All Semi-local Longest Common Subsequences in Subquadratic Time

Abstract

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 Θ (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. 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

“…Subquadratic distance multiplication algorithms for implicit simple (sub)unit-Monge matrices were given in [58,61], running in time O(n 1.5 ). We now show an even faster algorithm for this problem.…”

Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs

“…We consider a generalisation of the LCS problem, which we introduced in [14] as the all semi-local LCS problem. It consists in computing the LCS lengths on substrings of a and b as follows:…”

“…For completeness, in this section we restate (without motivation and proofs) the necessary definitions and results from [14]. In the following section, we give the proof of the key lemma, a significant part of which was omitted from [14] due to space restrictions.…”

“…In the following section, we give the proof of the key lemma, a significant part of which was omitted from [14] due to space restrictions.…”

“…In [14], we defined 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. In the same paper we introduced a relatively simple geometric framework, allowing to represent the problem's output by a data structure of size O(m + n), and to query an individual output length efficiently.…”