Simplexity of the cube

Hughes, Robert B.; Anderson, Michael R.

doi:10.1016/0012-365x(95)00075-8

Cited by 38 publications

(33 citation statements)

References 11 publications

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“…It should be possible to improve our bounds for covers by using more information about the kinds of simplices that can occur in the cube and how they fit together, similar to what Hughes [5] and Hughes and Anderson [6] did to study (D v , T v )-minimal triangulations. These programs rely on enumerating specific features of configuration classes of simplices in the cube, or on enumerating the classes themselves.…”

Section: Resultsmentioning

confidence: 99%

“…Hughes [5], Hughes and Anderson * [6], Smith [12] Our bounds Cottle * * [2] and Sallee * * [10] Dimension …”

Section: Triangulations Of Cubesmentioning

confidence: 99%

“…Instead, we develop a linear program whose optimal solution is a bound for the size of the minimal cover. The linear programming approach was initiated by Sallee [9], Hughes [5], and Hughes and Anderson [6] to study triangulations of the cube. However, the results of Hughes and Hughes and Anderson are not fully comparable with ours, because their methods do not apply to covers.…”

Section: Triangulations Of Cubesmentioning

confidence: 99%

“…We use these results to establish new lower bounds for C (I d ), and via Theorem 1, these yield the best known bounds for general triangulations in dimensions up through at least 12 (and possibly more dimensions-see the concluding section). For d = 4 through d = 12 these numbers are 16, 60, 252, 1143, 5104, 22616, 98183, 520865, 2.9276 × 10 6 .…”

Section: Introductionmentioning

confidence: 99%

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Lower Bounds for Simplicial Covers and Triangulations of Cubes

Bliss

2004

Discrete Comput Geom

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Abstract. We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces.

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Section: Resultsmentioning

confidence: 99%

“…Hughes [5], Hughes and Anderson * [6], Smith [12] Our bounds Cottle * * [2] and Sallee * * [10] Dimension …”

Section: Triangulations Of Cubesmentioning

confidence: 99%

Section: Triangulations Of Cubesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lower Bounds for Simplicial Covers and Triangulations of Cubes

Bliss

2004

Discrete Comput Geom

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“…They use an outer loop over epochs that maintains the hyperbox, and an inner loop over rounds, each of which performs the core SSB step. Note that although partitioning a hyperbox into a set of simplexes is trivial in two dimensions, it is a challenging and time-consuming task in higher ones, see [9] and [10].…”

Section: B Outer Loop: Maintaining a Hyperboxmentioning

confidence: 99%

The Revised Stochastic Simplex Bisection Algorithm and Particle Swarm Optimization

Samuelsson¹

2017

Annals of Computer Science and Information Systems

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Abstract-The stochastic simplex bisection (SSB) algorithm is evaluated against particle swarm optimization (PSO) on a prominent test set. The original SSB algorithm performs on par with the PSO algorithm and a revised version of the SSB algorithm outperforms both of them. Detailed analysis of the performance on select objective functions brings to light key properties of the three algorithms. The core SSB algorithm is here viewed as a sampling tool for an outer loop that employs statistical pattern recognition. This opens the door for a host of other schemes.

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Bounds on Delaunay tessellations

Shores

Wegman

2010

WIREs Computational Stats

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Motivated by applications in density estimation and data compression, this article considers the bounds on the number of tiles in a Delaunay tessellation as a function of both the number of tessellating points and the dimension. Results can also be interpreted for the dual of the Delaunay tessellation, the Voronoi diagram. Several theoretical lower and upper bounds are found in the combinatorics and computational geometry literatures and are brought together in this article. We make a comparison of these bounds with several empirically derived curves based on multivariate uniform and Gaussian generated random tessellating points. The upper bounds are found to be very conservative when compared with the empirically derived number of tiles, often off by many orders of magnitude.

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Simplexity of the cube

Cited by 38 publications

References 11 publications

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Lower Bounds for Simplicial Covers and Triangulations of Cubes

The Revised Stochastic Simplex Bisection Algorithm and Particle Swarm Optimization

Bounds on Delaunay tessellations

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