Beating O(nm) in approximate LZW-compressed pattern matching

TL;DR

This paper introduces new algorithms for approximate pattern matching in LZW compressed texts that outperform previous methods, achieving sub-quadratic time complexities for both Hamming and edit distances using a periodicity-based approach.

Contribution

The paper presents the first sub-quadratic algorithms for approximate pattern matching in LZW compressed texts, utilizing a novel periodicity-based technique applicable directly to compressed data.

Findings

Achieves O(nm^0.5k^2) time for Hamming distance

Achieves O(nm^0.5k^3) time for edit distance

Uses linear space and improves over previous O(nm) solutions

Abstract

Given an LZW/LZ78 compressed text, we want to find an approximate occurrence of a given pattern of length m. The goal is to achieve time complexity depending on the size n of the compressed representation of the text instead of its length. We consider two specific definitions of approximate matching, namely the Hamming distance and the edit distance, and show how to achieve O(nm^0.5k^2) and O(nm^0.5k^3) running time, respectively, where k is the bound on the distance. Both algorithms use just linear space. Even for very small values of k, the best previously known solutions required O(nm) time. Our main contribution is applying a periodicity-based argument in a way that is computationally effective even if we need to operate on a compressed representation of a string, while the previous solutions were either based on a dynamic programming, or a black-box application of tools developed for uncompressed strings.