Longest Common Abelian Factors and Large Alphabets

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(2) n log* n) time and O(n log(2) n) 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).