Steiner Trees, Coordinate Systems and NP-Hardness

  • J. F. Weng
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

Hexagonal 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.MathSciNetMATHCrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F.K. Hwang and J.F. Weng, Hexagonal coordinate systems and Steiner minimal trees, Discrete Math., Vol. 62 (1986), pp. 49–57.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    F.K. Hwang, D.S. Richard and P. Winter, The Steiner tree problem, North-Holland Publishing Co., Amsterdam, 1992.MATHGoogle Scholar
  7. [7]
    Z.A. Melzak, On the problem of Steiner, Canad. Math. Bull., Vol. 4 (1961), pp. 143–148.MathSciNetMATHCrossRefGoogle Scholar
  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.MathSciNetMATHCrossRefGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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

Personalised recommendations