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The algorithm can be adapted to the case which allows empty subarrays or to keep track of the starting and ending indices of the maximum subarray. This algorithm calculates the maximum subarray ending at each position from the maximum subarray ending at the previous position, so it can be viewed as a case of dynamic programming.
This algorithm is an improvement over previously known quadratic time algorithms. [1] The maximum scoring subsequence from the set produced by the algorithm is also a solution to the maximum subarray problem. The Ruzzo–Tompa algorithm has applications in bioinformatics, [4] web scraping, [5] and information retrieval. [6]
The greedy algorithm for maximum coverage chooses sets according to one rule: at each stage, choose a set which contains the largest number of uncovered elements. It can be shown that this algorithm achieves an approximation ratio of 1 − 1 e {\displaystyle 1-{\frac {1}{e}}} .
Starting from the initial state (0, 0), it is possible to use any graph search algorithm (e.g. BFS) to search the state (N, T). If the state is found, then by backtracking we can find a subset with a sum of exactly T. The run-time of this algorithm is at most linear in the number of states.
This algorithm is slower than Manacher's algorithm, but is a good stepping stone for understanding Manacher's algorithm. It looks at each character as the center of a palindrome and loops to determine the largest palindrome with that center. The loop at the center of the function only works for palindromes where the length is an odd number.
The basic form of the Bron–Kerbosch algorithm is a recursive backtracking algorithm that searches for all maximal cliques in a given graph G.More generally, given three disjoint sets of vertices R, P, and X, it finds the maximal cliques that include all of the vertices in R, some of the vertices in P, and none of the vertices in X.
Given a function that accepts an array, a range query (,) on an array = [,..,] takes two indices and and returns the result of when applied to the subarray [, …,].For example, for a function that returns the sum of all values in an array, the range query (,) returns the sum of all values in the range [,].
Let : be a convex function with domain . A classical subgradient method iterates (+) = () where () denotes any subgradient of at (), and () is the iterate of . If is differentiable, then its only subgradient is the gradient vector itself.