Dietmar Cieslik

Steiner Minimal Trees - Nonconvex Optimization and Its Applications Dietmar Cieslik 1st Ed. Softcover of Orig. Ed. 1998 edition

Marc Notes: Originally published: Dordrecht; London: Kluwer Academic, 1998.; This book is the result of 18 years of research into Steiner's problem and its relatives in theory and application. Starting with investigations of shortest networks for VLSI layout and, on the other hand, for certain facility location problems, the author has found many common properties for Steiner's problem in various spaces. Review Quotes: In summary, this is a well written book on an interesting and challenging range of problems but from a mathematician's viewpoint. As such it can be strongly recommended.' Journal of the Operational Research Society, 49:12 (1998) The book has an encyclopedic character, contains lots of information and seems a must for those interested in the subject.' Nieuw Archief voor Wiskunde, 5/1:1 (2000) Table of Contents: Preface. 1. Introduction. 2. SMT and MST in Metric Spaces - A Survey. 3. Fermat's Problem in Banach- Minkowski Spaces. 4. The Degrees of the Vertices in Shortest Trees. 5. 1-Steiner-Minimal-Trees. 6. Methods to Construct Shortest Trees. 7. The Steiner Ratio of Banach-Minkowski Spaces. 8. Generalizations. References. Index. Publisher Marketing: The problem of "Shortest Connectivity," which is discussed here, has a long and convoluted history. Many scientists from many fields as well as laymen have stepped on its stage. Usually, the problem is known as Steiner's Problem and it can be described more precisely in the following way: Given a finite set of points in a metric space, search for a network that connects these points with the shortest possible 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. Steiner's Problem seems disarmingly simple, but it is rich with possibilities and difficulties, even in the simplest case, the Euclidean plane. This is one of the reasons that an enormous volume of literature has been published, starting in 1 the seventeenth century and continuing until today. The difficulty is that we look for the shortest network overall. Minimum span ning networks have been well-studied and solved eompletely in the case where only the given points must be connected. The novelty of Steiner's Problem is that new points, the Steiner points, may be introduced so that an intercon necting network of all these points will be shorter. This also shows that it is impossible to solve the problem with combinatorial and geometric methods alone."