Dictionary of Algorithms and Data Structures

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This is a dictionary of algorithms, algorithmic techniques, data structures, archetypal problems, and related definitions. Algorithms include common functions, such as Ackermann's function. Problems include traveling salesman and Byzantine generals. Some entries have links to implementations and more information. Index pages list entries by area and by type. The two-level index has a total download 1/20 as big as this page.

Don't use this site to cheat. Teachers, contact us if we can help.

To define or correct terms, please contact Paul E. Black. We need help in automata theory, combinatorics, parallel or randomized algorithms, heuristics, and quantum computing. We do not include algorithms particular to business data processing, communications, operating systems or distributed algorithms, programming languages, AI, graphics, or numerical analysis: it is tough enough covering "general" algorithms and data structures. However, if you want to tackle one of these areas, we'll consider including them.

Some terms with a leading variable, such as n-way, m-dimensional, or p-branching, are under k-. You may find terms dealing with hardware, the computer industry, slang, etc., in the Free On-Line Dictionary Of Computing or in A Glossary of Computer Oriented Abbreviations and Acronyms.

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

α: see inverse Ackermann function ω Ω ρ-approximation algorithm Θ A absolute performance guarantee abstract data type (a,b)-tree accepting state Ackermann's function active data structure acyclic directed graph: see directed acyclic graph acyclic graph adaptive heap sort adaptive Huffman coding adaptive k-d tree adaptive sort address-calculation sort adjacency-list representation adjacency-matrix representation adjacent admissible vertex ADT: see abstract data type adversary Aho-Corasick algorithm algorithm BSTW algorithm FGK algorithmically solvable: see decidable problem all pairs shortest path all simple paths alphabet Alpha Skip Search algorithm alternating path alternating Turing machine alternation American flag sort amortized cost ancestor and ANSI antichain antisymmetric Apostolico-Crochemore Apostolico-Giancarlo algorithm approximate string matching: see string matching with errors approximation algorithm arborescence arc: see edge arithmetic coding array array index array merging array search articulation point: see cut vertex assignment problem association list: see dictionary associative associative array asymptotically tight bound asymptotic bound asymptotic lower bound asymptotic space complexity asymptotic time complexity asymptotic upper bound augmenting path automaton average case average-case cost AVL tree axiomatic semantics B backtracking bag balance balanced binary search tree balanced binary tree balanced k-way merge sort balanced merge sort balanced multiway merge balanced multiway tree: see B-tree balanced quicksort balanced tree balanced two-way merge sort BANG file Batcher sort: see bitonic sort Baum Welch algorithm BB(α) tree BBP algorithm BDD BD-tree Bellman-Ford algorithm Benford's law best case best-case cost best-first search biconnected component biconnected graph bidirectional bubble sort big-O notation binary function binary GCD binary heap binary insertion sort binary knapsack problem: see knapsack problem binary priority queue binary relation binary search binary search tree binary tree binary tree representation of trees bingo sort binomial heap binomial tree bin packing problem bin sort: see bucket sort bintree bipartite graph bipartite matching bisector bitonic sort bit vector Bk tree blind sort blind trie block block addressing index blocking flow block search: see jump search Bloom filter blossom bogosort boolean boolean expression boolean function border Boruvka's algorithm bottleneck traveling salesman bottom-up tree automaton boundary-based representation bounded error probability in polynomial time: see BPP bounded queue bounded stack Boyer-Moore Boyer-Moore-Horspool bozo sort B+-tree BPP Bradford's law branch and bound breadth-first search Bresenham's algorithm brick sort bridge British Museum technique brute force brute force string search brute force string search with mismatches BSP-tree B*-tree B-tree bubble sort bucket bucket array bucketing method bucket sort bucket trie buddy system buddy tree build-heap Burrows-Wheeler transform busy beaver BV-tree BWT: see Burrows-Wheeler transform Byzantine generals C cactus stack Calculus of Communicating Systems calendar queue