Localized Query: Color Spanning Variations

  • Ankush AcharyyaEmail author
  • Anil Maheshwari
  • Subhas C. Nandy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)


Let P be a set of n points and each of the points is colored with one of the k possible colors. We present efficient algorithms to pre-process P such that for a given query point q, we can quickly identify the smallest color spanning object of the desired type containing q. In this paper, we focus on (i) intervals, (ii) axis-parallel square, (iii) axis-parallel rectangle, (iv) equilateral triangle of fixed orientation, as our desired type of objects.


Color-spanning object Multilevel range searching Localized query 


  1. 1.
    Abellanas, M., et al.: Smallest color-spanning objects. In: auf der Heide, F.M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 278–289. Springer, Heidelberg (2001). Scholar
  2. 2.
    Augustine, J., Das, S., Maheshwari, A., Nandy, S.C., Roy, S., Sarvattomananda, S.: Localized geometric query problems. Comput. Geom. 46(3), 340–357 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Augustine, J., Das, S., Maheshwari, A., Nandy, S.C., Roy, S., Sarvattomananda, S.: Recognizing the largest empty circle and axis-parallel rectangle in a desired location. arXiv preprint arXiv:1004.0558 (2010)
  4. 4.
    Augustine, J., Putnam, B., Roy, S.: Largest empty circle centered on a query line. J. Discret. Algorithms 8(2), 143–153 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008). Scholar
  6. 6.
    Bint, G., Maheshwari, A., Nandy, S.C., Smid, M.: Partial enclosure range searching. PreprintGoogle Scholar
  7. 7.
    Chan, T.M.: Optimal partition trees. Discret. Comput. Geom. 47(4), 661–690 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, D.Z., Misiołek, E.: Algorithms for interval structures with applications. Theoret. Comput. Sci. 508, 41–53 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Das, S., Goswami, P.P., Nandy, S.C.: Smallest color-spanning object revisited. Int. J. Comput. Geom. Appl. 19(5), 457–478 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gester, M., Hähnle, N., Schneider, J.: Largest empty square queries in rectilinear polygons. In: Gervasi, O., et al. (eds.) ICCSA 2015. LNCS, vol. 9155, pp. 267–282. Springer, Cham (2015). Scholar
  11. 11.
    Hasheminejad, J., Khanteimouri, P., Mohades, A.: Computing the smallest color spanning equilateral triangle. In: Proceedings of 31st EuroCG, pp. 32–35 (2015)Google Scholar
  12. 12.
    Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of voronoi surfaces and its applications. Discret. Comput. Geom. 9, 267–291 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    JaJa, J., Mortensen, C.W., Shi, Q.: Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 558–568. Springer, Heidelberg (2004). Scholar
  14. 14.
    Jiang, M., Wang, H.: Shortest color-spanning intervals. Theoret. Comput. Sci. 609, 561–568 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kaminker, T., Sharir, M.: Finding the largest disk containing a query point in logarithmic time with linear storage. JoCG 6(2), 3–18 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kaplan, H., Sharir, M.: Finding the maximal empty rectangle containing a query point. arXiv preprint arXiv:1106.3628 (2011)
  17. 17.
    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). Scholar
  18. 18.
    Khanteimouri, P., Mohades, A., Abam, M.A., Kazemi, M.R.: Spanning colored points with intervals. In: CCCG (2013)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ankush Acharyya
    • 1
    Email author
  • Anil Maheshwari
    • 2
  • Subhas C. Nandy
    • 1
  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Carleton UniversityOttawaCanada

Personalised recommendations