Skip to main content
SpringerLink
Log in

On Steiner minimal trees withL p distance

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

LetL p be the plane with the distanced p (A 1 ,A 2 ) = (¦x 1 −x 2¦p + ¦y1 −y 2¦p)/1p wherex i andy i are the cartesian coordinates of the pointA i . LetP be a finite set of points inL p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < ∞, each Steiner point is of degree exactly three. Define the Steiner ratio ϱ p to be inf{L s (P)/L m (P)¦P⊂L p } whereL s (P) andL m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ϱ1 = 2/3. Chung and Graham proved ϱ2 > 0.842. We prove in this paper that ϱ{∞} = 2/3 and √(√2/2)ϱ1ϱ2 ≤ ϱp ≤ √3/2 for anyp.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. R. K. Chung and R. L. Graham, A new bound for euclidean Steiner minimal trees,Ann. N. Y. Acad. Sci,440 (1985), 328–346.

    Article  MathSciNet  Google Scholar 

  2. F. R. K. Chung and F. K. Hwang, A lower bound for the Steiner tree problem,SIAM J. Appl. Math.,34 (1978), 27–36.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Z. Du, On Steiner ratio conjectures,Proceedings of NATO Advanced Research Workshop on Topological Network Designs, Denmark, 1989.

  4. D. Z. Du and F. K. Hwang, A new bound for the Steiner ratio,Trans. Amer. Math. Soc.,278 (1983), 138–148.

    Google Scholar 

  5. D. Z. Du, F. K. Hwang, and E. N. Yao, The Steiner ratio conjecture is true for five points,J. Combin. Theory Ser. A,38 (1985), 230–240.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. N. Gilbert and H. O. Pollak, Steiner minimal trees,SIAM J. Appl. Math.,16 (1968), 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. L. Graham and F. K. Hwang, Remarks on Steiner minimal trees,Bull. Inst. Math. Acad. Sinica,4 (1976), 177–182.

    MATH  MathSciNet  Google Scholar 

  8. F. K. Hwang, On Steiner minimal trees with rectilinear distance,SIAM J. Appl. Math.,30 (1976), 104–114.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. O. Pollak, Some remarks on the Steiner problem,J. Combin. Theory Ser. A,24 (1978), 278–295.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. H. Rubinstein and D. Thomas, The Steiner ratio conjecture for six points, Manuscript.

  11. J. M. Smith and M. Gross, Steiner minimal trees and urban service networks,J. Soc. Econ. Plann.,16 (1982), 21–38.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Academia Sinica, Beijing, People's Republic of China

    Zi -Cheng Liu & Ding -Zhu Du

Authors
  1. Zi -Cheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ding -Zhu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by F. K. Hwang.

This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Z.C., Du, D.Z. On Steiner minimal trees withL p distance. Algorithmica 7, 179–191 (1992). https://doi.org/10.1007/BF01758757

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01758757

Key words

Search

Navigation