candidate consistency testing candidate verification canonical complexity class capacitated facility location capacity capacity constraint cartesian tree: see randomized binary search tree cascade merge sort Caverphone CCS cell probe model cell tree cellular automaton centroid certificate chain chaining child Chinese postman problem Chinese remainder theorem Christofides algorithm chromatic index chromatic number circuit circuit complexity circuit value problem circular list circular queue clique clique problem clustering clustering free coalesced chaining coarsening cocktail shaker sort: see bidirectional bubble sort codeword coding tree Collatz problem collective recursion collision collision resolution scheme Colussi combination comb sort Commentz-Walter Communicating Sequential Processes commutative compact DAWG compact trie comparison sort competitive analysis competitive ratio complement complete binary tree complete graph completely connected graph complete tree complexity complexity class computable concave function concurrent flow concurrent read, concurrent write concurrent read, exclusive write confluently persistent data structure conjunction connected components connected graph constant function continuous knapsack problem: see fractional knapsack problem Cook reduction Cook's theorem CORDIC counting sort covering CRC: see cyclic redundancy check CRCW: see concurrent read, concurrent write CREW: see concurrent read, exclusive write critical path problem CSP CTL cube root cuckoo hashing cut cutting plane cutting stock problem cutting theorem cut vertex cycle cyclic redundancy check D D-adjacent DAG: see directed acyclic graph DAG shortest paths data structure DAWG: see directed acyclic word graph decidable language decidable problem decimation: see prune and search decision problem decomposable searching problem degree dense graph depoissonization depth depth-first search deque derangement descendant deterministic deterministic algorithm deterministic finite automata string search deterministic finite automaton: see deterministic finite state machine deterministic finite state machine deterministic finite tree automaton deterministic pushdown automaton deterministic tree automaton Deutsch-Jozsa algorithm DFA: see deterministic finite state machine DFS: see depth-first search DFS forest DFTA: see deterministic finite tree automaton diagonalization diameter dichotomic search dictionary diet: see discrete interval encoding tree difference digital search tree digital tree digraph: see directed graph Dijkstra's algorithm diminishing increment sort dining philosophers direct chaining directed acyclic graph directed acyclic word graph directed graph discrete interval encoding tree discrete p-center disjoint set disjunction distributional complexity distribution sort distributive partitioning sort divide and conquer divide and marriage before conquest division method domain dominance tree sort don't care Doomsday rule double-direction bubble sort: see bidirectional bubble sort double-ended priority queue double hashing double left rotation double metaphone double right rotation doubly-chained tree: see binary tree representation of trees doubly-ended queue: see deque doubly linked list DPDA: see deterministic pushdown automaton D-tree dual dual linear program Dutch national flag dyadic tree: see binary tree dynamic dynamic array dynamic hashing dynamic Huffman coding: see adaptive Huffman coding dynamic programming dynamization transformation E easy split, hard merge edge edge coloring edge connectivity edge crossing edge-weighted graph: see weighted graph edit distance: see Levenshtein distance edit operation edit script efficiency 8 queens elastic-bucket trie element uniqueness end-of-string enfilade ERCW: see exclusive read, concurrent write EREW: see exclusive read, exclusive write Euclidean algorithm: see Euclid's algorithm Euclidean distance Euclidean Steiner tree Euclidean traveling salesman problem Euclid's algorithm Euler cycle Eulerian graph Eulerian path: see Euler cycle exact string matching: see string matching EXCELL exchange sort: see bubble sort exclusive or: see xor exclusive read, concurrent write exclusive read, exclusive write exhaustive search existential state expandable hashing expander graph exponential extended binary tree extended Euclid's algorithm extended k-d tree extendible hashing external chaining: see separate chaining external index external memory algorithm external memory data structure external merge external node: see leaf external quicksort external radix sort external sort extrapolation search: see interpolation