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.