Abstract
We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n⌞d/2⌟) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n⌜d/2⌝) time.
Article PDF
Similar content being viewed by others
References
Agarwal, P. K. Partitioning arrangements of lines, I: An efficient deterministic algorithm,Discrete Comput. Geom. 5 (1990), 449–483.
Brown, K. Q. Voronoi diagrams from convex hulls,Inform. Process. Lett. 9 (1979), 223–228.
Chazelle, B. Cutting hyperplanes for divide-and-conquer,Discrete Comput. Geom. 9 (1993), 145–158.
Chazelle, B., Friedman, J. A deterministic view of random sampling and its use in geometry,Combinatorica 10 (1990), 229–249.
Chazelle, B., Matoušek, J. Derandomizing an output-sensitive convex hull algorithm in three dimensions (submitted for publication).
Clarkson, K. L. A randomized algorithm for closest-point queries,SIAM J. Comput. 17 (1988), 830–847.
Clarkson, K. L. Randomized geometric algorithms, inEuclidean Geometry and Computers, D. Z. Du and F. K. Hwang, eds., World Scientific, to appear.
Clarkson, K. L., Shor, P. W. Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.
Edelsbrunner, H.Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
Edelsbrunner, H., Seidel, R. Voronoi diagrams and arrangements,Discrete Comput. Geom. 1 (1986), 25–44.
Graham, R. L. An efficient algorithm for determining the convex hull of a planar point set,Inform. Process. Lett. 1 (1972), 132–133.
Kirkpatrick, D. G., Seidel R. The ultimate planar convex hull algorithm?SIAM J. Comput. 15 (1986), 287–299.
Matoušek, J. Approximations and optimal geometric divide-and-conquer,Proc. 23rd Annual ACM Symp. on Theory of Computing, 1991, pp. 505–511.
Matoušek, J. Efficient partition trees,Proc. 7th Annual ACM Symp. on Computational Geometry, 1991, pp. 1–9.
Matoušek, J. Cutting hyperplane arrangements,Discrete Comput. Geom. 6 (1991), 385–406.
Matoušek, J. Linear optimization queries,J. Algorithms, to appear.
Preparata, F. P., Hong, S. J. Convex hulls of finite sets of points in two and three dimensions,Comm. ACM 20 (1977), 87–93.
Raghavan, P. Probabilistic construction of deterministic algorithms: approximating packing integer programs,J. Comput. System Sci. 37 (1988), 130–143.
Seidel, R. A convex hull algorithm optimal for point sets in even dimensions, Technical Report 81-14, University of British Columbia, 1981.
Seidel, R. Constructing higher-dimensional convex hulls at logarithmic cost per face,Proc. 18th Annual ACM Symp. on Theory of Computing, 1986, pp. 404–413.
Seidel, R. Small-dimensional linear programming and convex hulls made easy,Discrete Comput. Geom. 6 (1991), 423–434.
Spencer, J.Ten Lectures on the Probabilistic Method, CBMS-NSF, SIAM, Philadelphia, PA, 1987.
Vapnik, V. N., Chervonenkis, A. Ya. On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl. 16 (1971), 264–280.
Author information
Authors and Affiliations
Additional information
This research was supported in part by the National Science Foundation under Grant CCR-9002352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. A preliminary version of this paper has appeared in “An optimal convex hull algorithm and new results on cuttings”,Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science, October 1991, pp. 29–38. The convex hull algorithm given in the present paper, although similar in spirit, is considerably simpler than the one given in the proceedings.
Rights and permissions
About this article
Cite this article
Chazelle, B. An optimal convex hull algorithm in any fixed dimension. Discrete Comput Geom 10, 377–409 (1993). https://doi.org/10.1007/BF02573985
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02573985