search extremal extreme point F facility location factor: see substring factorial fast fourier transform fathoming feasible region feasible solution feedback edge set feedback vertex set Ferguson-Forcade algorithm FFT: see fast fourier transform Fibonaccian search Fibonacci heap Fibonacci number Fibonacci tree FIFO: see queue filial-heir chain: see binary tree representation of trees Find find kth least element: see select kth element finitary tree finite Fourier transform finite state automaton: see finite state machine finite state machine finite state machine minimization finite state transducer first child-next sibling binary tree: see binary tree representation of trees first come, first served first-in, first-out Fisher-Yates shuffle fixed-grid method flash sort flow flow conservation flow function flow network Floyd-Warshall algorithm Ford-Bellman: see Bellman-Ford algorithm Ford-Fulkerson method forest forest editing problem formal language: see language formal methods formal verification forward index fractional knapsack problem fractional solution free edge free tree free vertex frequency count heuristic full array full binary tree full inverted index fully dynamic graph problem fully persistent data structure fully polynomial approximation scheme function functional data structure: see active data structure G Galil-Giancarlo Galil-Seiferas gamma function GBD-tree GCD: see greatest common divisor geometric optimization problem global optimum gnome sort graph graph coloring graph concentration graph drawing graph isomorphism graph partition Gray code greatest common divisor greedy algorithm greedy heuristic grid drawing grid file Grover's algorithm H halting problem Hamiltonian cycle Hamiltonian path Hamming distance hard split, easy merge hashbelt hash function hash heap hash table hash table delete Hausdorff distance hB-tree head heap heapify heap property heapsort heaviest common subsequence: see longest common subsequence height height-balanced binary search tree height-balanced tree heuristic hidden Markov model highest common factor: see greatest common divisor histogram sort HMM: see hidden Markov model homeomorphic horizontal visibility map Horner's rule Horspool: see Boyer-Moore-Horspool hsadelta Huffman coding huge sparse array Hungarian algorithm: see Munkres' assignment algorithm hybrid algorithm hyperedge hypergraph I ideal merge implication implies inclusion-exclusion principle inclusive or: see or incompressible string incremental hashing: see linear hashing in-degree independent set index file information theoretic bound in-order traversal in-place sort insertion sort integer linear program integer multi-commodity flow integer polyhedron interactive proof system interior-based representation internal node internal sort interpolation search interpolation-sequential search interpolation sort: see histogram sort intersection interval tree intractable introsort: see introspective sort introspective sort inverse Ackermann function inverse suffix array inverted file index inverted index irreflexive isomorphic iteration J Jaro-Winkler Johnson's algorithm Johnson-Trotter JSort J sort jump list jump search K Karmarkar's algorithm Karnaugh map Karp-Rabin Karp reduction k-ary heap k-ary Huffman coding k-ary tree k-clustering k-coloring k-connected graph k-d-B-tree k-dimensional K-dominant match k-d tree key KMP: see Knuth-Morris-Pratt algorithm KmpSkip Search knapsack problem knight's tour Knuth-Morris-Pratt algorithm Königsberg bridges problem: see Euler cycle Kolmogorov complexity Kraft's inequality Kripke structure Kruskal's algorithm kth order Fibonacci numbers kth shortest path kth smallest element: see select kth element KV diagram: see Karnaugh map k-way merge k-way merge sort k-way tree: see k-ary tree L labeled graph language last-in, first-out Las Vegas algorithm lattice layered graph LCM: see least common multiple LCS leaf least common multiple leftist tree left rotation Lempel-Ziv-Welch level-order traversal Levenshtein distance lexicographical order LIFO: see stack linear linear congruential generator linear hash linear hashing linear insertion sort: see insertion sort linear order: see total order linear probing linear probing sort linear product linear program linear quadtree linear search link linked list list list contraction little-o notation Lm distance load factor local alignment locality-sensitive hashing local optimum logarithmic longest common subsequence longest common substring Lotka's law lower bound lower triangular matrix lowest common ancestor l-reduction lucky sort LZW compression: see Lempel-Ziv-Welch

