On Computational Aspects of a Regular n-Simplex Bisection

Aparicio, Guillermo; Casado, Leocadio G.; Hendrix, E.M.T.; Garcı́a, Inmaculada; G.-Tóth, Boglárka

doi:10.1109/3pgcic.2013.88

Cited by 5 publications

(9 citation statements)

References 15 publications

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“…Aparicio et al (2013) studies the number and classes of simplices for n 3 when a specific longest edge selection method, denoted by LEB α , based on the angles between the edges is used as SelectLE() in Algorithm 1. The question is whether it can be done better, i.e.…”

Section: Simplex Refinement Using Longest Edge Bisectionmentioning

confidence: 99%

“…Aparicio et al (2013) investigated the effect of the LEB rule on the number of generated simplices, their roundness and whether sub-simplices have similar shapes. A new selection method of the longest edge to be bisected, denoted by LEB α , was presented.…”

Section: Introductionmentioning

confidence: 99%

“…Initially, MinNC can be set on a value already found in earlier computations, or infinity otherwise. For instance, an upper bound of eight is already known for a 3-simplex (Aparicio et al, 2013).…”

mentioning

confidence: 99%

“…There should be no local maximum other than the global maximum for a regular tetrahedron and there should be no local minimum other than the global minimum for a degenerate tetrahedron. Aparicio et al (2013) elaborates this idea by defining a simplex quality index, SQ as the ratio between the arithmetic and geometric mean of the set of squared angles between edges of a simplex. Let α 1 , .…”

mentioning

confidence: 99%

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On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

Aparicio¹,

Casado²,

Hendrix³

et al. 2015

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Abstract. In several areas like Global Optimization using branch-and-bound methods, the unit n-simplex is refined by bisecting the longest edge such that a binary search tree appears. This process generates simplices belonging to different shape classes. Having less simplex shapes facilitates the prediction of the further workload from a node in the binary tree, because the same shape leads to the same sub-tree. Irregular sub-simplices generated in the refinement process may have more than one longest edge when n 3. The question is how to choose the longest edge to be bisected such that the number of shape classes is as small as possible. We develop a Branch-and-Bound (B&B) algorithm to find the minimum number of classes in the refinement process. The developed B&B algorithm provides a minimum number of eight classes for a regular 3-simplex. Due to the high computational cost of solving this combinatorial problem, future research focuses on using high performance computing to derive the minimum number of shapes in higher dimensions.

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Section: Simplex Refinement Using Longest Edge Bisectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the Minimum Number of Simplex Shapes in Longest Edge Bisection Refinement of a Regular <italic>n</italic>-Simplex

Aparicio¹,

Casado²,

Hendrix³

et al. 2015

Informatica

Self Cite

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“…A set of LE indices generating the minimum number of classes of simplices (Aparicio et al 2013), was presented in Aparicio et al (2015). Another research question in this study is whether that set also provides one of the smallest trees, given a value of the stopping criterion on the size of the subsimplices.…”

Section: Introductionmentioning

confidence: 99%

Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement

et al. 2015

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In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For n > 2, simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary tree depends on the specific sequence of bisected longest edges. The questions are how to calculate the size of one of the smallest binary trees generated by LEB and how to find the corresponding sequence of LEs to bisect, which can be represented by a set of LE indices. Algorithms answering these questions are presented here. We focus on sets of LE indices that are repeated at a level of the binary tree. A set of

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