Minimal Absent Words on Run-Length Encoded Strings

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

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Author Details

Tooru Akagi
  • Department of Informatics, Kyushu University, Fukuoka, Japan
Kouta Okabe
  • Department of Information Science and Technology, Kyushu University, Fukuoka, Japan
Takuya Mieno
  • Faculty of Information Science and Technology, Hokkaido University, Sapporo, Japan
Yuto Nakashima
  • Department of Informatics, Kyushu University, Fukuoka, Japan
Shunsuke Inenaga
  • Department of Informatics, Kyushu University, Fukuoka, Japan
  • PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan


We thank the anonymous referees for their comments.

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • string algorithms
  • combinatorics on words
  • minimal absent words
  • run-length encoding


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