M Malhotra-Kumar-Maheshwari blocking flow Manhattan distance many-one reduction map: see dictionary Markov chain Marlena marriage problem: see assignment problem Master theorem matched edge matched vertex matching matrix matrix-chain multiplication problem max-heap property maximal independent set maximally connected component Maximal Shift maximum bipartite matching: see bipartite matching maximum-flow problem MAX-SNP MBB: see minimum bounding box Mealy machine mean median meld memoization merge merge sort metaheuristic metaphone midrange Miller-Rabin min-heap property minimal perfect hashing minimax minimum bounding box minimum cut minimum path cover minimum spanning tree minimum vertex cut Minkowski distance: see Lm distance mixed integer linear program mode model checking model of computation moderately exponential MODIFIND monotone priority queue monotonically decreasing monotonically increasing Monte Carlo algorithm Moore machine Morris-Pratt algorithm move: see transition move-to-front heuristic move-to-root heuristic MST: see minimum spanning tree multi-commodity flow multigraph multikey Quicksort multilayer grid file multiplication method multiprefix multiprocessor model multi-set: see bag multi suffix tree multiway decision multiway merge multiway search tree multiway tree Munkres' assignment algorithm N naive string search: see brute force string search nand n-ary function NC many-one reducibility nearest neighbor negation network flow: see flow function network flow problem: see maximum-flow problem next state NFA: see nondeterministic finite state machine NFTA: see nondeterministic finite tree automaton NIST node nonbalanced merge nonbalanced merge sort nondeterministic nondeterministic algorithm nondeterministic finite automaton: see nondeterministic finite state machine nondeterministic finite state machine nondeterministic finite tree automaton nondeterministic polynomial time: see NP nondeterministic tree automaton nondeterministic Turing machine nonterminal node: see internal node nor not Not So Naive NP NP-complete NP-complete language NP-hard n queens nullary function: see 0-ary function null tree NYSIIS O O: see big-O notation OBDD objective function oblivious algorithm occurrence octree off-line algorithm offset omega omicron one-based indexing one-dimensional on-line algorithm open addressing optimal optimal cost: see best-case cost optimal hashing: see perfect hashing optimal merge optimal mismatch optimal polygon triangulation problem optimal polyphase merge optimal polyphase merge sort optimal solution optimal triangulation problem optimal value optimization problem or oracle set oracle tape oracle Turing machine order ordered array ordered binary decision diagram: see OBDD ordered linked list ordered tree order preserving hash: see linear hash order-preserving Huffman coding order preserving minimal perfect hashing oriented acyclic graph: see directed acyclic graph oriented graph: see directed graph oriented tree: see rooted tree orthogonal drawing orthogonal lists orthogonally convex rectilinear polygon oscillating merge sort out-degree P P packing padding argument pagoda pairing heap PAM: see point access method parallel computation thesis parallel prefix computation parallel random-access machine parametric searching parent partial function partially decidable problem partially dynamic graph problem partially ordered set: see poset partially persistent data structure partial order partial recursive function partition passive data structure path path cover path system problem Patricia tree pattern pattern element P-complete PCP: see Post's correspondence problem PDA: see pushdown automaton Pearson's hash perfect binary tree perfect hashing perfect k-ary tree perfect matching perfect shuffle performance guarantee performance ratio: see relative performance guarantee permutation persistent data structure phonetic coding pigeonhole sort pile pipelined divide and conquer planar graph planarization planar straight-line graph PLOP-hashing point access method pointer jumping pointer machine poissonization polychotomy polyhedron polylogarithmic polynomial polynomial approximation scheme polynomial hierarchy polynomial time polynomial-time reduction polyphase merge polyphase merge sort polytope poset postfix traversal: see postorder traversal Post machine postman's sort postorder traversal Post's correspondence problem potential function PRAM: see parallel random-access machine predicate prefix prefix code prefix computation prefix sums: see scan prefix traversal: see preorder traversal preorder traversal primary clustering primitive recursive Prim-Jarnik algorithm principle of optimality priority queue prisoner's dilemma PRNG: see pseudo-random number generator probabilistic algorithm probabilistically checkable proof probabilistic Turing machine probe sequence procedure process algebra proper proper binary tree: see full binary tree proper coloring proper subset property list: see dictionary prune and search pseudo-random number generator PTAS: see polynomial approximation scheme pth order Fibonacci numbers: see kth order Fibonacci numbers P-tree purely functional language pushdown automaton pushdown transducer p-way merge sort: see k-way merge sort Q qm sort q sort quadratic probing quadtree quadtree complexity theorem quad trie quantum computation queue quick search quicksort R Rabin-Karp: see Karp-Rabin radix sort radix tree: see Patricia tree ragged matrix Raita random access machine randomization randomized algorithm randomized binary search tree randomized complexity randomized polynomial time: see RP randomized rounding randomized search tree Randomized-Select random number generator random sampling random search range range sort rank rapid sort Ratcliff/Obershelp pattern recognition RBST: see randomized binary search tree reachable rebalance recognizer rectangular matrix rectilinear rectilinear Steiner tree recurrence equations: see recurrence relation recurrence relation recursion recursion termination recursion tree recursive recursive data structure recursive doubling: see pointer jumping recursive language: see decidable language