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Optimal LZ-End Parsing Is Hard

Authors Hideo Bannai , Mitsuru Funakoshi , Kazuhiro Kurita , Yuto Nakashima , Kazuhisa Seto , Takeaki Uno

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Author Details

Hideo Bannai
  • M&D Data Science Center, Tokyo Medical and Dental University, Japan
Mitsuru Funakoshi
  • Department of Informatics, Kyushu University, Fukuoka, Japan
  • Japan Society for the Promotion of Science, Tokyo, Japan
Kazuhiro Kurita
  • Nagoya University, Japan
Yuto Nakashima
  • Department of Informatics, Kyushu University, Fukuoka, Japan
Kazuhisa Seto
  • Faculty of Information Science and Technology, Hokkaido University, Sapporo, Japan
Takeaki Uno
  • National Institute of Informatics, Tokyo, Japan

Funding

This work was partially supported by JSPS KAKENHI Grant Numbers JP20H05964 (for YN).
  • Bannai, Hideo: JSPS KAKENHI Grant Number JP20H04141.
  • Funakoshi, Mitsuru: JSPS KAKENHI Grant Number JP20J21147.
  • Kurita, Kazuhiro: JSPS KAKENHI Grant Number JP21K17812, JP22H03549, JP21H05861, and JST ACT-X Grant Number JPMJAX2105.
  • Nakashima, Yuto: JSPS KAKENHI Grant Number JP21K17705, JP23H04386, and JST ACT-X Grant Number JPMJAX200K.
  • Seto, Kazuhisa: JSPS KAKENHI Grant Number JP21H05839.

Acknowledgements

We would like to thank Dominik Köppl for discussion. We also gratefully acknowledge the comments of anonymous reviewers for improving the manuscript.

Cite AsGet BibTex

Hideo Bannai, Mitsuru Funakoshi, Kazuhiro Kurita, Yuto Nakashima, Kazuhisa Seto, and Takeaki Uno. Optimal LZ-End Parsing Is Hard. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 3:1-3:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CPM.2023.3

Abstract

LZ-End is a variant of the well-known Lempel-Ziv parsing family such that each phrase of the parsing has a previous occurrence, with the additional constraint that the previous occurrence must end at the end of a previous phrase. LZ-End was initially proposed as a greedy parsing, where each phrase is determined greedily from left to right, as the longest factor that satisfies the above constraint [Kreft & Navarro, 2010]. In this work, we consider an optimal LZ-End parsing that has the minimum number of phrases in such parsings. We show that a decision version of computing the optimal LZ-End parsing is NP-complete by showing a reduction from the vertex cover problem. Moreover, we give a MAX-SAT formulation for the optimal LZ-End parsing adapting an approach for computing various NP-hard repetitiveness measures recently presented by [Bannai et al., 2022]. We also consider the approximation ratio of the size of greedy LZ-End parsing to the size of the optimal LZ-End parsing, and give a lower bound of the ratio which asymptotically approaches 2.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
Keywords
  • Data Compression
  • LZ-End
  • Repetitiveness measures

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References

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