Compressed Indexing with Signature Grammars

TL;DR

This paper introduces new compressed indexing data structures that efficiently support pattern matching queries by leveraging signature grammars and LZ77 parsing, significantly improving previous solutions in space and time complexity.

Contribution

It presents the first data structure capable of deciding pattern occurrence in O(m) time with compressed space, combining randomized grammar construction with 2D-range reporting techniques.

Findings

Supports pattern matching in O(m + occ (lg lg n + lg^ε z)) time

Uses O(z lg(n/z)) space, where z is the LZ77 parse size

Improves upon previous solutions in both space and time complexity

Abstract

The compressed indexing problem is to preprocess a string $S$ of length $n$ into a compressed representation that supports pattern matching queries. That is, given a string $P$ of length $m$ report all occurrences of $P$ in $S$. We present a data structure that supports pattern matching queries in $O(m + occ (\lg\lg n + \lg^\epsilon z))$ time using $O(z \lg(n / z))$ space where $z$ is the size of the LZ77 parse of $S$ and $\epsilon > 0$ is an arbitrarily small constant, when the alphabet is small or $z = O(n^{1 - \delta})$ for any constant $\delta > 0$. We also present two data structures for the general case; one where the space is increased by $O(z\lg\lg z)$, and one where the query time changes from worst-case to expected. These results improve the previously best known solutions. Notably, this is the first data structure that decides if $P$ occurs in $S$ in $O(m)$ time using $O(z\lg(n/z))$ space. Our results are mainly obtained by a novel combination of a randomized grammar construction algorithm with well known techniques relating pattern matching to 2D-range reporting.