An Iterative Algorithm for Computing Measures of Generalized Voronoi Regions

Abstract

We present and analyze a fast algorithm for directly computing measures of generalized Voronoi regions associated with generators of arbitrary codimension. The algorithm consists of solving one eikonal equation to construct a kernel-based operator whose iteration accumulates mass along the closest generator. In particular, the algorithm does not require the computation of the Voronoi diagram or the gradient of the solution to the eikonal equation. The algorithm is shown to be first order and converge very quickly. By discretizing the distance to the generators on the grid (and not the generators themselves), very accurate geometric information is used even for coarse grids. Several examples are presented, including the computation of the population influence associated with the Los Angeles County highway system. The method is also one of the key ingredients for the fast computation of centroidal Voronoi tessellations (CVTs) of general rigid objects (e.g., rigid curves and surfaces) in higher dimensions. We present a few simple examples of these generalized CVTs.