Decompressing Lempel-Ziv Compressed Text

TL;DR

This paper presents new algorithms for decompressing Lempel-Ziv 77 compressed text efficiently, achieving near-optimal space and time complexities, and improves pattern matching on compressed data.

Contribution

It introduces algorithms that decompress LZ77 compressed text in linear time with minimal working space, surpassing previous folklore solutions, especially for general alphabets.

Findings

Achieves $O(n)$ time and $O(z)$ space for constant alphabets.

Provides a trade-off between time and space for large alphabets.

Improves pattern matching efficiency on LZ77-compressed text.

Abstract

We consider the problem of decompressing the Lempel--Ziv 77 representation of a string $S$ of length $n$ using a working space as close as possible to the size $z$ of the input. The folklore solution for the problem runs in $O(n)$ time but requires random access to the whole decompressed text. Another folklore solution is to convert LZ77 into a grammar of size $O(z\log(n/z))$ and then stream $S$ in linear time. In this paper, we show that $O(n)$ time and $O(z)$ working space can be achieved for constant-size alphabets. On general alphabets of size $\sigma$, we describe (i) a trade-off achieving $O(n\log^\delta \sigma)$ time and $O(z\log^{1-\delta}\sigma)$ space for any $0\leq \delta\leq 1$, and (ii) a solution achieving $O(n)$ time and $O(z\log\log (n/z))$ space. The latter solution, in particular, dominates both folklore algorithms for the problem. Our solutions can, more generally, extract any specified subsequence of $S$ with little overheads on top of the linear running time and working space. As an immediate corollary, we show that our techniques yield improved results for pattern matching problems on LZ77-compressed text.