Geometriae Dedicata

, Volume 93, Issue 1, pp 57–70 | Cite as

Minimum Networks for Four Points in Space

  • J. H. Rubinstein
  • D. A. Thomas
  • J. Weng


The minimum network problem (Steiner tree problem) in space is much harder than the one in the Euclidean plane. The Steiner tree problem for four points in the plane has been well studied. In contrast, very few results are known on this simple Steiner problem in 3D-space. In the first part of this paper we analyze the difficulties of the Steiner problem in space. From this analysis we introduce a new concept — Simpson intersections, and derive a system of iteration formulae for computing Simpson intersections. Using Simpson intersections the Steiner points can be determined by solving quadratic equations. As well this new computational method makes it easy to check the impossibility of computing Steiner trees on 4-point sets by radicals. At the end of the first part we consider some special cases (planar and symmetric 3D-cases) that can be solved by radicals. The Steiner ratio problem is to find the minimum ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree. This ratio problem in the Euclidean plane was solved by D. Z. Du and F. K. Hwang in 1990, but the problem in 3D-space is still open. In 1995 W.D. Smith and J.M. Smith conjectured that the Steiner ratio for 4-point sets in 3D-space is achieved by regular tetrahedra. In the second part of this paper, using the variational method, we give a proof of this conjecture.

minimum network Steiner tree Steiner ratio Unslovability 


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  1. 1.
    Cieslik, D.: Steiner Minimal Trees, Kluwer Acad. Publ., Dordrecht, 1998.Google Scholar
  2. 2.
    Courant, R. and Robbins, H.: What is Mathematics, Oxford Univ. Press, New York, 1941.Google Scholar
  3. 3.
    Du, D. Z., and Hwang, F. K.: A proof of Gilbert and Pollak's conjecture on the Steiner ratio, Algorithmica, 7 (1992), 121–135.Google Scholar
  4. 4.
    Du, D. Z., Hwang, F. K., Song, G. D. and Ting, G. Y.: Steiner minimal trees for four points, Discrete Comput. Geom. 2 (1987), 401–414.Google Scholar
  5. 5.
    Du, D. Z., Yao, E. Y. and Hwang, F. K.: A short proof of a result of Pollak on Steiner minimal trees, J. Combin. Theory Ser. A, 32 (1982), 396–400.Google Scholar
  6. 6.
    Du, X. F., Du, D. Z., Gao, B. and Lixue, Q.: A simple proof for a result of Ollerenshaw on Steiner trees, In: Advances in Optimization and Approximation, Nonconvex Optim. Appl. 1, Kluwer Acad. Publ., Dordrecht, 1994, pp. 68–71.Google Scholar
  7. 7.
    Hwang, F. K.: A linear time algorithm for full Steiner trees, Oper. Res. Lett. 5 (1986), 235–237.Google Scholar
  8. 8.
    Hwang, F. K. and Weng, J.F.: Hexagonal coordinate systems and Steiner minimal trees, Discrete Math. 62 (1986), 49–57.Google Scholar
  9. 9.
    Hwang, F. K., Richard, D. S. and Winter, P.: The Steiner Tree Problem, North-Holland, Amsterdam, 1992.Google Scholar
  10. 10.
    Kuhn, H. W.: 'steiner's' problem revisited, In: G. B. Dantzig and B. C. Eaves (eds), Studies in Optimization, Math. Assoc. Amer., Washington, 1974, pp. 52–70.Google Scholar
  11. 11.
    Mehlhos, St.: Simple counter examples for the unsolvability of the Fermat-and Steiner- Weber-problem by compass and ruler, Contrib. Algebra Geom. 41 (2000), 151–158.Google Scholar
  12. 12.
    Melzak, Z. A.: On the Steiner problem, Canad. Math. Bull. 4 (1961), 143–148.Google Scholar
  13. 13.
    Ollerenshaw, K.: Minimum networks linking four points in a plane, Inst. Math. Appl. 15 (1978), 208–211.Google Scholar
  14. 14.
    Pollak, H. O.: Some results on the Steiner problem, J. Combin. Theory Ser. A 24 (1978), 278–295.Google Scholar
  15. 15.
    Rubinstein, J. H. and Thomas, D. A.: A variational approach to the Steiner network problem, Ann. Oper. Res. 33 (1991), 481–499.Google Scholar
  16. 16.
    Rubinstein, J. H. and Thomas, D. A.: The Steiner ratio conjecture for six points, J. Combin. Theory Ser. A 58 (1991), 54–77.Google Scholar
  17. 17.
    Rubinstein, J. H. and Thomas, D. A.: The Steiner ratio conjecture for cocircular points, J. Discrete Comput. Geom. 7 (1992), pp. 77–86.Google Scholar
  18. 18.
    Smith, W. D.: How to find Steiner minimal trees in Euclidean d-space, Algorithmica 7 (1992), 137–177.Google Scholar
  19. 19.
    Smith, W. D. and Smith, J. M.: On the Steiner ratio in 3D-space, J. Combin. Theory Ser. A 69 (1995), 301–332.Google Scholar
  20. 20.
    Weng, J. F.: Generalized Steiner problem and hexagonal coordinate system (in Chinese), Acta Math. Appl. Sinica 8 (1985), 383–397.Google Scholar
  21. 21.
    Weng, J. F.: Symmetrization theorem of full Steiner trees, J. Combin. Theory Ser. A 66 (1994), 185–191.Google Scholar
  22. 22.
    Weng, J. F.: Variational approach and Steiner minimal trees on four points, DiscreteMath. 132 (1994), 349–362.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • J. H. Rubinstein
    • 1
  • D. A. Thomas
    • 2
  • J. Weng
    • 2
  1. 1.Department of Mathematics and StatisticsMelbourne UniversityAustralia
  2. 2.ARC Special Research Centre for Ultra-Broadband Information Networks, Department of Electrical and Electronic EngineeringMelbourne UniversityAustralia

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