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

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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.

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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

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.

References

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