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On the Approximation Ratio of Lempel-Ziv Parsing

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LATIN 2018: Theoretical Informatics (LATIN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

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Abstract

Shannon’s entropy is a clear lower bound for statistical compression. The situation is not so well understood for dictionary-based compression. A plausible lower bound is b, the least number of phrases of a general bidirectional parse of a text, where phrases can be copied from anywhere else in the text. Since computing b is NP-complete, a popular gold standard is z, the number of phrases in the Lempel-Ziv parse of the text, where phrases can be copied only from the left. While z can be computed in linear time, almost nothing has been known for decades about its approximation ratio with respect to b. In this paper we prove that \(z=O(b\log (n/b))\), where n is the text length. We also show that the bound is tight as a function of n, by exhibiting a string family where \(z = \varOmega (b\log n)\). Our upper bound is obtained by building a run-length context-free grammar based on a locally consistent parsing of the text. Our lower bound is obtained by relating b with r, the number of equal-letter runs in the Burrows-Wheeler transform of the text. On our way, we prove other relevant bounds between compressibility measures.

Partially funded by Basal Funds FB0001, Conicyt, by Fondecyt Grants 1-171058 and 1-170048, Chile, and by the Danish Research Council DFF-4005-00267.

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Notes

  1. 1.

    https://en.wikipedia.org/wiki/Wikipedia:Size_of_Wikipedia.

  2. 2.

    For this case, we could have defined bordering in a stricter way, as the first or last block of a chunk.

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Acknowledgements

We thank the reviewers for their insightful comments, which helped us improve the presentation significantly.

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Authors and Affiliations

  1. EIT, Diego Portales University, Santiago, Chile

    Travis Gagie

  2. Center for Biotechnology and Bioengineering (CeBiB), Santiago, Chile

    Travis Gagie & Gonzalo Navarro

  3. Department of Computer Science, University of Chile, Santiago, Chile

    Gonzalo Navarro

  4. DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark

    Nicola Prezza

Authors
  1. Travis Gagie
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  2. Gonzalo Navarro
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  3. Nicola Prezza
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Corresponding author

Correspondence to Gonzalo Navarro .

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Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Gagie, T., Navarro, G., Prezza, N. (2018). On the Approximation Ratio of Lempel-Ziv Parsing. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_36

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_36

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