Pattern Matching in Lempel-Ziv Compressed Strings: Fast, Simple, and Deterministic

Abstract

Countless variants of the Lempel-Ziv compression are widely used in many real-life applications. This paper is concerned with a natural modification of the classical pattern matching problem inspired by the popularity of such compression methods: given an uncompressed pattern $p[1\mathinner{\ldotp\ldotp} m]$ and a Lempel-Ziv representation of a string $t[1\mathinner{\ldotp\ldotp} N]$, does p occur in t? Farach and Thorup [5] gave a randomized $\mathcal{O}(n\log^2\frac{N}{n}+m)$ time solution for this problem, where n is the size of the compressed representation of t. Building on the methods of [3] and [6], we improve their result by developing a faster and fully deterministic $\mathcal{O}(n\log\frac{N}{n}+m)$ time algorithm with the same space complexity. Note that for highly compressible texts, $\log\frac{N}{n}$ might be of order n, so for such inputs the improvement is very significant. A small fragment of our method can be used to give an asymptotically optimal solution for the substring hashing problem considered by Farach and Muthukrishnan [4].