it is the intended solution to every\nHard subset problem in this lesson's practice set.",[905,6097,6099,6341],{"className":6098},[908,909],[911,6100,6104],{"xmlns":913,"width":6101,"height":6102,"viewBox":6103},"308.030","142.429","-75 -75 231.023 106.821",[918,6105,6106,6127,6130,6137,6140,6147,6159,6162,6169,6189,6192,6199,6202,6209,6221,6224,6231,6258,6265,6306],{"stroke":920,"style":921},[918,6107,6108,6115,6121],{"stroke":925},[918,6109,6111],{"transform":6110},"translate(-43.776 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— enumerate each half, then binary-search the complement",[381,6447,6448,6449,6464,6465,6498,6499,6552,6553,6556,6557,6635,6636,553],{},"For each sum ",[420,6450,6452],{"className":6451},[423],[420,6453,6455],{"className":6454,"ariaHidden":428},[427],[420,6456,6458,6461],{"className":6457},[432],[420,6459],{"className":6460,"style":495},[436],[420,6462,385],{"className":6463},[441,480]," on the top we binary-search the sorted bottom list for 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The same split-and-recombine\nidea is the graph analog ",[388,6554,6555],{},"bidirectional search",": to find a shortest path, run\nBFS forward from the source and backward from the target simultaneously and stop\nwhen the two frontiers meet, exploring 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nodes instead of\n",[420,6637,6639],{"className":6638},[423],[420,6640,6642],{"className":6641,"ariaHidden":428},[427],[420,6643,6645,6649],{"className":6644},[432],[420,6646],{"className":6647,"style":6648},[436],"height:0.8491em;",[420,6650,6652,6655],{"className":6651},[441],[420,6653,580],{"className":6654},[441,480],[420,6656,6658],{"className":6657},[449],[420,6659,6661],{"className":6660},[453],[420,6662,6664],{"className":6663},[457],[420,6665,6667],{"className":6666,"style":6648},[461],[420,6668,6669,6672],{"style":464},[420,6670],{"className":6671,"style":469},[468],[420,6673,6675],{"className":6674},[473,474,475,476],[420,6676,6631],{"className":6677},[441,480,476],[905,6679,6681,6937],{"className":6680},[908,909],[911,6682,6686],{"xmlns":913,"width":6683,"height":6684,"viewBox":6685},"449.371","190.098","-75 -75 337.028 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",[420,6942,6944],{"className":6943},[423],[420,6945,6947],{"className":6946,"ariaHidden":428},[427],[420,6948,6950,6953,6956],{"className":6949},[432],[420,6951],{"className":6952,"style":1504},[436],[420,6954,6631],{"className":6955},[441,480],[420,6957,4546],{"className":6958},[441]," from ",[420,6961,6963],{"className":6962},[423],[420,6964,6966],{"className":6965,"ariaHidden":428},[427],[420,6967,6969,6972],{"className":6968},[432],[420,6970],{"className":6971,"style":495},[436],[420,6973,6975],{"className":6974},[441,480],"s",[420,6977,6979],{"className":6978},[423],[420,6980,6982],{"className":6981,"ariaHidden":428},[427],[420,6983,6985,6989],{"className":6984},[432],[420,6986],{"className":6987,"style":6988},[436],"height:0.6151em;",[420,6990,6992],{"className":6991},[441,480],"t"," meet in the middle, touching ",[420,6995,6997],{"className":6996},[423],[420,6998,7000,7012],{"className":6999,"ariaHidden":428},[427],[420,7001,7003,7006,7009],{"className":7002},[432],[420,7004],{"className":7005,"style":4526},[436],[420,7007,4530],{"className":7008},[653],[420,7010],{"className":7011,"style":649},[648],[420,7013,7015,7018,7021,7024],{"className":7014},[432],[420,7016],{"className":7017,"style":4564},[436],[420,7019,445],{"className":7020},[441],[420,7022],{"className":7023,"style":1830},[648],[420,7025,7027,7030],{"className":7026},[441],[420,7028,580],{"className":7029},[441,480],[420,7031,7033],{"className":7032},[449],[420,7034,7036],{"className":7035},[453],[420,7037,7039],{"className":7038},[457],[420,7040,7042],{"className":7041,"style":4564},[461],[420,7043,7044,7047],{"style":464},[420,7045],{"className":7046,"style":469},[468],[420,7048,7050],{"className":7049},[473,474,475,476],[420,7051,7053,7056],{"className":7052},[441,476],[420,7054,6631],{"className":7055},[441,480,476],[420,7057,4546],{"className":7058},[441,476]," nodes instead of one frontier of ",[420,7061,7063],{"className":7062},[423],[420,7064,7066],{"className":7065,"ariaHidden":428},[427],[420,7067,7069,7072],{"className":7068},[432],[420,7070],{"className":7071,"style":6648},[436],[420,7073,7075,7078],{"className":7074},[441],[420,7076,580],{"className":7077},[441,480],[420,7079,7081],{"className":7080},[449],[420,7082,7084],{"className":7083},[453],[420,7085,7087],{"className":7086},[457],[420,7088,7090],{"className":7089,"style":6648},[461],[420,7091,7092,7095],{"style":464},[420,7093],{"className":7094,"style":469},[468],[420,7096,7098],{"className":7097},[473,474,475,476],[420,7099,6631],{"className":7100},[441,480,476],[523,7102,7104],{"id":7103},"when-to-reach-for-which","When to reach for which",[381,7106,7107,7108],{},"The three pruning disciplines line up neatly along one axis: ",[392,7109,7110],{},"what justifies\ndiscarding a branch.",[7112,7113,7114,7123,7131],"ul",{},[7115,7116,7117,7119,7120,7122],"li",{},[388,7118,294],{}," prunes by ",[388,7121,414],{},": a partial solution that violates a\nconstraint can never be completed, so cut it.",[7115,7124,7125,7119,7128,7130],{},[388,7126,7127],{},"Branch and bound",[388,7129,410],{},": a partial solution whose optimistic\nbound cannot beat the incumbent is pointless to complete, so cut it.",[7115,7132,7133,7135,7136,7139,7140,7143,7144,7235,7236,7292],{},[388,7134,520],{}," prunes ",[392,7137,7138],{},"nothing","; it instead trades exponential time for\nthe ",[388,7141,7142],{},"square root"," of it, ",[420,7145,7147],{"className":7146},[423],[420,7148,7150,7194],{"className":7149,"ariaHidden":428},[427],[420,7151,7153,7156,7185,7188,7191],{"className":7152},[432],[420,7154],{"className":7155,"style":437},[436],[420,7157,7159,7162],{"className":7158},[441],[420,7160,445],{"className":7161},[441],[420,7163,7165],{"className":7164},[449],[420,7166,7168],{"className":7167},[453],[420,7169,7171],{"className":7170},[457],[420,7172,7174],{"className":7173,"style":437},[461],[420,7175,7176,7179],{"style":464},[420,7177],{"className":7178,"style":469},[468],[420,7180,7182],{"className":7181},[473,474,475,476],[420,7183,481],{"className":7184},[441,480,476],[420,7186],{"className":7187,"style":649},[648],[420,7189,6391],{"className":7190},[653],[420,7192],{"className":7193,"style":649},[648],[420,7195,7197,7200],{"className":7196},[432],[420,7198],{"className":7199,"style":4564},[436],[420,7201,7203,7206],{"className":7202},[441],[420,7204,445],{"className":7205},[441],[420,7207,7209],{"className":7208},[449],[420,7210,7212],{"className":7211},[453],[420,7213,7215],{"className":7214},[457],[420,7216,7218],{"className":7217,"style":4564},[461],[420,7219,7220,7223],{"style":464},[420,7221],{"className":7222,"style":469},[468],[420,7224,7226],{"className":7225},[473,474,475,476],[420,7227,7229,7232],{"className":7228},[441,476],[420,7230,481],{"className":7231},[441,480,476],[420,7233,4546],{"className":7234},[441,476],", paying with ",[420,7237,7239],{"className":7238},[423],[420,7240,7242],{"className":7241,"ariaHidden":428},[427],[420,7243,7245,7248,7251,7254,7289],{"className":7244},[432],[420,7246],{"className":7247,"style":5404},[436],[420,7249,4026],{"className":7250,"style":4025},[441,480],[420,7252,1902],{"className":7253},[1896],[420,7255,7257,7260],{"className":7256},[441],[420,7258,445],{"className":7259},[441],[420,7261,7263],{"className":7262},[449],[420,7264,7266],{"className":7265},[453],[420,7267,7269],{"className":7268},[457],[420,7270,7272],{"className":7271,"style":4564},[461],[420,7273,7274,7277],{"style":464},[420,7275],{"className":7276,"style":469},[468],[420,7278,7280],{"className":7279},[473,474,475,476],[420,7281,7283,7286],{"className":7282},[441,476],[420,7284,481],{"className":7285},[441,480,476],[420,7287,4546],{"className":7288},[441,476],[420,7290,2082],{"className":7291},[2078]," memory to\nstore the enumerated half.",[381,7294,7295,7296,7299,7300,7315,7316,7363,7364,7367],{},"Branch and bound shines when a cheap, tight optimistic bound exists (knapsack's LP\nfill, a ",[385,7297,7298],{"href":369},"TSP"," node's spanning-tree\nlower bound). Meet in the middle shines when no such bound exists but ",[420,7301,7303],{"className":7302},[423],[420,7304,7306],{"className":7305,"ariaHidden":428},[427],[420,7307,7309,7312],{"className":7308},[432],[420,7310],{"className":7311,"style":495},[436],[420,7313,481],{"className":7314},[441,480]," is small\nenough that ",[420,7317,7319],{"className":7318},[423],[420,7320,7322],{"className":7321,"ariaHidden":428},[427],[420,7323,7325,7328],{"className":7324},[432],[420,7326],{"className":7327,"style":4564},[436],[420,7329,7331,7334],{"className":7330},[441],[420,7332,445],{"className":7333},[441],[420,7335,7337],{"className":7336},[449],[420,7338,7340],{"className":7339},[453],[420,7341,7343],{"className":7342},[457],[420,7344,7346],{"className":7345,"style":4564},[461],[420,7347,7348,7351],{"style":464},[420,7349],{"className":7350,"style":469},[468],[420,7352,7354],{"className":7353},[473,474,475,476],[420,7355,7357,7360],{"className":7356},[441,476],[420,7358,481],{"className":7359},[441,480,476],[420,7361,4546],{"className":7362},[441,476]," is affordable. Both are ",[392,7365,7366],{},"exact","; neither changes the\nworst-case exponential complexity; both routinely turn an infeasible instance into\na feasible one.",[523,7369,7371],{"id":7370},"takeaways","Takeaways",[7112,7373,7374,7389,7399,7566,7600,7753],{},[7115,7375,7376,7378,7379,7381,7382,7384,7385,7388],{},[388,7377,7127],{}," is backtracking for optimization: maintain an\n",[388,7380,532],{}," (best complete solution so far) and a ",[388,7383,540],{}," (optimistic\nestimate per node), and ",[388,7386,7387],{},"prune"," any node whose bound cannot beat the\nincumbent.",[7115,7390,7391,7392,4497,7395,7398],{},"The method's power is all in the ",[388,7393,7394],{},"bound tightness",[388,7396,7397],{},"search order",": a\ntighter bound and an earlier strong incumbent each prune more of the tree.",[7115,7400,7401,7402,7405,7406,7501,7502,7505,7506,553],{},"For ",[388,7403,7404],{},"0\u002F1 knapsack",", order by density ",[420,7407,7409],{"className":7408},[423],[420,7410,7412],{"className":7411,"ariaHidden":428},[427],[420,7413,7415,7418,7458,7461],{"className":7414},[432],[420,7416],{"className":7417,"style":1504},[436],[420,7419,7421,7424],{"className":7420},[441],[420,7422,1275],{"className":7423,"style":1274},[441,480],[420,7425,7427],{"className":7426},[449],[420,7428,7430,7450],{"className":7429},[453,1282],[420,7431,7433,7447],{"className":7432},[457],[420,7434,7436],{"className":7435,"style":1289},[461],[420,7437,7438,7441],{"style":1292},[420,7439],{"className":7440,"style":469},[468],[420,7442,7444],{"className":7443},[473,474,475,476],[420,7445,1302],{"className":7446},[441,480,476],[420,7448,1307],{"className":7449},[1306],[420,7451,7453],{"className":7452},[457],[420,7454,7456],{"className":7455,"style":1314},[461],[420,7457],{},[420,7459,1548],{"className":7460},[441],[420,7462,7464,7467],{"className":7463},[441],[420,7465,1337],{"className":7466,"style":1336},[441,480],[420,7468,7470],{"className":7469},[449],[420,7471,7473,7493],{"className":7472},[453,1282],[420,7474,7476,7490],{"className":7475},[457],[420,7477,7479],{"className":7478,"style":1289},[461],[420,7480,7481,7484],{"style":1352},[420,7482],{"className":7483,"style":469},[468],[420,7485,7487],{"className":7486},[473,474,475,476],[420,7488,1302],{"className":7489},[441,480,476],[420,7491,1307],{"className":7492},[1306],[420,7494,7496],{"className":7495},[457],[420,7497,7499],{"className":7498,"style":1314},[461],[420,7500],{}," and bound by the ",[388,7503,7504],{},"LP-relaxation","\nfractional fill; branch and bound ",[388,7507,7508,7509,7533,7534,7549,7550,7565],{},"beats the ",[420,7510,7512],{"className":7511},[423],[420,7513,7515],{"className":7514,"ariaHidden":428},[427],[420,7516,7518,7521,7524,7527,7530],{"className":7517},[432],[420,7519],{"className":7520,"style":1504},[436],[420,7522,2856],{"className":7523},[441],[420,7525,1902],{"className":7526},[1896],[420,7528,2863],{"className":7529,"style":1390},[441,480],[420,7531,2082],{"className":7532},[2078]," DP when ",[420,7535,7537],{"className":7536},[423],[420,7538,7540],{"className":7539,"ariaHidden":428},[427],[420,7541,7543,7546],{"className":7542},[432],[420,7544],{"className":7545,"style":1386},[436],[420,7547,1391],{"className":7548,"style":1390},[441,480]," is huge\nbut ",[420,7551,7553],{"className":7552},[423],[420,7554,7556],{"className":7555,"ariaHidden":428},[427],[420,7557,7559,7562],{"className":7558},[432],[420,7560],{"className":7561,"style":495},[436],[420,7563,481],{"className":7564},[441,480]," is moderate",[7115,7567,7568,7571,7572,7596,7597,7599],{},[388,7569,7570],{},"Depth-first"," branch and bound finds an incumbent fast with ",[420,7573,7575],{"className":7574},[423],[420,7576,7578],{"className":7577,"ariaHidden":428},[427],[420,7579,7581,7584,7587,7590,7593],{"className":7580},[432],[420,7582],{"className":7583,"style":1504},[436],[420,7585,4026],{"className":7586,"style":4025},[441,480],[420,7588,1902],{"className":7589},[1896],[420,7591,481],{"className":7592},[441,480],[420,7594,2082],{"className":7595},[2078]," memory;\n",[388,7598,4039],{}," (priority queue on bound) targets the optimum with fewer\nexpansions but can hold an exponential frontier.",[7115,7601,7602,7604,7605,7652,7653,7715,7716,7749,7750,7752],{},[388,7603,520],{}," enumerates each of two halves (",[420,7606,7608],{"className":7607},[423],[420,7609,7611],{"className":7610,"ariaHidden":428},[427],[420,7612,7614,7617],{"className":7613},[432],[420,7615],{"className":7616,"style":4564},[436],[420,7618,7620,7623],{"className":7619},[441],[420,7621,445],{"className":7622},[441],[420,7624,7626],{"className":7625},[449],[420,7627,7629],{"className":7628},[453],[420,7630,7632],{"className":7631},[457],[420,7633,7635],{"className":7634,"style":4564},[461],[420,7636,7637,7640],{"style":464},[420,7638],{"className":7639,"style":469},[468],[420,7641,7643],{"className":7642},[473,474,475,476],[420,7644,7646,7649],{"className":7645},[441,476],[420,7647,481],{"className":7648},[441,480,476],[420,7650,4546],{"className":7651},[441,476]," subset sums) and\nrecombines by sorting + binary search, giving ",[420,7654,7656],{"className":7655},[423],[420,7657,7659],{"className":7658,"ariaHidden":428},[427],[420,7660,7662,7665,7668,7671,7706,7709,7712],{"className":7661},[432],[420,7663],{"className":7664,"style":5404},[436],[420,7666,4026],{"className":7667,"style":4025},[441,480],[420,7669,1902],{"className":7670},[1896],[420,7672,7674,7677],{"className":7673},[441],[420,7675,445],{"className":7676},[441],[420,7678,7680],{"className":7679},[449],[420,7681,7683],{"className":7682},[453],[420,7684,7686],{"className":7685},[457],[420,7687,7689],{"className":7688,"style":4564},[461],[420,7690,7691,7694],{"style":464},[420,7692],{"className":7693,"style":469},[468],[420,7695,7697],{"className":7696},[473,474,475,476],[420,7698,7700,7703],{"className":7699},[441,476],[420,7701,481],{"className":7702},[441,480,476],[420,7704,4546],{"className":7705},[441,476],[420,7707],{"className":7708,"style":1830},[648],[420,7710,481],{"className":7711},[441,480],[420,7713,2082],{"className":7714},[2078],", exact search up\nto ",[420,7717,7719],{"className":7718},[423],[420,7720,7722,7740],{"className":7721,"ariaHidden":428},[427],[420,7723,7725,7728,7731,7734,7737],{"className":7724},[432],[420,7726],{"className":7727,"style":4526},[436],[420,7729,481],{"className":7730},[441,480],[420,7732],{"className":7733,"style":649},[648],[420,7735,4530],{"className":7736},[653],[420,7738],{"className":7739,"style":649},[648],[420,7741,7743,7746],{"className":7742},[432],[420,7744],{"className":7745,"style":512},[436],[420,7747,4476],{"className":7748},[441],"; ",[388,7751,6555],{}," is the graph analog.",[7115,7754,7755,7756,7758,7759,7761,7762,7836],{},"One axis: backtracking prunes by ",[388,7757,414],{},", branch and bound by ",[388,7760,410],{},",\nmeet in the middle trades exponential time for ",[388,7763,7764,7835],{},[420,7765,7767],{"className":7766},[423],[420,7768,7770],{"className":7769,"ariaHidden":428},[427],[420,7771,7773,7777],{"className":7772},[432],[420,7774],{"className":7775,"style":7776},[436],"height:1.04em;vertical-align:-0.2395em;",[420,7778,7781],{"className":7779},[441,7780],"sqrt",[420,7782,7784,7826],{"className":7783},[453,1282],[420,7785,7787,7823],{"className":7786},[457],[420,7788,7791,7803],{"className":7789,"style":7790},[461],"height:0.8005em;",[420,7792,7796,7799],{"className":7793,"style":7795},[7794],"svg-align","top:-3em;",[420,7797],{"className":7798,"style":2127},[468],[420,7800],{"className":7801,"style":7802},[441],"padding-left:0.833em;",[420,7804,7806,7809],{"style":7805},"top:-2.7605em;",[420,7807],{"className":7808,"style":2127},[468],[420,7810,7814],{"className":7811,"style":7813},[7812],"hide-tail","min-width:0.853em;height:1.08em;",[911,7815,7820],{"xmlns":913,"width":7816,"height":7817,"viewBox":7818,"preserveAspectRatio":7819},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[923,7821],{"d":7822},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[420,7824,1307],{"className":7825},[1306],[420,7827,7829],{"className":7828},[457],[420,7830,7833],{"className":7831,"style":7832},[461],"height:0.2395em;",[420,7834],{}," of it"," at the cost of\nmemory.",[7838,7839,7842,7847],"section",{"className":7840,"dataFootnotes":376},[7841],"footnotes",[523,7843,7846],{"className":7844,"id":829},[7845],"sr-only","Footnotes",[7848,7849,7850,7864],"ol",{},[7115,7851,7853,7856,7857],{"id":7852},"user-content-fn-erickson-bb",[388,7854,7855],{},"Erickson",", Ch. — Backtracking: branch and bound as backtracking augmented with a value bound; the optimism of the bound is what makes pruning sound. ",[385,7858,7863],{"href":7859,"ariaLabel":7860,"className":7861,"dataFootnoteBackref":376},"#user-content-fnref-erickson-bb","Back to reference 1",[7862],"data-footnote-backref","↩",[7115,7865,7867,7870,7871],{"id":7866},"user-content-fn-skiena-bb",[388,7868,7869],{},"Skiena",", § — Combinatorial Search \u002F Heuristics: pruning a combinatorial search by bounding the best achievable completion, illustrated on knapsack-style problems. ",[385,7872,7863],{"href":7873,"ariaLabel":7874,"className":7875,"dataFootnoteBackref":376},"#user-content-fnref-skiena-bb","Back to reference 2",[7862],[7877,7878,7879],"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":7881},[7882,7886,7887,7888,7889],{"id":525,"depth":18,"text":526,"children":7883},[7884,7885],{"id":1234,"depth":24,"text":1235},{"id":3997,"depth":24,"text":3998},{"id":4423,"depth":18,"text":520},{"id":7103,"depth":18,"text":7104},{"id":7370,"depth":18,"text":7371},{"id":829,"depth":18,"text":7846},"The previous lessons built backtracking:\na depth-first walk over a tree of partial solutions that prunes a branch the\nmoment it becomes infeasible, such as a queen attacking another or a graph\ncoloring conflict. That kind of pruning asks a yes\u002Fno question: can this partial\nsolution still be completed at all? For optimization problems, maximize this\nor minimize that, we can ask a sharper question: even in the best case, can\ncompleting this partial solution beat the best answer I already have? If not, the\nentire subtree is dead, feasible or not. Pruning by value rather than mere\nfeasibility is the idea behind branch and bound, and it is often the\ndifference between minutes and millennia.","md",{"moduleNumber":202,"lessonNumber":24,"order":7893},903,[7895,7899,7902,7905],{"title":7896,"slug":7897,"difficulty":7898},"Partition Array Into Two Arrays to Minimize Sum Difference","partition-array-into-two-arrays-to-minimize-sum-difference","Hard",{"title":7900,"slug":7901,"difficulty":7898},"Closest Subsequence Sum","closest-subsequence-sum",{"title":7903,"slug":7904,"difficulty":7898},"Maximum Score Words Formed by Letters","maximum-score-words-formed-by-letters",{"title":7906,"slug":7907,"difficulty":7908},"Beautiful Arrangement","beautiful-arrangement","Medium","---\ntitle: Branch & Bound and Meet in the Middle\nmodule: Backtracking & Search\nmoduleNumber: 9\nlessonNumber: 3\norder: 903\nsummary: >-\n  Plain backtracking prunes a search tree by _feasibility_; for _optimization_\n  problems we can prune far more aggressively by _value_. **Branch and bound**\n  keeps the best complete solution found so far and discards any partial solution\n  whose optimistic bound cannot beat it. **Meet in the middle** splits the\n  instance in two, enumerates each half, and recombines by binary search — turning\n  $2^n$ into $O(2^{n\u002F2}\\,n)$ and pushing exact search out to $n \\approx 40$.\ntopics: [Backtracking]\nsources:\n  - book: Skiena\n    ref: \"§ — Combinatorial Search \u002F Heuristics\"\n  - book: CLRS\n    ref: \"Ch. 35 — Approximation (branch-and-bound context)\"\n  - book: Erickson\n    ref: \"Ch. — Backtracking\"\npractice:\n  - title: 'Partition Array Into Two Arrays to Minimize Sum Difference'\n    slug: partition-array-into-two-arrays-to-minimize-sum-difference\n    difficulty: Hard\n  - title: 'Closest Subsequence Sum'\n    slug: closest-subsequence-sum\n    difficulty: Hard\n  - title: 'Maximum Score Words Formed by Letters'\n    slug: maximum-score-words-formed-by-letters\n    difficulty: Hard\n  - title: 'Beautiful Arrangement'\n    slug: beautiful-arrangement\n    difficulty: Medium\n---\n\nThe previous lessons built [**backtracking**](\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals):\na depth-first walk over a tree of partial solutions that _prunes_ a branch the\nmoment it becomes infeasible, such as a queen attacking another or a graph\ncoloring conflict. That kind of pruning asks a yes\u002Fno question: _can this partial\nsolution still be completed at all?