Range-Clustering Queries

In a geometric $k$-clustering problem the goal is to partition a set of points in $\mathbb{R}^d$ 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 $S\setminus Q$. We obtain the following results. We present a general method to compute a $(1+\epsilon)$-approximation to a range-clustering query, where $\epsilon>0$ 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 $\mathbb{R}^1$, and in $\mathbb{R}^2$ for $k=2$ or 3, we present data structures that answer range-clustering queries exactly.