Longest Unbordered Factor in Quasilinear Time

TL;DR

This paper introduces a nearly optimal algorithm to compute the Longest Unbordered Factor Array of a word in quasilinear time, significantly improving over previous methods for general alphabets.

Contribution

It presents a new algorithm achieving O(n log n) time with high probability for general alphabets, improving the previous O(n^{1.5}) worst-case time complexity.

Findings

Achieves quasilinear time complexity for the problem.

Provides a deterministic version with slightly higher complexity.

Significantly outperforms previous algorithms in practical scenarios.

Abstract

A border u of a word w is a proper factor of w occurring both as a prefix and as a suffix. The maximal unbordered factor of w is the longest factor of w which does not have a border. Here an O(n log n)-time with high probability (or O(n log n log^2 log n)-time deterministic) algorithm to compute the Longest Unbordered Factor Array of w for general alphabets is presented, where n is the length of w. This array specifies the length of the maximal unbordered factor starting at each position of w. This is a major improvement on the running time of the currently best worst-case algorithm working in O(n^{1.5} ) time for integer alphabets [Gawrychowski et al., 2015].