# Category: Dynamic programming

Damerau–Levenshtein distance
In information theory and computer science, the Damerau–Levenshtein distance (named after Frederick J. Damerau and Vladimir I. Levenshtein) is a string metric for measuring the edit distance between t
Markov decision process
In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly
Shortest common supersequence problem
In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subs
Maximum subarray problem
In computer science, the maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional
Longest increasing subsequence
In computer science, the longest increasing subsequence problem is to find a subsequence of a given sequence in which the subsequence's elements are in sorted order, lowest to highest, and in which th
Floyd–Warshall algorithm
In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest path
Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensi
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a
Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields
Bellman equation
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of
WASP (cricket calculation tool)
Winning and Score Predictor (WASP) is a calculation tool used in cricket to predict scores and possible results of a limited overs match, e.g. One Day and Twenty 20 matches. The prediction is based up
Hamilton–Jacobi–Bellman equation
In optimal control theory, the Hamilton-Jacobi-Bellman (HJB) equation gives a necessary and sufficient condition for optimality of a control with respect to a loss function. It is, in general, a nonli
Differential dynamic programming
Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne and subsequently analysed in Jacobson and May
Earley parser
In computer science, the Earley parser is an algorithm for parsing strings that belong to a given context-free language, though (depending on the variant) it may suffer problems with certain nullable
Levenshtein distance
In information theory, linguistics, and computer science, the Levenshtein distance is a string metric for measuring the difference between two sequences. Informally, the Levenshtein distance between t
Line wrap and word wrap
Line breaking, also known as word wrapping, is breaking a section of text into lines so that it will fit into the available width of a page, window or other display area. In text display, line wrap is
Longest common subsequence problem
The longest common subsequence (LCS) problem is the problem of finding the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest com
Viscosity solution
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solut
Automatic basis function construction
In machine learning, automatic basis function construction (or basis discovery) is the mathematical method of looking for a set of task-independent basis functions that map the state space to a lower-
Bellman–Ford algorithm
The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph.It is slower than Dijkstra's algorithm for the sa
Knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total we
Longest alternating subsequence
In combinatorial mathematics, probability, and computer science, in the longest alternating subsequence problem, one wants to find a subsequence of a given sequence in which the elements are in altern
Longest common substring problem
In computer science, the longest common substring problem is to find a longest string that is a substring of two or more strings. The problem may have multiple solutions.Applications include data dedu
Partially observable Markov decision process
A partially observable Markov decision process (POMDP) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics
Matrix chain multiplication
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to
Approximate string matching
In computer science, approximate string matching (often colloquially referred to as fuzzy string searching) is the technique of finding strings that match a pattern approximately (rather than exactly)
Forward–backward algorithm
The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations/emissions , i.e.
Held–Karp algorithm
The Held–Karp algorithm, also called Bellman–Held–Karp algorithm, is a dynamic programming algorithm proposed in 1962 independently by Bellman and by Held and Karp to solve the traveling salesman prob
Needleman–Wunsch algorithm
The Needleman–Wunsch algorithm is an algorithm used in bioinformatics to align protein or nucleotide sequences. It was one of the first applications of dynamic programming to compare biological sequen
Optimal stopping
In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an ex
Overlapping subproblems
In computer science, a problem is said to have overlapping subproblems if the problem can be broken down into subproblems which are reused several times or a recursive algorithm for the problem solves
Bitonic tour
In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the
Hunt–Szymanski algorithm
In computer science, the Hunt–Szymanski algorithm, also known as Hunt–McIlroy algorithm, is a solution to the longest common subsequence problem. It was one of the first non-heuristic algorithms used
Value function
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. In a controlled dynamical system,
Stochastic dynamic programming
Originally introduced by Richard E. Bellman in, stochastic dynamic programming is a technique for modelling and solving problems of decision making under uncertainty. Closely related to stochastic pro
Dynamic time warping
In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be det
Backward induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a
Dynamic discrete choice
Dynamic discrete choice (DDC) models, also known as discrete choice models of dynamic programming, model an agent's choices over discrete options that have future implications. Rather than assuming ob
Optimal substructure
In computer science, a problem is said to have optimal substructure if an optimal solution can be constructed from optimal solutions of its subproblems. This property is used to determine the usefulne
Ruzzo–Tompa algorithm
The Ruzzo–Tompa algorithm is a linear-time algorithm for finding all non-overlapping, contiguous, maximal scoring subsequences in a sequence of real numbers. This algorithm is an improvement over prev
Hirschberg's algorithm
In computer science, Hirschberg's algorithm, named after its inventor, Dan Hirschberg, is a dynamic programming algorithm that finds the optimal sequence alignment between two strings. Optimality is m
Recursive economics
Recursive economics is a branch of modern economics based on a paradigm of individuals making a series of two-period optimization decisions over time.
Smith–Waterman algorithm
The Smith–Waterman algorithm performs local sequence alignment; that is, for determining similar regions between two strings of nucleic acid sequences or protein sequences. Instead of looking at the e
Viterbi algorithm
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results i
Subset sum problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any su