A Sausage Heuristic for Steiner Minimal Trees in Three-Dimensional Euclidean Space

Given a set V of size N≥4 vertices in a metric space, how can one interconnect them with the possible use of a set S of size M vertices not in the set V, but in the same metric space, so that the cumulative cost of the inter-connections between all the vertices is a minimum? When one uses the Euclidean metric to compute these inter-connections, this is referred to as the Euclidean Steiner Minimal Tree Problem. This is an NP-hard problem. The Steiner Ratio ρ of a vertex set is the length of this Steiner Minimal Tree (SMT), divided by the length of the Minimum Spanning Tree (MST), and is a popular and tractable measure of solution quality.The ?-Sausage heuristic described in this paper employs a decomposition technique to explore the point set. The fixed vertices of the set are connected to a set of centroid vertices of Delaunay tetrahedrons. The path topology is preserved as far as possible, together with a cycle prevention rule, where junctions, and deviations from the ?-Sausage structure occur. Furthermore, repeated sweeps, with different root vertices are accommodated.The computational complexity of the heuristic is shown to be O(N²). Experimental results with thousands of vertices are presented. Comparisons with an exponential running time Branch and Bound algorithm are also shown.