Advertisement

Discrete & Computational Geometry

, Volume 2, Issue 4, pp 401–414 | Cite as

Steiner minimal trees on sets of four points

  • D. Z. Du
  • F. K. Hwang
  • G. D. Song
  • G. Y. Ting
Article

Abstract

LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree if
AOD>3 (<) 90°, and both are Steiner minimal trees if
AOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) If
AOD≥90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) If
AOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.

Keywords

Span Tree Minimal Span Tree Equilateral Triangle Discrete Comput Geom Steiner Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    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.MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. Z. Du, F. K. Hwang, and J. F. Weng, Steiner minimal trees on zigzag lines,Trans. Amer. Math. Soc. 278 (1983), 149–156.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Z. Du and F. K. Hwang, Steiner minimal trees for bar waves, to appear.Google Scholar
  4. 4.
    M. R. Garey, R. L. Graham, and D. S. Johnson, The complexity of computing Steiner minimal tress,SIAM J. Appl. Math. 32 (1977), 835–859.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. N. Gilbert and H. O. Pollak, Steiner minimal trees,SIAM J. Appl. Math. 16 (1968), 1–29.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Z. A. Melzak, On the problem of Steiner,Canad. Math. Bull. 4 (1960), 143–148.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dame K. Ollerenshaw, Minimum networks linking four points in a plane,Inst. Math. Appl. 15 (1978), 208–211.MathSciNetGoogle Scholar
  8. 8.
    H. O. Pollak, Some remarks on the Steiner problem,J. Combin. Theor. Ser. A 24 (1978), 278–295.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • D. Z. Du
    • 1
    • 2
  • F. K. Hwang
    • 3
  • G. D. Song
    • 4
  • G. Y. Ting
    • 5
  1. 1.University of CaliforniaSanta BarbaraUSA
  2. 2.Academia SinicaBejingChina
  3. 3.AT&T Bell LaboratoriesMurray HillUSA
  4. 4.Quiquihaer Light Engineering CollegeHeilungjiangChina
  5. 5.Quiquihaer Teachers' CollegeHeilungjiangChina

Personalised recommendations