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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 (Formula presented.), 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 LEs was presented in Aparicio et al. (Informatica 26(1):17–32, 2015), for (Formula presented.). An additional question is whether this set is the best one under the so-called (Formula presented.)-valid condition.

@article{salmeron_generating_2017, title = {Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement}, volume = {33}, issn = {1382-6905}, doi = {10.1007/s10878-015-9970-y}, abstract = {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 (Formula presented.), 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 LEs was presented in Aparicio et al. (Informatica 26(1):17–32, 2015), for (Formula presented.). An additional question is whether this set is the best one under the so-called (Formula presented.)-valid condition.}, language = {English}, number = {2}, journal = {Journal of Combinatorial Optimization}, author = {Salmerón, J. M. G. and Aparicio, G. and Casado, L. G. and García, I. and Hendrix, E. M. T. and Tóth, B. G.}, year = {2017}, keywords = {Bisection sequence, Branch-and-bound, Combinatorial optimization, Longest edge bisection, Regular simplex}, pages = {389--402} }

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