The projector algorithm: A simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has found numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell (actually, even just portions of one cell) can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.