We give the first near-linear time $(1+\eps)$-approximation algorithm for $k$-median clustering of polygonal trajectories under the discrete Fr\'{e}chet distance, and the first polynomial time $(1+\eps)$-approximation algorithm for $k$-median clustering of finite point sets under the Hausdorff distance, provided the cluster centers, ambient dimension, and $k$ are bounded by a constant. The main technique is a general framework for solving clustering problems where the cluster centers are restricted to come from a \emph{simpler} metric space. We precisely characterize conditions on the simpler metric space of the cluster centers that allow faster $(1+\eps)$-approximations for the $k$-median problem. We also show that the $k$-median problem under Hausdorff distance is \textsc{NP-Hard}.

中文翻译：

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离散 Fr\'{e}chet 和 Hausdorff 距离下的 k 中值聚类

我们给出了离散Fr\'{e}chet距离下多边形轨迹的$k$-中值聚类的第一个近线性时间$(1+\eps)$-近似算法，以及第一个多项式时间$(1+ \eps)$-近似算法，用于在 Hausdorff 距离下对有限点集进行 $k$-中值聚类，前提是聚类中心、环境维度和 $k$ 以常数为界。主要技术是用于解决聚类问题的通用框架，其中聚类中心被限制为来自 \emph {simpler} 度量空间。我们在聚类中心的更简单的度量空间上精确地描述了条件，这些条件允许更快地 $(1+\eps)$-近似值来解决 $k$-中值问题。我们还表明，Hausdorff 距离下的 $k$-中值问题是 \textsc{NP-Hard}。