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Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

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SOFSEM 2020: Theory and Practice of Computer Science (SOFSEM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12011))

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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Notes

  1. 1.

    At least one of w[2..] and \(w[..|w|\, - \,1]\) is absent from T[i..j], because \(w \not \in \mathsf {MAW}\) (T[i..j]).

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Authors and Affiliations

  1. Department of Informatics, Kyushu University, Fukuoka, Japan

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

  2. Japan Society for Promotion of Science, Tokyo, Japan

    Yuta Fujishige

  3. PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan

    Shunsuke Inenaga

Authors
  1. Takuya Mieno
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  2. Yuki Kuhara
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  3. Tooru Akagi
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  4. Yuta Fujishige
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  5. Yuto Nakashima
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  6. Shunsuke Inenaga
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  7. Hideo Bannai
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  8. Masayuki Takeda
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Corresponding author

Correspondence to Takuya Mieno .

Editor information

Editors and Affiliations

  1. University of Macedonia, Thessaloniki, Greece

    Alexander Chatzigeorgiou

  2. University of Bergamo, Bergamo, Italy

    Riccardo Dondi

  3. Cyprus University of Technology, Limassol, Cyprus

    Herodotos Herodotou

  4. Carnegie Mellon University Qatar, Doha, Qatar

    Christos Kapoutsis

  5. Open University of Cyprus, Nicosia, Cyprus

    Yannis Manolopoulos

  6. University of Cyprus, Nicosia, Cyprus

    George A. Papadopoulos

  7. Paris Dauphine University, Paris, France

    Florian Sikora

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Mieno, T. et al. (2020). Minimal Unique Substrings and Minimal Absent Words in a Sliding Window. In: Chatzigeorgiou, A., et al. SOFSEM 2020: Theory and Practice of Computer Science. SOFSEM 2020. Lecture Notes in Computer Science(), vol 12011. Springer, Cham. https://doi.org/10.1007/978-3-030-38919-2_13

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  • DOI: https://doi.org/10.1007/978-3-030-38919-2_13

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