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, Volume 19, Issue 3, pp 265–281 | Cite as

Enumerating Order Types for Small Point Sets with Applications

  • Oswin Aichholzer
  • Franz Aurenhammer
  • Hannes Krasser
Article

Abstract

Order types are a means to characterize the combinatorial properties of a finite point configuration. In particular, the crossing properties of all straight-line segments spanned by a planar n-point set are reflected by its order type. We establish a complete and reliable data base for all possible order types of size n=10 or less. The data base includes a realizing point set for each order type in small integer grid representation. To our knowledge, no such project has been carried out before.

We substantiate the usefulness of our data base by applying it to several problems in computational and combinatorial geometry. Problems concerning triangulations, simple polygonalizations, complete geometric graphs, and k-sets are addressed. This list of applications is not meant to be exhaustive. We believe our data base to be of value to many researchers who wish to examine their conjectures on small point configurations.

computational geometry order types planar straight-line graphs realizable matroids 

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References

  1. 1.
    Aichholzer, O.: The path of a triangulation, in Proc. 15th Ann. ACM Sympos. Computational Geometry, Miami Beach, USA, 1999, pp. 14-23.Google Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F. and Krasser, H.: Enumerating order types for small point sets with applications, In: Proc. 17th Ann. ACM Sympos. Computational Geometry, Medford, USA, 2001, pp. 11-18.Google Scholar
  3. 3.
    Aichholzer, O., Aurenhammer, F., Hurtado, F. and Krasser, H.: Towards compatible triangulations, In: Proc. 7th Ann. Int. Computing and Combinatorics Conf. CoCOON-2001, Guilin, China, Lecture Notes in Comput. Sci. 2108, Springer, New York, 2001, pp. 101–110. To appear in Theoret. Comput. Sci.Google Scholar
  4. 4.
    Aichholzer, O., Hurtado, F. and Noy, M.: On the number of triangulations every planar point set must have, In: Proc. 13th Ann. Canadian Conference on Computational Geometry CCCG 2001, Waterloo, Canada, 2001, pp. 13-16.Google Scholar
  5. 5.
    Aichholzer, O. and Krasser, H.: The point set order type data base: A collection of applications and results, In: Proc. 13th Ann. Canadian Conference on Computational Geometry CCCG 2001, Waterloo, Canada, 2001, pp. 17-20.Google Scholar
  6. 6.
    Arkin, E., Fekete, S., Hurtado, F., Mitchell, J., Noy, M., Sacristán, V. and Sethia, S.: On the reflexivity of point sets, In: Proc. 10th Ann. Fall Workshop on Computational Geometry, Stony Brook, NY, 2000.Google Scholar
  7. 7.
    Aronov, B., Seidel, R. and Souvaine, D.: On compatible triangulations of simple polygons, Comput. Geom. 3 (1993), 27–35.Google Scholar
  8. 8.
    Aurenhammer, F.: Voronoi diagrams-a survey of a fundamental geometric data structure, ACM Comput. Surveys 23 (1991), 345–405.Google Scholar
  9. 9.
    Aronov, B., Erdös, P., Goddard, W., Kleitman, D. J., Klugerman, M., Pach, J. and Schulman, L. J.: Crossing families, Combinatorica 14 (1994), 127–134.Google Scholar
  10. 10.
    Avis, D. and Fukuda, K.: Reverse search for enumeration, Discrete Appl. Math. 65 (1996), 618–632.Google Scholar
  11. 11.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N. and Ziegler, G.: Oriented Matroids, Cambridge Univ. Press, 1993.Google Scholar
  12. 12.
    Bokowski, J. and Guedes de Oliveira, A.: On the generation of oriented matroids, Discrete Comput. Geom. 24 (2000), 197–208.Google Scholar
  13. 13.
    Brodsky, A., Durocher, S. and Gether, E.: The rectilinear crossing number of K 10 is 62, Electron. J. Combinatorics 8 (2001), Research Paper 23.Google Scholar
  14. 14.
    de Berg, M., van Krefeld, M., Overmars, M. and Schwarzkopf, O.: Computational Geometry-Algorithms and Applications, Springer, Berlin, 1997.Google Scholar
  15. 15.
    Dey, T. K.: Improved bounds for planar k-sets and related problems, Discrete Comput. Geom. 19 (1998), 373–382.Google Scholar
  16. 16.
    Erdös, P. and Guy, R. K.: Crossing number problems, Amer. Math. Monthly 88 (1973), 52–58.Google Scholar
  17. 17.
    Felsner, S.: On the number of arrangements of pseudolines, Discrete Comput. Geom. 18 (1997), 257–267.Google Scholar
  18. 18.
    Finschi, L. and Fukuda, K.: Generation of oriented matroids-a graph theoretical approach, Discrete Comput. Geom. 27 (2002), 117–136.Google Scholar
  19. 19.
    Galtier, J., Hurtado, F., Noy, M., Perennes, S. and Urrutia, J.: Simultaneous edge flipping in triangulations, Manuscript, Universitat Politecnica de Catalunya, Barcelona, Spain, 2000. http://www-ma2.upc.es/~hurtado/flipcorner.htmlGoogle Scholar
  20. 20.
    García, A., Noy, M. and Tejel, J.: Lower bounds on the number of crossing-free subgraphs of K N, Comput. Geom. 16 (2000), 211–221.Google Scholar
  21. 21.
    Goodman, J. E.: Pseudoline arrangements, In: J. E. Goodman and J. O'Rourke (eds), Handbook of Discrete and Computational Geometry, CRC Press LLC, Boca Raton, NY, 1997.Google Scholar
  22. 22.
    Goodman, J. E. and Pollack, R.: Multidimensional sorting, SIAM J. Comput. 12 (1983), 484–507.Google Scholar
  23. 23.
    Goodman, J. E., Pollack, R. and Sturmfels, B.: Coordinate representation of order types requires exponential storage, In: Proc. 21st Ann. ACM Sympos. Theory of Computing, 1989, pp. 405-410.Google Scholar
  24. 24.
    Hayward, R. B.: A lower bound for the optimal crossing-free Hamiltonian cycle problem, Discrete Comput. Geom. 2 (1987), 327–343.Google Scholar
  25. 25.
    Kranakis, E. and Urrutia, J.: Isomorphic triangulations with small number of Steiner points, Intertat. J. Comput. Geom. Appl. 9 (1999), 171–180.Google Scholar
  26. 26.
    Krasser, H.: Kompatible Triangulierungen ebener Punktmengen, MS thesis, IGI-TU Graz, Austria, 1999.Google Scholar
  27. 27.
    Saalfeld, A.: Joint triangulations and triangulation maps, In: Proc. 3rd Ann. ACM Sympos. Computational Geometry, Waterloo, Canada, 1987, pp. 195-204.Google Scholar
  28. 28.
    Santos, F. and Seidel, R.: A better bound on the number of triangulations of a planar point set, Manuscript, 2000.Google Scholar
  29. 29.
    Tóth, G. and Valtr, P.: Note on Erdös-Szekeres theorem, Discrete Comput. Geom. 19 (1998), 457–459.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Franz Aurenhammer
    • 1
  • Hannes Krasser
    • 1
  1. 1.Institute for Theoretical Computer ScienceGraz University of TechnologyGrazAustria

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