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 the subset of S inside 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 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^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.
|Title of host publication||33rd International Symposium on Computational Geometry (SoCG 2017), 14-17 July 2017, Brisbane, Australia|
|Place of Publication||Dagstuhl|
|Publisher||Schloss Dagstuhl - Leibniz-Zentrum für Informatik|
|Number of pages||16|
|Publication status||Published - 2017|
|Name||Leibniz International Proceedings in Informatics (LIPIcs)|
- Geometric data structures, clustering, k-center problem
Abrahamsen, M., de Berg, M. T., Buchin, K. A., Mehr, M., & Mehrabi, A. D. (2017). Range-clustering queries. In 33rd International Symposium on Computational Geometry (SoCG 2017), 14-17 July 2017, Brisbane, Australia (pp. 1-16).  (Leibniz International Proceedings in Informatics (LIPIcs)). Schloss Dagstuhl - Leibniz-Zentrum für Informatik.