基于简化的文本-模式距离近似算法
Approximating Text-to-Pattern Distance via Dimensionality Reduction
摘要：
文本到模式的距离是字符串匹配中的一个基本问题，给定一个长度为$m$的模式和一个长度为$n$的文本，在一个整数字母表上，我们被要求计算模式和文本之间的距离。距离函数可以是例如针对某个参数$p > 0$的汉明距离或$\ell_p$距离。在过去的$\sim 40$年中，几乎所有最先进的精确和近似算法都使用FFT作为黑盒。在这项工作中，我们提出了$\widetilde{O}(n/\varepsilon^2)$时间算法的$(1\pm\varepsilon)$ -近似的$\ell_2$距离，和$\widetilde{O}(n/\varepsilon^3)$算法近似的汉明和$\ell_1$距离，所有不使用FFT。这与Chan等人最近的发展无关。[STOC 2020]，其中提出了不使用FFT的汉明距离的$O(n/\varepsilon^2)$算法-尽管他们的算法更具“组合性”，但我们的技术适用于汉明以外的其他范数。
英文摘要：
Text-to-pattern distance is a fundamental problem in string matching, where given a pattern of length $m$ and a text of length $n$, over an integer alphabet, we are asked to compute the distance between pattern and the text at every location. The distance function can be e.g. Hamming distance or $\ell_p$ distance for some parameter $p > 0$. Almost all state-of-the-art exact and approximate algorithms developed in the past $\sim 40$ years were using FFT as a black-box. In this work we present $\widetilde{O}(n/\varepsilon^2)$ time algorithms for $(1\pm\varepsilon)$-approximation of $\ell_2$ distances, and $\widetilde{O}(n/\varepsilon^3)$ algorithm for approximation of Hamming and $\ell_1$ distances, all without use of FFT. This is independent to the very recent development by Chan et al. [STOC 2020], where $O(n/\varepsilon^2)$ algorithm for Hamming distances not using FFT was presented -- although their algorithm is much more "combinatorial", our techniques apply to other norms than Hamming.
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