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The Union of Colorful Simplices Spanned by a Colored Point Set

  • André Schulz
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6508)

Abstract

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d ∈ ℕ, the maximum combinatorial complexity of the colorful union of n colored points in ℝ d is between \(\Omega(n^{(d-1)^2})\) and \(O(n^{(d-1)^2}\log n)\). For d = 2, the upper bound is known to be O(n), and for d = 3 we present an upper bound of O(n 4 α(n)), where α(·) is the extremely slowly growing inverse Ackermann function. We also prove several structural properties of the colorful union. In particular, we show that the boundary of the colorful union is covered by O(n d − 1) hyperplanes, and the colorful union is the union of d + 1 star-shaped polyhedra. These properties lead to efficient data structures for point inclusion queries in the colorful union.

Keywords

Convex Hull Combinatorial Complexity Relative Interior Colorful Union Colored Point 
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.

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References

  1. 1.
    Afshani, P., Chan, T.M.: On approximate range counting and depth. Discrete Comput. Geom. 42(1), 3–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P.K.: Range searching. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 36, pp. 809–838. CRC Press, Boca Raton (2004)Google Scholar
  3. 3.
    Agarwal, P.K., Matoušek, J.: Ray shooting and parametric search. SlAM J. Comput. 22, 794–806 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agarwal, P.K., Sharir, M.: Davenport-Schinzel sequences and their geometric applications. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, ch. 1, pp. 1–47. Elsevier, Amsterdam (2000)Google Scholar
  5. 5.
    Akiyama, J., Alon, N.: Disjoint simplices and geometric hypergraphs. In: Proceedings of the Third International Conference on Combinatorial Mathematics, pp. 1–3. Academy of Sciences, New York (1989)Google Scholar
  6. 6.
    Alon, N., Bárány, I., Füredi, Z., Kleitman, D.J.: Point selections and weak ε-nets for convex hulls. Combin. Probab. Comput. 1(3), 189–200 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arocha, J.L., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142–154 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aronov, B., Sharir, M.: Castles in the air revisited. Discrete Comput. Geom. 12(1), 119–150 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2-3), 141–152 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bárány, I., Onn, S.: Carathéodory’s theorem, colourful and applicable. Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest 6, 11–21 (1997)zbMATHGoogle Scholar
  11. 11.
    Bárány, I., Onn, S.: Colourful linear programming and its relatives. Math. Oper. Res. 22(3), 550–567 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boissonnat, J.-D., Devillers, O., Preparata, F.P.: Computing the union of 3-colored triangles. Intern. J. Comput. Geom. Appl. 1, 187–196 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cheng, A.Y., Ouyang, M.: On algorithms for simplicial depth. In: Canadian Conf. Comput. Geom., pp. 53–56 (2001)Google Scholar
  14. 14.
    de Berg, M.: Ray Shooting, Depth Orders and Hidden Surface Removal. LNCS, vol. 703. Springer, Heidelberg (1993); ch. 5, Ray shooting from a fixed point, pp. 53–65CrossRefzbMATHGoogle Scholar
  15. 15.
    Dey, T.K., Pach, J.: Extremal problems for geometric hypergraphs. Discrete Comput. Geom. 19, 473–484 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Deza, A., Huang, S., Stephen, T., Terlaky, T.: Colourful simplicial depth. Discrete Comput. Geom. 35(4), 597–615 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Deza, A., Huang, S., Stephen, T., Terlaky, T.: The colourful feasibility problem. Discrete Appl. Math. 156(11), 2166–2177 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Edelsbrunner, H.: The upper envelope of piecewise linear functions: Tight bounds on the number of faces. Discrete Comput. Geom. 4(1), 337–343 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ezra, E., Aronov, B., Sharir, M.: Improved bound for the union of fat triangles (2010) (manuscript)Google Scholar
  20. 20.
    Ezra, E., Sharir, M.: Almost tight bound for the union of fat tetrahedra in three dimensions. In: Proc. 48th Sympos. on Foundations of Comp. Sci. (FOCS), pp. 525–535. IEEE, Los Alamitos (2007)Google Scholar
  21. 21.
    Gil, J., Steiger, W., Wigderson, A.: Geometric medians. Discrete Math. 108(1-3), 37–51 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Holmsen, A.F., Pach, J., Tverberg, H.: Points surrounding the origin. Combinatorica 28(6), 633–644 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khuller, S., Mitchell, J.S.B.: On a triangle counting problem. Inf. Process. Lett. 33(6), 319–321 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Math, vol. 212. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Matoušek, J., Pach, J., Sharir, M., Sifrony, S., Welzl, E.: Fat triangles determine linearly many holes. SIAM J. Comput. 23, 154–169 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pach, J., Safruti, I., Sharir, M.: The union of congruent cubes in three dimensions. Discrete Comput. Geom. 30, 133–160 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pach, J., Sharir, M.: The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis. Discrete Comput. Geom. 4, 291–310 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pach, J., Tardos, G.: On the boundary complexity of the union of fat triangles. SIAM J. Comput. 31, 1745–1760 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pellegrini, M.: Ray shooting and lines in space. In: Goodman, J., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 36, pp. 809–838. CRC Press, Boca Raton (2004)Google Scholar
  30. 30.
    Tagansky, B.: A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Discrete Comput. Geom. 16(4), 455–479 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Živaljević, R.T., Vrećica, S.T.: The colored Tverberg’s problem and complexes of injective functions. J. Combin. Theory Ser. A 61(2), 309–318 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wiernik, A., Sharir, M.: Planar realization of nonlinear Davenport Schinzel sequences by segments. Discrete Comput. Geom. 3, 15–47 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • André Schulz
    • 1
  • Csaba D. Tóth
    • 2
  1. 1.Institut für Mathematsche Logik und GrundlagenforschungUniversität MünsterGermany
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCanada

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