Steiner Tree Heuristics in Euclidean d-Space

We present a class of heuristics for the Euclidean Steiner tree problem in a d-dimensional space, d ≥ 3. These heuristics identify small subsets with few, geometrically close, terminals using Delaunay tessellations and minimum spanning trees. Low cost spanning trees of these subsets are determined by applying the exact algorithm for the Euclidean Steiner minimal tree in d-space as well as its heuristic modifications. These low cost spanning trees are sorted according to an appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. Computational experiments indicate that problem instances with 80,000 terminals in R, 41,000 terminals in R, 15,000 terminals in R, and 3,800 terminals in R, can be solved within 1 minute while producing high quality solutions.