The "Runs" Theorem

TL;DR

This paper introduces a new Lyndon word-based characterization of string repetitions, proving the runs conjecture with a simple proof, and presents a linear-time algorithm for finding all runs without using Lempel-Ziv factorization.

Contribution

It provides a novel Lyndon word-based characterization of runs, a simplified proof of the runs conjecture, and a new linear-time algorithm that does not rely on Lempel-Ziv factorization.

Findings

Proved the runs conjecture: maximum number of runs is less than string length.

Established an upper bound of 3n for the sum of run exponents.

Developed a simple linear-time algorithm for computing all runs.

Abstract

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov \& Kucherov (FOCS '99)), which states that the maximum number of runs $\rho(n)$ in a string of length $n$ is less than $n$. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. In addition, we obtain an upper bound of $3n$ for the maximum sum of exponents $\sigma(n)$ of runs in a string of length $n$, improving on the best known bound of $4.1n$ by Crochemore et al. (JDA 2012), as well as other improved bounds on related problems. The characterization also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string. We also establish a relationship between runs and nodes of the Lyndon tree, which gives a simple optimal solution to the 2-Period Query problem that was recently solved by Kociumaka et al. (SODA 2015).