Balanced Islands in Two Colored Point Sets in the Plane

Let P be a set of n points in general position in the plane, r of which are red and b of which are blue. In this paper we prove that there exist: for every α ∈ [ 0, 1 2 ] , a convex set containing exactly dαre red points and exactly dαbe blue points of P ; a convex set containing exactly ⌈ r+1 2 ⌉ red points and exactly ⌈ b+1 2 ⌉ blue points of P . Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an O(n) time algorithm and an O(n logn) time algorithm in the second case. Finally, if dαre + dαbe is small, that is, not much larger than 1 3 n, we improve the running time to O(n logn). ∗Oswin Aichholzer and Birgit Vogtenhuber are supported by the ESF EUROCORES programme EuroGIGA–ComPoSe, Austrian Science Fund (FWF): I 648-N18. Pablo PerezLantero is partially supported by projects CONICYT FONDECYT/Iniciación 11110069 (Chile), and Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 (Chile). †Institute for Software Technology, Graz University of Technology, Graz, Austria.[oaich|bvogt]@ist.tugraz.at ‡Universidad de Sevilla, Spain. [email protected] §Departamento de Matemática Aplicada II, Universidad de Sevilla, Spain. [email protected] ¶Departamento de Matemáticas, Cinvestav, D.F., México. [email protected] ‖Departamento de Matemáticas, Facultad de Ciencias, UNAM, Mexico. [email protected] ∗∗Escuela de Ingenieŕıa Civil en Informática, Universidad de Valparáıso, Chile. [email protected]. ††Instituto de Matemáticas, UNAM, México. [email protected] 1 ar X iv :1 51 0. 01 81 9v 1 [ cs .C G ] 7 O ct 2 01 5