A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k⌈(d+1)/2⌉+1n⌊(d+1)/2⌋) and space complexityO(k⌈(d+1)/2⌉n⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

[1]  D. T. Lee,et al.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[2]  Jean-Daniel Boissonnat,et al.  The hierarchical representation of objects: the Delaunay tree , 1986, SCG '86.

[3]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[4]  Leonidas J. Guibas,et al.  Randomized incremental construction of Delaunay and Voronoi diagrams , 1990, Algorithmica.

[5]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[6]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[7]  Raimund Seidel,et al.  Constructing Arrangements of Lines and Hyperplanes with Applications , 1986, SIAM J. Comput..

[8]  Mariette Yvinec,et al.  Applications of random sampling to on-line algorithms in computational geometry , 1992, Discret. Comput. Geom..

[9]  Jean-Daniel Boissonnat,et al.  On the Randomized Construction of the Delaunay Tree , 1993, Theor. Comput. Sci..

[10]  Kenneth L. Clarkson,et al.  New applications of random sampling in computational geometry , 1987, Discret. Comput. Geom..

[11]  Bernard Chazelle,et al.  An Improved Algorithm for Constructing k th-Order Voronoi Diagrams , 1987, IEEE Trans. Computers.

[12]  Rex A. Dwyer.  Higher-dimensional voronoi diagrams in linear expected time , 1991, Discret. Comput. Geom..

[13]  Rex A. Dwyer.  Higher-dimensional voronoi diagrams in linear expected time , 1989, SCG '89.

[14]  Ketan Mulmuley,et al.  On levels in arrangements and voronoi diagrams , 1991, Discret. Comput. Geom..

[15]  Leonidas J. Guibas,et al.  A linear-time algorithm for computing the voronoi diagram of a convex polygon , 1989, Discret. Comput. Geom..

[16]  Kurt Mehlhorn,et al.  On the construction of abstract voronoi diagrams , 1990, STACS.

[17]  Der-Tsai Lee.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[18]  Ketan Mulmuley.  On obstructions in relation to a fixed viewpoint , 1989, 30th Annual Symposium on Foundations of Computer Science.

[19]  Bernard Chazelle,et al.  An Improved Algorithm for Constructing kth-Order Voronoi Diagrams , 1985, IEEE Transactions on Computers.

[20]  Robin Sibson,et al.  Computing Dirichlet Tessellations in the Plane , 1978, Comput. J..