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

TL;DR

This paper presents a mathematical formula linking the length of a full Euclidean Steiner tree to terminal coordinates using roots of unity, and extends the result to Weber networks with similar topology.

Contribution

It introduces a novel formula for Steiner tree length based on terminal coordinates and roots of unity, extending to Weber networks with equivalent topology.

Findings

Length of a Steiner tree expressed as a sum involving roots of unity.

Extension of the formula to Weber networks with similar topology.

Provides a new mathematical characterization of Steiner tree lengths.

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.