On Counting Range Maxima Points in Plane

  • Anil Kishore Kalavagattu
  • Jatin Agarwal
  • Ananda Swarup Das
  • Kishore Kothapalli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)

Abstract

We consider the problem of reporting and counting maximal points in a given orthogonal query range in two-dimensions. Our model of computation is the pointer machine model. Let P be a static set of n points in ℝ2. A point is maximal in P if it is not dominated by any other point in P. We propose a linear space data structure that can support counting the number of maximal points inside a 3-sided orthogonal query rectangle unbounded on its right in O(logn) time. For counting the number of maximal points in a 4-sided orthogonal query rectangle, we propose an O(n logn) space data structure that can be constructed in O(n logn) time and queried upon in O(logn) time. This work proposes the first data structure for counting the number of maximal points in a query range. Das et al. proposed a data structure for the counting version in the word RAM model [WALCOM 2012].

For the corresponding reporting versions, we propose a linear size data structure for reporting maximal points inside a 3-sided query range in time O(logn + k), where k is the size of the output. We propose an O(n logn) size data structure for reporting the maximal points in a 4-sided orthogonal query range in time O(logn + k), where k is the size of the output. The methods we propose for reporting maximal points are simpler than previous methods and meet the best known bounds.

Keywords

Maxima Plane Orthogonal range Reporting Counting 

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References

  1. 1.
    Bentley, J.L.: Multidimensional divide-and-conquer. Communications of the ACM 23(4), 214–229 (1980)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    JáJá, 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)CrossRefGoogle Scholar
  3. 3.
    Chan, C.-Y., Jagadish, H.V., Tan, K.-L., Tung, A.K.H., Zhang, Z.: Finding k-dominant skylines in high dimensional space. In: Proceedings of the ACM SIGMOD International Conference on Management of Data (2006)Google Scholar
  4. 4.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. Journal of the ACM 22(4), 469–476 (1975)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Frederickson, G.N., Rodger, S.H.: A New Approach to the Dynamic Maintenance of Maximal Points in a Plane. Discrete & Comp. Geom. 5, 365–374 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Janardan, R.: On the Dynamic Maintenance of Maximal Points in the Plane. Information Processing Letters 40(2), 59–64 (1991)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Brodal, G.S., Tsakalidis, K.: Dynamic Planar Range Maxima Queries. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 256–267. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Kalavagattu, A.K., Das, A.S., Kothapalli, K., Srinathan, K.: On Finding Skyline Points for Range Queries in Plane. In: Proceedings of 23rd Canadian Conference on Computational Geometry (CCCG), pp. 343–346 (2011)Google Scholar
  9. 9.
    Bayer, R.: Symmetric Binary B-Trees: Data Structure and Maintenance Algorithms. Acta Informatica 1, 290–306 (1972)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Sarnak, N., Tarjan, R.E.: Planar point location using persistent search trees. Communications of the ACM 29, 669–679 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Das, A.S., Gupta, P., Kalavagattu, A.K., Agarwal, J., Srinathan, K., Kothapalli, K.: Range Aggregate Maximal Points in the Plane. In: Rahman, M. S., Nakano, S.-i. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 52–63. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer (2000) ISBN 3-540-65620-0Google Scholar
  13. 13.
    Yu, C.C., Hon, W.K., Wang, B.F.: Improved Data Structures for Orthogonal Range Successor Queries. Computational Geometry: Theory and Applications 44, 148–159 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anil Kishore Kalavagattu
    • 1
  • Jatin Agarwal
    • 1
  • Ananda Swarup Das
    • 1
  • Kishore Kothapalli
    • 1
  1. 1.International Institute of Information TechnologyHyderabadIndia

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