License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2018.46
URN: urn:nbn:de:0030-drops-84894
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8489/
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Kosolobov, Dmitry

Relations Between Greedy and Bit-Optimal LZ77 Encodings

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LIPIcs-STACS-2018-46.pdf (0.5 MB)


Abstract

This paper investigates the size in bits of the LZ77 encoding, which is the most popular and efficient variant of the Lempel-Ziv encodings used in data compression. We prove that, for a wide natural class of variable-length encoders for LZ77 phrases, the size of the greedily constructed LZ77 encoding on constant alphabets is within a factor O(log n / log log log n) of the optimal LZ77 encoding, where n is the length of the processed string. We describe a series of examples showing that, surprisingly, this bound is tight, thus improving both the previously known upper and lower bounds. Further, we obtain a more detailed bound O(min{z, log n / log log z}), which uses the number z of phrases in the greedy LZ77 encoding as a parameter, and construct a series of examples showing that this bound is tight even for binary alphabet. We then investigate the problem on non-constant alphabets: we show that the known O(log n) bound is tight even for alphabets of logarithmic size, and provide tight bounds for some other important cases.

BibTeX - Entry

@InProceedings{kosolobov:LIPIcs:2018:8489,
  author =	{Dmitry Kosolobov},
  title =	{{Relations Between Greedy and Bit-Optimal LZ77 Encodings}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8489},
  URN =		{urn:nbn:de:0030-drops-84894},
  doi =		{10.4230/LIPIcs.STACS.2018.46},
  annote =	{Keywords: Lempel-Ziv, LZ77 encoding, greedy LZ77, bit optimal LZ77}
}

Keywords: Lempel-Ziv, LZ77 encoding, greedy LZ77, bit optimal LZ77
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018


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