Faster fully compressed pattern matching by recompression

TL;DR

This paper introduces a faster algorithm for fully compressed pattern matching using recompression techniques, significantly improving over previous methods by reducing the complexity to nearly linear in the size of compressed inputs.

Contribution

The paper presents a novel recompression-based algorithm for fully compressed pattern matching with improved time complexity.

Findings

Achieves O((n+m)log M) runtime for compressed pattern matching.

Outperforms the previous O(n^2m) algorithm by Lifshits.

Provides an efficient method for handling compressed strings in pattern matching.

Abstract

In this paper, a fully compressed pattern matching problem is studied. The compression is represented by straight-line programs (SLPs), i.e. a context-free grammars generating exactly one string; the term fully means that both the pattern and the text are given in the compressed form. The problem is approached using a recently developed technique of local recompression: the SLPs are refactored, so that substrings of the pattern and text are encoded in both SLPs in the same way. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. This technique yields an O((n+m)log M) algorithm for compressed pattern matching, assuming that M fits in O(1) machine words, where n (m) is the size of the compressed representation of the text (pattern, respectively), while M is the size of the decompressed pattern. If only m+n fits in O(1) machine words, the running time increases to O((n+m)log M log(n+m)). The previous best algorithm due to Lifshits had O(n^2m) running time.