Region Counting Distances and Region Counting Circles

The region counting distances, introduced by Demaine, Iacono and Langerman [5], associate to any pair of points p, q the number of items of a dataset S contained in a region R(p, q) surrounding p, q. We define region counting disks and circles, and study the complexity of these objects. In particular, we prove that for some wide class of regions R(p, q), the complexity of a region counting circle of radius k is either at least as large as the complexity of the k-level in an arrangement of lines, or is linear in |S|. We give a complete characterization of regions falling into one of these two cases. Algorithms to compute -approximations of region counting distances and approximations of region counting circles are presented.