Skip to main content
SpringerLink
Log in

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.

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.K. Hwang D.S. Richards P. Winter (1992) The Steiner Tree Problem , Ann Discrete Math 53 Elsevier Amsterdam

    Google Scholar 

  2. M.R. Garey R.L. Graham D.S. Johnson (1977) ArticleTitleThe complexity of computing Steiner minimal trees SIAM J. Appl. Math 32 835–859 Occurrence Handle10.1137/0132072

    Article  Google Scholar 

  3. J.H. Rubinstein D.A. Thomas J.F. Weng (2002) ArticleTitleMinimum networks for four points in space Geom. Dedicata 93 57–70 Occurrence Handle10.1023/A:1020389712969

    Article  Google Scholar 

  4. F.K. Hwang (1986) ArticleTitleA linear time algorithm for full Steiner trees Oper. Res. Lett 5 235–237 Occurrence Handle10.1016/0167-6377(86)90008-8

    Article  Google Scholar 

  5. J.F. Weng (1997) ArticleTitleExpansion of linear Steiner trees Algorithmica 19 318–330

    Google Scholar 

  6. Rubinstein, J. H., Weng, J. F. and Wormald, N. C.: Lower bound on the length of Euclidean minimal Steiner trees, Preprint.

  7. J. H. Rubinstein D. A. Thomas (1991) ArticleTitleA variational approach to the Steiner network problem Ann. Oper. Res. 33 481–499

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria, 3010, Australia

    M. Brazil, D. A. Thomas & J. F. Weng

Authors
  1. M. Brazil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. A. Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. F. Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Brazil.

Additional information

Supported by a grant from the Australia Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brazil, M., Thomas, D.A. & Weng, J.F. Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space. Geom Dedicata 109, 107–119 (2004). https://doi.org/10.1007/s10711-004-1528-6

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-004-1528-6

Keywords

Search

Navigation