Relations Between Greedy and Bit-Optimal LZ77 Encodings

TL;DR

This paper analyzes the relationship between greedy and optimal LZ77 encodings, establishing tight bounds on their size difference for various alphabet sizes and providing new theoretical insights into data compression efficiency.

Contribution

It proves tight bounds on the size ratio between greedy and optimal LZ77 encodings for constant and non-constant alphabets, improving previous bounds and offering detailed analysis.

Findings

Greedy LZ77 encoding is within a factor of $O(rac{ ext{log} n}{ ext{log log log n}})$ of optimal on constant alphabets.

The bound $O( ext{min}igrace{z, rac{ ext{log} n}{ ext{log log z}}igrace}$ is tight for binary alphabets.

For non-constant alphabets, the $O( ext{log} n)$ bound is tight even for logarithmic alphabet sizes.

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.