A Tight Bound for β-Skeleton of Minimum Weight Triangulations

  • Cao An Wang
  • Boting Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)


In this paper, we prove a tight bound for β value \( \left( {\beta = \frac{{\sqrt {2\sqrt 3 + 9} }} {3}} \right) \) ) such that being less than this value,the β-skeleton of a planar point set may not belong to the minimum weight triangulation of this set, while being equal to or greater than this value, the β-skeleton always belongs to the minimum weight triangulation. Thus, we settled the conjecture of the tight bound for β-skeleton of minimum weight triangulation by Mark Keil. We also present a new sufficient condition for identifying a subgraph of minimum weight triangulation of a planar n-point set. The identified subgraph could be different from all the known subgraphs, and the subgraph can be found in O(n 2 log n) time.


Line Segment Convex Hull Computational Geometry Internal Edge Primal Plane 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AC93]
    E. Anagnostou and D. Corneil, Polynomial time instances of the minimum weight triangulation problem, Computational Geometry: Theory and applications, vol. 3, pp. 247–259, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [CGT95]
    S.-W. Cheng, M. Golin and J. Tsang, Expected case analysis of b-skeletons with applications to the construction of minimum weight triangulations, CCCG Conference Proceedings, P.Q., Canada, pp. 279–284, 1995.Google Scholar
  3. [BDE96]
    P. Bose, L. Devroye, and W. Evens, Diamonds are not a minimum weight triangulation’s best friend, Proceedings of 8th CCCG, 1996, Ottawa, pp. 68–73.Google Scholar
  4. [CX96]
    S.-W. Cheng and Y.-F. Xu, Approaching the largest β-skeleton within the minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.Google Scholar
  5. [DK98]
    Y. Dai and N. Katoh, On computing new classes of optimal triangulations with angular constraints, Proceedings on 4th annual international conference of Computing and Combinatorics, LNCS 1449, pp.15–24.Google Scholar
  6. [DM96]
    M. T. Dickerson, M. H. Montague, The exact minimum weight triangulation, Proc. 12th Ann. Symp. Computational Geometry, Philadelphia, Association for Computing Machinery, 1996.Google Scholar
  7. [Gi79]
    P. D. Gilbert, New results in planar triangulations, TR-850, University of Illinois Coordinated science Lab, 1979.Google Scholar
  8. [GJ79]
    M. Garey and D. Johnson, Computer and Intractability. A guide to the theory of NP-completeness, Freeman, 1979.Google Scholar
  9. [Ke94]
    J. M. Keil, Computing a subgraph of the minimum weight triangulation, Computational Geometry: Theory and Applications pp. 13–26, 4 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [KR85]
    D. Kirkpatrick and J. Radke, A framework for computational morphology, in G. Toussaint, ed., Computational Geometry, Elsevier, 1985, pp. 217–248.Google Scholar
  11. [Kl80]
    G. Klinesek, Minimal triangulations of polygonal domains, Ann. Discrete Math., pp. 121–123, 9 (1980).MathSciNetCrossRefGoogle Scholar
  12. [MR92]
    H. Meijer and D. Rappaport, Computing the minimum weight triangulation of a set of linearly ordered points, Information Processing Letters, vol. 42, pp. 35–38, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [O’R93]
    J. O’Rourke, Computational Geometry In C, Cambridge University Press, 1993.Google Scholar
  14. [MWX96]
    A. Mirzain, C. Wang and Y. Xu, On stable line segments in triangulations, Proceedings of 8th CCCG, Ottawa, 1996, pp.68–73.Google Scholar
  15. [WX96]
    C. Wang and Y. Xu, Minimum weight triangulations with convex layers constraint, to appear J. of Global Optimization.Google Scholar
  16. [WCX97]
    C. Wang, F. Chin, and Y. Xu, A new subgraph of Minimum weight triangulations, J. of Combinational Optimization Vol 1, No. 2, pp. 115–127.Google Scholar
  17. [XZ96]
    Y. Xu, D. Zhou, Improved heuristics for the minimum weight triangulation, Acta Mathematics Applicatae Sinica, vol. 11, no. 4, pp. 359–368, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Yan95]
    B. Yang, A better subgraph of the minimum weight triangulation, The IPL, Vol.56, pp. 255–258.Google Scholar
  19. [YXY94]
    B. Yang, Y. Xu and Z. You, A chain decomposition algorithm for the proof of a property on minimum weight triangulations, Proc. 5th International Symposium on Algorithms and Computation (ISAAC’94), LNCS 834, Springer-Verlag, pp. 423–427, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Cao An Wang
    • 1
  • Boting Yang
    • 1
  1. 1.Department of Computer ScienceMemorial University of NewfoundlandSt.John’s, NFLDCanada

Personalised recommendations