Amortized Bounds for Dynamic Orthogonal Range Reporting

  • Bryan T. Wilkinson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)


We consider the fundamental problem of 2-D dynamic orthogonal range reporting for 2- and 3-sided queries in the standard word RAM model. While many previous dynamic data structures use O(logn /loglogn) update time, we achieve faster O(log1/2 + ε n) and O(log2/3 + ε n) update times for 2- and 3-sided queries, respectively. Our data structures have optimal O(logn / loglogn) query time. Only Mortensen [14] had previously lowered the update time convincingly below O(logn), with 3- and 4-sided data structures supporting updates in O(log5/6 + ε n) and O(log7/8 + ε n) time, respectively. In practice, fast updates are often as important as fast queries, so we make a step forward for an important problem that has not seen any progress in recent years. We also obtain new results for the special case of 3-sided insertion-only emptiness, showing that the difference in complexity between fully dynamic and partially dynamic 2-D orthogonal range reporting can be significant (i.e., Ω(polylog n) factor differences). In particular, we achieve O((logn loglogn)2/3) update time and O((logn loglogn)1/3) query time. At the other end of our update/query trade-off curve, we achieve O(logn/loglogn) update time and O(loglogn) query time. In contrast, in the pointer machine model, there are only O(loglogn) factor differences between the complexities of fully dynamic and partially dynamic 2-D orthogonal range reporting.


Query Range Query Time External Memory Lower Common Ancestor Secondary Node 
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: Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 534–544 (1998)Google Scholar
  2. 2.
    Andersson, A., Thorup, M.: Dynamic ordered sets with exponential search trees. Journal of the ACM (JACM) 54(3) (2007)Google Scholar
  3. 3.
    Arge, L., Samoladas, V., Vitter, J.S.: On two-dimensional indexability and optimal range search indexing. In: Proceedings of the ACM Symposium on Principles of Database Systems (PODS), pp. 346–357 (1999)Google Scholar
  4. 4.
    Bender, M.A., Cole, R., Demaine, E.D., Farach-Colton, M., Zito, J.: Two simplified algorithms for maintaining order in a list. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 152–164. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Bentley, J.L.: Multidimensional divide-and-conquer. Communications of the ACM (CACM) 23(4), 214–229 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dietz, P.F., Sleator, D.D.: Two algorithms for maintaining order in a list. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 365–372 (1987)Google Scholar
  7. 7.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Information Processing Letters (IPL) 6(3), 80–82 (1977)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fredman, M.L., Willard, D.E.: Blasting through the information theoretic barrier with fusion trees. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 1–7 (1990)Google Scholar
  9. 9.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal of Computing 13(2), 338–355 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Itai, A., Konheim, A.G., Rodeh, M.: A sparse table implementation of priority queues. In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, pp. 417–431. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  11. 11.
    Kumar, V., Schwabe, E.J.: Improved algorithms and data structures for solving graph problems in external memory. In: Proceedings of the Annual IEEE Symposium on Parallel and Distributed Processing (SPDP), pp. 169–176 (1996)Google Scholar
  12. 12.
    McCreight, E.M.: Priority search trees. SIAM Journal of Computing 14(2), 257–276 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Mehlhorn, K., Näher, S.: Dynamic fractional cascading. Algorithmica 5(2), 215–241 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Mortensen, C.W.: Fully dynamic orthogonal range reporting on RAM. SIAM Journal of Computing 35(6), 1494–1525 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Willard, D.E.: Examining computational geometry, van Emde Boas trees, and hashing from the perspective of the fusion tree. SIAM Journal of Computing 29(3), 1030–1049 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Bryan T. Wilkinson
    • 1
  1. 1.MADALGOAarhus UniversityAarhusDenmark

Personalised recommendations