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# Steiner Trees, Coordinate Systems and NP-Hardness

• J. F. Weng
Chapter
Part of the Combinatorial Optimization book series (COOP, volume 6)

## Abstract

Given a set A of points a 1, a 2,..., in the Euclidean plane, the Steiner tree problem asks for a minimum network T(A) (or T if A is not necessarily mentioned) interconnecting A with some additional points to shorten the network [6]. The given points are referred to as terminals and the additional points are referred to as Steiner points. Trivially, T is a tree, called the Euclidean Steiner minimal tree (ESMT) for A. It is well known that Steiner minimal trees satisfy an angle condition: all angles at the vertices of Steiner minimal trees are not less than 120° [6]. A tree satisfying this angle condition is called a Steiner tree. Therefore, a Steiner minimal tree must be a Steiner tree.

## Keywords

Steiner Tree Integer Point Steiner Point Steiner Tree Problem Negative Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. [1]
M. Brazil, D.A. Thomas and J.F. Weng, Gradient-constrained minimal Steiner trees, DIMACS Series in Discrete Math. and Theoretical Comput. Science, Vol. 40, pp. 23–38.Google Scholar
2. [2]
M. Brazil, D.A. Thomas and J.F. Weng, On the complexity of the Steiner problem, preprint.Google Scholar
3. [3]
D.Z. Du and F.K. Hwang, A proof of the Gilbert-Pollak conjecture on the Steiner ratio, Algorithmica, Vol. 7 (1992), pp. 121–135.
4. [4]
M.R. Garey, R.L. Graham and D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math., Vol. 32 (1977), pp. 835–859.
5. [5]
F.K. Hwang and J.F. Weng, Hexagonal coordinate systems and Steiner minimal trees, Discrete Math., Vol. 62 (1986), pp. 49–57.
6. [6]
F.K. Hwang, D.S. Richard and P. Winter, The Steiner tree problem, North-Holland Publishing Co., Amsterdam, 1992.
7. [7]
Z.A. Melzak, On the problem of Steiner, Canad. Math. Bull., Vol. 4 (1961), pp. 143–148.
8. [8]
J.H. Rubinstein, D.A. Thomas and N.C. Wormald, Steiner trees for terminals constrained to curves, SIAM J. Discrete Math., Vol. 10 (1997), pp. 1–17.
9. [9]
J.F. Weng, Generalized Steiner problem and hexagonal coordinate system (Chinese, English summary), Acta. Math. Appl. Sinica, Vol. 8 (1985), pp. 383–397.
10. [10]
J.F. Weng, Steiner problem in hexagonal metric, unpublished manuscript.Google Scholar
11. [11]
J.F. Weng, A new model of generalized Steiner trees and 3-coordinate systems, DIMACS Series in Discrete Math. and Theoretical Comput. Science, Vol. 40, pp. 415–424.Google Scholar

## Copyright information

© Springer Science+Business Media Dordrecht 2000

## Authors and Affiliations

• J. F. Weng
• 1
1. 1.Department of Mathematics and Statistics and Department of Electrical and Electronic EngineeringThe University of MelbourneParkvilleAustralia