# Maximal Unbordered Factors of Random Strings

## TL;DR

This paper proves that the expected maximum length of unbordered factors in a random string is close to the string length, confirming a conjecture and enabling linear-time average-case algorithms.

## Contribution

It confirms a conjecture by precisely characterizing the expected maximum unbordered factor length in random strings and analyzes the average-case complexity of finding such factors.

## Key findings

- Expected maximum unbordered factor length is n - Θ(σ^{-1})
- Maximum unbordered factor can be found in linear time on average
- Average-case complexity is between Ω(√n) and O(√n log_σ n)

## Abstract

A border of a string is a non-empty prefix of the string that is also a suffix of the string, and a string is unbordered if it has no border other than itself. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the following: If we pick a string of length $n$ from a fixed non-unary alphabet uniformly at random, then the expected maximum length of its unbordered factors is $n - O(1)$. We confirm this conjecture by proving that the expected value is, in fact, ${n - \Theta(\sigma^{-1})}$, where $\sigma$ is the size of the alphabet. This immediately implies that we can find such a maximal unbordered factor in linear time on average. However, we go further and show that the optimum average-case running time is in $\Omega (\sqrt{n}) \cap O (\sqrt{n \log_\sigma n})$ due to analogous bounds by Czumaj and G\k{a}sieniec [CPM 2000] for the problem of computing the shortest period of a uniformly random string.