List of algorithms

An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.

Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology.^[1]

The following is a list of well-known algorithms along with one-line descriptions for each.

Automated planning

Combinatorial algorithms

General combinatorial algorithms

Brent's algorithm: finds a cycle in function value iterations using only two iterators^[2]
Floyd's cycle-finding algorithm: finds a cycle in function value iterations^[3]
Gale–Shapley algorithm: solves the stable marriage problem^{[citation needed]}
Pseudorandom number generators (uniformly distributed—see also List of pseudorandom number generators for other PRNGs with varying degrees of convergence and varying statistical quality):^{[citation needed]}

Graph algorithms

Coloring algorithm: Graph coloring algorithm.
Hopcroft–Karp algorithm: convert a bipartite graph to a maximum cardinality matching
Hungarian algorithm: algorithm for finding a perfect matching
Prüfer coding: conversion between a labeled tree and its Prüfer sequence
Tarjan's off-line lowest common ancestors algorithm: computes lowest common ancestors for pairs of nodes in a tree
Topological sort: finds linear order of nodes (e.g. jobs) based on their dependencies.

Graph drawing

Force-based algorithms (also known as force-directed algorithms or spring-based algorithm)
Spectral layout

Network theory

Network analysis
- Link analysis
  - Girvan–Newman algorithm: detect communities in complex systems
  - Web link analysis
    - Hyperlink-Induced Topic Search (HITS) (also known as Hubs and authorities)
    - PageRank
    - TrustRank
Flow networks
- Dinic's algorithm: is a strongly polynomial algorithm for computing the maximum flow in a flow network.
- Edmonds–Karp algorithm: implementation of Ford–Fulkerson
- Ford–Fulkerson algorithm: computes the maximum flow in a graph
- Karger's algorithm: a Monte Carlo method to compute the minimum cut of a connected graph
- Push–relabel algorithm: computes a maximum flow in a graph

Routing for graphs

Edmonds' algorithm (also known as Chu–Liu/Edmonds' algorithm): find maximum or minimum branchings
Euclidean minimum spanning tree: algorithms for computing the minimum spanning tree of a set of points in the plane
Longest path problem: find a simple path of maximum length in a given graph
Minimum spanning tree
Nonblocking minimal spanning switch say, for a telephone exchange
Shortest path problem
- Bellman–Ford algorithm: computes shortest paths in a weighted graph (where some of the edge weights may be negative)
- Dijkstra's algorithm: computes shortest paths in a graph with non-negative edge weights
- Floyd–Warshall algorithm: solves the all pairs shortest path problem in a weighted, directed graph
- Johnson's algorithm: all pairs shortest path algorithm in sparse weighted directed graph
Transitive closure problem: find the transitive closure of a given binary relation
Traveling salesman problem
- Christofides algorithm
- Nearest neighbour algorithm
Warnsdorff's rule: a heuristic method for solving the Knight's tour problem

Graph search

A*: special case of best-first search that uses heuristics to improve speed
B*: a best-first graph search algorithm that finds the least-cost path from a given initial node to any goal node (out of one or more possible goals)
Backtracking: abandons partial solutions when they are found not to satisfy a complete solution
Beam search: is a heuristic search algorithm that is an optimization of best-first search that reduces its memory requirement
Beam stack search: integrates backtracking with beam search
Best-first search: traverses a graph in the order of likely importance using a priority queue
Bidirectional search: find the shortest path from an initial vertex to a goal vertex in a directed graph
Breadth-first search: traverses a graph level by level
Brute-force search: an exhaustive and reliable search method, but computationally inefficient in many applications
D*: an incremental heuristic search algorithm
Depth-first search: traverses a graph branch by branch
Dijkstra's algorithm: a special case of A* for which no heuristic function is used
General Problem Solver: a seminal theorem-proving algorithm intended to work as a universal problem solver machine.
Iterative deepening depth-first search (IDDFS): a state space search strategy
Jump point search: an optimization to A* which may reduce computation time by an order of magnitude using further heuristics
Lexicographic breadth-first search (also known as Lex-BFS): a linear time algorithm for ordering the vertices of a graph
Uniform-cost search: a tree search that finds the lowest-cost route where costs vary
SSS*: state space search traversing a game tree in a best-first fashion similar to that of the A* search algorithm

Subgraphs

Cliques
- Bron–Kerbosch algorithm: a technique for finding maximal cliques in an undirected graph
- MaxCliqueDyn maximum clique algorithm: find a maximum clique in an undirected graph
Strongly connected components
Subgraph isomorphism problem

Sequence algorithms

Approximate sequence matching

Bitap algorithm: fuzzy algorithm that determines if strings are approximately equal.
Phonetic algorithms
- Daitch–Mokotoff Soundex: a Soundex refinement which allows matching of Slavic and Germanic surnames
- Double Metaphone: an improvement on Metaphone
- Match rating approach: a phonetic algorithm developed by Western Airlines
- Metaphone: an algorithm for indexing words by their sound, when pronounced in English
- NYSIIS: phonetic algorithm, improves on Soundex
- Soundex: a phonetic algorithm for indexing names by sound, as pronounced in English
String metrics: computes a similarity or dissimilarity (distance) score between two pairs of text strings
- Damerau–Levenshtein distance: computes a distance measure between two strings, improves on Levenshtein distance
- Dice's coefficient (also known as the Dice coefficient): a similarity measure related to the Jaccard index
- Hamming distance: sum number of positions which are different
- Jaro–Winkler distance: is a measure of similarity between two strings
- Levenshtein edit distance: computes a metric for the amount of difference between two sequences
Trigram search: search for text when the exact syntax or spelling of the target object is not precisely known

Selection algorithms

Sequence search

Linear search: locates an item in an unsorted sequence
Selection algorithm: finds the kth largest item in a sequence
Ternary search: a technique for finding the minimum or maximum of a function that is either strictly increasing and then strictly decreasing or vice versa
Sorted lists
- Binary search algorithm: locates an item in a sorted sequence
- Fibonacci search technique: search a sorted sequence using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers
- Jump search (or block search): linear search on a smaller subset of the sequence
- Predictive search: binary-like search which factors in magnitude of search term versus the high and low values in the search. Sometimes called dictionary search or interpolated search.
- Uniform binary search: an optimization of the classic binary search algorithm
- Eytzinger binary search: cache friendly binary search algorithm ^[4]

