Faster Clustering via Preprocessing

We examine the efficiency of clustering a set of points, when the encompassing metric space may be preprocessed in advance. In computational problems of this genre, there is a first stage of preprocessing, whose input is a collection of points M ; the next stage receives as input a query set Q ⊂ M , and should report a clustering of Q according to some objective, such as 1-median, in which case the answer is a point a ∈ M minimizing ∑ q∈Q dM (a, q). We design fast algorithms that approximately solve such problems under standard clustering objectives like p-center and p-median, when the metric M has low doubling dimension. By leveraging the preprocessing stage, our algorithms achieve query time that is near-linear in the query size n = |Q|, and is (almost) independent of the total number of points m = |M |. This work was supported in part by The Israel Science Foundation (grant #452/08), by a US-Israel BSF grant #2010418, and by the Citi Foundation.