Extracting Powers and Periods in a String from Its Runs Structure

This paper uses Lyndon words and introduces the Lyndon structure of runs as a useful tool when computing powers and presents an efficient algorithm for testing primitivity of factors of a string and computing their primitive roots.

Abstract

A breakthrough in the field of text algorithms was the discovery of the fact that the maximal number of runs in a string of length n is O(n) and that they can all be computed in O(n) time. We study some applications of this result. New simpler O(n) time algorithms are presented for a few classical string problems: computing all distinct kth string powers for a given k, in particular squares for k = 2, and finding all local periods in a given string of length n. Additionally, we present an efficient algorithm for testing primitivity of factors of a string and computing their primitive roots. Applications of runs, despite their importance, are underrepresented in existing literature (approximately one page in the paper of Kolpakov & Kucherov, 1999). In this paper we attempt to fill in this gap. We use Lyndon words and introduce the Lyndon structure of runs as a useful tool when computing powers. In problems related to periods we use some versions of the Manhattan skyline problem.