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 $\sigma$, compute the Hamming distance between the pattern and the text at every location. Several $(1+\epsilon)$-approximation algorithms have been proposed in the literature, with running time of the form $O(\epsilon^{-O(1)}n\log n\log m)$, all using fast Fourier transform (FFT). We describe a simple $(1+\epsilon)$-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results: - We obtain the first linear-time approximation algorithm; the running time is $O(\epsilon^{-2}n)$. - We obtain a faster exact algorithm computing all Hamming distances up to a given threshold k; its running time improves previous results by logarithmic factors and is linear if $k\le\sqrt m$. - We obtain approximation algorithms with better $\epsilon$-dependence using rectangular matrix multiplication. The time-bound is $\~O(n)$ when the pattern is sufficiently long: $m\ge \epsilon^{-28}$. Previous algorithms require $\~O(\epsilon^{-1}n)$ time. - When k is not too small, we obtain a truly sublinear-time algorithm to find all locations with Hamming distance approximately (up to a constant factor) less than k, in $O((n/k^{\Omega(1)}+occ)n^{o(1)})$ time, where occ is the output size. The algorithm leads to a property tester, returning true if an exact match exists and false if the Hamming distance is more than $\delta m$ at every location, running in $\~O(\delta^{-1/3}n^{2/3}+\delta^{-1}n/m)$ time. - We obtain a streaming algorithm to report all locations with Hamming distance approximately less than k, using $\~O(\epsilon^{-2}\sqrt k)$ space. Previously, streaming algorithms were known for the exact problem with \~O(k) space or for the approximate problem with $\~O(\epsilon^{-O(1)}\sqrt m)$ space.