Journal of Global Optimization

, Volume 35, Issue 4, pp 573–592 | Cite as

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

Article

Abstract

We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images. The lower bound is equal to the length when the topology is full. We use the lower bound to prove bounds on the “error” e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tree from 120°. Such bounds are useful for validating algorithms computing minimal Steiner trees. In addition we give a number of examples illustrating features of the relationship between e and d, and make a conjecture which, if true, would somewhat strengthen our bounds on the error.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S. (1996), Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Sciences, pp. 2–13.Google Scholar
  2. 2.
    Beasley, E.J. 1992A heuristic for Euclidean and rectilinear Steiner problemsEuropean Journal of Operational Research58299327Google Scholar
  3. 3.
    Brazil, M., Thomas, D.A., Weng, J.F. 2004Upper and lower bounds for the lengths of Steiner trees in 3-spaceGeometriae Dedicata109107119CrossRefGoogle Scholar
  4. 4.
    Chang, S.K. 1992The generation of minimal trees with a Steiner topologyJournal of ACM19699711CrossRefGoogle Scholar
  5. 5.
    Hwang, F.K., Richards, D.S., Winter, P. 1992The Steiner Tree Problem; Annals of Discrete Mathematics 53Elsevier Science PublishersB.V., AmsterdamGoogle Scholar
  6. 6.
    Lin, A., Han, S.-P. 2002On the distance between two ellipsoidsSIAM Journal on Optimization13298308CrossRefGoogle Scholar
  7. 7.
    Smith, W.D. 1992How to find Steiner minimal trees in Euclidean d-spaceAlgorithmica7137177CrossRefGoogle Scholar
  8. 8.
    Zelikovsky, A.Z. 1993An 11/6-approximation algorithm for the network Steiner problemAlgorithmica9463470CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MelbourneAustralia
  2. 2.ARC Special Research Center for Ultra-Broadband Information Networks (CUBIN) Department of Electrical and Electronic EngineeringUniversity of MelbourneAustralia
  3. 3.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations