Hamming Distance Oracle

TL;DR

This paper introduces data structures for efficiently computing Hamming distances between substrings of two strings, providing bounds and lower bounds for constant and general alphabets.

Contribution

It presents new data structures with specific time bounds for the Hamming distance oracle problem and establishes conditional lower bounds based on fast matrix multiplication.

Findings

Data structure with (nm/x) preprocess and O(x) query time for constant alphabet.

Data structure with (nm/) preprocess and O(x) query time for general alphabet.

Conditional lower bounds showing limits of data structure efficiency unless fast matrix multiplication is possible.

Abstract

In this paper, we present and study the \emph{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 answer queries regarding the Hamming distance between a substring of $S$ and a substring of $T$. For a constant size alphabet strings, we show that for every $x\le nm$ there is a data structure with $\tilde{O}(nm/x)$ preprocess time and $O(x)$ query time. We also provide a combinatorial conditional lower bound, showing that for every $\varepsilon > 0$ and $x \le nm$ there is no data structure with query time $O(x)$ and preprocess time $O((\frac{nm}{x})^{1-\varepsilon})$ unless combinatorial fast matrix multiplication is possible. For strings over general alphabet, we present a data structure with $\tilde{O}(nm/\sqrt{x})$ preprocess time and $O(x)$ query time for every $x \le nm$.