Computing the Center Region and Its Variants

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10167)

Abstract

We present an \(O(n^2\log ^4 n)\)-time algorithm for computing the center region of a set of n points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes \(O(n^{2+\epsilon })\) time for any \(\epsilon > 0\). It is known that the complexity of the center region is \(\varOmega (n^2)\), thus our algorithm is almost tight.

The second problem we consider is computing a colored version of the center region in the two-dimensional Euclidean space. We present an \(O(n\log ^4 n)\)-time algorithm for this problem.

References

  1. 1.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: Smallest color-spanning objects. In: Heide, F.M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 278–289. Springer, Heidelberg (2001). doi: 10.1007/3-540-44676-1_23 CrossRefGoogle Scholar
  2. 2.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: The farthest color Voronoi diagram and related problems. Technical report, University of Bonn (2006)Google Scholar
  3. 3.
    Agarwal, P.K., Sharir, M., Welzl, E.: Algorithms for center and tverberg points. ACM Trans. Algorithms 5(1), 1–20 (2008)MathSciNetGoogle Scholar
  4. 4.
    Chan, T.M.: An optimal randomized algorithm for maximum Tukey depth. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 430–436 (2004)Google Scholar
  5. 5.
    Jadhav, S., Mukhopadhyay, A.: Computing a centerpoint of a finite planar set of points in linear time. Discrete Comput. Geom. 12(3), 291–312 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Khanteimouri, P., Mohades, A., Abam, M.A., Kazemi, M.R.: Computing the smallest color-spanning axis-parallel square. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 634–643. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-45030-3_59 CrossRefGoogle Scholar
  7. 7.
    Langerman, S., Steiger, W.: Optimization in arrangements. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 50–61. Springer, Heidelberg (2003). doi: 10.1007/3-540-36494-3_6 CrossRefGoogle Scholar
  8. 8.
    Matousek, J.: Computing the center of a planar point set. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year. American Mathematical Society (1991)Google Scholar
  9. 9.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringPOSTECHPohangSouth Korea

Personalised recommendations