A fast and simple $O (z \log n)$-space index for finding approximately longest common substrings

TL;DR

This paper introduces a space-efficient index for large texts that enables approximate longest common substring searches with high probability, using only $O(z \, \log n)$ space where $z$ is the LZ77 parse size.

Contribution

It presents a novel index structure that efficiently supports approximate LCS queries with sublinear space proportional to the LZ77 parse size.

Findings

Index uses $O(z \log n)$ space.

Query time is $O(m \log \log z + \mathrm{polylog}(m+z))$ with high probability.

Achieves near-linear approximation of the longest common substring.

Abstract

We describe how, given a text $T [1..n]$ and a positive constant $\epsilon$, we can build a simple $O (z \log n)$-space index, where $z$ is the number of phrases in the LZ77 parse of $T$, such that later, given a pattern $P [1..m]$, in $O (m \log \log z + \mathrm{polylog} (m + z))$ time and with high probability we can find a substring of $P$ that occurs in $T$ and whose length is at least a $(1 - \epsilon)$-fraction of the length of a longest common substring of $P$ and $T$.