Randomized incremental construction of Delaunay and Voronoi diagrams

In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

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

[2]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites. , 1908 .

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

[4]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

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

[6]  Bruce W. Weide,et al.  Optimal Expected-Time Algorithms for Closest Point Problems , 1980, TOMS.

[7]  Steven Fortune.  A note on Delaunay diagonal flips , 1993, Pattern Recognit. Lett..

[8]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .

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

[10]  Jorge Urrutia,et al.  A combinatorial result about points and balls in euclidean space , 1989, Discret. Comput. Geom..

[11]  Raimund Seidel,et al.  Circles through two points that always enclose many points , 1989 .

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

[13]  G. Andrews.  The Theory of Partitions: Frontmatter , 1976 .

[14]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[15]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[16]  Roberto Tamassia,et al.  Fully dynamic techniques for point location and transitive closure in planar structures , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[17]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[18]  Shô Iseki,et al.  A generalization of a functional equation related to the theory of partitions , 1960 .

[19]  Alok Aggarwal,et al.  Fining k points with minimum spanning trees and related problems , 1989, SCG '89.

[20]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[21]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[22]  Ketan Mulmuley,et al.  A fast planar partition algorithm. I , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[23]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[24]  Alok Aggarwal,et al.  Finding k Points with Minimum Diameter and Related Problems , 1991, J. Algorithms.

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