Range-Clustering Queries

In a geometric k-clustering problem the goal is to partition a set of points in R 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 Lp-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, and in R for k = 2 or 3, we present data structures that answer range-clustering queries exactly.