Sequence merging

Simple merge algorithm
k-way merge algorithm
Union (merge, with elements on the output not repeated)

Sequence permutations

Fisher–Yates shuffle (also known as the Knuth shuffle): randomly shuffle a finite set
Schensted algorithm: constructs a pair of Young tableaux from a permutation
Steinhaus–Johnson–Trotter algorithm (also known as the Johnson–Trotter algorithm): generates permutations by transposing elements
Heap's permutation generation algorithm: interchange elements to generate next permutation

Sequence combinations

Sequence alignment

Dynamic time warping: measure similarity between two sequences which may vary in time or speed
Hirschberg's algorithm: finds the least cost sequence alignment between two sequences, as measured by their Levenshtein distance
Needleman–Wunsch algorithm: find global alignment between two sequences
Smith–Waterman algorithm: find local sequence alignment

Sequence sorting

Exchange sorts
- Bubble sort: for each pair of indices, swap the items if out of order
- Cocktail shaker sort or bidirectional bubble sort, a bubble sort traversing the list alternately from front to back and back to front
- Comb sort
- Gnome sort
- Odd–even sort
- Quicksort: divide list into two, with all items on the first list coming before all items on the second list.; then sort the two lists. Often the method of choice
Humorous or ineffective
Hybrid
- Flashsort
- Introsort: begin with quicksort and switch to heapsort when the recursion depth exceeds a certain level
- Timsort: adaptative algorithm derived from merge sort and insertion sort. Used in Python 2.3 and up, and Java SE 7.
Insertion sorts
- Insertion sort: determine where the current item belongs in the list of sorted ones, and insert it there
- Library sort
- Patience sorting
- Shell sort: an attempt to improve insertion sort
- Tree sort (binary tree sort): build binary tree, then traverse it to create sorted list
- Cycle sort: in-place with theoretically optimal number of writes
Merge sorts
- Merge sort: sort the first and second half of the list separately, then merge the sorted lists
- Slowsort
- Strand sort
Non-comparison sorts
- Bead sort
- Bucket sort
- Burstsort: build a compact, cache efficient burst trie and then traverse it to create sorted output
- Counting sort
- Pigeonhole sort
- Postman sort: variant of Bucket sort which takes advantage of hierarchical structure
- Radix sort: sorts strings letter by letter
Selection sorts
- Heapsort: convert the list into a heap, keep removing the largest element from the heap and adding it to the end of the list
- Selection sort: pick the smallest of the remaining elements, add it to the end of the sorted list
- Smoothgamersort
Other
Unknown class
- Samplesort

Subsequences

Longest common subsequence problem: Find the longest subsequence common to all sequences in a set of sequences
Longest increasing subsequence problem: Find the longest increasing subsequence of a given sequence
Ruzzo–Tompa algorithm: Find all non-overlapping, contiguous, maximal scoring subsequences in a sequence of real numbers
Shortest common supersequence problem: Find the shortest supersequence that contains two or more sequences as subsequences

Substrings

Kadane's algorithm: finds the contiguous subarray with largest sum in an array of numbers
Longest common substring problem: find the longest string (or strings) that is a substring (or are substrings) of two or more strings
Substring search
- Aho–Corasick string matching algorithm: trie based algorithm for finding all substring matches to any of a finite set of strings
- Boyer–Moore string-search algorithm: amortized linear (sublinear in most times) algorithm for substring search
- Boyer–Moore–Horspool algorithm: Simplification of Boyer–Moore
- Knuth–Morris–Pratt algorithm: substring search which bypasses reexamination of matched characters
- Rabin–Karp string search algorithm: searches multiple patterns efficiently
- Zhu–Takaoka string matching algorithm: a variant of Boyer–Moore
Ukkonen's algorithm: a linear-time, online algorithm for constructing suffix trees
Matching wildcards
- Rich Salz' wildmat: a widely used open-source recursive algorithm
- Krauss matching wildcards algorithm: an open-source non-recursive algorithm

Computational mathematics

Abstract algebra

Chien search: a recursive algorithm for determining roots of polynomials defined over a finite field
Schreier–Sims algorithm: computing a base and strong generating set (BSGS) of a permutation group
Todd–Coxeter algorithm: Procedure for generating cosets.

Computer algebra

Buchberger's algorithm: finds a Gröbner basis
Cantor–Zassenhaus algorithm: factor polynomials over finite fields
Faugère F4 algorithm: finds a Gröbner basis (also mentions the F5 algorithm)
Gosper's algorithm: find sums of hypergeometric terms that are themselves hypergeometric terms
Knuth–Bendix completion algorithm: for rewriting rule systems
Multivariate division algorithm: for polynomials in several indeterminates
Pollard's kangaroo algorithm (also known as Pollard's lambda algorithm ): an algorithm for solving the discrete logarithm problem
Polynomial long division: an algorithm for dividing a polynomial by another polynomial of the same or lower degree
Risch algorithm: an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives)

Geometry

Closest pair problem: find the pair of points (from a set of points) with the smallest distance between them
Collision detection algorithms: check for the collision or intersection of two given solids
Cone algorithm: identify surface points
Convex hull algorithms: determining the convex hull of a set of points
- Graham scan
- Quickhull
- Gift wrapping algorithm or Jarvis march
- Chan's algorithm
- Kirkpatrick–Seidel algorithm
Euclidean distance transform: computes the distance between every point in a grid and a discrete collection of points.
Geometric hashing: a method for efficiently finding two-dimensional objects represented by discrete points that have undergone an affine transformation
Gilbert–Johnson–Keerthi distance algorithm: determining the smallest distance between two convex shapes.
Jump-and-Walk algorithm: an algorithm for point location in triangulations
Laplacian smoothing: an algorithm to smooth a polygonal mesh
Line segment intersection: finding whether lines intersect, usually with a sweep line algorithm
- Bentley–Ottmann algorithm
- Shamos–Hoey algorithm
Minimum bounding box algorithms: find the oriented minimum bounding box enclosing a set of points
Nearest neighbor search: find the nearest point or points to a query point
Nesting algorithm: make the most efficient use of material or space
Point in polygon algorithms: tests whether a given point lies within a given polygon
Point set registration algorithms: finds the transformation between two point sets to optimally align them.
Rotating calipers: determine all antipodal pairs of points and vertices on a convex polygon or convex hull.
Shoelace algorithm: determine the area of a polygon whose vertices are described by ordered pairs in the plane
Triangulation
- Delaunay triangulation
  - Ruppert's algorithm (also known as Delaunay refinement): create quality Delaunay triangulations
  - Chew's second algorithm: create quality constrained Delaunay triangulations
- Marching triangles: reconstruct two-dimensional surface geometry from an unstructured point cloud
- Polygon triangulation algorithms: decompose a polygon into a set of triangles
- Voronoi diagrams, geometric dual of Delaunay triangulation
  - Bowyer–Watson algorithm: create voronoi diagram in any number of dimensions
  - Fortune's Algorithm: create voronoi diagram
- Quasitriangulation

