Linear Time Runs over General Ordered Alphabets

TL;DR

This paper proves Kosolobov's conjecture by presenting an $ ext{O}(n)$ time algorithm for finding all runs in strings over general ordered alphabets, using properties of the Lyndon array.

Contribution

It introduces an $ ext{O}(n)$ time algorithm for computing runs over general ordered alphabets, confirming the conjecture and improving upon previous $ ext{O}(n ext{alpha}(n))$ solutions.

Findings

Achieved linear time complexity for runs over general ordered alphabets.

Utilized combinatorial properties of the Lyndon array.

Proved Kosolobov's conjecture.

Abstract

A run in a string is a maximal periodic substring. For example, the string $\texttt{bananatree}$ contains the runs $\texttt{anana} = (\texttt{an})^{3/2}$ and $\texttt{ee} = \texttt{e}^2$. There are less than $n$ runs in any length-$n$ string, and computing all runs for a string over a linearly-sortable alphabet takes $\mathcal{O}(n)$ time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an $\mathcal{O}(n\alpha(n))$ time algorithm (where $\alpha(n)$ is the extremely slowly growing inverse Ackermann function). We show how to achieve $\mathcal{O}(n)$ time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov's conjecture.