Note on the Greedy Parsing Optimality for Dictionary-Based Text Compression

TL;DR

This paper proves the optimality of greedy parsing in dictionary-based text compression schemes, including bounded variants of LZ77, by introducing the suffix-closed property of dynamic dictionaries.

Contribution

It introduces the suffix-closed property for dynamic dictionaries and proves greedy parsing optimality for all LZ77-based schemes, including bounded dictionaries.

Findings

Greedy parsing is optimal for unbounded dictionaries (known)

Bounded LZ77 variants satisfy the suffix-closed property

Greedy parsing optimality is proven for bounded dictionary schemes

Abstract

Dynamic dictionary-based compression schemes are the most daily used data compression schemes since they appeared in the foundational papers of Ziv and Lempel in 1977, commonly referred to as LZ77. Their work is the base of Deflate, gZip, WinZip, 7Zip and many others compression software. All of those compression schemes use variants of the greedy approach to parse the text into dictionary phrases. Greedy parsing optimality was proved by Cohn et al. (1996) for fixed length code and unbounded dictionaries. The optimality of the greedy parsing was never proved for bounded size dictionary which actually all of those schemes require. We define the suffix-closed property for dynamic dictionaries and we show that any LZ77-based dictionary, including the bounded variants, satisfy this property. Under this condition we prove the optimality of the greedy parsing as a variant of the proof by Cohn et al.