Number theoretic algorithms

Binary GCD algorithm: Efficient way of calculating GCD.
Booth's multiplication algorithm
Chakravala method: a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation
Discrete logarithm:
Euclidean algorithm: computes the greatest common divisor
Extended Euclidean algorithm: also solves the equation ax + by = c
Integer factorization: breaking an integer into its prime factors
Multiplication algorithms: fast multiplication of two numbers
Modular square root: computing square roots modulo a prime number
Odlyzko–Schönhage algorithm: calculates nontrivial zeroes of the Riemann zeta function
Lenstra–Lenstra–Lovász algorithm (also known as LLL algorithm): find a short, nearly orthogonal lattice basis in polynomial time
Primality tests: determining whether a given number is prime

Numerical algorithms

Differential equation solving

Euler method
Backward Euler method
Trapezoidal rule (differential equations)
Linear multistep methods
Runge–Kutta methods
- Euler integration
Multigrid methods (MG methods), a group of algorithms for solving differential equations using a hierarchy of discretizations
Partial differential equation:
- Finite difference method
- Crank–Nicolson method for diffusion equations
- Lax–Wendroff for wave equations
Verlet integration (French pronunciation: [vɛʁˈlɛ]): integrate Newton's equations of motion

Elementary and special functions

Computation of π:
- Borwein's algorithm: an algorithm to calculate the value of 1/π
- Gauss–Legendre algorithm: computes the digits of pi
- Chudnovsky algorithm: a fast method for calculating the digits of π
- Bailey–Borwein–Plouffe formula: (BBP formula) a spigot algorithm for the computation of the nth binary digit of π
Division algorithms: for computing quotient and/or remainder of two numbers
- Long division
- Restoring division
- Non-restoring division
- SRT division
- Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q.
- Goldschmidt division
Hyperbolic and Trigonometric Functions:
- BKM algorithm: computes elementary functions using a table of logarithms
- CORDIC: computes hyperbolic and trigonometric functions using a table of arctangents
Exponentiation:
- Addition-chain exponentiation: exponentiation by positive integer powers that requires a minimal number of multiplications
- Exponentiating by squaring: an algorithm used for the fast computation of large integer powers of a number
Montgomery reduction: an algorithm that allows modular arithmetic to be performed efficiently when the modulus is large
Multiplication algorithms: fast multiplication of two numbers
- Booth's multiplication algorithm: a multiplication algorithm that multiplies two signed binary numbers in two's complement notation
- Fürer's algorithm: an integer multiplication algorithm for very large numbers possessing a very low asymptotic complexity
- Karatsuba algorithm: an efficient procedure for multiplying large numbers
- Schönhage–Strassen algorithm: an asymptotically fast multiplication algorithm for large integers
- Toom–Cook multiplication: (Toom3) a multiplication algorithm for large integers
Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal).
- Newton's method
Rounding functions: the classic ways to round numbers
Spigot algorithm: a way to compute the value of a mathematical constant without knowing preceding digits
Square and Nth root of a number:
- Alpha max plus beta min algorithm: an approximation of the square-root of the sum of two squares
- Methods of computing square roots
- nth root algorithm
Summation:
- Binary splitting: a divide and conquer technique which speeds up the numerical evaluation of many types of series with rational terms
- Kahan summation algorithm: a more accurate method of summing floating-point numbers
Unrestricted algorithm

Geometric

Filtered back-projection: efficiently computes the inverse 2-dimensional Radon transform.
Level set method (LSM): a numerical technique for tracking interfaces and shapes

Interpolation and extrapolation

Birkhoff interpolation: an extension of polynomial interpolation
Cubic interpolation
Hermite interpolation
Lagrange interpolation: interpolation using Lagrange polynomials
Linear interpolation: a method of curve fitting using linear polynomials
Monotone cubic interpolation: a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.
Multivariate interpolation
- Bicubic interpolation: a generalization of cubic interpolation to two dimensions
- Bilinear interpolation: an extension of linear interpolation for interpolating functions of two variables on a regular grid
- Lanczos resampling ("Lanzosh"): a multivariate interpolation method used to compute new values for any digitally sampled data
- Nearest-neighbor interpolation
- Tricubic interpolation: a generalization of cubic interpolation to three dimensions
Pareto interpolation: a method of estimating the median and other properties of a population that follows a Pareto distribution.
Polynomial interpolation
- Neville's algorithm
Spline interpolation: Reduces error with Runge's phenomenon.
- De Boor algorithm: B-splines
- De Casteljau's algorithm: Bézier curves
Trigonometric interpolation

Linear algebra

Krylov methods (for large sparse matrix problems; third most-important numerical method class of the 20th century as ranked by SISC; after fast-fourier and fast-multipole)
Eigenvalue algorithms
Gram–Schmidt process: orthogonalizes a set of vectors
Matrix multiplication algorithms
- Cannon's algorithm: a distributed algorithm for matrix multiplication especially suitable for computers laid out in an N × N mesh
- Coppersmith–Winograd algorithm: square matrix multiplication
- Freivalds' algorithm: a randomized algorithm used to verify matrix multiplication
- Strassen algorithm: faster matrix multiplication