_ For **optimization** problems, maximize this\nor minimize that, we can ask a sharper question: _even in the best case, can\ncompleting this partial solution beat the best answer I already have?_ If not, the\nentire subtree is dead, feasible or not. Pruning by **value** rather than mere\n**feasibility** is the idea behind branch and bound, and it is often the\ndifference between minutes and millennia.\n\nWhen even aggressive pruning is not enough, when the search tree is genuinely\n$2^n$-shaped and $n$ is, say, $36$, a second technique buys a square root of the\nrunning time outright. **Meet in the middle** splits the instance, enumerates\neach half independently, and stitches the halves back together with sorting and\nbinary search. Both techniques attack the same enemy, the exponential, from\ndifferent sides.\n\n## Branch and bound: pruning by value\n\nBranch and bound is backtracking with two extra pieces of bookkeeping. The first\nis the **incumbent**: the value (and witness) of the best _complete_ solution\nfound anywhere in the search so far. The second is a **bound** computed at every\nnode, an _optimistic_ estimate of the best objective achievable by any\ncompletion of that partial solution. For a maximization problem the bound is an\n**upper bound** (no completion can do better than this); for minimization it is a\n**lower bound**.\n\n> **Remark (Pruning rule).** At a node with bound $b$ and incumbent value $z^*$:\n> if $b \\le z^*$ (maximization) — or $b \\ge z^*$ (minimization) — then _no_\n> completion of this partial solution can improve on the incumbent, so prune the\n> entire subtree. Otherwise branch on the next decision and recurse.\n\nCorrectness is immediate: the bound is optimistic, so $b \\le z^*$ means even the\nbest descendant is no better than a solution we already hold; discarding it loses\nnothing.[^erickson-bb] The _power_ of the method lives entirely in two design\nchoices. A **tighter bound** prunes more nodes; a bound equal to the true\noptimum would prune everything but the answer. And a **better search order**,\nfinding a strong incumbent early, raises $z^*$ sooner, which retroactively\nprunes more of the tree. The two interact: a good incumbent makes a mediocre\nbound effective.\n\n> **Intuition.** Backtracking explores until a branch is _impossible_. Branch and\n> bound explores until a branch is _pointless_. The incumbent is a moving\n> floor; the bound is a ceiling on each subtree; whenever a subtree's ceiling\n> drops below the floor, the subtree is swept away.\n\n$$\n% caption: Prune any node whose optimistic bound can't beat the incumbent\n\\begin{tikzpicture}[\n  every node\u002F.style={circle, draw, minimum size=9mm, inner sep=1pt, font=\\small},\n  level distance=14mm, level 1\u002F.style={sibling distance=34mm},\n  level 2\u002F.style={sibling distance=17mm}, >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node (r) {$42$}\n    child {node[draw=acc, very thick] (a) {$40$}\n      child {node (aa) {$31$}}\n      child {node[draw=acc, very thick] (ab) {$38$}}\n    }\n    child {node[draw=red!75!black, text=red!75!black, dashed] (b) {$29$}\n      child {node[draw=red!75!black, text=red!75!black, dashed] (ba) {$27$} edge from parent[draw=red!75!black, dashed]}\n      child {node[draw=red!75!black, text=red!75!black, dashed] (bb) {$22$} edge from parent[draw=red!75!black, dashed]}\n      edge from parent[draw=red!75!black, dashed]\n    };\n  \\draw[acc, very thick] (r) -- (a);\n  \\draw[acc, very thick] (a) -- (ab);\n  \\node[draw=none, right=2mm of r, font=\\footnotesize] {bound};\n  \\node[draw=none, font=\\footnotesize, text=red!75!black] at (3.6,-1.4) {$\\le 30 = z^*$};\n  \\node[draw=none, font=\\footnotesize, text=red!75!black] at (3.8,-2.8) {$\\times$ pruned};\n  \\node[draw=none, acc, font=\\footnotesize] at (-4.2,-2.0) {incumbent $z^*=38$};\n\\end{tikzpicture}\n$$\n\nThe right subtree carries bound $29 \\le z^* = 38$, so it is pruned without ever\nbeing expanded; the blue path is the active best completion that established the\nincumbent.\n\n### Worked example: 0\u002F1 knapsack by branch and bound\n\nWe have $n$ items with values $v_i$ and weights $w_i$ and a capacity $W$;\nchoose a subset of maximum total value with total weight $\\le W$. The decision\ntree is binary, take item $i$ or skip it, so it has $2^n$ leaves. To bound a\nnode, we use the **LP relaxation**: relax the integrality constraint and allow a\n_fraction_ of the next item. First order all items by **value density**\n$v_i \u002F w_i$, descending. At a node that has fixed a prefix of decisions, with\naccumulated value $V$ and remaining capacity $c$, greedily fill $c$ with the\nnot-yet-decided items in density order, taking the last one fractionally:\n\n$$\n\\text{bound} \\;=\\; V \\;+\\; \\sum_{i \\in F_{\\text{full}}} v_i\n\\;+\\; \\Bigl(c - \\!\\!\\sum_{i \\in F_{\\text{full}}}\\!\\! w_i\\Bigr)\\cdot \\frac{v_j}{w_j},\n$$\n\nwhere $F_{\\text{full}}$ are the remaining items that fit wholly and $j$ is the\nfirst item that overflows (filled fractionally). This fractional fill is the\noptimal solution to the _relaxed_ problem, so it can only over-estimate the\nintegral optimum, exactly the optimism a bound needs.