On k-Median clustering in high dimensions

We study approximation algorithms for k-median clustering. We obtain small coresets for k-median clustering in metric spaces as well as in Euclidean spaces. Specifically, in IR, those coresets are of size with only polynomial dependency on d. This leads to a (1 + ε)-approximation algorithm for kmedian clustering in IR, with running time O(ndk + 2 O(1) dn), for any σ > 0. This is an improvement over previous results [7, 20, 21]. We also provide fast constant factor approximation algorithms for kmedian clustering in finite metric spaces. We use those coresets to compute (1 + )approximation k-median clustering in the streaming model of computation, using only O(kd −2 log n) space, where the points are taken from IR. This is the first streaming algorithm, for this problem, that has space complexity with only polynomial dependency on the dimension.