Budgeted Steiner Networks: Three Terminals with Equal Path Weights

Given a set of terminals in 2D/3D, the network with the shortest total length that connects all terminals is a Steiner tree. At the other extreme, with enough total length budget, every terminal can be connected to every other terminal via a straight line, yielding a complete graph over all terminals that connects every pair of terminals with a shortest path. In this work, we study a generalization of Steiner trees, asking what happens between these two extremes. For a given total length budget, we seek a network structure that minimizes the sum of the weighted distances between pairs of terminals. Focusing on three terminals with equal pairwise path weights, we characterize the full evolutionary pathway between the Steiner tree and the complete graph, which contains interesting intermediate structures.