Approximating LZ77 via Small-Space Multiple-Pattern Matching

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 2z phrases in $\mathcal{O}(n\log n)$ time, and a parse of at most (1 + ε)z phrases in $\mathcal{O}(n\log^{2}n)$ for any constant ε > 0. As previous quasilinear-time algorithms for LZ77 use Ω(n/polylogn) space, but z can be exponentially small in n, these improvements in space consumption are substantial.