Geometric conditions for Euclidean Steiner trees inℜd

Geometric conditions for Euclidean Steiner trees in ℜd

We present geometric conditions that can be used to restrict or eliminate candidate topologes for Euclidean Steiner minimal trees in , d ≥ 2. Our emphasis is on conditions that are not restricted to the planar case (d = 2). For trees with a Steiner topology we give restrictions on terminal-Steiner connections that are based on the Voronoi diagram associated with the set of terminal nodes. We th...

Approximate Euclidean Steiner Trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120◦. This notion arises in numerical approximations of minimum Steiner trees. We investigate theworst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It ha...

Bottleneck Distances and Steiner Trees in the Euclidean d-Space

Some of the most efficient heuristics for the Euclidean Steiner minimal trees in the d-dimensional space, d ≥ 2, use Delaunay tessellations and minimum spanning trees to determine small subsets of geometrically close terminals. Their low-cost Steiner trees are determined and concatenated in a greedy fashion to obtain low cost trees spanning all terminals. The weakness of this approach is that o...

Computing Steiner minimal trees in Euclidean d-space

In this paper, we propose modifications on Smith’s branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in R. At each node of the B&B tree, we solve a convex programming problem in conic form to obtain a lower bound on the minimal Steiner tree length for a given topology. We also use conic formulation to obtain bounds on the child problems at a given node, that are applied on a st...

Steiner Trees and Steiner Forests