Heuristics to Reduce the Number of Simplices in Longest Edge Bisection Refinement of a Regular n-Simplex

  • Guillermo Aparicio
  • Leocadio G. Casado
  • Boglárka G-Tóth
  • Eligius M. T. Hendrix
  • Inmaculada García
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8580)


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. The refinement usually selects the first longest edge and ends when the size of the sub-simplices generated in the refinement is smaller than a given accuracy. Irregular sub-simplices may have more than one longest edge only for n ≥ 3. The question is how to choose the longest edge to be bisected such that the number of sub-simplices in the generated binary tree is minimal. The difficulty of this Combinatorial Optimization problem increases with n. Therefore, heuristics are studied that aim to minimize the number of generated simplices.


Regular Simplex Longest Edge Bisection Complete Binary Tree 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, A.: On the Bisection Method for Triangles. Mathematics of Computation 40(162), 571–574 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aparicio, G., Casado, L.G., Hendrix, E.M.T., G.-Tóth, B., García, I.: On the minimum number of simplex classes in longest edge bisection refinement of a regular n-simplex. Informatica (submitted)Google Scholar
  3. 3.
    Aparicio, G., Casado, L.G., Hendrix, E.M.T., García, I., G.-Tóth, B.: On computational aspects of a regular n-simplex bisection. In: Proceedings of the 2013 Eighth International Conference on P2P, Parallel, Grid, Cloud and Internet Computing, Compiegne, France, pp. 513–518 (October 2013)Google Scholar
  4. 4.
    Ashayeri, J., van Eijs, A., Nederstigt, P.: Blending modelling in a process manufacturing: A case study. European Journal of Operational Research 72(3), 460–468 (1994)CrossRefGoogle Scholar
  5. 5.
    Casado, L.G., García, I., G.-Tóth, B., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. Journal of Global Optimization 50(4), 645–655 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hannukainen, A., Korotov, S., Křížek, M.: On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions. Science of Computer Programming 90(A), 34–41 (2014)CrossRefGoogle Scholar
  7. 7.
    Horst, R.: On generalized bisection of n-simplices. Mathematics of Computation 66(218), 691–698 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Mitchell, W.F.: A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15(4), 326–347 (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Guillermo Aparicio
    • 1
  • Leocadio G. Casado
    • 2
  • Boglárka G-Tóth
    • 3
  • Eligius M. T. Hendrix
    • 4
  • Inmaculada García
    • 4
  1. 1.TIC 146: Supercomputing-Algorithms Research GroupUniversidad de Almería, Agrifood Campus of International Excellence (ceiA3)Spain
  2. 2.Department of InformaticsUniversity of Almería, Agrifood Campus of International Excellence (ceiA3)Spain
  3. 3.Department of Differential EquationsBudapest University of Technology and EconomicsBudapestHungary
  4. 4.Department of Computer ArchitectureUniversidad de Málaga, Campus de TeatinosSpain

Personalised recommendations