An Algorithm for Convex Polytopes

Chand, Donald R.; Kapur, S. S.

doi:10.1145/321556.321564

Cited by 219 publications

(92 citation statements)

References 2 publications

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“…Constructing a (d + l)-dimensional convex hull is a viable approach to the problem of constructing a d-dimensional Voronoi diagram. The gift-wrapping algorithm [7,3,23] may be used in Q(n(S n + 5*)) time. Or SeidePs shelling algorithm [21] may be used in 0(n 2 + (5 n + 5*) log n), where S n is the number of nearest-site simplices and 5* the number of furthest-site simplices in the result.…”

Section: Introductionmentioning

confidence: 99%

Higher-dimensional voronoi diagrams in linear expected time

Dwyer

1991

Discrete Comput Geom

149

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A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram of n independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of simplices of the Voronoi diagram of points chosen from the uniform distribution on the interior of a d-dimensional ball; it is shown that in this case, the complexity of the diagram is 0(n) for fixed d. An algorithm for constructing the Voronoi diagram is presented and analyzed using the new method. The algorithm is shown to require only ©(n) time on average for random points from a ci-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense.

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Section: Introductionmentioning

confidence: 99%

Higher-dimensional voronoi diagrams in linear expected time

Dwyer

1991

Discrete Comput Geom

149

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“…( e ) The Gift-Wrapping Algorithm of Chand and Kapur [7]. Analogous statements apply for the Gift-Wrapping Algorithm, which is the subject of our study.…”

Section: O(m + E(#v) 2 + E(#~) Log M) (16)mentioning

confidence: 75%

Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Borgwardt

1997

Discrete Comput Geom

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Abstract. This paper presents an algorithm and its probabilistic analysis for constructing the convex hull ofm given points in Rn, the n-dimensional Euclidean space. The algorithm under consideration combines the Gift-Wrapping concept with the so-called Throw-Away Principle (introduced by Aki and Toussaint [1 ] and later by Devroye [10]) for nonextremal points. The latter principle had been used for a convex-hull-construction algorithm in R 2 and for its probabilistic analysis in a recent paper by Borgwardt et al. [5]. There, the considerations remained much simpler, because in R2 the construction of the convex hull essentially requires recognition of the extremal points and of their order only.In this paper the Simplex method is used to organize a walk over the surface of the convex hull. During this walk all facets are discovered. Under the condition of general position this information is sufficient, because the whole face lattice can simply be deduced when the set of facets is available.Exploiting the advantages of the revised Simplex method reduces the update effort to an n x n matrix and the number of calculated quotients for the pivot search to the points which are not thrown away.For this algorithm a probabilistic analysis can be carried out. We assume that our m random points are distributed identically, independently, and symmetrically under rotations in R n. Then the calculation of the expected effort becomes possible for a whole parametrical class of distributions over the unit ball. The results mean a progress in three directions:--a parametrization of the expected effort can be given; --the dependency on n--the dimension of the space---can be evaluated; --the additional work of preprocessing for detecting the vertices can be avoided without losing its advantages. 80K.H. Borgwardt

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“…The software package iB4e [21] implements this algorithm. Another algorithm converting the oracle representation to the convex hull and halfspace representation is "gift-wrapping" [6].…”

Section: Polynomials and Polytopesmentioning

confidence: 99%

Newton Polytopes and Witness Sets

Hauenstein

Sottile

2014

Math.Comput.Sci.

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Abstract. We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in C n using numerical computation. The first algorithm assumes that we may only compute values of f -this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes the Newton polytope of H using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when H is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face.

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An Algorithm for Convex Polytopes

Cited by 219 publications

References 2 publications

Higher-dimensional voronoi diagrams in linear expected time

Higher-dimensional voronoi diagrams in linear expected time

Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Newton Polytopes and Witness Sets

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