Length of a Full Steiner Tree as a Function of Terminal Coordinates

Abstract

Given the coordinates of the terminals $\{(x_j,y_j){\}}_{j=1}^n$ of the full Euclidean Steiner tree, its length equals $$\left|\sum _{j=1}^n{z}_j{U}_j\right|,$$ where $\{z_j:=x_j+\mathbf{i}{y}_j{\}}_{j=1}^n$ and $\{U_j{\}}_{j=1}^n$ are suitably chosen $6$th roots of unity. We also extend this result for the cost of the optimal Weber networks which are topologically equivalent to some full Steiner trees.

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References in corpus (1)

• Some Analytics for Steiner Minimal Tree Problem for Four Terminals