A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
- 1.5k Downloads
Virtually no additional storage is required beyond the input data.
The output list produced is free of duplicates.
The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.
The running time is output sensitive for nondegenerate inputs.
The algorithm is easy to parallelize efficiently.
For example, the algorithm finds thev vertices of a polyhedron inRd defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inRd, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inRd can be found inO(n2dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
KeywordsConvex Hull Span Tree Convex Polyhedron Enumeration Tree Hyperplane Arrangement
- 4.B. Chazelle, An Optimal Convex Hull Algorithm and New Results on Cuttings,Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, pp. 29–38, 1991.Google Scholar
- 10.K. Fukuda, Oriented Matroid Programming, Ph.D. Thesis, University of Waterloo, 1982.Google Scholar
- 11.K. Fukuda and T. Matsui, On the Finiteness of the Criss-Cross Method,European J. Oper. Res., to appear.Google Scholar
- 12.M. E. Houle, H. Imai, K. Imai, J.-M. Robert, and P. Yamamoto, Orthogonal Weighted LinearL 1 andL ∞ Approximation and Applications, Manuscript, September 1990.Google Scholar
- 14.T. S. Motzkin, H. Raiffa, G. L. Thompson, and R. M. Thrall,The Double Description Method, Annals of Mathematical Studies, vol. 8, Princeton University Press, Princeton, NJ, 1953.Google Scholar
- 16.G. Rote, Degenerate Convex Hulls in High Dimensions Without Extra Storage,Proc. 8th Annual Symposium on Computational Geometry, ACM Press, New York, pp. 26–32, 1992.Google Scholar
- 17.R. Seidel, A Convex Hull Algorithm Optimal for Point Sets in Even Dimensions, Report 81-14, Department of Computer Science, University of British Columbia, 1981.Google Scholar
- 18.R. Seidel, Constructing Higher-Dimensional Convex Hulls at Logarithmic Cost per Face,Proc. 1986 Symposium on the Theory of Computing, pp. 404–413.Google Scholar
- 22.Z. Wang, A Conformal Elimination Free Algorithm for Oriented Matroid Programming,Chinese Ann. Math., Ser. B, vol. 8, p. 1, 1987.Google Scholar
- 23.D. Avis and K. Fukuda, Reverse Search for Enumeration, Research Report 92-5, GSSM, University of Tsukuba, April 1992.Google Scholar