Approximating LZ77 via Small-Space Multiple-Pattern Matching

TL;DR

This paper introduces a space-efficient algorithm for approximate LZ77 parsing using generalized multiple-pattern matching, significantly reducing space requirements while maintaining near-linear time complexity.

Contribution

It generalizes Karp-Rabin string matching to handle multiple patterns in small space and applies this to approximate LZ77 parsing with improved space efficiency.

Findings

Achieves $ ext{O}(n ext{log} n + m)$ time with $ ext{O}(s)$ space for multiple pattern matching.

Provides an approximation of LZ77 parse with at most $(1+ ext{varepsilon})z$ phrases in $ ext{O}( ext{varepsilon}^{-1} n ext{log} n)$ time.

Reduces space from previous $ ext{Omega}(n/ ext{polylog} n)$ to $ ext{O}(z)$, where $z$ can be exponentially small.

Abstract

We generalize Karp-Rabin string matching to handle multiple patterns in $\mathcal{O}(n \log n + m)$ time and $\mathcal{O}(s)$ space, where $n$ is the length of the text and $m$ is the total length of the $s$ patterns, returning correct answers with high probability. As a prime application of our algorithm, we show how to approximate the LZ77 parse of a string of length $n$. If the optimal parse consists of $z$ phrases, using only $\mathcal{O}(z)$ working space we can return a parse consisting of at most $(1+\varepsilon)z$ phrases in $\mathcal{O}(\varepsilon^{-1}n\log n)$ time, for any $\varepsilon\in (0,1]$. As previous quasilinear-time algorithms for LZ77 use $\Omega(n/\textrm{polylog }n)$ space, but $z$ can be exponentially small in $n$, these improvements in space are substantial.