Minimum Steiner Trees in Normed Planes

Approximating Minimum Steiner Point Trees in Minkowski

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance differe...

Approximating minimum Steiner point trees in Minkowski planes

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance differe...

Approximating Steiner Trees and Forests with Minimum Number of Steiner Points

Let R be a finite set of terminals in a metric space (M,d). We consider finding a minimum size set S ⊆ M of additional points such that the unit-disc graph G[R ∪ S] of R ∪ S satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem G[R ∪ S] should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-M...

The Steiner ratio for the dual normed plane

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimal possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the metric space. Du et al. (1993) conjectured that the Steiner ratio on a normed plan...

The local Steiner problem in normed planes