Computing Runs on a General Alphabet

TL;DR

This paper presents a RAM algorithm for finding all maximal repetitions in a string over a general alphabet with sub-quadratic time complexity, outperforming previous solutions for large alphabets.

Contribution

It introduces a new algorithm that computes all runs in $O(n ext{log}^{2/3} n)$ time, improving over existing methods for large alphabet sizes.

Findings

Algorithm runs in $O(n ext{log}^{2/3} n)$ time

Outperforms previous solutions for $\sigma = n^{ ext{Omega}(1)}$

Conjecture of a possible linear time algorithm for all runs

Abstract

We describe a RAM algorithm computing all runs (maximal repetitions) of a given string of length $n$ over a general ordered alphabet in $O(n\log^{\frac{2}3} n)$ time and linear space. Our algorithm outperforms all known solutions working in $\Theta(n\log\sigma)$ time provided $\sigma = n^{\Omega(1)}$, where $\sigma$ is the alphabet size. We conjecture that there exists a linear time RAM algorithm finding all runs.