Solving systems of linear equations
- Biconjugate gradient method: solves systems of linear equations
- Conjugate gradient: an algorithm for the numerical solution of particular systems of linear equations
- Gaussian elimination
- Gauss–Jordan elimination: solves systems of linear equations
- Gauss–Seidel method: solves systems of linear equations iteratively
- Levinson recursion: solves equation involving a Toeplitz matrix
- Stone's method: also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations
- Successive over-relaxation (SOR): method used to speed up convergence of the Gauss–Seidel method
- Tridiagonal matrix algorithm (Thomas algorithm): solves systems of tridiagonal equations
Sparse matrix algorithms
- Cuthill–McKee algorithm: reduce the bandwidth of a symmetric sparse matrix
- Minimum degree algorithm: permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition
- Symbolic Cholesky decomposition: Efficient way of storing sparse matrix

Monte Carlo

Gibbs sampling: generates a sequence of samples from the joint probability distribution of two or more random variables
Hybrid Monte Carlo: generates a sequence of samples using Hamiltonian weighted Markov chain Monte Carlo, from a probability distribution which is difficult to sample directly.
Metropolis–Hastings algorithm: used to generate a sequence of samples from the probability distribution of one or more variables
Wang and Landau algorithm: an extension of Metropolis–Hastings algorithm sampling

Numerical integration

MISER algorithm: Monte Carlo simulation, numerical integration

Root finding

Bisection method
False position method: and Illinois method: 2-point, bracketing
Halley's method: uses first and second derivatives
ITP method: minmax optimal and superlinear convergence simultaneously
Muller's method: 3-point, quadratic interpolation
Newton's method: finds zeros of functions with calculus
Ridder's method: 3-point, exponential scaling
Secant method: 2-point, 1-sided

Optimization algorithms

Hybrid Algorithms

Alpha–beta pruning: search to reduce number of nodes in minimax algorithm
Branch and bound
Bruss algorithm: see odds algorithm
Chain matrix multiplication
Combinatorial optimization: optimization problems where the set of feasible solutions is discrete
- Greedy randomized adaptive search procedure (GRASP): successive constructions of a greedy randomized solution and subsequent iterative improvements of it through a local search
- Hungarian method: a combinatorial optimization algorithm which solves the assignment problem in polynomial time
Constraint satisfaction
- General algorithms for the constraint satisfaction
- Chaff algorithm: an algorithm for solving instances of the Boolean satisfiability problem
- Davis–Putnam algorithm: check the validity of a first-order logic formula
- Davis–Putnam–Logemann–Loveland algorithm (DPLL): an algorithm for deciding the satisfiability of propositional logic formula in conjunctive normal form, i.e. for solving the CNF-SAT problem
- Exact cover problem
  - Algorithm X: a nondeterministic algorithm
  - Dancing Links: an efficient implementation of Algorithm X
Cross-entropy method: a general Monte Carlo approach to combinatorial and continuous multi-extremal optimization and importance sampling
Differential evolution
Dynamic Programming: problems exhibiting the properties of overlapping subproblems and optimal substructure
Ellipsoid method: is an algorithm for solving convex optimization problems
Evolutionary computation: optimization inspired by biological mechanisms of evolution
- Evolution strategy
- Gene expression programming
- Genetic algorithms
  - Fitness proportionate selection – also known as roulette-wheel selection
  - Stochastic universal sampling
  - Truncation selection
  - Tournament selection
- Memetic algorithm
- Swarm intelligence
  - Ant colony optimization
  - Bees algorithm: a search algorithm which mimics the food foraging behavior of swarms of honey bees
  - Particle swarm
Frank-Wolfe algorithm: an iterative first-order optimization algorithm for constrained convex optimization
Golden-section search: an algorithm for finding the maximum of a real function
Gradient descent
Grid Search
Harmony search (HS): a metaheuristic algorithm mimicking the improvisation process of musicians
Interior point method
Linear programming
- Benson's algorithm: an algorithm for solving linear vector optimization problems
- Dantzig–Wolfe decomposition: an algorithm for solving linear programming problems with special structure
- Delayed column generation
- Integer linear programming: solve linear programming problems where some or all the unknowns are restricted to integer values
  - Branch and cut
  - Cutting-plane method
- Karmarkar's algorithm: The first reasonably efficient algorithm that solves the linear programming problem in polynomial time.
- Simplex algorithm: an algorithm for solving linear programming problems
Line search
Local search: a metaheuristic for solving computationally hard optimization problems
- Random-restart hill climbing
- Tabu search
Minimax used in game programming
Nearest neighbor search (NNS): find closest points in a metric space
- Best Bin First: find an approximate solution to the nearest neighbor search problem in very-high-dimensional spaces
Newton's method in optimization
Nonlinear optimization
- BFGS method: a nonlinear optimization algorithm
- Gauss–Newton algorithm: an algorithm for solving nonlinear least squares problems
- Levenberg–Marquardt algorithm: an algorithm for solving nonlinear least squares problems
- Nelder–Mead method (downhill simplex method): a nonlinear optimization algorithm
Odds algorithm (Bruss algorithm): Finds the optimal strategy to predict a last specific event in a random sequence event
Random Search
Simulated annealing
Stochastic tunneling
Subset sum algorithm
A hybrid HS-LS conjugate gradient algorithm (see https://doi.org/10.1016/j.cam.2023.115304)
A hybrid BFGS-Like method (see more https://doi.org/10.1016/j.cam.2024.115857)
Conjugate gradient methods (see more https://doi.org/10.1016/j.jksus.2022.101923)

Computational science

Astronomy

Doomsday algorithm: day of the week
Zeller's congruence is an algorithm to calculate the day of the week for any Julian or Gregorian calendar date
various Easter algorithms are used to calculate the day of Easter

Bioinformatics

Basic Local Alignment Search Tool also known as BLAST: an algorithm for comparing primary biological sequence information
Kabsch algorithm: calculate the optimal alignment of two sets of points in order to compute the root mean squared deviation between two protein structures.
Velvet: a set of algorithms manipulating de Bruijn graphs for genomic sequence assembly
Sorting by signed reversals: an algorithm for understanding genomic evolution.
Maximum parsimony (phylogenetics): an algorithm for finding the simplest phylogenetic tree to explain a given character matrix.
UPGMA: a distance-based phylogenetic tree construction algorithm.
Bloom Filter: probabilistic data structure used to test for the existence of an element within a set. Primarily used in bioinformatics to test for the existence of a k-mer in a sequence or sequences.

