Understanding maximal repetitions in strings

TL;DR

This paper provides a simple proof that the number of maximal repetitions (runs) in a string is linear in the string length, reinforcing the foundation for efficient algorithms computing repetitions.

Contribution

It offers a straightforward proof of the linear bound on the number of runs and derives the linearity of the sum of their exponents as a consequence.

Findings

Number of runs in a string is O(n)

Sum of exponents of all runs is linear in string length

Simplifies understanding of string repetitions

Abstract

The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach, the stronger result concerning the linearity of the sum of exponents of all runs follows easily.