### Abstract

Refinement of the unit simplex by iterative longest edge bisection (LEB) up to sub-simplices have a size smaller or equal to a given accuracy, generates a binary tree. For a dimension higher than three, the size of the generated tree depends on the bisected LE. There may exist more than one selection sequence of LE that solves the Smallest Binary Tree Size Problem (SBTSP). Solving SBTSP by full enumeration requires considering every possible LE bisection in each sub-simplex. This is an irregular Combinatorial Optimization problem with an increasing computational burden in the dimension and the stopping criterion. Therefore, parallel computing is appealing to find the minimum size for hard instances in a reasonable time.

The aim of this study is to develop and compare threaded algorithms running on multicore systems to solve the SBTS problem. Versions running on multicore systems with a static number of threads using TBB, and a dynamic number of threads using Pthread are compared. Interestingly, TBB scales better than the Pthread implementations for lower dimensional problems. However, when the problem dimension is higher than six, the Pthread approach with a dynamic number of threads finds a solution, where the TBB version fails. This is caused by the smaller memory footprint of the Pthread version, as it traverses deeper branches of the tree than the TBB work-stealing approach.

The aim of this study is to develop and compare threaded algorithms running on multicore systems to solve the SBTS problem. Versions running on multicore systems with a static number of threads using TBB, and a dynamic number of threads using Pthread are compared. Interestingly, TBB scales better than the Pthread implementations for lower dimensional problems. However, when the problem dimension is higher than six, the Pthread approach with a dynamic number of threads finds a solution, where the TBB version fails. This is caused by the smaller memory footprint of the Pthread version, as it traverses deeper branches of the tree than the TBB work-stealing approach.

Original language | English |
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Pages (from-to) | 166-178 |

Journal | Journal of Parallel and Distributed Computing |

Volume | 112 |

DOIs | |

Publication status | Published - 1 Feb 2018 |

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### Keywords

- Regular simplex
- Longest edge bisection
- Binary tree
- TBB
- Pthreads
- Dynamic number of threads
- Shared memory

### Cite this

Aparicio, G., Salmerón, J. M. G., Casado, L. G., Asenjo, R., & Hendrix, E. M. T. (2018). Parallel algorithms for computing the smallest binary tree size in unit simplex refinement.

*Journal of Parallel and Distributed Computing*,*112*, 166-178. https://doi.org/10.1016/j.jpdc.2017.05.016