An improved algorithm for computing Steiner minimal trees in Euclidean -space (original) (raw)

An improved algorithm for computing Steiner minimal trees in Euclidean d-space

2008

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Insight into the computation of Steiner minimal trees in Euclidean space of general dimension

Discrete Applied Mathematics

We present well known properties related to the topology of Steiner minimal trees and to the geometric position of Steiner points, and investigate their application in the main exact algorithms that have been proposed for the Euclidean Steiner problem. We discuss the difficulty in the application of properties that were very successfully applied to solve the problem in the plane, when the dimension of the space increases, and point out that the application of some geometric conditions for Steiner points is hindered when the well known implicit enumeration scheme proposed by Smith in 1992 is considered. Finally, we mention mathematicaloptimization methods as a direction to explore in the search for good formulations of inequalities that would allow the application of these restrictive geometric conditions.

A partition-based relaxation for Steiner trees

Mathematical Programming, 2011

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals R ⊆ V , and non-negative costs c e for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The nodes V \R are called Steiner nodes.

A fast algorithm for Steiner trees

Acta Informatica, 1981

Given an undirected distance graph G = (V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S___ V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(ISI ]V] 2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2 (1-1) times that of the optimal tree, where l is the number of leaves in the optimal tree.

Rectilinear steiner trees: Efficient special-case algorithms

Networks, 1977

A minimal rectizinear Steiner tree for a set A of points i n the p l u m is a tree which interconnects A using horyizontal om? vertical lines of shortest possible total length. Such trees have potential application t o Wire layout for printed

On-line steiner trees in the Euclidean plane

Discrete & Computational Geometry, 1993

Suppose we are given a sequence of n points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph, (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio is O(log n) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(log n/ log log n) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.

Exact and heuristic algorithms for the Euclidean Steiner tree problem

2010

In this thesis, we study the Euclidean Steiner tree problem (ESTP) which arises in the field of combinatorial optimization. The ESTP asks for a network of minimal total edge length spanning a set of given terminal points in d with the ability to add auxiliary connecting points (Steiner points) to decrease the overall length of the network. The graph theory literature contains extensive studies of exact, approximation, and heuristic algorithms for ESTP in the plane, but less is known in higher dimensions. The contributions of this thesis include a heuristic algorithm and enhancements to an exact algorithm for solving the ESTP. We present a local search heuristic for the ESTP in d for d ≥ 2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to create a network on the inserted points, and second order cone programming to optimize the locations of Steiner points. Unlike other ESTP heuristics relying on the Delaunay triangulation, the algorithm inserts Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. The algorithm extends effectively into higher dimensions, and we present computational results on benchmark test problems in d for 2 ≤ d ≤ 5. We develop new geometric conditions derived from properties of Steiner trees which bound below the number of Steiner points on paths between terminals in the Steiner minimal tree. We describe conditions for trees with a Steiner topology and show how these conditions relate to the Voronoi diagram. We describe more restrictive conditions for trees with a full Steiner topology and their implementation into the algorithm of Smith (1992). We present computational results on benchmark test problems in d for 2 ≤ d ≤ 5. iii 5.2 Geometric conditions for SMTs with a FST .