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Published in:
2020 | OriginalPaper | Chapter
Minimal Unique Substrings and Minimal Absent Words in a Sliding Window
Authors : Takuya Mieno, Yuki Kuhara, Tooru Akagi, Yuta Fujishige, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda
Published in: SOFSEM 2020: Theory and Practice of Computer Science
Publisher: Springer International Publishing
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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).