Separating a Voronoi Diagram via Local Search

Given a set P of n points in R, we show how to insert a set Z of O ( n1−1/d ) additional points, such that P can be broken into two sets P1 and P2, of roughly equal size, such that in the Voronoi diagram V(P ∪ Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1, P2) of P , we present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, I.1.2 Algorithms, I.3.5 Computational Geometry and Object Modeling