Euclidean Steiner minimal trees with obstacles and Steiner visibility graphs

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...

Steiner Minimal Trees ∗

While a spanning tree spans all vertices of a given graph, a Steiner tree spans a given subset of vertices. In the Steiner minimal tree problem, the vertices are divided into two parts: terminals and nonterminal vertices. The terminals are the given vertices which must be included in the solution. The cost of a Steiner tree is defined as the total edge weight. A Steiner tree may contain some no...

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

Steiner Minimal Trees in Simple Polygons

An O(n log n) time and O(n) space algorithm for the Euclidean Steiner tree problem with four terminals in a simple polygon with n vertices is given. Its applicability to the problem of determining good quality solutions for any number of terminals is discussed.