Hamming Distance Oracle
<|reference_start|>Hamming Distance Oracle: 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 $x \le nm$.<|reference_end|>
@article{boneh2024hamming,
    title={Hamming Distance Oracle},
    author={Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus},
    journal={arXiv preprint arXiv:2407.05430},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.05430},
    primaryClass={cs.DS}
}