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

02/05/2021

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by   Alexei Yu. Uteshev, et al.

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Given the coordinates of the terminals {(x_j,y_j)}_j=1^n of the full Euclidean Steiner tree, its length equals | ∑_j=1^n z_j U_j | , where {z_j:=x_j+ 𝐢 y_j}_j=1^n and {U_j}_j=1^n are suitably chosen 6th 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.