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

DOI：

 作者 单位 杨承磊 山东大学,计算机科学与技术学院,山东,济南,250100 汪嘉业 山东大学,计算机科学与技术学院,山东,济南,250100 孟祥旭 山东大学,计算机科学与技术学院,山东,济南,250100

多边形的Voronoi图在路径规划、碰撞检测等方面有着广泛的应用,其顶点和边数在这些应用算法的复杂度分析方面起着重要作用.Held证明了一个简单多边形的内部Voronoi图最多有n+k-2个顶点和2(n+k)-3条边,其中nk分别是多边形的顶点和内尖点数.但其结论不能适用于多连通多边形.对多连通多边形进行研究,通过将其Voronoi图转化为有根树,并利用有根树的性质,给出了其内部Voronoi图的顶点和边数上界的估计,并对Voronoi区域的边界所包含顶点和边数的平均值进行了讨论."SDU数字博物馆"系统所采用的基于Voronoi图的可见性算法的复杂度分析,就利用了所得出的结论.

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.
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