The State of Art on the Steiner Ratio Value in R³

Authors

  • R. Mondaini
  • N.V. Oliveira

DOI:

https://doi.org/10.5540/tema.2004.05.02.0249

Abstract

Our aim in this work is to make a brief review of the results related to the search of the Infimum and Supremum Values of the Steiner Ratio for point sets in R3. We show the fundamental achievements which were obtained in a research period of 35 years. We also comment on a recently proposed new upper bound value which is an improvement of Smith and Mac Gregor Smith’s bound.

References

[1] F.R.K. Chung and E.N. Gilbert, Steiner Trees for the Regular Simplex, Bull. Inst. Math. Academia Sinica, 4, No. 2 (1976), 313-325.

F.R.K. Chung and F.K. Hwang, A Lower Bound for the Steiner Tree Problem, SIAM J. Appl. Math., 34 (1978), 27-36.

F.R.K. Chung and R.L. Graham, A New Bound for Euclidean Steiner Minimal Trees, Ann. N.Y. Acad. Sci., 440 (1985), 328-346.

D.-Z. Du, On Steiner Ratio Conjectures, Ann. Oper. Res., 33 (1991), 437-451.

D.-Z. Du and F.K. Hwang, The Steiner Ratio Conjecture of Gilbert and Pollak is true, Proc. Natl. Acad. Sci. USA, 87 (1990), 9464-9466.

D.-Z. Du and F.K. Hwang, A New Bound for the Steiner Ratio, Trans. Amer. Math. Soc., 278, No. 1 (1983), 137-148.

D.-Z. Du and F.K. Hwang, A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratio, Algorithmica, 7 (1992), 121-135.

D.-Z. Du and W.D. Smith, Disproofs of Generalized Gilbert-Pollak Conjecture on the Steiner Ratio in Three or More Dimensions, J. Comb. Theor., A74 (1996), 115-130.

E.N. Gilbert and H.O. Pollak, Steiner Minimal Trees, SIAM J. Appl. Math., 16 (1968), 1-29.

R.L. Graham and F.K. Hwang, A Remark on Steiner Minimal Trees, Bull. Inst. Math. Academia Sinica, 4, No. 1 (1976), 177-182.

R. Mondaini, The Disproof of a Conjecture on the Steiner Ratio in E3 and its consequences for a Full Geometric Description of Macromolecular Chiralit, em “Proc. 2nd Braz. Symp. Math. Comp. Biol.”, pp. 101–177, e-papers ed., Rio de Janeiro, RJ, 2003.

R. Mondaini, F. Montenegro and D.F. Mondaini, “Modelling a Steiner Point Distriution of a Special Helix Point Configuration”, Technical Report ES-398/96, PESC/COPPE-UFRJ, 1996.

R. Mondaini, The Steiner Ratio and the Homochirality of Biomacromolecular Structures, Nonconvex Optimization and its Applications Series, 74 (2004), 373-390.

R. Mondaini, The Euclidean Steiner Ratio and the Measure of Chirality of Biomacromolecules, Gen. Mol. Biol., (2003), to appear.

W.D. Smith, How to find Steiner Minimal Trees in d-Space, Algorithmica, 7 (1992), 137-177.

W.D. Smith and J. Mac Gregor Smith, On the Steiner Ratio in 3-Space, J. Comb. Theor., A69 (1995), 301-332.

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Published

2004-06-01

How to Cite

Mondaini, R., & Oliveira, N. (2004). The State of Art on the Steiner Ratio Value in R³. Trends in Computational and Applied Mathematics, 5(2), 249–257. https://doi.org/10.5540/tema.2004.05.02.0249

Issue

Vol. 5 No. 2 (2004)

Section

Original Article

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