Longest Unbordered Factors on Run-Length Encoded Strings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918101496365056 |
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| author | Sekizaki, Shoma Mieno, Takuya |
| author_facet | Sekizaki, Shoma Mieno, Takuya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Longest Unbordered Factors on Run-Length Encoded Strings Sekizaki, Shoma Mieno, Takuya Data Structures and Algorithms 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. |
| title | Longest Unbordered Factors on Run-Length Encoded Strings |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.16285 |