Discrete & Computational Geometry

, Volume 16, Issue 4, pp 339–359 | Cite as

Triangulations intersect nicely

  • O. Aichholzer
  • F. Aurenhammer
  • Siu-Wing Cheng
  • N. Katoh
  • G. Rote
  • M. Taschwer
  • Yin-Feng Xu


We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.


Bipartite Graph Short Edge Adjacency List Identical Edge Planar Triangulation 
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.


  1. [AART] O. Aichholzer, F. Aurenhammer, G. Rote, and M. Taschwer, Triangulations intersect nicely,Proc. 11th Ann. Symp. on Computational Geometry, Vancouver, British Columbia, June 1995, pp. 220–229.Google Scholar
  2. [AARX] O. Aichholzer, F. Aurenhammer, G. Rote, and Y.-F. Xu, New greedy triangulation algorithms, in preparation.Google Scholar
  3. [B1] B. Bollobás,Graph Theory. An Introductory Course, Springer-Verlag, Berlin, 1979.zbMATHGoogle Scholar
  4. [B2] R. A. Brualdi, Comments on bases in dependence structures,Bull. Austral. Math. Soc. 1 (1969), 161–167.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [B3] T. H. Brylawski, Some properties of basic families of subsets,Discrete Math. 6 (1973), 333–341.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CX1] S.-W. Cheng and Y.-F. Xu, Constrained independence system and triangulations of planar point sets, in: D.-Z. Du and Ming Li, (eds.),Computing and Combinatorics, Proc. First Ann. Internat. Conf., COCOON'95, Xi'an, China, August 1995. Lecture Notes in Computer Science, Vol. 959, Springer-Verlag, Berlin, 1995, pp. 41–50.CrossRefGoogle Scholar
  7. [CX2] S.-W. Cheng and Y.-F. Xu, Approaching the largest β-skeleton within a minimum-weight triangulation,Proc. 12th Ann. Symp. on Computational Geometry, Philadelphia, PA, 1996, pp. 196–203.Google Scholar
  8. [DJ] G. Das and D. Joseph, Which triangulations approximate the complete graph,Proc. Internat. Symp. on Optimal Algorithms, Lecture Notes in Computer Science, Vol. 401, Springer-Verlag, Berlin, 1989, pp. 168–192.Google Scholar
  9. [DDMW] M. Dickerson, R. L. Drysdale, S. McElfresh, and E. Welzl, Fast greedy triangulation algorithms,Proc. 10th Ann. Symp. on Computational Geometry, 1994, pp. 211–220.Google Scholar
  10. [DDS] M. Dickerson, R. L. Drysdale, and J.-R. Sack, Simple algorithms for enumerating interpoint distances and findingk nearest neighbors,Internat. J. Comput. Geom. Appl. 3 (1992), 221–239.CrossRefMathSciNetGoogle Scholar
  11. [DM] M. T. Dickerson and M. H. Montague, A. (usually?) connected subgraph of the minimum weight triangulation,Proc. 12th Ann. Symp. on Computational Geometry, Philadelphia, PA, 1996, pp. 204–213.Google Scholar
  12. [DRA] R. L. Drysdale, G. Rote, and O. Aichholzer, A simple linear time greedy triangulation algorithm for uniformly distributed points, Report IIG-408, Institutes for Information Processing, Technische Universität Graz, February 1995, 16 pages.Google Scholar
  13. [GJ] M. Garey and D. Johnson,Computers and Intractability. A Guide to the Theory of NP-Completeness, Freeman, San Francisco, CA, 1979.zbMATHGoogle Scholar
  14. [G] S. Goldman, A space efficient greedy triangulation algorithm,Inform. Process. Lett. 31 (1989), 191–196.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [HK] J. E. Hopcroft and R. Karp, Ann 5/2 algorithm for maximum matchings in bipartite graphs,SIAM J. Comput. 2 (1973), 225–231.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [HNU] F. Hurtado, M. Noy, and J. Urrutia, Flipping edges in triangulations,Proc. 12th Ann. Symp. on Computational Geometry, Philadelphia, PA, 1996, pp. 214–223.Google Scholar
  17. [K] M. Keil, Computing a subgraph of the minimum weight triangulation,Comput. Geom. Theory Appl. 4 (1994), 13–26.zbMATHMathSciNetGoogle Scholar
  18. [L] E. Lawler,Combinatorial Optimization: Networks and Matroids, Holt, Rinehart, and Winston, New York, 1976.zbMATHGoogle Scholar
  19. [LK1] C. Levcopoulos and D. Krznaric, The greedy triangulation can be computed from the Delaunay in linear time, Tech. Report LU-CS-TR:94-136, Dept. of Computer Science and Numer. Analysis, Lund University, Lund, Sweden, 1994.Google Scholar
  20. [LK2] C. Levcopoulos and D. Krznaric, Quasi-greedy triangulations approximating the minimum weight triangulation,Proc. 7th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), 1996, pp. 392–401.Google Scholar
  21. [T] R. E. Tarjan,Data Structures and Network Algorithms, SIAM, Philadelphia, PA, 1987.Google Scholar
  22. [X] Y.-F. Xu, Minimum weight triangulation problem of a planar point set, Ph.D. Thesis, Institute of Applied Mathematics, Academia Sinica, Beijing, 1992.Google Scholar
  23. [YXY] B.-T. Yang, Y.-F. Xu, and Z.-Y. You, A chain decomposition algorithm for the proof of a property on minimum weight triangulations,Proc. 5th Internat. Symp. on Algorithms and Computation (ISAAC'94), Lecture Notes in Computer Science, Vol. 834, Springer-Verlag, Berlin, 1994, pp. 423–427.Google Scholar
  24. [Y] P. Yoeli, Compilation of data for computer-assisted relief cartography, in: J. C. Davis and M. J. McCullagh (eds.),Display and Analysis of Spatial Data, Wiley, New York, 1975, pp. 352–367.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • O. Aichholzer
    • 1
  • F. Aurenhammer
    • 1
  • Siu-Wing Cheng
    • 2
  • N. Katoh
    • 3
  • G. Rote
    • 4
  • M. Taschwer
    • 1
  • Yin-Feng Xu
    • 5
  1. 1.Institute for Theoretical Computer ScienceGraz University of TechnologyGrazAustria
  2. 2.Department of Computer ScienceHong Kong University of Science and TechnologyClear Water BayHong Kong
  3. 3.Department of Management ScienceKobe University of CommerceKobeJapan
  4. 4.Institut für MathematikTechnische Universität GrazGrazAustria
  5. 5.School of ManagementXi'an Jiaotong UniversityXi'an ShaanxiPeople's Republic of China

Personalised recommendations