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We give a sampling-based algorithm for the -Median problem, with running time ( log), where  is the desired number of clusters and ∈ is a confidence parameter. This is the first -Median algorithm with fully polynomial running time that is independent of , the size of the data set. It gives a solution that is, with high probability, an (1)-approximation, if each cluster in some optimal solution has Ω points. We also give weakly-polynomial-time algorithms for this problem and a relaxed version of -Median in which a small fraction of outliers can be excluded. We give near-matching lower bounds showing that this assumption about cluster size is necessary. We also present a related algorithm for finding a clustering that excludes a small number of outliers.

A k-Median Algorithm with Running Time Independent of Data Size