Range-clustering queries

In a geometric k -clustering problem the goal is to partition a set of points in R d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S : given a query box Q and an integer k>2 , compute an optimal k -clustering for S∖Q . We obtain the following results. We present a general method to compute a (1+ϵ) -approximation to a range-clustering query, where ϵ>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k -center clustering in any L p -metric and a variant of k -center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes. We extend our method to deal with capacitated k -clustering problems, where each of the clusters should not contain more than a given number of points. For the special cases of rectilinear k -center clustering in R 1 , and in R 2 for k=2 or 3, we present data structures that answer range-clustering queries exactly.