Longest Common Abelian Factors and Large Alphabets

Abstract

Two strings X and Y are considered Abelian equal if the letters of X can be permuted to obtain Y (and vice versa). Recently, Alatabbi et al. (2015) considered the longest common Abelian factor problem in which we are asked to find the length of the longest Abelian-equal factor present in a given pair of strings. They provided an algorithm that uses $O(\sigma n^2)$ time and $O(\sigma n)$ space, where n is the length of the pair of strings and $\sigma$ is the alphabet size. In this paper we describe an algorithm that uses $O(n^2\log ^2n\log ^*n)$ time and $O(n\log ^2n)$ space, significantly improving Alatabbi et al.’s result unless the alphabet is small. Our algorithm makes use of techniques for maintaining a dynamic set of strings under split, join, and equality testing (Melhorn et al., Algorithmica 17(2), 1997).