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Link to original content: https://doi.org/10.4230/LIPIcs.ESA.2024.18
Height-Bounded Lempel-Ziv Encodings
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Height-Bounded Lempel-Ziv Encodings

Authors Hideo Bannai , Mitsuru Funakoshi , Diptarama Hendrian , Myuji Matsuda, Simon J. Puglisi

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Author Details

Hideo Bannai
  • M&D Data Science Center, Tokyo Medical and Dental University (TMDU), Japan
Mitsuru Funakoshi
  • NTT Communication Science Laboratories, Kyoto, Japan
Diptarama Hendrian
  • M&D Data Science Center, Tokyo Medical and Dental University (TMDU), Japan
Myuji Matsuda
  • Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University (TMDU), Japan
Simon J. Puglisi
  • Department of Computer Science, University of Helsinki, Finland

Funding

  • Bannai, Hideo: JSPS KAKENHI Grant Number JP24K02899.
  • Puglisi, Simon J.: Academy of Finland grants 339070 and 351150.

Cite As Get BibTex

Hideo Bannai, Mitsuru Funakoshi, Diptarama Hendrian, Myuji Matsuda, and Simon J. Puglisi. Height-Bounded Lempel-Ziv Encodings. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.18

Abstract

We introduce height-bounded LZ encodings (LZHB), a new family of compressed representations that are variants of Lempel-Ziv parsings with a focus on bounding the worst-case access time to arbitrary positions in the text directly via the compressed representation. An LZ-like encoding is a partitioning of the string into phrases of length 1 which can be encoded literally, or phrases of length at least 2 which have a previous occurrence in the string and can be encoded by its position and length. An LZ-like encoding induces an implicit referencing forest on the set of positions of the string. An LZHB encoding is an LZ-like encoding where the height of the implicit referencing forest is bounded. An LZHB encoding with height constraint h allows access to an arbitrary position of the underlying text using O(h) predecessor queries.
While computing the optimal (i.e., smallest) LZHB encoding efficiently seems to be difficult [Cicalese & Ugazio 2024, arXiv, to appear at DLT 2024], we give the first linear time algorithm for strings over a constant size alphabet that computes the greedy LZHB encoding, i.e., the string is processed from beginning to end, and the longest prefix of the remaining string that can satisfy the height constraint is taken as the next phrase. Our algorithms significantly improve both theoretically and practically, the very recently and independently proposed algorithms by Lipták et al. (CPM 2024).
We also analyze the size of height bounded LZ encodings in the context of repetitiveness measures, and show that there exists a constant c such that the size ẑ_{HB(clog n)} of the optimal LZHB encoding whose height is bounded by clog n for any string of length n is O(ĝ_{rl}), where ĝ_{rl} is the size of the smallest run-length grammar. Furthermore, we show that there exists a family of strings such that ẑ_{HB(clog n)} = o(ĝ_{rl}), thus making ẑ_{HB(clog n)} one of the smallest known repetitiveness measures for which O(polylog n) time access is possible using linear (O(ẑ_{HB(clog n)})) space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
Keywords
  • Lempel-Ziv parsing
  • data compression

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