A Note on Efficient Computation of All Abelian Periods in a String

TL;DR

This paper presents a new, efficient method for computing all Abelian periods in a string by reducing the problem to the already solved problem of Abelian squares, offering improved algorithms and insights.

Contribution

It introduces a novel reduction from Abelian periods to Abelian squares and provides new algorithms for efficiently finding all Abelian periods and squares.

Findings

Linear time algorithm for long Abelian periods

Alternative efficient algorithm for Abelian squares

Reduction technique offers new insights into Abelian period problems

Abstract

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all `long' Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems.