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.
Similar content being viewed by others
References
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.
E.J. Beasley (1992) ArticleTitleA heuristic for Euclidean and rectilinear Steiner problems European Journal of Operational Research 58 299–327
M. Brazil D.A. Thomas J.F. Weng (2004) ArticleTitleUpper and lower bounds for the lengths of Steiner trees in 3-space Geometriae Dedicata 109 107–119 Occurrence Handle10.1007/s10711-004-1528-6
S.K. Chang (1992) ArticleTitleThe generation of minimal trees with a Steiner topology Journal of ACM 19 699–711 Occurrence Handle10.1145/321724.321733
F.K. Hwang D.S. Richards P. Winter (1992) The Steiner Tree Problem; Annals of Discrete Mathematics 53 Elsevier Science Publishers B.V., Amsterdam
A. Lin S.-P. Han (2002) ArticleTitleOn the distance between two ellipsoids SIAM Journal on Optimization 13 298–308 Occurrence Handle10.1137/S1052623401396510
W.D. Smith (1992) ArticleTitleHow to find Steiner minimal trees in Euclidean d-space Algorithmica 7 137–177 Occurrence Handle10.1007/BF01758756
A.Z. Zelikovsky (1993) ArticleTitleAn 11/6-approximation algorithm for the network Steiner problem Algorithmica 9 463–470 Occurrence Handle10.1007/BF01187035
Author information
Authors and Affiliations
Corresponding author
Additional information
J. H. Rubinstein, J. Weng: Research supported by the Australian Research Council
N. Wormald: Research supported by the Australian Research Council and the Canada Research Chairs Program. Research partly carried out while the author was in the Department of Mathematics and Statistics, University of Melbourne
Rights and permissions
About this article
Cite this article
Rubinstein, J.H., Weng, J. & Wormald, N. Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees. J Glob Optim 35, 573–592 (2006). https://doi.org/10.1007/s10898-005-4207-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-005-4207-8