On Maximal Unbordered Factors

TL;DR

This paper studies the properties of the longest unbordered substring in a string, proving it is nearly as long as the string itself for large strings over alphabets of size five or more, and introduces a new algorithm for finding it.

Contribution

It establishes a high expected length of the maximal unbordered factor for large strings over certain alphabets and proposes a new efficient algorithm for its computation.

Findings

Expected maximal unbordered factor length is at least 0.99 n for large n and alphabet size ≥ 5.

The result applies to strings over alphabets of size five or more.

A new algorithm for computing the maximal unbordered factor is proposed.

Abstract

Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. In this work we investigate the relationship between $n$ and the length of the maximal unbordered factor of $S$. We prove that for the alphabet of size $\sigma \ge 5$ the expected length of the maximal unbordered factor of a string of length~$n$ is at least $0.99 n$ (for sufficiently large values of $n$). As an application of this result, we propose a new algorithm for computing the maximal unbordered factor of a string.