Approximating hexagonal Steiner minimal trees by fast optimal layout of minimum spanning trees

We study algorithms for approximating a Steiner minimal tree interconnecting n points under hexagonal routing. We prove that: (1) every minimum spanning tree is separable; (2) a minimum spanning tree with maximum node degree no more than 5 can be computed in O (n log n) time; (3) an optimal L-shaped layout of a given minimum spanning tree can be computed in O(n) time; (4) an optimal stair-shaped layout of a given minimum spanning tree can be computed in O(n²) time. Computational results on standard benchmarks show that our algorithm compares favorably to the current best algorithms