Longest Unbordered Factors on Run-Length Encoded Strings

A border of a string is a non-empty proper prefix of the string that is also a suffix. A string is unbordered if it has no border. The longest unbordered factor is a fundamental notion in stringology, closely related to string periodicity. This paper addresses the longest unbordered factor problem: given a string of length $n$, the goal is to compute its longest factor that is unbordered. While recent work has achieved subquadratic and near-linear time algorithms for this problem, the best known worst-case time complexity remains $O(n \log n)$ [Kociumaka et al., ISAAC 2018]. In this paper, we investigate the problem in the context of compressed string processing, particularly focusing on run-length encoded (RLE) strings. We first present a simple yet crucial structural observation relating unbordered factors and RLE-compressed strings. Building on this, we propose an algorithm that solves the problem in $O(m^{1.5} \log^2 m)$ time and $O(m \log^2 m)$ space, where $m$ is the size of the RLE-compressed input string. To achieve this, our approach simulates a key idea from the $O(n^{1.5})$-time algorithm by [Gawrychowski et al., SPIRE 2015], adapting it to the RLE setting through new combinatorial insights. When the RLE size $m$ is sufficiently small compared to $n$, our algorithm may show linear-time behavior in $n$, potentially leading to improved performance over existing methods in such cases.