Geoscience

Vincenty's formulae: a fast algorithm to calculate the distance between two latitude/longitude points on an ellipsoid
Geohash: a public domain algorithm that encodes a decimal latitude/longitude pair as a hash string

Linguistics

Lesk algorithm: word sense disambiguation
Stemming algorithm: a method of reducing words to their stem, base, or root form
Sukhotin's algorithm: a statistical classification algorithm for classifying characters in a text as vowels or consonants

Medicine

ESC algorithm for the diagnosis of heart failure
Manning Criteria for irritable bowel syndrome
Pulmonary embolism diagnostic algorithms
Texas Medication Algorithm Project

Physics

Constraint algorithm: a class of algorithms for satisfying constraints for bodies that obey Newton's equations of motion
Demon algorithm: a Monte Carlo method for efficiently sampling members of a microcanonical ensemble with a given energy
Featherstone's algorithm: computes the effects of forces applied to a structure of joints and links
Ground state approximation
- Variational method
  - Ritz method
n-body problems
- Barnes–Hut simulation: Solves the n-body problem in an approximate way that has the order $O(n log n)$ instead of $O(n 2)$ as in a direct-sum simulation.
- Fast multipole method (FMM): speeds up the calculation of long-ranged forces
Rainflow-counting algorithm: Reduces a complex stress history to a count of elementary stress-reversals for use in fatigue analysis
Sweep and prune: a broad phase algorithm used during collision detection to limit the number of pairs of solids that need to be checked for collision
VEGAS algorithm: a method for reducing error in Monte Carlo simulations
Glauber dynamics: a method for simulating the Ising Model on a computer

Statistics

Algorithms for calculating variance: avoiding instability and numerical overflow
Approximate counting algorithm: allows counting large number of events in a small register
Bayesian statistics
- Nested sampling algorithm: a computational approach to the problem of comparing models in Bayesian statistics
Clustering Algorithms
- Average-linkage clustering: a simple agglomerative clustering algorithm
- Canopy clustering algorithm: an unsupervised pre-clustering algorithm related to the K-means algorithm
- Complete-linkage clustering: a simple agglomerative clustering algorithm
- DBSCAN: a density based clustering algorithm
- Expectation-maximization algorithm
- Fuzzy clustering: a class of clustering algorithms where each point has a degree of belonging to clusters
  - Fuzzy c-means
  - FLAME clustering (Fuzzy clustering by Local Approximation of MEmberships): define clusters in the dense parts of a dataset and perform cluster assignment solely based on the neighborhood relationships among objects
- KHOPCA clustering algorithm: a local clustering algorithm, which produces hierarchical multi-hop clusters in static and mobile environments.
- k-means clustering: cluster objects based on attributes into partitions
- k-means++: a variation of this, using modified random seeds
- k-medoids: similar to k-means, but chooses datapoints or medoids as centers
- Linde–Buzo–Gray algorithm: a vector quantization algorithm to derive a good codebook
- Lloyd's algorithm (Voronoi iteration or relaxation): group data points into a given number of categories, a popular algorithm for k-means clustering
- OPTICS: a density based clustering algorithm with a visual evaluation method
- Single-linkage clustering: a simple agglomerative clustering algorithm
- SUBCLU: a subspace clustering algorithm
- Ward's method: an agglomerative clustering algorithm, extended to more general Lance–Williams algorithms
- WACA clustering algorithm: a local clustering algorithm with potentially multi-hop structures; for dynamic networks
Estimation Theory
- Expectation-maximization algorithm A class of related algorithms for finding maximum likelihood estimates of parameters in probabilistic models
  - Ordered subset expectation maximization (OSEM): used in medical imaging for positron emission tomography, single-photon emission computed tomography and X-ray computed tomography.
- Odds algorithm (Bruss algorithm) Optimal online search for distinguished value in sequential random input
- Kalman filter: estimate the state of a linear dynamic system from a series of noisy measurements
False nearest neighbor algorithm (FNN) estimates fractal dimension
Hidden Markov model
- Baum–Welch algorithm: computes maximum likelihood estimates and posterior mode estimates for the parameters of a hidden Markov model
- Forward-backward algorithm: a dynamic programming algorithm for computing the probability of a particular observation sequence
- Viterbi algorithm: find the most likely sequence of hidden states in a hidden Markov model
Partial least squares regression: finds a linear model describing some predicted variables in terms of other observable variables
Queuing theory
- Buzen's algorithm: an algorithm for calculating the normalization constant G(K) in the Gordon–Newell theorem
RANSAC (an abbreviation for "RANdom SAmple Consensus"): an iterative method to estimate parameters of a mathematical model from a set of observed data which contains outliers
Scoring algorithm: is a form of Newton's method used to solve maximum likelihood equations numerically
Yamartino method: calculate an approximation to the standard deviation σθ of wind direction θ during a single pass through the incoming data
Ziggurat algorithm: generates random numbers from a non-uniform distribution

Computer science

Computer architecture

Tomasulo algorithm: allows sequential instructions that would normally be stalled due to certain dependencies to execute non-sequentially

Computer graphics

Clipping
- Line clipping
- Polygon clipping
Contour lines and Isosurfaces
- Marching cubes: extract a polygonal mesh of an isosurface from a three-dimensional scalar field (sometimes called voxels)
- Marching squares: generates contour lines for a two-dimensional scalar field
- Marching tetrahedrons: an alternative to Marching cubes
Discrete Green's Theorem: is an algorithm for computing double integral over a generalized rectangular domain in constant time. It is a natural extension to the summed area table algorithm
Flood fill: fills a connected region of a multi-dimensional array with a specified symbol
Global illumination algorithms: Considers direct illumination and reflection from other objects.
Hidden-surface removal or Visual surface determination
- Newell's algorithm: eliminate polygon cycles in the depth sorting required in hidden-surface removal
- Painter's algorithm: detects visible parts of a 3-dimensional scenery
- Scanline rendering: constructs an image by moving an imaginary line over the image
- Warnock algorithm
Line Drawing: graphical algorithm for approximating a line segment on discrete graphical media.
- Bresenham's line algorithm: plots points of a 2-dimensional array to form a straight line between 2 specified points (uses decision variables)
- DDA line algorithm: plots points of a 2-dimensional array to form a straight line between 2 specified points (uses floating-point math)
- Xiaolin Wu's line algorithm: algorithm for line antialiasing.
Midpoint circle algorithm: an algorithm used to determine the points needed for drawing a circle
Ramer–Douglas–Peucker algorithm: Given a 'curve' composed of line segments to find a curve not too dissimilar but that has fewer points
Shading
- Gouraud shading: an algorithm to simulate the differing effects of light and colour across the surface of an object in 3D computer graphics
- Phong shading: an algorithm to interpolate surface normal-vectors for surface shading in 3D computer graphics
Slerp (spherical linear interpolation): quaternion interpolation for the purpose of animating 3D rotation
Summed area table (also known as an integral image): an algorithm for computing the sum of values in a rectangular subset of a grid in constant time
Binary space partitioning

