An all-substrings common subsequence algorithm

Abstract

Given two strings A and B of lengths n a and n b , n a n b , respectively, the all-substrings longest common subsequence (ALCS) problem obtains, for every substring B of B, the length of the longest string that is a subsequence of both A and B . The ALCS problem has many applications, such as finding approximate tandem repeats in strings, solving the circular alignment of two strings and finding the alignment of one string with several others that have a common substring. We present an algorithm to prepare the basic data structure for ALCS queries that takes O(n a n b ) time and O(n a + n b ) space. After this preparation, it is possible to build a matrix of size O(n 2 b ) that allows any LCS length to be retrieved in constant time. Some trade-offs between the space required and the querying time are discussed. To our knowledge, this is the first algorithm in the literature for the ALCS problem.

Semi-local String Comparison: Algorithmic Techniques and Applications

“…Based on the ideas of Schmidt [101], Alves et al [7] gave an algorithm for the string-substring LCS problem that runs in time O(mn), which therefore extends the functionality of the standard dynamic programming algorithm, while matching its asymptotic running time. In the course of the computation, the string-substring LCS problem is solved for all prefixes of a against all prefixes of b.…”

“…The LCS score of string a against substring b 4 : 11 = "cabcaba" is 5; this score is realised by a common subsequence "abcba". This example, which will serve as a running example for this chapter, is borrowed from Alves et al [7].…”

Efficient all path score computations on grid graphs

“…In the same paper, Schmidt showed that IGAPS can be solved in O(Cn 2 ) time. An O(n 2 ) algorithm based on a similar approach for the BGAPS problem was also given by Alves et al [1] and Tiskin [23]. Tiskin [22, p. 60] gave an O(n 2 (log log n/ log n) 2 ) time algorithm for a special case of BGAPS, in which the grid graph corresponds to an LCS problem on two strings.…”

“…It has since been studied in several additional papers [1,2,7,[11][12][13][14][15][21][22][23]. Schmidt [21] showed that the GAPS problem can be solved in O(n 2 log n) time.…”