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Covering point sets with two disjoint disks or squares |
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| Authors: | Sergio Cabello, J. Miguel Dí az-B ez, Carlos Seara, J. Antoni Sellar s, Jorge Urrutia,Inmaculada Ventura |
| Affiliation: | aDepartment of Mathematics, IMFM, Slovenia bDepartment of Mathematics, FMF, University of Ljubljana, Slovenia cDepartamento de Matemática Aplicada II, Universidad de Sevilla, Spain dDepartament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Spain eInstitut d'Informàtica i Aplicacions, Universitat de Girona, Spain fInstituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico gDepartamento de Matemáticas, Universidad de Huelva, Spain |
| Abstract: | We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n8/3log2n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares SR and SB instead of unit disks can be solved in O(nlogn) 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(n3logn) time. |
| Keywords: | Covering Facility location Geometric optimization Disks Squares |
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