Cryptography

Asymmetric (public key) encryption:
Digital signatures (asymmetric authentication):
- DSA, and its variants:
  - ECDSA and Deterministic ECDSA
  - EdDSA (Ed25519)
- RSA
Cryptographic hash functions (see also the section on message authentication codes):
- BLAKE
- MD5 – Note that there is now a method of generating collisions for MD5
- RIPEMD-160
- SHA-1 – Note that there is now a method of generating collisions for SHA-1
- SHA-2 (SHA-224, SHA-256, SHA-384, SHA-512)
- SHA-3 (SHA3-224, SHA3-256, SHA3-384, SHA3-512, SHAKE128, SHAKE256)
- Tiger (TTH), usually used in Tiger tree hashes
- WHIRLPOOL
Cryptographically secure pseudo-random number generators
- Blum Blum Shub – based on the hardness of factorization
- Fortuna, intended as an improvement on Yarrow algorithm
- Linear-feedback shift register (note: many LFSR-based algorithms are weak or have been broken)
- Yarrow algorithm
Key exchange
- Diffie–Hellman key exchange
- Elliptic-curve Diffie–Hellman (ECDH)
Key derivation functions, often used for password hashing and key stretching
- bcrypt
- PBKDF2
- scrypt
- Argon2
Message authentication codes (symmetric authentication algorithms, which take a key as a parameter):
- HMAC: keyed-hash message authentication
- Poly1305
- SipHash
Secret sharing, Secret Splitting, Key Splitting, M of N algorithms
- Blakey's Scheme
- Shamir's Scheme
Symmetric (secret key) encryption:
- Advanced Encryption Standard (AES), winner of NIST competition, also known as Rijndael
- Blowfish
- Twofish
- Threefish
- Data Encryption Standard (DES), sometimes DE Algorithm, winner of NBS selection competition, replaced by AES for most purposes
- IDEA
- RC4 (cipher)
- Tiny Encryption Algorithm (TEA)
- Salsa20, and its updated variant ChaCha20
Post-quantum cryptography
Proof-of-work algorithms

Digital logic

Boolean minimization
- Quine–McCluskey algorithm: also called as Q-M algorithm, programmable method for simplifying the Boolean equations
- Petrick's method: another algorithm for Boolean simplification
- Espresso heuristic logic minimizer: a fast algorithm for Boolean function minimization

Machine learning and statistical classification

Almeida–Pineda recurrent backpropagation: Adjust a matrix of synaptic weights to generate desired outputs given its inputs
ALOPEX: a correlation-based machine-learning algorithm
Association rule learning: discover interesting relations between variables, used in data mining
Boosting (meta-algorithm): Use many weak learners to boost effectiveness
- AdaBoost: adaptive boosting
- BrownBoost: a boosting algorithm that may be robust to noisy datasets
- LogitBoost: logistic regression boosting
- LPBoost: linear programming boosting
Bootstrap aggregating (bagging): technique to improve stability and classification accuracy
Computer Vision
- Grabcut based on Graph cuts
Decision Trees
- C4.5 algorithm: an extension to ID3
- ID3 algorithm (Iterative Dichotomiser 3): use heuristic to generate small decision trees
Clustering: a class of unsupervised learning algorithms for grouping and bucketing related input vector
k-nearest neighbors (k-NN): a non-parametric method for classifying objects based on closest training examples in the feature space
Linde–Buzo–Gray algorithm: a vector quantization algorithm used to derive a good codebook
Locality-sensitive hashing (LSH): a method of performing probabilistic dimension reduction of high-dimensional data
Neural Network
- Backpropagation: a supervised learning method which requires a teacher that knows, or can calculate, the desired output for any given input
- Hopfield net: a Recurrent neural network in which all connections are symmetric
- Perceptron: the simplest kind of feedforward neural network: a linear classifier.
- Pulse-coupled neural networks (PCNN): Neural models proposed by modeling a cat's visual cortex and developed for high-performance biomimetic image processing.
- Radial basis function network: an artificial neural network that uses radial basis functions as activation functions
- Self-organizing map: an unsupervised network that produces a low-dimensional representation of the input space of the training samples
Random forest: classify using many decision trees
Reinforcement learning:
- Q-learning: learns an action-value function that gives the expected utility of taking a given action in a given state and following a fixed policy thereafter
- State–Action–Reward–State–Action (SARSA): learn a Markov decision process policy
- Temporal difference learning
Relevance-Vector Machine (RVM): similar to SVM, but provides probabilistic classification
Supervised learning: Learning by examples (labelled data-set split into training-set and test-set)
Support Vector Machine (SVM): a set of methods which divide multidimensional data by finding a dividing hyperplane with the maximum margin between the two sets
- Structured SVM: allows training of a classifier for general structured output labels.
Winnow algorithm: related to the perceptron, but uses a multiplicative weight-update scheme

Programming language theory

C3 linearization: an algorithm used primarily to obtain a consistent linearization of a multiple inheritance hierarchy in object-oriented programming
Chaitin's algorithm: a bottom-up, graph coloring register allocation algorithm that uses cost/degree as its spill metric
Hindley–Milner type inference algorithm
Rete algorithm: an efficient pattern matching algorithm for implementing production rule systems
Sethi-Ullman algorithm: generates optimal code for arithmetic expressions

