arXiv 1802.091042018-02cs.DScs.ITmath.IT

求最近向量对的一种新算法

A New Algorithm for Finding Closest Pair of Vectors

Ning Xie, Shuai Xu, Yekun Xu

摘要：

给定$\{0,1\}^{m}$中的$n$向量$x_0, x_1, \ldots, x_{n-1}$，如何找到两个向量的汉明距离最小？此问题称为\emph{Closest Pair Problem}。如果这些向量是均匀随机生成的，除了其中两个与皮尔逊相关系数$ρ$相关，则该问题称为\emph{Light Bulb Problem}。在这项工作中，我们提出了一种新的编码为基础的计划最接近的对问题。我们设计的随机和确定性算法，实现最知名的运行时间时，输入向量$m$的长度是小的，最小距离是非常小的$m$相比。具体来说，我们的随机算法的运行时间是$O(n\log^{2}n\cdot 2^{c m} \cdot \mathrm{poly}(m))$，我们的确定性算法的运行时间是$O(n\log{n}\cdot 2^{c' m} \cdot \mathrm{poly}(m))$，其中$c$和$c'$是常数，仅取决于最近对的（相对）距离。当应用到灯泡问题，我们的结果产生国家的最先进的确定性运行时间时，皮尔逊相关系数$ρ$是非常大的。具体来说，当使用$ρ\geq 0.9933$时，我们的确定性算法比以前最好的确定性算法运行得更快（Alman，SOSA 2019）。

英文摘要：

Given $n$ vectors $x_0, x_1, \ldots, x_{n-1}$ in $\{0,1\}^{m}$, how to find two vectors whose pairwise Hamming distance is minimum? This problem is known as the \emph{Closest Pair Problem}. If these vectors are generated uniformly at random except two of them are correlated with Pearson-correlation coefficient $ρ$, then the problem is called the \emph{Light Bulb Problem}. In this work, we propose a novel coding-based scheme for the Closest Pair Problem. We design both randomized and deterministic algorithms, which achieve the best-known running time when the length of input vectors $m$ is small and the minimum distance is very small compared to $m$. Specifically, the running time of our randomized algorithm is $O(n\log^{2}n\cdot 2^{c m} \cdot \mathrm{poly}(m))$ and the running time of our deterministic algorithm is $O(n\log{n}\cdot 2^{c' m} \cdot \mathrm{poly}(m))$, where $c$ and $c'$ are constants depending only on the (relative) distance of the closest pair. When applied to the Light Bulb Problem, our result yields state-of-the-art deterministic running time when the Pearson-correlation coefficient $ρ$ is very large. Specifically, when $ρ\geq 0.9933$, our deterministic algorithm runs faster than the previously best deterministic algorithm (Alman, SOSA 2019).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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