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Faster Clustering via Preprocessing

Abstract

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 MM; the next stage receives as input a query set QMQ\subset M, and should report a clustering of QQ according to some objective, such as 1-median, in which case the answer is a point aMa\in M minimizing qQdM(a,q)\sum_{q\in Q} d_M(a,q). We design fast algorithms that approximately solve such problems under standard clustering objectives like pp-center and pp-median, when the metric MM has low doubling dimension. By leveraging the preprocessing stage, our algorithms achieve query time that is near-linear in the query size n=Qn=|Q|, and is (almost) independent of the total number of points m=Mm=|M|.Comment: 24 page

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