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

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(n√mk2 and O(n√mk3) running time, respectively, where k is the bound on the distance, both in linear space. Even for very small values of k, the best previously known solutions required Ω(nm) time. Our main contribution is applying a periodicity-based argument in a way that is computationally effective even if we 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. © 2013 Springer-Verlag.