The Steiner Ratio

1. Front Matter

   Pages i-xi

   PDF

2. The Historical Genesis

   • Dietmar Cieslik

   Pages 1-19

3. Networks, Spaces and Algorithms

   • Dietmar Cieslik

   Pages 21-58

4. Shortest Trees in Metric Spaces — A Survey

   • Dietmar Cieslik

   Pages 59-78

5. The Steiner Ratio of Metric Spaces

   • Dietmar Cieslik

   Pages 79-98

6. The Steiner Ratio of Banach-Minkowski Spaces

   • Dietmar Cieslik

   Pages 99-130

7. Euclidean Spaces

   • Dietmar Cieslik

   Pages 131-139

8. The Steiner Ratio of Neighboured Spaces

   • Dietmar Cieslik

   Pages 141-156

9. Banach-Minkowski Planes

   • Dietmar Cieslik

   Pages 157-169

10. The Steiner Ratio and the Embedding of Spaces

    • Dietmar Cieslik

    Pages 171-181

11. The Steiner Ratio and Discrete Geometry

    • Dietmar Cieslik

    Pages 183-195

12. The Dependence of the Steiner Ratio on the Dimension

    • Dietmar Cieslik

    Pages 197-203

13. Related Questions

    • Dietmar Cieslik

    Pages 205-221

14. Back Matter

    Pages 223-244

    PDF

Steiner's Problem concerns finding a shortest interconnecting network for a finite set of points in a metric space. A solution must be a tree, which is called a Steiner Minimal Tree (SMT), and may contain vertices different from the points which are to be connected. Steiner's Problem is one of the most famous combinatorial-geometrical problems, but unfortunately it is very difficult in terms of combinatorial structure as well as computational complexity. However, if only a Minimum Spanning Tree (MST) without additional vertices in the interconnecting network is sought, then it is simple to solve. So it is of interest to know what the error is if an MST is constructed instead of an SMT. The worst case for this ratio running over all finite sets is called the Steiner ratio of the space.
The book concentrates on investigating the Steiner ratio. The goal is to determine, or at least estimate, the Steiner ratio for many different metric spaces. The author shows that the description of the Steiner ratio contains many questions from geometry, optimization, and graph theory.
Audience: Researchers in network design, applied optimization, and design of algorithms.

• Finite
• Graph theory
• addition
• algorithms
• complexity
• geometry
• network
• networks
• optimization
• sets
• vertices

• Institute of Mathematics and C.S., University of Greifswald, Germany

  Dietmar Cieslik

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