Algorithms for computing Abelian periods of words

Constantinescu and Ilie [S. Constantinescu, L. Ilie. Fine and Wilf’s theorem for abelian periods, Bulletin of the European Association for Theoretical Computer Science 89 (2006) 167–170] introduced the notion of an Abelian period   of a word. A word of length nn over an alphabet of size σσ can have Θ(n2)Θ(n2) distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time O(n2×σ)O(n2×σ) using O(n×σ)O(n×σ) space. We present an offline algorithm based on a select function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present online algorithms that also enable us to compute all the Abelian periods of all the prefixes of ww.