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

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 havemore 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 B E. M. T. Hendrix [email protected] J. M. G. Salmerón [email protected] G. Aparicio [email protected] L. G. Casado [email protected] I. García [email protected] B. G.-Tóth [email protected] 1 Department of Computer Science, University of Almería (ceiA3), Almería, Spain 2 Computer Architecture Department, Universidad de Málaga, Málaga, Spain 3 University of Wageningen, Wageningen, The Netherlands 4 Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary