Subquadratic-Time Algorithms for Abelian Stringology Problems

Abstract

We propose the first subquadratic-time algorithms to a number of natural problems inabelian pattern matching (also called jumbled pattern matching) for strings over a constant-sized alphabet.Two strings are considered equivalent in this model if the numbers of occurrences of respectivesymbols in both of them, specified by their so-called Parikh vectors, are the same. We consider theproblems of computing abelian squares, abelian periods, abelian runs, abelian covers, and abelian borders.This work can be viewed as a continuation of a recent very active line of research on subquadraticspace and time jumbled indexing for binary and constant-sized alphabets (e.g., Moosa and Rahman,2012). Our algorithms apply the so-called four Russian technique in a fancy way and work in linearspace. The purpose of this work is to show that breaking the O(n2) barrier is possible for the aforementionedproblems. In this paper we extend our previous work, published in a preliminary form atMACIS 2015 conference, with effcient computation of abelian runs.