Constrained independence system and triangulations of planar point sets

  • Siu-Wing Chengl
  • Yin-Feng Xu
Session 2A: Computational Geometry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)


We propose and study a new constrained independence system. We obtain a sequence of results, including a matching theorem for bases of the system and introducing a set of light elements which give a lower bound for the objective function of a minimization problem in the system. We then demonstrate that the set of triangulations of a planar point set can be modeled as constrained independence systems. The corresponding minimization problem in the system is the well-known minimum weight triangulation problem. Thus, we obtain two matching theorems for triangulations and a set of light edges (or light triangles) that give a lower bound for the minimum weight triangulation. We also prove directly a third matching theorem for triangulations. We show that the set of light edges is a superset of some subsets of edges of a minimum weight triangulation that were studied before.


Minimization Problem Bipartite Graph Perfect Match Algorithm Greedy Greedy Solution 
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  1. 1.
    O. Aichholzer, F. Aurenhammer, M. Taschwer, and G. Rote, Triangulations intersect nicely, in Proc. 11th Annual Symposium on Computational Geometry, 1995.Google Scholar
  2. 2.
    B. Bollobás, Graph Theory. An Introductory Course, Springer-Verlag, 1979.Google Scholar
  3. 3.
    S. Cheng and Y. Xu, Approaching the largest β-skeleton within a minimum weight triangulation. Manuscript, 1995.Google Scholar
  4. 4.
    H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, 1987.Google Scholar
  5. 5.
    M. Garey and D. Johnson, Computers and Intractability. A guide to the Theory of NP-completeness, Freeman, 1979.Google Scholar
  6. 6.
    M. Keil, Computing a subgraph of the minimum weight triangulation, Computational Geometry: Theory and Applications, 4 (1994), pp. 13–26.Google Scholar
  7. 7.
    D. Kirkpatrick and J. Radke, A framework for computational morphology, in Computational Geometry, G. Toussaint, ed., Elsevier, Amsterdam, 1985, pp. 217–248.Google Scholar
  8. 8.
    D. Plaisted and J. Hong, A heuristic triangulation algorithm, Journal of Algorithms, 8 (1987), pp. 405–437.CrossRefGoogle Scholar
  9. 9.
    G. Rote. personal communication.Google Scholar
  10. 10.
    D. Welsh, Matroid Theory, Academic Press, New York, 1976.Google Scholar
  11. 11.
    Y. Xu, Minimum weight triangulation problem of a planar point set, PhD thesis, Institute of Applied Mathematics, Academia Sinica, Beijing, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Siu-Wing Chengl
    • 1
  • Yin-Feng Xu
    • 2
  1. 1.Department of Computer ScienceThe Hong Kong University of Science & TechnologyClear Water BayHong Kong
  2. 2.School of ManagementXi'an Jiaotong UniversityXi'an, ShaanxiPRC

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