Abstract
We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:
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(a)
Virtually no additional storage is required beyond the input data.
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(b)
The output list produced is free of duplicates.
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(c)
The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.
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(d)
The running time is output sensitive for nondegenerate inputs.
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(e)
The algorithm is easy to parallelize efficiently.
For example, the algorithm finds thev vertices of a polyhedron inR d defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inR d, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inR d can be found inO(n 2 dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
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The work of David Avis was performed while visiting the laboratory of Professor Masakazu Kojima of Tokyo Institute of Technology, supported by the JSPS/NSERC bilateral exchange programs.
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Avis, D., Fukuda, K. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput Geom 8, 295–313 (1992). https://doi.org/10.1007/BF02293050
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DOI: https://doi.org/10.1007/BF02293050