Algorithmica

, Volume 7, Issue 1–6, pp 193–218 | Cite as

Graham's problem on shortest networks for points on a circle

  • J. H. Rubinstein
  • D. A. Thomas
Article

Abstract

Suppose a configurationX consists ofn points lying on a circle of radiusr. If at most one of the edges joining neighboring points has length strictly greater thanr, then the Steiner treeS consists of all these edges with a longest edge removed. In order to showS is, in fact, just the minimal spanning treeT, a variational approach is used to show the Steiner ratio for this configuration is at least one and equals one only ifS andT coincide. The variational approach greatly reduces the number of possible Steiner trees that need to be considered.

Key words

Graham's conjecture Steiner trees Spanning trees Variational approach Cocircular points 

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References

  1. [1]
    D. Z. Du, F. K. Hwang, and S. C. Chao, Steiner Minimal Tree for Points on a Circle,Proc. Amer. Math. Soc.,95 (1985), 613–618.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. R. Garey, R. L. Graham, and D. S. Johnson, The Complexity of Computing Steiner Minimal Trees,SIAM J. Appl. Math.,32 (1977), 835–859.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. N. Gilbert and H. O. Pollak, Steiner Minimal Trees,SIAM J. Appl. Math.,16 (1968), 1–29.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    R. L. Graham, Some Results on Steiner Minimal Trees, Unpublished manuscript dated 11 May 1967.Google Scholar
  5. [5]
    Z. A. Melzak, On the Problem of Steiner,Canad. Math. Bull,4 (1961), 143–148.MATHMathSciNetGoogle Scholar
  6. [6]
    H. O. Pollak, Some Remarks on the Steiner Problem,J. Combin. Theory Ser. A,24 (1978), 278–295.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J. H. Rubinstein and D. A. Thomas, A Variational Approach to the Steiner Network Problem, Preprint.Google Scholar
  8. [8]
    J. H. Rubinstein and D. A. Thomas, Critical Points for the Steiner Ratio Conjecture, Preprint.Google Scholar
  9. [9]
    J. H. Rubinstein and D. A. Thomas, The Steiner Ratio Conjecture for Cocircular Points, Preprint.Google Scholar
  10. [10]
    J. H. Rubinstein and D. A. Thomas, The Steiner Ratio Conjecture for Six Points, Preprint.Google Scholar
  11. [11]
    J. H. Rubinstein, D. A. Thomas, and J. F. Weng, Degree Five Steiner Points Cannot Reduce Network Costs for Planar Sets, Preprint.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • J. H. Rubinstein
    • 1
  • D. A. Thomas
    • 2
  1. 1.Institute of Advanced Study, and Mathematics DepartmentUniversity of MelbourneParkvilleAustralia
  2. 2.University of MelbourneParkvilleAustralia

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