Parsing

CYK algorithm: an O(n³) algorithm for parsing context-free grammars in Chomsky normal form
Earley parser: another O(n³) algorithm for parsing any context-free grammar
GLR parser: an algorithm for parsing any context-free grammar by Masaru Tomita. It is tuned for deterministic grammars, on which it performs almost linear time and O(n³) in worst case.
Inside-outside algorithm: an O(n³) algorithm for re-estimating production probabilities in probabilistic context-free grammars
LL parser: a relatively simple linear time parsing algorithm for a limited class of context-free grammars
LR parser: A more complex linear time parsing algorithm for a larger class of context-free grammars. Variants:
Packrat parser: a linear time parsing algorithm supporting some context-free grammars and parsing expression grammars
Recursive descent parser: a top-down parser suitable for LL(k) grammars
Shunting-yard algorithm: converts an infix-notation math expression to postfix
Pratt parser
Lexical analysis

Quantum algorithms

Deutsch–Jozsa algorithm: criterion of balance for Boolean function
Grover's algorithm: provides quadratic speedup for many search problems
Shor's algorithm: provides exponential speedup (relative to currently known non-quantum algorithms) for factoring a number
Simon's algorithm: provides a provably exponential speedup (relative to any non-quantum algorithm) for a black-box problem

Theory of computation and automata

Hopcroft's algorithm, Moore's algorithm, and Brzozowski's algorithm: algorithms for minimizing the number of states in a deterministic finite automaton
Powerset construction: algorithm to convert nondeterministic automaton to deterministic automaton.
Tarski–Kuratowski algorithm: a non-deterministic algorithm which provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy

Information theory and signal processing

Coding theory

Error detection and correction

BCH Codes
BCJR algorithm: decoding of error correcting codes defined on trellises (principally convolutional codes)
Forward error correction
Gray code
Hamming codes
- Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits
- Hamming distance: sum number of positions which are different
- Hamming weight (population count): find the number of 1 bits in a binary word
Redundancy checks
- Adler-32
- Cyclic redundancy check
- Damm algorithm
- Fletcher's checksum
- Longitudinal redundancy check (LRC)
- Luhn algorithm: a method of validating identification numbers
- Luhn mod N algorithm: extension of Luhn to non-numeric characters
- Parity: simple/fast error detection technique
- Verhoeff algorithm

Lossless compression algorithms

Burrows–Wheeler transform: preprocessing useful for improving lossless compression
Context tree weighting
Delta encoding: aid to compression of data in which sequential data occurs frequently
Dynamic Markov compression: Compression using predictive arithmetic coding
Dictionary coders
- Byte pair encoding (BPE)
- Deflate
- Lempel–Ziv
  - LZ77 and LZ78
  - Lempel–Ziv Jeff Bonwick (LZJB)
  - Lempel–Ziv–Markov chain algorithm (LZMA)
  - Lempel–Ziv–Oberhumer (LZO): speed oriented
  - Lempel–Ziv–Stac (LZS)
  - Lempel–Ziv–Storer–Szymanski (LZSS)
  - Lempel–Ziv–Welch (LZW)
  - LZWL: syllable-based variant
  - LZX
  - Lempel–Ziv Ross Williams (LZRW)
Entropy encoding: coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols
- Arithmetic coding: advanced entropy coding
  - Range encoding: same as arithmetic coding, but looked at in a slightly different way
- Huffman coding: simple lossless compression taking advantage of relative character frequencies
  - Adaptive Huffman coding: adaptive coding technique based on Huffman coding
  - Package-merge algorithm: Optimizes Huffman coding subject to a length restriction on code strings
- Shannon–Fano coding
- Shannon–Fano–Elias coding: precursor to arithmetic encoding^[5]
Entropy coding with known entropy characteristics
- Golomb coding: form of entropy coding that is optimal for alphabets following geometric distributions
- Rice coding: form of entropy coding that is optimal for alphabets following geometric distributions
- Truncated binary encoding
- Unary coding: code that represents a number n with n ones followed by a zero
- Universal codes: encodes positive integers into binary code words
  - Elias delta, gamma, and omega coding
  - Exponential-Golomb coding
  - Fibonacci coding
  - Levenshtein coding
Fast Efficient & Lossless Image Compression System (FELICS): a lossless image compression algorithm
Incremental encoding: delta encoding applied to sequences of strings
Prediction by partial matching (PPM): an adaptive statistical data compression technique based on context modeling and prediction
Run-length encoding: lossless data compression taking advantage of strings of repeated characters
SEQUITUR algorithm: lossless compression by incremental grammar inference on a string

Lossy compression algorithms

3Dc: a lossy data compression algorithm for normal maps
Audio and Speech compression
- A-law algorithm: standard companding algorithm
- Code-excited linear prediction (CELP): low bit-rate speech compression
- Linear predictive coding (LPC): lossy compression by representing the spectral envelope of a digital signal of speech in compressed form
- Mu-law algorithm: standard analog signal compression or companding algorithm
- Warped Linear Predictive Coding (WLPC)
Image compression
- Block Truncation Coding (BTC): a type of lossy image compression technique for greyscale images
- Embedded Zerotree Wavelet (EZW)
- Fast Cosine Transform algorithms (FCT algorithms): computes Discrete Cosine Transform (DCT) efficiently
- Fractal compression: method used to compress images using fractals
- Set Partitioning in Hierarchical Trees (SPIHT)
- Wavelet compression: form of data compression well suited for image compression (sometimes also video compression and audio compression)
Transform coding: type of data compression for "natural" data like audio signals or photographic images
Video compression
Vector quantization: technique often used in lossy data compression

Digital signal processing

Adaptive-additive algorithm (AA algorithm): find the spatial frequency phase of an observed wave source
Discrete Fourier transform: determines the frequencies contained in a (segment of a) signal
Fast folding algorithm: an efficient algorithm for the detection of approximately periodic events within time series data
Gerchberg–Saxton algorithm: Phase retrieval algorithm for optical planes
Goertzel algorithm: identify a particular frequency component in a signal. Can be used for DTMF digit decoding.
Karplus-Strong string synthesis: physical modelling synthesis to simulate the sound of a hammered or plucked string or some types of percussion

