k-Median clustering under discrete Fréchet and Hausdorff distances
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We give the first near-linear time (1+)-approximation algorithm for k-median clustering of polygonal trajectories under the discrete Fréchet distance, and the first polynomial time (1+)-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 simpler metric space. We precisely characterize conditions on the simpler metric space of the cluster centers that allow faster (1+)-approximations for the k-median problem. We also show that the k-median problem under Hausdorff distance is NP-Hard.
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