In the Euclidean $k$-center problem in sliding window model, input points are given in a data stream and the goal is to find the $k$ smallest congruent balls whose union covers the $N$ most recent points of the stream. In this model, input points are allowed to be examined only once and the amount of space that can be used to store relative information is limited. Cohen-Addad et al.~\cite{cohen-2016} gave a $(6+\epsilon)$-approximation for the metric $k$-center problem using O($k/\epsilon \log \alpha$) points, where $\alpha$ is the ratio of the largest and smallest distance and is assumed to be known in advance. In this paper, we present a $(3+\epsilon)$-approximation algorithm for the Euclidean $1$-center problem using O($1/\epsilon \log \alpha$) points. We present an algorithm for the Euclidean $k$-center problem that maintains a coreset of size $O(k)$. Our algorithm gives a $(c+2\sqrt{3} + \epsilon)$-approximation for the Euclidean $k$-center problem using O($k/\epsilon \log \alpha$) points by using any given $c$-approximation for the coreset where $c$ is a positive real number. For example, by using the $2$-approximation~\cite{feder-greene-1988} of the coreset, our algorithm gives a $(2+2\sqrt{3} + \epsilon)$-approximation ($\approx 5.465$) using $O(k\log k)$ time. This is an improvement over the approximation factor of $(6+\epsilon)$ by Cohen-Addad et al.~\cite{cohen-2016} with the same space complexity and smaller update time per point. Moreover we remove the assumption that $\alpha$ is known in advance. Our idea can be adapted to the metric diameter problem and the metric $k$-center problem to remove the assumption. For low dimensional Euclidean space, we give an approximation algorithm that guarantees an even better approximation.

中文翻译：

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在滑动窗口上计算欧几里得 k 中心

在滑动窗口模型中的欧几里得 $k$-center 问题中，输入点在数据流中给出，目标是找到 $k$ 最小的全等球，其联合覆盖了流的 $N$ 最近点。在这个模型中，输入点只允许检查一次，可用于存储相关信息的空间量是有限的。Cohen-Addad 等人~\cite{cohen-2016} 使用 O($k/\epsilon \log \alpha$) 点给出了度量 $k$-center 问题的 $(6+\epsilon)$-近似值，其中 $\alpha$ 是最大和最小距离的比值，假设事先已知。在本文中，我们提出了一种使用 O($1/\epsilon \log \alpha$) 点的欧几里得 $1$-center 问题的 $(3+\epsilon)$-近似算法。我们提出了一种用于欧几里得 $k$-center 问题的算法，该算法维护一个大小为 $O(k)$ 的核心集。我们的算法使用 O($k/\epsilon \log \alpha$) 点通过使用任何给定的 $ 为欧几里得 $k$-center 问题提供 $(c+2\sqrt{3} + \epsilon)$-近似值核心集的 c$ 近似值，其中 $c$ 是正实数。例如，通过使用核心集的 $2$-approximation~\cite{feder-greene-1988}，我们的算法给出了 $(2+2\sqrt{3} + \epsilon)$-approximation ($\approx 5.465 $) 使用 $O(k\log k)$ 时间。这是对 Cohen-Addad 等人的 $(6+\epsilon)$ 近似因子的改进。~\cite{cohen-2016} 具有相同的空间复杂度和更小的每点更新时间。此外，我们删除了 $\alpha$ 事先已知的假设。我们的想法可以适用于公制直径问题和公制$k$-center 问题以消除假设。对于低维欧几里得空间，我们给出了一个近似算法，保证了更好的近似。