Skip to main content
SpringerLink
Log in

On Space Efficient Two Dimensional Range Minimum Data Structures

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The two dimensional range minimum query problem is to preprocess a static m by n matrix (two dimensional array) A of size N=m⋅n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using a data structure of size O(N/c) bits requires Ω(c) query time, for any c where 1≤c≤N. This lower bound holds for arrays of any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits which can be preprocessed in O(N) time to support O(clog 2 c) query time. For c=O(1), this is the first O(1) query time algorithm using a data structure of optimal size O(N) bits. For the case where queries can not probe A, we give a data structure of size O(N⋅min {m,log n}) bits with O(1) query time, assuming m≤n. This leaves a gap to the space lower bound of Ω(Nlog m) bits for this version of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abouelhoda, M.I., Kurtz, S., Ohlebusch, E.: Replacing suffix trees with enhanced suffix arrays. J. Discrete Algorithms 2(1), 53–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aho, A., Hopcroft, J., Ullman, J.: On finding lowest common ancestors in trees. In: Proc. 5th Annual ACM Symposium on Theory of Computing, pp. 253–265. ACM, New York (1973)

    Chapter  Google Scholar 

  3. Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest common ancestors: a survey and a new distributed algorithm. In: Proc. 14th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 258–264. ACM, New York (2002)

    Google Scholar 

  4. Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Proc. 18th Annual Symposium on Combinatorial Pattern Matching. Lecture Notes in Computer Science, vol. 4580, pp. 286–294. Springer, Berlin (2007)

    Chapter  Google Scholar 

  5. Atallah, M.J., Yuan, H.: Data structures for range minimum queries in multidimensional arrays. In: Proc. 20th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 150–160. SIAM, Philadelphia (2010)

    Google Scholar 

  6. Bender, M., Farach-Colton, M.: The LCA problem revisited. In: Proc. 4th Latin American Theoretical Informatics Symposium. Lecture Notes in Computer Science, vol. 1776, pp. 88–94. Springer, Berlin (2000)

    Google Scholar 

  7. Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. J. Algorithms 57(2), 75–94 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bentley, J.L.: Decomposable searching problems. Inf. Process. Lett. 8(5), 244–251 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berkman, O., Galil, Z., Schieber, B., Vishkin, U.: Highly parallelizable problems. In: Proc. 21st Annual ACM Symposium on Theory of Computing, pp. 309–319. ACM, New York (1989)

    Google Scholar 

  10. Chazelle, B., Rosenberg, B.: Computing partial sums in multidimensional arrays. In: Proc. 5th Annual Symposium on Computational Geometry, pp. 131–139. ACM, New York (1989)

    Chapter  Google Scholar 

  11. Chen, G., Puglisi, S.J., Smyth, W.F.: Lempel-Ziv factorization using less time and space. Math. Comput. Sci. 1, 605–623 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Demaine, E.D., Landau, G.M., Weimann, O.: On Cartesian trees and range minimum queries. In: Proc. 36th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 5555, pp. 341–353. Springer, Berlin (2009)

    Chapter  Google Scholar 

  13. Fischer, J.: Optimal succinctness for range minimum queries. In: Proc. 9th Latin American Theoretical Informatics Symposium. Lecture Notes in Computer Science, vol. 6034, pp. 158–169. Springer, Berlin (2010)

    Google Scholar 

  14. Fischer, J., Heun, V.: Theoretical and practical improvements on the rmq-problem, with applications to lca and lce. In: Proc. 17th Annual Symposium on Combinatorial Pattern Matching. Lecture Notes in Computer Science, vol. 4009, pp. 36–48. Springer, Berlin (2006)

    Chapter  Google Scholar 

  15. Fischer, J., Heun, V.: A new succinct representation of rmq-information and improvements in the enhanced suffix array. In: Proc. 1st International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies. Lecture Notes in Computer Science, vol. 4614, pp. 459–470. Springer, Berlin (2007)

    Chapter  Google Scholar 

  16. Fischer, J., Mäkinen, V., Navarro, G.: An(other) entropy-bounded compressed suffix tree. In: Proc. 19th Annual Symposium on Combinatorial Pattern Matching. Lecture Notes in Computer Science, vol. 5029, pp. 152–165. Springer, Berlin (2008)

    Chapter  Google Scholar 

  17. Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proc. 16th Annual ACM Symposium on Theory of Computing, pp. 135–143. ACM, New York (1984)

    Google Scholar 

  18. Gál, A., Miltersen, P.B.: The cell probe complexity of succinct data structures. Theor. Comput. Sci. 379(3), 405–417 (2007)

    Article  MATH  Google Scholar 

  19. Georgiadis, L., Tarjan, R.E.: Finding dominators revisited: extended abstract. In: Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 869–878. SIAM, Philadelphia (2004)

    Google Scholar 

  20. Golynski, A.: Optimal lower bounds for rank and select indexes. Theor. Comput. Sci. 387(3), 348–359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Iliopoulos, C.S., Crochemore, M., Kubica, M., Rahman, M.S., Walen, T.: Improved algorithms for the range next value problem and applications. In: Proc. 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics, vol. 1, pp. 205–216. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Leibniz (2008)

    Google Scholar 

  23. Miltersen, P.B.: Cell probe complexity—a survey. Advances in Data Structures Workshop (Pre-conference Workshop of Foundations of Software Technology and Theoretical Computer Science) (1999). http://www.daimi.au.dk/~bromille/Papers/survey3.ps

  24. Muthukrishnan, S.: Efficient algorithms for document retrieval problems. In: Proc. 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 657–666. SIAM, Philadelphia (2002)

    Google Scholar 

  25. Poon, C.K.: Dynamic orthogonal range queries in OLAP. Theory Comput. Syst. 296(3), 487–510 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sadakane, K.: Compressed suffix trees with full functionality. Theory Comput. Syst. 41(4), 589–607 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sadakane, K.: Succinct data structures for flexible text retrieval systems. J. Discrete Algorithms 5(1), 12–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saxena, S.: Dominance made simple. Inf. Process. Lett. 109(9), 419–421 (2009)

    Article  MATH  Google Scholar 

  29. Schieber, B., Vishkin, U.: On finding lowest common ancestors: simplification and parallelization. SIAM J. Comput. 17(6), 1253–1262 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Välimäki, N., Mäkinen, V.: Space-efficient algorithms for document retrieval. In: Proc. 18th Annual Symposium on Combinatorial Pattern Matching. Lecture Notes in Computer Science, vol. 4580, pp. 205–215. Springer, Berlin (2007)

    Chapter  Google Scholar 

  31. Vuillemin, J.: A unifying look at data structures. Commun. ACM 23(4), 229–239 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MADALGO (Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation), Department of Computer Science, Aarhus University, IT Parken, Åbogade 34, 8200, Aarhus N, Denmark

    Gerth Stølting Brodal & Pooya Davoodi

  2. School of Computer Science and Engineering, Seoul National University, 599 Gwanakro, Gwanak-Gu, Seoul, 151-744, South Korea

    S. Srinivasa Rao

Authors
  1. Gerth Stølting Brodal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pooya Davoodi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Srinivasa Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Srinivasa Rao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brodal, G.S., Davoodi, P. & Rao, S.S. On Space Efficient Two Dimensional Range Minimum Data Structures. Algorithmica 63, 815–830 (2012). https://doi.org/10.1007/s00453-011-9499-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9499-0

Keywords

Search

Navigation