Online Computation of Abelian Runs

TL;DR

This paper introduces an efficient online algorithm for identifying all abelian runs with a specific period in a word, achieving optimal time and space complexities based on word length and alphabet size.

Contribution

The paper presents the first online algorithm for computing all abelian runs of a given period in a string with optimal time and space complexity.

Findings

Algorithm runs in O(n * |P|) time

Uses O(σ + |P|) space

Effectively finds all abelian runs in linear time

Abstract

Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period $\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having abelian period $\mathcal{P}$. We give an algorithm that finds all the abelian runs of period $\mathcal{P}$ in a word of length $n$ in time $O(n\times |\mathcal{P}|)$ and space $O(\sigma+|\mathcal{P}|)$.