A semidynamic construction of higher-order voronoi diagrams and its randomized analysis
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
Thek-Delaunay tree extends the Delaunay tree introduced in , and . 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+k 3n) 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⌉+1 n ⌊(d+1)/2⌋) and space complexityO(k ⌈(d+1)/2⌉ n ⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.
Supplementary Material (0)
- J. D. Boissonnat and M. Teillaud. A hierarchical representation of objects: the Delaunay Tree. InProceedings of the Second ACM Symposium on Computational Geometry, Yorktown Heights, pp. 260–268, June 1986.
- J. D. Boissonnat and M. Teillaud. On the randomized construction of the Delaunay tree.Theoretical Computer Science. To be published. Full paper available as Technical Report INRIA 1140.
- M. I. Shamos and D. Hoey. Closest-point problems. InProceedings of the 16th IEEE Symposium on Foundations of Computer Science, pp. 151–162, October 1975.
- D. T. Lee. Onk-nearest neighbor Voronoi diagrams in the plane.IEEE Transactions on Computers, 31:478–487, 1982.
- A. Aggarwal, L. J. Guibas, J. Saxe, and P. W. Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon.Discrete and Computational Geometry, 4:591–604, 1989.
- B. Chazelle and H. Edelsbrunner. An improved algorithm for constructingkth-order Voronoi diagrams. InProceedings of the First ACM Symposium on Computational Geometry, Baltimore, pp. 228–234, June 1985.
- K. L. Clarkson. New applications of random sampling in computational geometry.Discrete and Computational Geometry, 2:195–222, 1987.
- L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams.Algorithmica, 7:381–413, 1992.
- K. Mehlhorn, S. Meiser, and C. Ó'Dunlaing. On the construction of abstract Voronoi diagrams.Discrete and Computational Geometry, 6:211–224, 1991.
- K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II.Discrete and Computational Geometry, 4:387–421, 1989.
- K. Mulmuley. On obstruction in relation to a fixed viewpoint. InProceedings of the 30th IEEE Symposium on Foundations of Computer Science, pp. 592–597, 1989.
- R. A. Dwyer. Higher-dimensional Voronoi diagrams in linear expected time.Discrete and Computational Geometry, 6:343–367, 1991.
- H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications.SIAM Journal on Computing, 15:341–363, 1986.
- K. Mulmuley. On levels in arrangements and Voronoi diagrams.Discrete and Computational Geometry, 6:307–338, 1991.
- F. P. Preparata and M. I. Shamos.Computational Geometry: An Introduction. Springer-Verlag, New York, 1985.
- P. J. Green and R. Sibson. Computing Dirichlet tesselations in the plane.The Computer Journal, 21:168–173, 1978.
- J. D. Boissonnat, O. Devillers, R. Schott, M. Teillaud, and M. Yvinec. Applications of Random Sampling to On-line Algorithms in Computational Geometry.Discrete and Computational Geometry, 8:51–71, 1992.
About this Article
- A semidynamic construction of higher-order voronoi diagrams and its randomized analysis
Volume 9, Issue 4 , pp 329-356
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Computational geometry
- Dynamic algorithm
- Randomized complexity analysis
- Orderk Voronoi diagram
- Industry Sectors