recursively enumerable language recursively solvable: see decidable problem red-black tree reduced basis reduced digraph reduced ordered binary decision diagram reduction reflexive regular decomposition rehashing: see double hashing relation relational structure relative performance guarantee relaxation relaxed balance repeated squaring rescalable restricted universe sort result cache Reverse Colussi Reverse Factor R-file Rice's method right rotation right-threaded tree RNG: see random number generator ROBDD: see reduced ordered binary decision diagram Robin Hood hashing root root balance: see balance rooted tree rotate left: see left rotation rotate right: see right rotation rotation rough graph RP R+-tree R*-tree R-tree run time S saguaro stack: see cactus stack SAM: see spatial access method saturated edge SBB tree scan scapegoat tree scatter storage: see hash table Schorr-Waite graph marking algorithm search search tree search tree property secant search secondary clustering segment Select select and partition selection problem: see select kth element selection sort select kth element select mode self-loop self-organizing heuristic self-organizing list self-organizing sequential search: see transpose sequential search semidefinite programming separate chaining separation theorem sequential search: see linear search set set cover set packing shadow heap shadow merge shadow merge insert shaker sort: see bidirectional bubble sort Shannon-Fano coding shared memory Shell sort Shift-Or Shor's algorithm shortcutting: see pointer jumping shortest common supersequence shortest common superstring shortest path shortest spanning tree: see minimum spanning tree shuffle: see permutation shuffle sort sibling sieve of Eratosthenes sift up signature Simon's algorithm simple merge simple path simple uniform hashing simplex simulated annealing simulation theorem single-destination shortest-path problem single-pair shortest-path problem: see shortest path single program multiple data single-source shortest-path problem singly linked list: see linked list singularity analysis sink sinking sort: see bubble sort skd-tree skew symmetry skip list skip search slope selection Smith algorithm Smith-Waterman algorithm smoothsort solvable sort sorted array sorted list sort in place: see in-place sort soundex source space-constructible function space ordering method spanning tree sparse graph sparse matrix sparsification sparsity spatial access method spiral storage splay tree SPMD: see single program multiple data square matrix square root SST: see minimum spanning tree stable stack stack tree star encoding star-shaped polygon start state state state machine state transition: see transition static static Huffman coding: see Huffman coding s-t cut st-digraph Steiner point Steiner ratio Steiner tree Steiner vertex Steinhaus-Johnson-Trotter: see Johnson-Trotter Stirling's approximation Stirling's formula stooge sort straight-line drawing strand sort strictly decreasing strictly increasing strictly lower triangular matrix strictly upper triangular matrix string string editing problem string matching string matching on ordered alphabets string matching with errors string matching with mismatches string searching: see string matching strip packing strongly connected component strongly connected graph strongly NP-hard subadditive ergodic theorem subgraph subgraph isomorphism sublinear time algorithm subsequence subset substring subtree suffix suffix array suffix automaton suffix tree superimposed code superset supersink supersource symmetric symmetrically linked list: see doubly linked list symmetric binary B-tree: see red-black tree symmetric set difference symmetry breaking T tail tail recursion target temporal logic terminal terminal node: see leaf ternary search tree text text searching: see string matching theta: see Θ threaded binary tree threaded tree three-dimensional three-way merge sort three-way radix quicksort: see multikey Quicksort time-constructible function time/space complexity top-down radix sort top-down tree automaton topological order topological sort topology tree total function totally decidable language: see decidable language totally decidable problem: see decidable problem totally undecidable problem total order tour: see Hamiltonian cycle tournament tournament sort towers of Hanoi tractable transition transition function transitive transitive closure transitive reduction transpose sequential search traveling salesman treap tree tree automaton tree contraction tree editing problem treesort (1) treesort (2) tree traversal triangle inequality triconnected graph trie trinary function tripartition TSP: see traveling salesman TST: see ternary search tree Turbo-BM Turbo Reverse Factor Turing machine Turing reduction Turing transducer twin grid file 2-choice hashing two-dimensional 2-left hashing two-level grid file 2-3-4 tree 2-3 tree Two Way algorithm two-way linked list: see doubly linked list two-way merge sort U UB-tree UKP: see unbounded knapsack problem unary function unbounded knapsack problem uncomputable function uncomputable problem undecidable language undecidable problem undirected graph uniform circuit complexity uniform circuit family uniform hashing uniform matrix union union of automata universal B-tree: see UB-tree universal hashing universal state universal Turing machine universe UnShuffle sort unsolvable problem unsorted list upper triangular matrix V van Emde-Boas priority queue vehicle routing problem Veitch diagram: see Karnaugh map Venn diagram vertex vertex coloring vertex connectivity vertex cover vertical visibility map virtual hashing visibility map visible Viterbi algorithm Vitter's algorithm VRP: see vehicle routing problem W walk weak-heap weak-heap sort weight-balanced tree: see BB(α) tree weighted, directed graph weighted graph window witness work work-depth model work-efficient work-preserving worst case worst-case cost worst-case minimum access X xor Y Yule distribution: see Zipfian distribution Z Zeller's congruence 0-ary function 0-based indexing 0-1 knapsack problem: see knapsack problem Zhu-Takaoka Zipfian distribution Zipf's law zipper ZPP

