Hamming Distance Oracle
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911948235341824 |
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| author | Boneh, Itai Fried, Dvir Golan, Shay Kraus, Matan |
| author_facet | Boneh, Itai Fried, Dvir Golan, Shay Kraus, Matan |
| contents | 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 $x \le nm$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hamming Distance Oracle Boneh, Itai Fried, Dvir Golan, Shay Kraus, Matan Data Structures and Algorithms 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 $x \le nm$. |
| title | Hamming Distance Oracle |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.05430 |