Semi-local longest common subsequences in subquadratic time

Abstract

For two strings a, b of lengths m, n, respectively, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS. In this paper, we define a generalisation, called "the all semi-local LCS problem", where each string is compared against all substrings of the other string, and all prefixes of each string are compared against all suffixes of the other string. An explicit representation of the output lengths is of size ((m + n) 2 ). We show that the output can be represented implicitly by a geometric data structure of size O (m + n), allowing efficient queries of the individual output lengths. The currently best all string-substring LCS algorithm by Alves et al., based on previous work by Schmidt, can be adapted to produce the output in this form. We also develop the first all semi-local LCS algorithm, running in time o(mn) when m and n are reasonably close. Compared to a number of previous results, our approach presents an improvement in algorithm functionality, output representation efficiency, and/or running time.

Fast Distance Multiplication of Unit-Monge Matrices

“…In [60,61], we introduced the semi-local LCS problem, described its connections with unit-Monge matrices and seaweed braids, and gave a number of its algorithmic applications. Many of these applications use distance multiplication of simple unitMonge matrices as a subroutine.…”

“…Motivated by applications to string comparison, we introduced (using different terminology) in [60,61] the following subclass of Monge matrices. A matrix is called unit-Monge, if its density matrix is a permutation matrix; we further restrict our attention to a subclass of simple unit-Monge matrices, which satisfy a straightforward boundary condition.…”

“…A matrix is called unit-Monge, if its density matrix is a permutation matrix; we further restrict our attention to a subclass of simple unit-Monge matrices, which satisfy a straightforward boundary condition. In [60,61], we gave an algorithm for distance multiplication of such matrices, running in time O(n 1.5 ). Landau [43] asked (again using different terminology) whether this problem can be solved in linear time.…”

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.…”