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DOI: 10.4230/LIPIcs.CPM.2022.27

Minimal Absent Words on Run-Length Encoded Strings

Authors: Tooru Akagi, Kouta Okabe, Takuya Mieno, Yuto Nakashima, and Shunsuke Inenaga

Published in: LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Abstract

A string w is called a minimal absent word for another string T if w does not occur (as a substring) in T and all proper substrings of w occur in T. State-of-the-art data structures for reporting the set MAW(T) of MAWs from a given string T of length n require O(n) space, can be built in O(n) time, and can report all MAWs in O(|MAW(T)|) time upon a query. This paper initiates the problem of computing MAWs from a compressed representation of a string. In particular, we focus on the most basic compressed representation of a string, run-length encoding (RLE), which represents each maximal run of the same characters a by a^p where p is the length of the run. Let m be the RLE-size of string T. After categorizing the MAWs into five disjoint sets ℳ₁, ℳ₂, ℳ₃, ℳ₄, ℳ₅ using RLE, we present matching upper and lower bounds for the number of MAWs in ℳ_i for i = 1,2,4,5 in terms of RLE-size m, except for ℳ₃ whose size is unbounded by m. We then present a compact O(m)-space data structure that can report all MAWs in optimal O(|MAW(T)|) time.

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Tooru Akagi, Kouta Okabe, Takuya Mieno, Yuto Nakashima, and Shunsuke Inenaga. Minimal Absent Words on Run-Length Encoded Strings. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{akagi_et_al:LIPIcs.CPM.2022.27,
  author =	{Akagi, Tooru and Okabe, Kouta and Mieno, Takuya and Nakashima, Yuto and Inenaga, Shunsuke},
  title =	{{Minimal Absent Words on Run-Length Encoded Strings}},
  booktitle =	{33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-234-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{223},
  editor =	{Bannai, Hideo and Holub, Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.27},
  URN =		{urn:nbn:de:0030-drops-161545},
  doi =		{10.4230/LIPIcs.CPM.2022.27},
  annote =	{Keywords: string algorithms, combinatorics on words, minimal absent words, run-length encoding}
}

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Minimal Absent Words on Run-Length Encoded Strings

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