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arXiv · 1909.02804

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Takuya Mieno·Yuki Kuhara·Tooru Akagi·Yuta Fujishige·Yuto Nakashima·Shunsuke Inenaga·Hideo Bannai·Masayuki Takeda
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Abstract

A substring $u$ of a string $T$ is called a minimal unique substring (MUS) of $T$ if $u$ occurs exactly once in $T$ and any proper substring of $u$ occurs at least twice in $T$. A string $w$ is called a minimal absent word (MAW) of $T$ if $w$ does not occur in $T$ and any proper substring of $w$ occurs in $T$. In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string $T$. We first show how the set of MUSs can change in a sliding window over $T$, and present an $O(n\log\sigma)$-time and $O(d)$-space algorithm to compute MUSs in a sliding window of width $d$ over $T$, where $\sigma$ is the maximum number of distinct characters in every window. We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over $T$. Our bounds improve on the previous results in [Crochemore et al., 2017].

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Takuya Mieno, Yuki Kuhara, Tooru Akagi, Yuta Fujishige, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda. 2019-09-06. Minimal Unique Substrings and Minimal Absent Words in a Sliding Window. https://arxiv.org/abs/1909.02804

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