[^skiena-bb]\n\n$$\n% caption: The LP-relaxation bound: greedily fill remaining capacity $c$ in density order;\n%          the overflowing item $j$ is sliced fractionally (the part beyond $c$ is the\n%          relaxation's over-estimate)\n\\begin{tikzpicture}[\n  >=stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-0.4,1.55) rectangle (10.4,-1.6);\n  % capacity bar outline (width c -> 6 units), height 0.8\n  % whole items that fit: widths 2.2 and 1.8\n  \\fill[acc!15] (0,0) rectangle (2.2,0.8);\n  \\fill[acc!15] (2.2,0) rectangle (4.0,0.8);\n  % fractional slice of item j inside capacity (width 2.0)\n  \\fill[acc!15] (4.0,0) rectangle (6.0,0.8);\n  % capacity boundary\n  \\draw[acc, very thick] (6.0,-0.2) -- (6.0,1.0);\n  \\node[draw=none, acc, font=\\scriptsize] at (6.0,1.25) {capacity $c$};\n  % the overflow part of item j (beyond capacity) -- not actually taken (integrally)\n  \\draw[red!75!black, dashed] (6.0,0) rectangle (7.6,0.8);\n  % item separators + labels\n  \\draw[black!55] (0,0) rectangle (6.0,0.8);\n  \\draw[black!55] (2.2,0)--(2.2,0.8);\n  \\draw[black!55] (4.0,0)--(4.0,0.8);\n  \\node[draw=none, font=\\scriptsize] at (1.1,0.4) {item $1$};\n  \\node[draw=none, font=\\scriptsize] at (3.1,0.4) {item $2$};\n  \\node[draw=none, font=\\scriptsize, acc] at (5.0,0.4) {$\\tfrac{c'}{w_j}$ of $j$};\n  \\node[draw=none, font=\\scriptsize, text=red!75!black] at (6.8,0.4) {spills};\n  % brackets \u002F labels under\n  \\draw[black!55] (0,-0.25)--(0,-0.4)--(4.0,-0.4)--(4.0,-0.25);\n  \\node[draw=none, font=\\scriptsize] at (2.0,-0.7) {fit wholly: $\\sum_{i\\in F_{\\text{full}}} v_i$};\n  \\draw[acc] (4.0,-0.25)--(4.0,-0.4)--(6.0,-0.4)--(6.0,-0.25);\n  \\node[draw=none, font=\\scriptsize, acc] at (5.0,-0.7) {fractional value};\n  % the over-estimate callout\n  \\node[draw=none, font=\\scriptsize, text=red!75!black, align=center] at (8.9,0.4)\n    {item $j$'s tail is\\\\ never integrally\\\\ taken \\-- the bound\\\\ over-estimates};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{KnapsackBnB}$ — maximize value within capacity $W$ (items sorted by $v_i\u002Fw_i$)\n$z^* \\gets 0$ \u002F\u002F incumbent value\n$\\textsc{Expand}(i = 0,\\ V = 0,\\ \\text{weight} = 0)$:\n  if $\\text{weight} > W$ then return \u002F\u002F infeasible\n  if $V > z^*$ then $z^* \\gets V$ \u002F\u002F new incumbent\n  if $i = n$ then return\n  if $\\textsc{Bound}(i, V, \\text{weight}) \\le z^*$ then return \u002F\u002F prune by value\n  $\\textsc{Expand}(i+1,\\ V + v_i,\\ \\text{weight} + w_i)$ \u002F\u002F take item $i$\n  $\\textsc{Expand}(i+1,\\ V,\\ \\text{weight})$ \u002F\u002F skip item $i$\n\n$\\textsc{Bound}(i, V, \\text{weight})$:\n  $b \\gets V;\\quad c \\gets W - \\text{weight}$\n  for $j \\gets i$ to $n-1$ do\n    if $w_j \\le c$ then $b \\gets b + v_j;\\ c \\gets c - w_j$\n    else return $b + c \\cdot v_j \u002F w_j$ \u002F\u002F fractional fill\n  return $b$\n```\n\nBranching on _take_ before _skip_ tends to find a heavy, valuable incumbent\nearly, which makes the bound bite sooner. Note what this buys us against the\n[**dynamic-programming**](\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack) solution. The\nDP runs in $\\Theta(nW)$ time, _pseudo_-polynomial, because $W$ enters as a\nmagnitude, not a bit-length. When $W$ is enormous (say weights are $9$-digit\nnumbers) the DP table is hopeless, yet if $n$ is moderate the branch-and-bound\ntree, ferociously pruned, finishes quickly. The two methods are complementary:\n**DP wins when $W$ is small; branch and bound wins when $W$ is huge but $n$ is\nmoderate.**\n\n$$\n% caption: Knapsack B\\&B ($W{=}6$; items\n%          $A{:}\\langle10,2\\rangle,B{:}\\langle12,4\\rangle,C{:}\\langle6,3\\rangle$ by\n%          density). The skip-$A$ node's LP bound $16\\le z^*{=}22$, so that subtree is\n%          pruned (red)\n\\begin{tikzpicture}[\n  every node\u002F.style={draw, rounded corners, align=center, minimum size=8mm, inner sep=3pt, font=\\small},\n  level distance=16mm,\n  level 1\u002F.style={sibling distance=46mm},\n  level 2\u002F.style={sibling distance=23mm},\n  edgelbl\u002F.style={draw=none, font=\\scriptsize},\n  dead\u002F.style={fill=red!8, draw=red!70, dashed},\n  edge from parent\u002F.style={draw, ->, >=stealth}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node {$V{=}0$\\\\$b{=}22$}\n    child {node[draw=acc, very thick] {$V{=}10$\\\\$b{=}22$}\n      child {node[fill=acc!15, draw=acc, very thick] {$z^*{=}22$\\\\$\\{A,B\\}$} edge from parent[draw=acc] node[edgelbl,left]{take $B$}}\n      child {node {$V{=}10$\\\\$b{=}16$} edge from parent node[edgelbl,right]{skip $B$}}\n      edge from parent[draw=acc] node[edgelbl,left]{take $A$}}\n    child {node[dead] {$V{=}0$\\\\$b{=}16\\,\\times$}\n      edge from parent[draw=red!70,dashed] node[edgelbl,right]{skip $A$}};\n  \\node[draw=none, align=left, font=\\scriptsize, text=red!75!black] at (5.6,-2.0) {$b\\le z^*{=}22$\\\\ pruned};\n\\end{tikzpicture}\n$$\n\nBranching _take_-first dives straight to the incumbent $\\{A,B\\}$ with value\n$z^*{=}22$. The skip-$A$ subtree's optimistic LP bound is only $16$ (take $B$\nwhole, then $\\tfrac23$ of $C$: $12 + 2\\cdot\\tfrac{6}{3} = 16$), which cannot beat\n$22$, so the entire right half is discarded before a single completion is built.\n\n### Search order: depth-first vs best-first\n\nThe skeleton above is **depth-first** branch and bound: it recurses to a leaf\nfast, so it finds _some_ complete solution, an incumbent, almost immediately,\nand it uses only $O(n)$ stack. The cost is that the first incumbent may be poor,\nweakening early pruning. The alternative is **best-first** search: keep a\n[priority queue](\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort) of live nodes keyed by\ntheir bound, and always expand the node with the most promising bound. Best-first\ntends to drive toward the optimum with the fewest _expansions_ and, for many\nproblems, expands the optimal node first, but it can hold an exponential frontier\nof live nodes in the queue, so its **memory** is the liability. The practical\ncompromise is to seed the incumbent with a quick greedy solution, then run\ndepth-first with strong bounds: cheap memory, and a floor high enough that the\nbound prunes hard from the start.