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...

We consider the problem of finding a shortest rectilinear Steiner tree for a given set of points in the plane in the presence of rectilinear obstacles that must be avoided. We extend the Steiner ratio to the obstacle-avoiding case and show that it is equal to the Steiner ratio for the obstacle-free case.

Let G be a graph. The Steiner distance of $$W\subseteq V(G)$$ is the minimum size connected subgraph containing W. Such necessarily tree called W-tree. set $$A\subseteq k-Steiner general position if $$V(T_B)\cap A = B$$ holds for every $$B\subseteq A$$ cardinality k, and B-tree $$T_B$$ . number $$\mathrm{sgp}_k(G)$$ largest in G. cliques are introduced used to bound from below. determined trees...

Consider an undirected graph G=(V,E) model for a communication network, where each edge is owned by selfish agent, who reports the cost offering use of her edge. Note that agent may misreport own benefit. In such non-cooperative setting, we aim at designing approximately truthful mechanism establishing Steiner tree, minimum tree spanning over all terminals. We present truthful-in-expectation ac...

In this paper, a parallel algorithm for the Steiner tree problem is presented. The algorithm is based on the well-known multi-start paradigm the GRASP and the well-known proximity structures from computational geometry. The main contribution of this paper is the O(n log n+log n log( n log n )) parallel algorithm for computing Steiner tree on the Euclidean plane. The parallel algorithm used prox...

The Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. In this paper, we define a special class of minimum Gilbert networks, called pseudo-Gilbert–Steiner trees, and we show that it can be constructed by Gilbert’s generalization of Melzak’s method. Besides, a counterexample, a pseudo-Gilbert–Steiner tree, is con...

We first show that we can assume without loss of generality that G is a complete graph. In particular, let G denote the metric completion of G: the cost of an edge (u, v) in G for all u, v ∈ V is given by the length of the shortest path between u and v in G. As an exercise verify that costs in G form a metric. Lemma 2.1.1 Let T be a tree spanning R in G, then we can find a subtree in G, T ′ tha...

This paper proposes a new problem, which we call the Dynamic Steiner Tree Problem. This is related to multipoint connection routing in communications networks, where the set of nodes to be connected changes over time. This problem can be divided into two cases, one in which rearrangement of existing routes is not allowed and a second in which rearrangement is allowed. In the first case, we show...

In optimizing the area of Very Large Scale Integrated (VLSI) layouts, circuit interconnections should generally be realized with minimum total interconnect. This chapter addresses several variations of the corresponding fundamental Steiner minimal tree (SMT) problem, where a given set of pins is to be connected using minimum total wirelength. Steiner trees are important in global routing and wi...