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Approximation Ratios of \(\mathsf {RePair}\), \(\mathsf {LongestMatch}\) and \(\mathsf {Greedy}\) on Unary Strings

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String Processing and Information Retrieval (SPIRE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11811))

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Abstract

A grammar-based compressor computes for a given input w a context-free grammar that produces only w. So-called global grammar-based compressors (\(\mathsf {RePair}\), \(\mathsf {LongestMatch}\) and \(\mathsf {Greedy}\)) achieve impressive practical compression results, but the recursive character of those algorithms makes it hard to achieve strong theoretical results. To this end, this paper studies the approximation ratio of those algorithms for unary input strings, which is strongly related to the field of addition chains. We show that in this setting, \(\mathsf {RePair}\) and \(\mathsf {LongestMatch}\) produce equal size grammars that are by a factor of at most \(\log _2(3)\) larger than a smallest grammar. We also provide a matching lower bound. The main result of this paper is a new lower bound for \(\mathsf {Greedy}\) of 1.348..., which improves the best known lower bound for arbitrary (not necessarily unary) input strings.

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Notes

  1. 1.

    While LZ78 was not introduced as a grammar-based compressor, it is straightforward to compute from the LZ78-factorization of w an SLP for w of roughly the same size.

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Authors and Affiliations

  1. University of Siegen, Siegen, Germany

    Danny Hucke

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  1. Danny Hucke
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Correspondence to Danny Hucke .

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Editors and Affiliations

  1. University of A Coruña, A Coruña, Spain

    Nieves R. Brisaboa

  2. University of Helsinki, Helsinki, Finland

    Simon J. Puglisi

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Hucke, D. (2019). Approximation Ratios of \(\mathsf {RePair}\), \(\mathsf {LongestMatch}\) and \(\mathsf {Greedy}\) on Unary Strings. In: Brisaboa, N., Puglisi, S. (eds) String Processing and Information Retrieval. SPIRE 2019. Lecture Notes in Computer Science(), vol 11811. Springer, Cham. https://doi.org/10.1007/978-3-030-32686-9_1

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  • DOI: https://doi.org/10.1007/978-3-030-32686-9_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-32685-2

  • Online ISBN: 978-3-030-32686-9

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