Abstract
Geometric optimization formulations are presented for the Steiner Minimal Tree problem with N = 3, 4, 5, 6 vertices in E 3. The geometric optimization problems are based on a dual formulations of the primal Steiner Minimal Tree problem in E 3. The algorithm for small point sets in E 3 is useful in any optimal seeking or heuristic approach to the geometric Steiner problem. The dual construction is an important concept because it yields a lower bound on the Steiner problem important for certain applications in molecular modelling.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bellare, M. and P. Rogaway, 1995. “The Complexity of Approximating a Nonlinear Program,” Mathematical Programming, 69, 429–441.
Chung, F.R.K. and F.K. Hwang, 1978,“A Lower Bound for the Steiner Tree Problem,” SIAM J. of Appl. Math 34 27–36.
Cockayne, E. 1970. “On the Efficiency of the Algorithm for Steiner Minimal Trees,” SIAM J. Appl. Math. 18 (1), 150–159.
Coxeter, H.S.M., 1961. Introduction to Geometry. Wiley.
Crippen, G.M., 1981. Distance Geometry and Conformatinal Calculations. Chichester, England: Research Studies Press.
Du, D.Z. “On Steiner Ratio Conjectures,” Annals of Operations Research, 33, 437–449.
Gilbert, E. and H.O. Pollak, 1968. “Steiner Minimal Trees.” SIAM J. Appl. Math. 16 (1), 1–29.
Personal communication. Gilbert, E.
Hendrickson, B.A., 1990. “The Molecule Problem: Determining Conformation from Pairwise Distances,” Ph.D. Thesis #90–1159. Department of Computer Science, Cornell University, Ithaca, NY 14853–7501.
Melzak, Z.A., 1961 “On the Problem of Steiner,” Canadian Math. Bulletin 4 (2), 143–148.
Saxe, J.B., 1979. “Embeddability of weighted graphs in k—space is strongly N7’—Hard,” 17th Allerton Conference on Communication, Control and Computing,480–489.
Schorn, Peter. 1993. “Exact and Approximate Solutions to the Embedding Problem in Distance Geometry.” Paper presented at the Computational Geometry Meeting, Raleigh, North Carolina, October 1993.
Smith, J. MacGregor, R. Weiss, and M. Patel, 1995. “An 0(N 2 ) Heuristic for Steiner Minimal Trees in E 3, ” Networks, 25, 273–289.
Smith, J. MacGregor, R. Weiss, B. Toppur, and N. Maculan, 1999. “On the Use of Lower Bounds from Steiner Trees to Characterize Proteins and Other Minimum Energy Configurations,” Working Paper, Department of Mechanical and Industrial Engineering, UNiversity of Massachusetts, Amherst, MA 01003.
Smith, Warren D., 1992. “How to find Steiner minimal trees in Euclidean d-space,” Algorithmica 7137–177.
Smith, Warren D. and J. MacGregor Smith, 1995. “On the Steiner Ratio in 3-Space,” Journal of Combinatorial Theory, Series A, 69 (2), 301–332.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Smith, J.M. (2000). Geometric Optimization Problems for Steiner Minimal Trees in E 3 . In: Pardalos, P.M. (eds) Approximation and Complexity in Numerical Optimization. Nonconvex Optimization and Its Applications, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3145-3_26
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3145-3_26
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4829-8
Online ISBN: 978-1-4757-3145-3
eBook Packages: Springer Book Archive