Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C-R and C-B with disjoint interiors such that the number of red points covered by C-R plus the number of blue points covered by C-B is maximized. We give an algorithm to solve this problem in O(n(8/3)log(2) n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S-R and S-B instead of unit disks can be solved in O(n log n) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n(3) log n) time. (c) 2007 Elsevier B.V. All rights reserved.