Improving the Construction of the Visibility–Voronoi Diagram

Ron Wein et al. [4] introduced the Visibility-Voronoi diagram for clearance c, denoted by V V , which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. They also proposed an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV-complex. As [4] used a straightforward approach for constructing VV-diagram, its construction time is O(n log n), which is not optimum. In this paper we improve this time to O(k log n), where k is the number of visibility edges, based on the method used for the preprocessing of the VV-complex in [4].