Algorithms for Computing 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 called 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, focusing specifically 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. Furthermore, we propose another algorithm for the problem that runs in average $O(m \log^2 m)$ time and $O(m\log m)$ space: The average-case bound holds under an exponent-bounded random-input model, in which the input is chosen uniformly from the strings of RLE size $m$ whose run exponents are at most a given bound $E\geq 2$. Our approaches extend key ideas from [Gawrychowski et al., SPIRE 2015] to the RLE setting through new combinatorial insights. When the RLE size $m$ is sufficiently small compared to $n$, our algorithms achieve linear-time behavior in $n$, potentially leading to improved performance over existing methods in such cases.