# 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 language English Relations Between Greedy and Bit-Optimal LZ77 Encodings Rolf Niedermeier, Brigitte Vallée 14 Dagstuhl Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2018 46:1-46:14 46 978-3-95977-062-0 https://doi.org/10.4230/LIPIcs.STACS.2018.46 Published - 2018 A4 Article in conference proceedings Symposium on Theoretical Aspects of Computer Science - Caen, FranceDuration: 28 Feb 2018 → 3 Mar 2018Conference number: 35

### Publication series

Name Leibniz International Proceedings in Informatics (LIPIcs) Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik 96 1868-8969

## Fields of Science

• 113 Computer and information sciences