Subquadratic-Time Algorithms for Abelian Stringology Problems

Abstract

We propose the first subquadratic-time algorithms to a number of natural problems in abelian 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 respective symbols in both of them, specified by their so-called Parikh vectors, are the same. We propose the following algorithms for a string of length n:

• Counting and finding longest/shortest abelian squares in $O(n^2/\log ^2n)$ time. Abelian squares were first considered by Erdös (1961); Cummings and Smyth (1997) proposed an $O(n^2)$-time algorithm for computing them.

• Computing all shortest (general) abelian periods in $O(n^2/\sqrt{\log n})$ time. Abelian periods were introduced by Constantinescu and Ilie (2006) and the previous, quadratic-time algorithms for their computation were given by Fici et al. (2011) for a constant-sized alphabet and by Crochemore et al. (2012) for a general alphabet.

• Finding all abelian covers in $O(n^2/\log n)$ time. Abelian covers were defined by Matsuda et al. (2014).

• Computing abelian border array in $O(n^2/\log ^2n)$ time.

This work can be viewed as a continuation of a recent very active line of research on subquadratic space and time jumbled indexing for binary and constant-sized alphabets (e.g., Moosa and Rahman, 2012). All our algorithms work in linear space.

T. Kociumaka—Supported by Polish budget funds for science in 2013–2017 as a research project under the ‘Diamond Grant’ program.

J. Radoszewski—Supported by the Polish Ministry of Science and Higher Education under the ‘Iuventus Plus’ program in 2015–2016 grant no 0392/IP3/2015/73.

J. Radoszewski—Newton International Fellow at King’s College London.