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We thank those who contributed definitions as well as many others who offered suggestions and corrections.

Here are some references on algorithms and data structures.

The Stony Brook Algorithm Repository, which has algorithms organized by type, succinct, illustrated definitions, and ratings of sites with implementations.

Data Structures and Algorithms is a wonderful site with illustrations, explanations, analysis, and code taking the student from arrays and lists through trees, graphs, and intractable problems.

Eric Weisstein's World of Mathematics or MathWorld.

The Computing Research Repository (CoRR).

Bibliography

[AS98] Pankaj K. Agarwal and Micha Sharir, Efficient Algorithms for Geometric Optimization, ACM Computing Surveys, 30(4):412-458, December 1998.

[ATCH99] Algorithms and Theory of Computation Handbook, Mikhail J. Atallah, ed., CRC Press LLC, 1999.

[CLR90] Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to Algorithms, MIT Press, 1990.

[GBY91] Gaston H. Gonnet and Ricardo Baeza-Yates, Handbook of Algorithms and Data Structures -- in Pascal and C, 2^nd edition, Addison-Wesley, 1991.

[GCG92] P. Gupta, P. P. Chakrabarti, and S. Ghose, The Towers of Hanoi: Generalizations, Specializations, and Algorithms, Intern. J. Computer Math., 46:149-161, Gordon and Breach Science Publishers S.A., 1992.

[GG98] Volker Gaede and Oliver Günther, Multidimensional Access Methods, ACM Computing Surveys, 30(2):170-231, June 1998.

[GT97] Michael T. Goodrich and Roberto Tamassia, Data Structures and Algorithms in Java, 2^nd edition, John Wiley & Sons, 1997.

[Graef06] Goetz Graefe, Implementing Sorting in Database Systems, ACM Computing Surveys, 38(3), Article 10, September 2006.

[Hirv01] Mika Hirvensalo, Quantum Computing, Springer-Verlag, 2001.

[HS83] Ellis Horowitz and Sartaj Sahni, Fundamentals of Data Structures, Computer Science Press, 1983.

[Knuth97] Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, volumes 1 and 2, 2^nd edition, 1997.

[Knuth98] Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, volume 3, 2^nd edition, 1998.

[Leda98] LEDA (accessed 4 December 2002).

[Sedge97] Robert Sedgewick, Algorithms in C, Addison-Wesley, 1997.

[Stand98] Thomas Standish, Data Structures in Java, Addison-Wesley, 1998.

[Sund98] Daniel M. Sunday, A Very Fast Substring Search Algorithm, Communications of the ACM, 33(8):132-142, August 1998.

[Vitt01] Jeffrey Scott Vitter, External Memory Algorithms and Data Structures: Dealing with Massive Data, ACM Computing Surveys, 33(2):209-271, June 2001.

[Wier98] Roel Wieringa, A Survey of Structured and Object-Oriented Software Specification Methods and Techniques, ACM Computing Surveys, 30(4):459-527, December 1998.

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Here are citation examples and an explanation of credit.

Robots, please index all term pages, including spelling variants.

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