Approximating Text-to-Pattern Hamming Distances

Abstract

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size σ, compute the Hamming distance (i.e., the number of mismatches) between the pattern and the text at every location. Several randomized (1 + ε)-approximation algorithms have been proposed in the literature (e.g., by Karloff (Inf. Proc. Lett., 1993), Indyk (FOCS 1998), and Kopelowitz and Porat (SOSA 2018)), with running time of the form O(ε −O(1) n log n log m), all using fast Fourier transform (FFT). We describe a simple randomized (1 + ε)-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results (all Monte-Carlo randomized) in different settings: