Identifying all abelian periods of a string in quadratic time and relevant problems

TL;DR

This paper presents two quadratic-time algorithms for identifying all abelian periods of a string using different mapping techniques, and introduces a faster logarithmic algorithm for weak abelian periods.

Contribution

It introduces two novel quadratic algorithms for abelian period detection and a more efficient logarithmic algorithm for weak abelian periods.

Findings

Quadratic algorithms for abelian periods using sum and prime product mappings

A logarithmic algorithm for weak abelian periods

Comparison of algorithm efficiencies and problem solutions

Abstract

Abelian periodicity of strings has been studied extensively over the last years. In 2006 Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters. We show two O(|y|^2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.