Hamming Distance Oracles

In this paper, we present and study the Hamming distance oracle problem. In this problem, the task is to preprocess two strings S and T of lengths n and m, respectively, to obtain a data structure that is able to return the Hamming distance between a substring of S and a substring of T. For strings over a constant-size alphabet, we show that for every x ≤ min{n, m} there is a data structure with Õ(nm/x) preprocessing time and O(x) query time. We also provide a conditional lower bound, showing that for every ε > 0 there is no combinatorial data structure with query time O(x) and preprocessing time O(nm/x )^1−ε) unless combinatorial fast matrix multiplication is possible. For strings over a general alphabet, we present a data structure with Õ(nm/√x) pre-processing time and O(x) query time for every x ≤ min{n, m}. Moreover, for every ε > 0 we provide a data structure with a preprocessing time of Õ(n+m/ε3 ) that returns with high probability a (1 ± ε) approximation of the Hamming distance of two input substrings. The query time of the approximation data structure is Õ(1/ε²).