Relations Between Greedy and Bit-Optimal LZ77 Encodings

Dmitry Kosolobov

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

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 $O(\frac{\log n}{\log\log\log n})$ 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 $O(\min\{z, \frac{\log n}{\log\log z}\})$, 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.
Original languageEnglish
Title of host publicationRelations Between Greedy and Bit-Optimal LZ77 Encodings
EditorsRolf Niedermeier, Brigitte Vallée
Number of pages14
Place of PublicationDagstuhl
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Publication date2018
Pages46:1-46:14
Article number46
ISBN (Print)978-3-95977-062-0
DOIs
Publication statusPublished - 2018
MoE publication typeA4 Article in conference proceedings
EventSymposium on Theoretical Aspects of Computer Science - Caen, France
Duration: 28 Feb 20183 Mar 2018
Conference number: 35

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Volume96
ISSN (Electronic)1868-8969

Fields of Science

  • 113 Computer and information sciences

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