In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to ζ(3) = 1/13 + 1/23 + 1/33 + · · · as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log2 log n+ω(1), where ω(1) is any function going to ∞ with n, then th...

Motivated by the reconstruction of phylogenetic tree in biology, we study the full Steiner tree problem in this paper. Given a complete graph G = (V; E) with a length function on E and a proper subset R ⊂ V , the problem is to 4nd a full Steiner tree of minimum length in G, which is a kind of Steiner tree with all the vertices of R as its leaves. In this paper, we show that this problem is NP-c...

We present a polynomial time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n log n) can be constructed that contains a (1 + ǫ)-approximation of the optimal tree. Then one can find an approximately optim...

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−2 for any 2 ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where ...

Steiner tree problem on the graph is an NPComplete problem and has no exact solution in polynomial time. Since this problem is practically useful, there are more attentions to heuristic and approximation approaches rather than exact ones. By using heuristic algorithms, the near optimum answers are obtained in polynomial time that this is faster than exact approaches. The goal of Steiner Tree pr...

In the design of wireless networks, techniques for improving energy efficiency and extending network lifetime have great importance, particularly for defense and civil/rescue applications where resupplying transmitters with new batteries is not feasible. In this paper we study a method for improving the lifetime of wireless networks by minimizing the length of the longest edge in the interconne...

An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented. The algorithm is based on a decomposition approach which first partitions the vertex set into triangles via the Delaunay triangulation, then "recomposes" the suboptimal Steiner Minimal Tree (SMT) according to the Voronoi diagram and Minimum Spanning Tree (MST) of the point set. The ESMT algorithm was imp...

Steiner’s Problem is the ”Problem of shortest connectivity”, that means, given a finite set of points in a Banach space X, search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we ...