Advertisement

A Simple Linear-Space Data Structure for Constant-Time Range Minimum Query

  • Stephane Durocher
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8066)

Abstract

We revisit the range minimum query problem and present a new O(n)-space data structure that supports queries in O(1) time. Although previous data structures exist whose asymptotic bounds match ours, our goal is to introduce a new solution that is simple, intuitive, and practical without increasing asymptotic costs for query time or space.

Keywords

Minimum Element Query Range Query Time Recursive Call Range Minimum 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest commmon ancestors: a survey and a new algorithms for a distributed environment. Theory of Computing Systems 37(3), 441–456 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amir, A., Fischer, J., Lewenstein, M.: Two-dimensional range minimum queries. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 286–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Arlazarov, V.L., Dinic, E.A., Kronrod, M.A., Faradžev, I.A.: On economical construction of the transitive closure of a directed graph. Soviet Mathematics—Doklady 11(5), 1209–1210 (1970)zbMATHGoogle Scholar
  4. 4.
    Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. Journal of Algorithms 57(2), 75–94 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berkman, O., Vishkin, U.: Recursive star-tree parallel data structures. SIAM Journal on Computing 22(2), 221–242 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    Brodal, G.S., Davoodi, P., Srinivasa Rao, S.: Path minima queries in dynamic weighted trees. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 290–301. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Brodal, G.S., Davoodi, P., Rao, S.S.: On space efficient two dimensional range minimum data structures. Algorithmica 63(4), 815–830 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brodnik, A., Carlsson, S., Demaine, E.D., Munro, J.I., Sedgewick, R.D.: Resizable arrays in optimal time and space. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 37–48. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. In: Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS). Leibniz International Proceedings in Informatics, vol. 14, pp. 291–301 (2012)Google Scholar
  12. 12.
    Chan, T.M., Durocher, S., Larsen, K.G., Morrison, J., Wilkinson, B.T.: Linear-space data structures for range mode query in arrays. Theory of Computing Systems (to appear, 2013)Google Scholar
  13. 13.
    Davoodi, P.: Data Structures: Range Queries and Space Efficiency. PhD thesis, Aarhus University (2011)Google Scholar
  14. 14.
    Davoodi, P., Raman, R., Satti, S.R.: Succinct representations of binary trees for range minimum queries. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 396–407. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Demaine, E.D., Landau, G.M., Weimann, O.: On Cartesian trees and range minimum queries. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 341–353. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Information Processing Letters 6(3), 80–82 (1977)CrossRefzbMATHGoogle Scholar
  17. 17.
    Fischer, J.: Optimal succinctness for range minimum queries. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 158–169. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Fischer, J., Heun, V.: Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE. In: Lewenstein, M., Valiente, G. (eds.) CPM 2006. LNCS, vol. 4009, pp. 36–48. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Fischer, J., Heun, V.: A new succinct representation of RMQ-information and improvements in the enhanced suffix array. In: Chen, B., Paterson, M., Zhang, G. (eds.) ESCAPE 2007. LNCS, vol. 4614, pp. 459–470. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Fischer, J., Heun, V.: Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM Journal on Computing 40(2), 465–492 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Golin, M., Iacono, J., Krizanc, D., Raman, R., Rao, S.S.: Encoding 2D range maximum queries. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 180–189. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing 13(2), 338–355 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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
  24. 24.
    Munro, J.I.: Tables. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 37–42. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  25. 25.
    Pasailă, D.: Range minimum query and lowest common ancestor, http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=lowestCommonAncestor
  26. 26.
    Sadakane, K.: Succinct data structures for flexible text retrieval systems. Journal of Discrete Algorithms 5, 12–22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schieber, B., Vishkin, U.: On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing 17(6), 1253–1262 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Skala, M.: Array range queries. In: Brodnik, A., Lopez-Ortiz, A., Raman, V., Viola, A. (eds.) Munro Festschrift. LNCS, vol. 8066, pp. 337–354. Springer, Heidelberg (2013)Google Scholar
  29. 29.
    Yuan, H., Atallah, M.J.: Data structures for range minimum queries. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 150–160 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stephane Durocher
    • 1
  1. 1.University of ManitobaWinnipegCanada

Personalised recommendations

Cite chapter

Buy options