For a real number p ≥ 2, an integer k > 0 and a set of terminals X in the plane, the Euclidean power-p Steiner tree problem asks for a tree interconnecting X and at most k Steiner points such that the sum of the p-th powers of the edge lengths is minimised. We show that this problem is in the complexity subclass exp-APX (but not poly-APX) of NPO. We then demonstrate that the approximation algor...

The minimum Steiner tree problem, a classical combinatorial optimization problem with a long history, is a NP-complete problem. Due to its wide application, study of heuristic algorithm about Steiner tree problem has important practical and theoretical significance. In this paper we first review one of the existing algorithms for solving the Steiner problem in graphs, Minimum Spanning Tree Heur...

This work presents a Steiner tree construction procedure, Maximum delay violation Elmore routing tree, to meet specified sink arrival time constraints. It is shown that the optimal tree requires the use of non-Hanan points. The procedure works in two phases: a minimum-delay Steiner Elmore routing tree is first constructed using a minor variant of the Steiner Elmore routing tree procedure, after...

Let P be a set of n points in a metric space. A Steiner Minimal Tree (SMT) on P is a shortest network interconnecting P while a Minimum Spanning Tree (MST) is a shortest network interconnecting P with all edges between points of P . The Steiner ratio is the infimum over P of ratio of the length of SMT over that of MST. Steiner ratio problem is to determine the value of the ratio. In this paper ...

In communication networks, many applications, such as video on demand and video conferencing, must establish a communications tree that spans a subset K in a vertex set. The source node can then send identical data to all nodes in set K along this tree. This kind of communication is known as multicast communication. A network optimization problem, called the Steiner tree problem (STP), is prese...

For point sets in the rectilinear plane we consider the following five measures of the interconnect length and prove bounds on the worst-case ratio: minimum Steiner tree, minimum star, clique, minimum spanning tree, and bounding box. In particular, we prove that for any set of n points: (n − 1) times the shortest Steiner tree is less or equal to the clique unless n = 4; and the minimum spanning...

The Steiner problem is an NP-hard optimization problem which consists of finding the minimal-length tree connecting a set of N points in the Euclidean plane. Exact methods of resolution currently available are exponential in N , making exact minimal trees accessible for only small size problems (up to N ≈ 100). An acceptable suboptimal solution is provided by the minimum spanning tree (MST) whi...

We introduce a flow-dependent version of the quadratic Steiner tree problem in the plane. An instance of the problem on a set of embedded sources and a sink asks for a directed tree T spanning these nodes and a bounded number of Steiner points, such that ∑

The planar rectilinear Steiner tree problem has been extensively studied. The common formulation ignores circuit fabrication issues such as multiple routing layers, preferred routing directions, and vias between layers. In this paper, the authors extend a previously presented planar rectilinear Steiner tree heuristic to consider layer assignment, preferred routing direction restrictions, and vi...

We consider the problem of finding a minimum spanning and Steiner tree for a set of points in the plane where the orientations of edge segments are restricted to uniformly distributed orientations, , and where the coordinate system can be rotated around the origin by an arbitrary angle. The most important cases with applications in VLSI design arise when or . In the former, so-called rectilinea...