Image processing

Contrast Enhancement
- Histogram equalization: use histogram to improve image contrast
- Adaptive histogram equalization: histogram equalization which adapts to local changes in contrast
Connected-component labeling: find and label disjoint regions
Dithering and half-toning
Elser difference-map algorithm: a search algorithm for general constraint satisfaction problems. Originally used for X-Ray diffraction microscopy
Feature detection
- Canny edge detector: detect a wide range of edges in images
- Generalised Hough transform
- Hough transform
- Marr–Hildreth algorithm: an early edge detection algorithm
- SIFT (Scale-invariant feature transform): is an algorithm to detect and describe local features in images.
- SURF (Speeded Up Robust Features): is a robust local feature detector, first presented by Herbert Bay et al. in 2006, that can be used in computer vision tasks like object recognition or 3D reconstruction. It is partly inspired by the SIFT descriptor. The standard version of SURF is several times faster than SIFT and claimed by its authors to be more robust against different image transformations than SIFT.^[6]^[7]
Richardson–Lucy deconvolution: image de-blurring algorithm
Blind deconvolution: image de-blurring algorithm when point spread function is unknown.
Median filtering
Seam carving: content-aware image resizing algorithm
Segmentation: partition a digital image into two or more regions
- GrowCut algorithm: an interactive segmentation algorithm
- Random walker algorithm
- Region growing
- Watershed transformation: a class of algorithms based on the watershed analogy

Software engineering

Cache algorithms
CHS conversion: converting between disk addressing systems
Double dabble: Convert binary numbers to BCD
Hash Function: convert a large, possibly variable-sized amount of data into a small datum, usually a single integer that may serve as an index into an array
- Fowler–Noll–Vo hash function: fast with low collision rate
- Pearson hashing: computes 8 bit value only, optimized for 8 bit computers
- Zobrist hashing: used in the implementation of transposition tables
Unicode Collation Algorithm
Xor swap algorithm: swaps the values of two variables without using a buffer

Database algorithms

Algorithms for Recovery and Isolation Exploiting Semantics (ARIES): transaction recovery
Join algorithms
The Chase

Distributed systems algorithms

Clock synchronization
Consensus (computer science): agreeing on a single value or history among unreliable processors
Detection of Process Termination
- Dijkstra-Scholten algorithm
- Huang's algorithm
Lamport ordering: a partial ordering of events based on the happened-before relation
Leader election: a method for dynamically selecting a coordinator
- Bully algorithm
Mutual exclusion
Snapshot algorithm: record a consistent global state for an asynchronous system
- Chandy–Lamport algorithm
Vector clocks: generate a partial ordering of events in a distributed system and detect causality violations

Memory allocation and deallocation algorithms

Buddy memory allocation: an algorithm to allocate memory such with less fragmentation
Garbage collectors
- Cheney's algorithm: an improvement on the Semi-space collector
- Generational garbage collector: Fast garbage collectors that segregate memory by age
- Mark-compact algorithm: a combination of the mark-sweep algorithm and Cheney's copying algorithm
- Mark and sweep
- Semi-space collector: an early copying collector
Reference counting

Networking

Karn's algorithm: addresses the problem of getting accurate estimates of the round-trip time for messages when using TCP
Luleå algorithm: a technique for storing and searching internet routing tables efficiently
Network congestion
- Exponential backoff
- Nagle's algorithm: improve the efficiency of TCP/IP networks by coalescing packets
- Truncated binary exponential backoff

Operating systems algorithms

Banker's algorithm: algorithm used for deadlock avoidance
Page replacement algorithms: for selecting the victim page under low memory conditions
- Adaptive replacement cache: better performance than LRU
- Clock with Adaptive Replacement (CAR): a page replacement algorithm with performance comparable to adaptive replacement cache

Process synchronization

Scheduling

Earliest deadline first scheduling
Fair-share scheduling
Least slack time scheduling
List scheduling
Multi level feedback queue
Rate-monotonic scheduling
Round-robin scheduling
Shortest job next
Shortest remaining time
Top-nodes algorithm: resource calendar management

I/O scheduling

Disk scheduling

Elevator algorithm: Disk scheduling algorithm that works like an elevator.
Shortest seek first: Disk scheduling algorithm to reduce seek time.

References

^ "algorithm". LII / Legal Information Institute. Retrieved 2023-10-26.
^ Gegenfurtner, Karl R. (1992-12-01). "PRAXIS: Brent's algorithm for function minimization". Behavior Research Methods, Instruments, & Computers. 24 (4): 560–564. doi:10.3758/BF03203605. ISSN 1532-5970.
^ "richardshin.com | Floyd's Cycle Detection Algorithm". 2013-09-30. Retrieved 2023-10-26.
^ "Eytzinger Binary Search - Algorithmica". Retrieved 2023-04-09.
^ "Shannon-Fano-Elias Coding" (PDF). my.ece.msstate.edu. Archived from the original (PDF) on 2021-02-28. Retrieved 2023-10-11.
^ "Archived copy" (PDF). www.vision.ee.ethz.ch. Archived from the original (PDF) on 21 February 2007. Retrieved 13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
^ "Archived copy" (PDF). Archived from the original (PDF) on 2013-10-06. Retrieved 2013-10-05.{{cite web}}: CS1 maint: archived copy as title (link)

[1] "algorithm". LII / Legal Information Institute. Retrieved 2023-10-26.

[2] Gegenfurtner, Karl R. (1992-12-01). "PRAXIS: Brent's algorithm for function minimization". Behavior Research Methods, Instruments, & Computers. 24 (4): 560–564. doi:10.3758/BF03203605. ISSN 1532-5970.

[3] "richardshin.com | Floyd's Cycle Detection Algorithm". 2013-09-30. Retrieved 2023-10-26.

[4] "Eytzinger Binary Search - Algorithmica". Retrieved 2023-04-09.

[5] "Shannon-Fano-Elias Coding" (PDF). my.ece.msstate.edu. Archived from the original (PDF) on 2021-02-28. Retrieved 2023-10-11.

[6] "Archived copy" (PDF). www.vision.ee.ethz.ch. Archived from the original (PDF) on 21 February 2007. Retrieved 13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)

[7] "Archived copy" (PDF). Archived from the original (PDF) on 2013-10-06. Retrieved 2013-10-05.{{cite web}}: CS1 maint: archived copy as title (link)

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