一致分布点集Delaunay三角化最佳期望时间算法

Optimal Expected-Time Algorithm for Delaunay Triangulation of Uniform Sites

对文献(Dwyer R A.Higher-dimensional Voronoi diagrams in linear expected time.Discrete&ComputationalGeometry,1991,6(4):342-367)给出的对d≥2维空间站点集合构造Delaunay超三角形算法做了改进,提高了其计算效率,并把站点的分布从限于单位球体扩展成d≥2维空间中任意凸的超多面体.证明了如果站点是独立地从一致分布在凸的超多面体的点集中取出,在线性期望时间内可对站点集实现Delaunay三角化.该证明方法比较直观.虽然这类算法对输入点集有一致分布的要求,但在很多实际应用情况下这种要求常是被满足的,此时使用这类算法便可体现文中算法快速和易于实现的优点.

This paper presents an algorithm that improves the results in reference（Dwyer R A.Higher-dimensional Voronoi diagrams in linear expected time.Discrete Computational Geometry,1991,6（4）： 342-367）.The improvements are improving the efficiency of the algorithm,extending the region of the sites distributed from a unit sphere to any convex polyhedron.The main results are as following.If the sites are chosen independently from a uniform distribution on a convex d≥2 dimension polyhedron,the modified algorithm can create its Delaunay triangulation in linear time.Although this kind of algorithm requires that sites are drawn independently from a uniform distribution,this requirement is often satisfied by many applications,in which the advantage of simplicity and efficiency of the algorithm can be expressed very well.