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 |
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Title of host publication | Relations Between Greedy and Bit-Optimal LZ77 Encodings |
Editors | Rolf Niedermeier, Brigitte Vallée |
Number of pages | 14 |
Place of Publication | Dagstuhl |
Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Publication date | 2018 |
Pages | 46:1-46:14 |
Article number | 46 |
ISBN (Print) | 978-3-95977-062-0 |
DOIs | |
Publication status | Published - 2018 |
MoE publication type | A4 Article in conference proceedings |
Event | Symposium on Theoretical Aspects of Computer Science - Caen, France Duration: 28 Feb 2018 → 3 Mar 2018 Conference number: 35 |
Publication series
Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Publisher | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik |
Volume | 96 |
ISSN (Electronic) | 1868-8969 |
Fields of Science
- 113 Computer and information sciences