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.

中文翻译：

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完整的Steiner树的长度与终端坐标的关系

给定完整欧氏Steiner树的末端$ \ {（x_j，y_j）\} _ {j = 1} ^ n $的坐标，其长度等于$$ \ left | \ sum_ {j = 1} ^ n z_j U_j \ right | \，$$其中$ \ {z_j：= x_j + \ mathbf我y_j \} _ {J = 1} ^ N $和$ \ {U_j \} _ {J = 1} ^ N $被适当选择的$ 6 $团结的根基。我们还将此结果扩展为最佳Weber网络的成本，该拓扑在拓扑上等同于某些完整的Steiner树。