\n\n$$\n% caption: Two search orders. Depth-first dives to a leaf for an early incumbent with\n%          $O(n)$ stack; best-first expands the highest-bound live node — fewer\n%          expansions, but an exponential frontier\n\\begin{tikzpicture}[\n  >=stealth, font=\\small,\n  nd\u002F.style={draw, circle, minimum size=6mm, inner sep=0},\n  hl\u002F.style={draw=acc, very thick},\n  lbl\u002F.style={draw=none, font=\\scriptsize},\n  edge from parent\u002F.style={draw, ->, >=stealth}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\begin{scope}[level distance=11mm, level 1\u002F.style={sibling distance=16mm}, level 2\u002F.style={sibling distance=8mm}]\n    \\node[nd,hl] (df) at (0,0) {}\n      child {node[nd,hl] {}\n        child {node[nd,hl] {} edge from parent[draw=acc,very thick]}\n        child {node[nd] {}}\n        edge from parent[draw=acc,very thick]}\n      child {node[nd] {}\n        child {node[nd] {}}\n        child {node[nd] {}}};\n  \\end{scope}\n  \\node[lbl] at (0,-4.0) {depth-first: dive to leaf};\n  \\node[lbl, acc] at (0,-4.7) {early incumbent, $O(n)$ memory};\n  \\begin{scope}[xshift=58mm, level distance=11mm, level 1\u002F.style={sibling distance=16mm}, level 2\u002F.style={sibling distance=8mm}]\n    \\node[nd] (bf) at (0,0) {}\n      child {node[nd,hl] {37}\n        child {node[nd] {} edge from parent}\n        child {node[nd] {}}\n        edge from parent[draw=acc,very thick]}\n      child {node[nd] {29}\n        child {node[nd] {}}\n        child {node[nd] {}}};\n  \\end{scope}\n  \\node[lbl] at (5.8,-4.0) {best-first: expand top bound};\n  \\node[lbl, acc] at (5.8,-4.7) {fewer expansions, big frontier};\n\\end{tikzpicture}\n$$\n\nDepth-first (left) follows one accented path to a leaf, banking an incumbent fast\nwhile holding only the current root-to-node stack. Best-first (right) instead pops\nthe live node of highest bound ($37 > 29$) from a priority queue, steering toward\nthe optimum in fewer expansions at the cost of keeping the whole frontier in\nmemory.\n\n## Meet in the middle\n\nSome problems resist pruning entirely: the bound is weak, the structure\nsymmetric, every branch genuinely live. If the instance is a subset problem over\n$n$ items and $n \\le 40$, **meet in the middle** sidesteps pruning and attacks the\nexponent directly. Split the items into two halves $A$ and $B$ of size $\\approx\nn\u002F2$. Enumerate *all* $2^{n\u002F2}$ subset sums of $A$ into a list $S_A$, and\nlikewise all subset sums of $B$ into $S_B$. Every subset of the whole is one\nchoice from $A$ paired with one from $B$, so the full answer is recovered by\n**combining one element of $S_A$ with one of $S_B$**, but we do that\ncombination cleverly, not by trying all $2^{n\u002F2}\\cdot 2^{n\u002F2} = 2^n$ pairs.\n\nFor the canonical task, find a subset whose sum is **closest to a target $T$**\n(this is the minimum-partition-difference problem with $T = (\\sum_i x_i)\u002F2$), sort\n$S_B$, then for each $a \\in S_A$ binary-search $S_B$ for the value nearest\n$T - a$. Each query is $O(\\log 2^{n\u002F2}) = O(n)$, so the whole combine is\n$O(2^{n\u002F2}\\,n)$.\n\n```algorithm\ncaption: $\\textsc{MeetInTheMiddle}$ — subset sum closest to target $T$\nsplit items into halves $A$ (size $\\lceil n\u002F2\\rceil$) and $B$\n$S_A \\gets$ all $2^{|A|}$ subset sums of $A$\n$S_B \\gets$ all $2^{|B|}$ subset sums of $B$\nsort $S_B$\n$best \\gets \\infty$\nfor each $a$ in $S_A$ do\n  $r \\gets T - a$ \u002F\u002F complement from $B$\n  $s \\gets$ value in $S_B$ nearest $r$ (binary search: floor and ceiling of $r$)\n  $best \\gets \\min(best,\\ |\\,a + s - T\\,|)$\nreturn $best$\n```\n\nThe enumeration is $O(2^{n\u002F2})$ per half, the sort is $O(2^{n\u002F2}\\,n)$, and the\ncombine is $O(2^{n\u002F2}\\,n)$, so the whole algorithm is $O(2^{n\u002F2}\\,n)$, a quadratic\nimprovement over the $O(2^n)$ brute force. Concretely, $2^{40}$ is about $10^{12}$\n(out of reach) while $2^{20}$ is about $10^6$ (instant). The technique is exact,\nno approximation and no pruning luck, and it is the intended solution to every\nHard subset problem in this lesson's practice set.\n\n$$\n% caption: $2^n \\to O(2^{n\u002F2})$ — enumerate each half, then binary-search the complement\n\\begin{tikzpicture}[\n  cell\u002F.style={draw, minimum width=8mm, minimum height=6mm, inner sep=1pt, font=\\small},\n  >=stealth]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[draw=none, font=\\footnotesize] at (-1.3,0) {$S_A$:};\n  \\node[cell] (a0) at (0,0) {$0$};\n  \\node[cell] (a1) at (0.8,0) {$3$};\n  \\node[cell, draw=acc, very thick] (a2) at (1.6,0) {$5$};\n  \\node[cell] (a3) at (2.4,0) {$8$};\n  \\node[draw=none, font=\\footnotesize] at (-1.3,-2.2) {$S_B$:};\n  \\node[cell] (b0) at (0,-2.2) {$1$};\n  \\node[cell] (b1) at (0.8,-2.2) {$4$};\n  \\node[cell, draw=acc, very thick] (b2) at (1.6,-2.2) {$6$};\n  \\node[cell] (b3) at (2.4,-2.2) {$9$};\n  \\node[draw=none, font=\\footnotesize] at (5.0,-1.1) {target $T = 11$};\n  \\node[draw=none, font=\\footnotesize] at (4.6,-2.2) {(sorted)};\n  \\draw[->, acc, very thick] (a2.south) .. controls (1.6,-1.1) and (1.6,-1.1) .. node[right, draw=none, font=\\footnotesize] {$T-5=6$} (b2.north);\n  \\node[draw=none, acc, font=\\footnotesize] at (1.6,-3.0) {$5 + 6 = 11$};\n\\end{tikzpicture}\n$$\n\nFor each sum $a$ on the top we binary-search the sorted bottom list for $T - a$;\nthe blue pair $5 + 6 = 11$ hits the target exactly. The same split-and-recombine\nidea is the graph analog **bidirectional search**: to find a shortest path, run\nBFS forward from the source and backward from the target simultaneously and stop\nwhen the two frontiers meet, exploring $\\approx 2\\cdot b^{d\u002F2}$ nodes instead of\n$b^d$.