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Steiner Minimal Trees

  • Book
  • © 1998

Overview

Authors:
  1. Dietmar Cieslik

    1. Ernst-Moritz-Arndt University, Greifswald, Germany

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Part of the book series: Nonconvex Optimization and Its Applications (NOIA, volume 23)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Dietmar Cieslik
    Pages 1-48

  3. SMT and MST in Metric Spaces — A Survey

    • Dietmar Cieslik
    Pages 49-90

  4. Fermat’s Problem in Banach-Minkowski Spaces

    • Dietmar Cieslik
    Pages 91-123

  5. The Degrees of the Vertices in Shortest Trees

    • Dietmar Cieslik
    Pages 125-162

  6. 1-Steiner-Minimal-Trees

    • Dietmar Cieslik
    Pages 163-175

  7. Methods to Construct Shortest Trees

    • Dietmar Cieslik
    Pages 177-233

  8. The Steiner Ratio of Banach-Minkowski Spaces

    • Dietmar Cieslik
    Pages 235-269

  9. Generalizations

    • Dietmar Cieslik
    Pages 271-285

  10. Back Matter

    Pages 287-322
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Keywords

About this book

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.

Reviews

`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)

Authors and Affiliations

  • Ernst-Moritz-Arndt University, Greifswald, Germany

    Dietmar Cieslik

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