Guaranteed Voronoi Diagrams of Uncertain Sites

In this paper we investigate the Voronoi diagram that is induced by a set of sites in the plane, where each site’s precise location is uncertain but is known to be within a particular region, and the cells of this diagram contain those points guaranteed to be closest to a particular site. We then examine the diagram for sites with disc-shaped regions of uncertainty, prove that it has linear complexity, and provide an optimal O(n log n) algorithm for its construction. We also examine the diagram for polygonal regions of uncertainty, and prove that it has linear complexity as well. We then describe a generalization of these diagrams, in which each Voronoi cell is associated with a subset of the sites, and each point in a cell is guaranteed to be closest to some site in the subset associated with the cell.