\n\n$$\n% caption: Bidirectional search: two BFS frontiers of radius $d\u002F2$ from $s$ and $t$ meet\n%          in the middle, touching $\\approx 2\\,b^{d\u002F2}$ nodes instead of one frontier of\n%          $b^d$\n\\begin{tikzpicture}[\n  >=stealth, font=\\small]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\useasboundingbox (-2.6,2.2) rectangle (8.8,-2.6);\n  % --- top: one-sided BFS, big disc of radius d ---\n  \\begin{scope}[shift={(-0.5,0.9)}, scale=0.42]\n    \\fill[red!12] (0,0) circle (3.0);\n    \\draw[red!75!black] (0,0) circle (3.0);\n    \\node[draw, circle, fill=acc!15, minimum size=4mm, inner sep=0, font=\\scriptsize] at (0,0) {$s$};\n    \\node[draw, circle, minimum size=4mm, inner sep=0, font=\\scriptsize] at (3.0,0) {$t$};\n  \\end{scope}\n  \\node[draw=none, font=\\scriptsize, align=left, text=red!75!black] at (3.6,0.9)\n    {one frontier of radius $d$:\\\\ $\\approx b^{d}$ nodes};\n  % --- bottom: two small discs of radius d\u002F2 meeting ---\n  \\begin{scope}[shift={(-0.9,-1.4)}, scale=0.42]\n    \\fill[acc!15] (0,0) circle (1.6);\n    \\draw[acc] (0,0) circle (1.6);\n    \\fill[acc!15] (3.0,0) circle (1.6);\n    \\draw[acc] (3.0,0) circle (1.6);\n    \\node[draw, circle, fill=acc!15, minimum size=4mm, inner sep=0, font=\\scriptsize] at (0,0) {$s$};\n    \\node[draw, circle, fill=acc!15, minimum size=4mm, inner sep=0, font=\\scriptsize] at (3.0,0) {$t$};\n    % meeting point\n    \\fill[acc] (1.5,0) circle (0.12);\n  \\end{scope}\n  \\node[draw=none, font=\\scriptsize, align=left, acc] at (3.9,-1.4)\n    {two frontiers of radius $\\tfrac{d}{2}$ meet:\\\\ $\\approx 2\\,b^{d\u002F2}$ nodes};\n\\end{tikzpicture}\n$$\n\n## When to reach for which\n\nThe three pruning disciplines line up neatly along one axis: _what justifies\ndiscarding a branch._\n\n- **Backtracking** prunes by **feasibility**: a partial solution that violates a\n  constraint can never be completed, so cut it.\n- **Branch and bound** prunes by **value**: a partial solution whose optimistic\n  bound cannot beat the incumbent is pointless to complete, so cut it.\n- **Meet in the middle** prunes _nothing_; it instead trades exponential time for\n  the **square root** of it, $2^n \\to 2^{n\u002F2}$, paying with $O(2^{n\u002F2})$ memory to\n  store the enumerated half.\n\nBranch and bound shines when a cheap, tight optimistic bound exists (knapsack's LP\nfill, a [TSP](\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness) node's spanning-tree\nlower bound). Meet in the middle shines when no such bound exists but $n$ is small\nenough that $2^{n\u002F2}$ is affordable. Both are _exact_; neither changes the\nworst-case exponential complexity; both routinely turn an infeasible instance into\na feasible one.\n\n## Takeaways\n\n- **Branch and bound** is backtracking for optimization: maintain an\n  **incumbent** (best complete solution so far) and a **bound** (optimistic\n  estimate per node), and **prune** any node whose bound cannot beat the\n  incumbent.\n- The method's power is all in the **bound tightness** and **search order**: a\n  tighter bound and an earlier strong incumbent each prune more of the tree.\n- For **0\u002F1 knapsack**, order by density $v_i\u002Fw_i$ and bound by the **LP-relaxation**\n  fractional fill; branch and bound **beats the $\\Theta(nW)$ DP when $W$ is huge\n  but $n$ is moderate**.\n- **Depth-first** branch and bound finds an incumbent fast with $O(n)$ memory;\n  **best-first** (priority queue on bound) targets the optimum with fewer\n  expansions but can hold an exponential frontier.\n- **Meet in the middle** enumerates each of two halves ($2^{n\u002F2}$ subset sums) and\n  recombines by sorting + binary search, giving $O(2^{n\u002F2}\\,n)$, exact search up\n  to $n \\approx 40$; **bidirectional search** is the graph analog.\n- One axis: backtracking prunes by **feasibility**, branch and bound by **value**,\n  meet in the middle trades exponential time for **$\\sqrt{}$ of it** at the cost of\n  memory.\n\n[^erickson-bb]: **Erickson**, Ch. — Backtracking: branch and bound as backtracking augmented with a value bound; the optimism of the bound is what makes pruning sound.\n[^skiena-bb]: **Skiena**, § — Combinatorial Search \u002F Heuristics: pruning a combinatorial search by bounding the best achievable completion, illustrated on knapsack-style problems.\n[^clrs-bb]: **CLRS**, Ch. 35 — Approximation Algorithms: the LP relaxation as an optimistic bound on an integer program, the same relaxation used to bound knapsack nodes.\n",{"text":7911,"minutes":7912,"time":7913,"words":7914},"9 min read",8.81,528600,1762,{"title":302,"description":7890},[7917,7919,7922],{"book":7869,"ref":7918},"§ — Combinatorial Search \u002F Heuristics",{"book":7920,"ref":7921},"CLRS","Ch. 35 — Approximation (branch-and-bound context)",{"book":7855,"ref":7923},"Ch. — 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