The length of minimal filling for a five-point metric space

Dual Linear Programming Problem and One-Dimensional Gromov Minimal Fillings of Finite Metric Spaces

“….. But the formulas obtained by the authors [1] and B. Bednov [10] together with the computational results of the present paper, see below, show that the sequence {k n } of maximal multiplicities of the multi-tours corresponding to the vertices of the polyhedrons Λ G for the binary trees G with n ≥ 3 boundary vertices starts as 1, 1, 1, 2, 2, . .…”

“…The weight of minimal parametric filling of this type equals to the maximum of those four expressions (the half-perimeters of the corresponding tours depicted in Figure 1). This formula can be easily obtained from results of B. Bednov [10], and its particular cases were obtained by students of Mechanical and Mathematical Faculty of Lomonosov Moscow Sate University E. Zaval'nuk, Z. Ovsyannikov, O. Rubleva in their term theses.…”

“…A filling of the type G having the least possible weight is called a minimal parametric filling of the type G. Passing to the infimum over all possible types G leads to the concept of minimal filling. Minimal Fillings Theory of finite metric spaces is actively evolved, see for example [5], [6], [11], and [10]. It turns out to be related not only with geometry of finite metric spaces, but with structure of optimal networks also.…”

“…In the present paper this method is used to obtain an alternative proof of the non-trivial formula of the weight of a parametric minimal filling obtained by A. Eremin in [6] in terms of multi-tours by a rater tricky technique, see below, and essentially improve the estimate on the multiplicity of the multitours standing in the formula. We also derived explicit formulas for the weight of a minimal parametric fillings of any fixed type for five-point and six-point metric spaces (the case of five-point metric spaces was considered in [10], and the case of six-point spaces is a new one) and, in fact, we give an algorithm permitting to obtain such formulas for any finite metric space and any fixed type of a filling.…”