Upper Bounds on the Size of Inner Voronoi Diagrams of Multiply Connected Polygons

Abstract

The Voronoi diagram (VD) of a planar polygon has many applications, from path planning in robotics to collision detection in virtual reality. To study the complexity of algorithms based on Voronoi diagram, it is important to estimate the numbers of vertices and edges of a VD. Held proves that the inner Voronoi diagram of a simple polygon has at most n+k−2 vertices and 2(n+k)−3 edges, where n is the number of the polygon's vertices and k is the number of reflex vertices. But this conclusion holds not for a multiply-connected polygon, i.e. a polygon with holes. In this paper, by constructing a rooted tree from a VD, and based on some properties of the rooted tree, new upper bounds on the numbers of vertices and edges in an inner Voronoi diagram of a multiply-connected polygon are proved. The average numbers of Voronoi vertices and edges on the boundary of a VD are also presented. The result of this paper has been used to analyze the complexity of VD-based visibility computing algorithm in SDU Virtual Museum.