Algorithms for Computing Abelian Periods of Words

TL;DR

This paper introduces efficient algorithms for computing Abelian periods of words, improving practical performance over brute-force methods while maintaining similar worst-case theoretical complexity.

Contribution

It presents an off-line algorithm based on a extbackslash sel function and on-line algorithms for all prefixes, enhancing practical efficiency in Abelian period computation.

Findings

Off-line algorithm matches brute-force worst-case complexity but is faster in practice.

On-line algorithms compute Abelian periods for all prefixes efficiently.

Algorithms handle words over any alphabet size with optimal theoretical bounds.

Abstract

Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\sigma$ can have $\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \times \sigma)$ using $O(n \times \sigma)$ space. We present an off-line algorithm based on a $\sel$ function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present on-line algorithms that also enable to compute all the Abelian periods of all the prefixes of $w$.