Dynamic Range Selection in Linear Space

  • Meng He
  • J. Ian Munro
  • Patrick K. Nicholson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


Given a set S of n points in the plane, we consider the problem of answering range selection queries on S: that is, given an arbitrary x-range Q and an integer k > 0, return the k-th smallest y-coordinate from the set of points that have x-coordinates in Q. We present a linear space data structure that maintains a dynamic set of n points in the plane with real coordinates, and supports range selection queries in \(O((\lg n / \lg \lg n)^2)\) time, as well as insertions and deletions in \(O((\lg n / \lg \lg n)^2)\) amortized time. The space usage of this data structure is an \(\Theta(\lg n / \lg \lg n)\) factor improvement over the previous best result, while maintaining asymptotically matching query and update times. We also present a succinct data structure that supports range selection queries on a dynamic array of n values drawn from a bounded universe.


Internal Node Matrix Structure Query Range Query Time Ranking Tree 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alstrup, S., Husfeldt, T., Rauhe, T.: Marked ancestor problems. In: Proc. 39th Annual Symposium on Foundations of Computer Science, pp. 534–543. IEEE (1998)Google Scholar
  2. 2.
    Arge, L., Vitter, J.S.: Optimal external memory interval management. SIAM J. Comput. 32(6), 1488–1508 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bose, P., Kranakis, E., Morin, P., Tang, Y.: Approximate Range Mode and Range Median Queries. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 377–388. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Brodal, G., Jørgensen, A.: Data Structures for Range Median Queries. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 822–831. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Brodal, G., Gfeller, B., Jorgensen, A., Sanders, P.: Towards optimal range medians. Theoretical Computer Science (2010)Google Scholar
  6. 6.
    Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms, 2nd edn. McGraw-Hill Higher Education (2001)Google Scholar
  7. 7.
    Gagie, T., Puglisi, S.J., Turpin, A.: Range Quantile Queries: Another Virtue of Wavelet Trees. In: Karlgren, J., Tarhio, J., Hyyrö, H. (eds.) SPIRE 2009. LNCS, vol. 5721, pp. 1–6. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Gfeller, B., Sanders, P.: Towards Optimal Range Medians. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 475–486. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Gil, J., Werman, M.: Computing 2-d min, median, and max filters. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(5), 504–507 (1993)CrossRefGoogle Scholar
  10. 10.
    Har-Peled, S., Muthukrishnan, S.: Range Medians. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 503–514. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    He, M., Munro, J.I.: Succinct Representations of Dynamic Strings. In: Chavez, E., Lonardi, S. (eds.) SPIRE 2010. LNCS, vol. 6393, pp. 334–346. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    He, M., Munro, J.I.: Space Efficient Data Structures for Dynamic Orthogonal Range Counting. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 500–511. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Jacobson, G.: Space-efficient static trees and graphs. In: Proc. SFCS, pp. 549–554 (1989)Google Scholar
  14. 14.
    Jørgensen, A., Larsen, K.: Range selection and median: Tight cell probe lower bounds and adaptive data structures. In: Proc. SODA (2011)Google Scholar
  15. 15.
    Krizanc, D., Morin, P., Smid, M.: Range mode and range median queries on lists and trees. Nordic Journal of Computing 12, 1–17 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Larsen, K.: The cell probe complexity of dynamic range counting. Arxiv preprint arXiv:1105.5933 (2011)Google Scholar
  17. 17.
    Pǎtraşcu, M.: Lower bounds for 2-dimensional range counting. In: Proc. 39th ACM Symposium on Theory of Computing (STOC), pp. 40–46 (2007)Google Scholar
  18. 18.
    Petersen, H.: Improved Bounds for Range Mode and Range Median Queries. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 418–423. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Petersen, H., Grabowski, S.: Range mode and range median queries in constant timeand sub-quadratic space. Inf. Process. Lett. 109, 225–228 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Raman, R., Raman, V., Rao, S.S.: Succinct Dynamic Data Structures. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 426–437. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Meng He
    • 1
  • J. Ian Munro
    • 2
  • Patrick K. Nicholson
    • 2
  1. 1.Faculty of Computer ScienceDalhousie UniversityCanada
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada

Personalised recommendations