Computing the Smallest Color-Spanning Axis-Parallel Square

  • Payam Khanteimouri
  • Ali Mohades
  • Mohammad Ali Abam
  • Mohammad Reza Kazemi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning axis-parallel square. First, for a dynamic set of colored points on the real line, we propose a dynamic structure with O(log2 n) update time per insertion and deletion for maintaining the smallest color-spanning interval. Next, we use this result to compute the smallest color-spanning square. Although we show there could be Ω(kn) minimal color-spanning squares, our algorithm runs in O(nlog2 n) time and O(n) space.


Computational Geometry Algorithm Color-Spanning Objects 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: Smallest Color-Spanning Objects. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 278–289. Springer, Heidelberg (2001)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. Tech. Report. University of Bonn (2006)Google Scholar
  3. 3.
    Consuegra, M.E., Narasimhan, G., Tanigawa, S.: Geometric Avatar Problems. Tech. Report TR-2013-02-25. Florida International University (2013)Google Scholar
  4. 4.
    Das, S., Goswami, P.P., Nandy, S.C.: Smallest Color-Spanning Object Revisited. Int. J. Comput. Geometry Appl. 19, 457–478 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fan, C., Ju, W., Luo, J., Zhu, B.: On Some Geometric Problems of Color-Spanning Sets. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 113–124. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Fan, C., Luo, J., Zhong, F., Zhu, B.: Expected Computations on Color Spanning Sets. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 130–141. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Gupta, P., Janardan, R., Smid, M.H.M.: Further Results on Generalized Intersection Searching Problems: Counting, Reporting, and Dynamization. J. Algorithms 19(2), 282–317 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huttenlocher, D.P., Kedem, K., Sharir, M.: The Upper Envelope of Voronoi Surfaces and Its Applications. Discrete Computational Geometry 9, 267–291 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Löffler, M.: Data Imprecision in Computational Geometry. Ph.D. Thesis. Utrecht University (2009)Google Scholar
  10. 10.
    Matoušek, J.: On Enclosing k Points by a Circle. Information Processing Letters 53(4), 217–221 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Overmars, M.H., van Leeuwen, J.: Maintenance of Configurations in the Plane. J. Comput. Syst. Sci. 23, 166–204 (1981)CrossRefzbMATHGoogle Scholar
  12. 12.
    Smid, M.H.M.: Finding k Points With a Smallest Enclosing Square. MPI-I-92-152, Max-Planck-Institut Inform., Saarbrücken, Germany (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Payam Khanteimouri
    • 1
  • Ali Mohades
    • 1
  • Mohammad Ali Abam
    • 2
  • Mohammad Reza Kazemi
    • 1
  1. 1.Tehran PolytechnicAmirkabir University of TechnologyTehranIran
  2. 2.Sharif University of TechnologyTehranIran

Personalised recommendations