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INFORMATICA, , Vol. 26, No. 1, 17-32
© Institute of Mathematics and Informatics,
DOI: http://dx.doi.org/10.15388/Informatica.2015.36

ISSN 0868-4952

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

Guillermo APARICIO, Leocadio G. CASADO1, Eligius M.T. HENDRIX, Boglárka G.-TÓTH, Inmaculada GARCIA

Research Group TIC146: High Performance Computing - Algorithms University of Almería, Spain Informatics Department, University of Almería (ceiA3), Spain Department of Computer Architecture, Universidad de Málaga, Spain Department of Differential Equations, Budapest University of Technology and Economics Hungary E-mail: guillermoaparicio@ual.es, leo@ual.es, eligius@uma.es, igarciaf@uma.es, bog@math.bme.hu

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\geqslant 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.

Keywords:

regular simplex, longest edge bisection, branch-and-bound, combinatorial optimization, simplex shape


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