Sublinear Time Lempel-Ziv (LZ77) Factorization

Abstract

The Lempel-Ziv (LZ77) factorization of a string is a widely-used algorithmic tool that plays a central role in data compression and indexing. For a length-n string over integer alphabet $$[0, \sigma )$$ with $${\sigma = n^{\mathcal O(1)}}$$, and on a word RAM of width $$w = \varTheta (\log n)$$, it can be computed in $$\mathcal O(n)$$ time. However, the packed representation of the string occupies only $$\varTheta (n \log \sigma )$$ bits or equivalently $$\varTheta (n / \log _\sigma n)$$ words of space, and hence we can hope for algorithms that run in $$\mathcal O(n / \log _\sigma n)$$ time and words of space. Kempa showed how to compute the LZ77 factorization with overlaps in $$\mathcal O(n / \log _\sigma n + z \log ^{11} n)$$ time and $$\mathcal O(n / \log _\sigma n + z \log ^{10} n)$$ words of space, where z is the number of phrases in the LZ77 factorization (SODA 2019). We significantly improve this result by achieving $$\mathcal O(n / \log _\sigma n + z \log ^{3+\epsilon } z)$$ time with overlaps, and $$\mathcal O(n / \log _\sigma n + z \log ^{23/5+\epsilon } z)$$ without overlaps (for any constant $$\epsilon \in \mathbb R^+$$ ). In both cases, we require only $$\mathcal O(n / \log _\sigma n)$$ words of space. One ingredient of the solution is a novel approximation algorithm that computes an LZ-like parsing of at most 3z phrases in $$\mathcal O(n / \log _\sigma n)$$ time and words of space. All algorithms are deterministic.