eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-02-27
46:1
46:14
10.4230/LIPIcs.STACS.2018.46
article
Relations Between Greedy and Bit-Optimal LZ77 Encodings
Kosolobov, Dmitry
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol096-stacs2018/LIPIcs.STACS.2018.46/LIPIcs.STACS.2018.46.pdf
Lempel-Ziv
LZ77 encoding
greedy LZ77
bit optimal LZ77