On (Dynamic) Range Minimum Queries in External Memory

  • Lars Arge
  • Johannes Fischer
  • Peter Sanders
  • Nodari Sitchinava
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

We study the one-dimensional range minimum query (RMQ) problem in the external memory model. We provide the first space-optimal solution to the batched static version of the problem. On an instance with N elements and Q queries, our solution takes Θ(sort(N + Q)) = Θ(\(N+Q \over B\) logM /B\(N+Q \over B\)) I/O complexity and O(N + Q) space, where M is the size of the main memory and B is the block size. This is a factor of O(logM /BN) improvement in space complexity over the previous solutions. We also show that an instance of the batched dynamic RMQ problem with N updates and Q queries can be solved in O (\(N+Q \over B\)\(\log^{2}_{M /B}\)\(N+Q \over B\)) I/O complexity and O(N + Q) space.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Communications of the ACM 31(9), 1116–1127 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arge, L.: The buffer tree: A technique for designing batched external data structures. Algorithmica 37(1), 1–24 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arge, L., Goodrich, M.T., Nelson, M.J., Sitchinava, N.: Fundamental parallel algorithms for private-cache chip multiprocessors. In: SPAA, pp. 197–206 (2008)Google Scholar
  4. 4.
    Arge, L., Goodrich, M.T., Sitchinava, N.: Parallel external memory graph algorithms. In: IPDPS, pp. 1–11 (2010)Google Scholar
  5. 5.
    Arge, L., Procopiuc, O., Ramaswamy, S., Suel, T., Vitter, J.S.: Theory and practice of I/O-efficient algorithms for multidimensional batched searching problems. In: SODA, pp. 685–694 (1998)Google Scholar
  6. 6.
    Arge, L., Toma, L., Zeh, N.: I/O-efficient topological sorting of planar DAGs. In: SPAA, pp. 85–93. ACM Press (2003)Google Scholar
  7. 7.
    Chiang, Y.J., Goodrich, M.T., Grove, E.F., Tamassia, R., Vengroff, D.E., Vitter, J.S.: External-memory graph algorithms. In: SODA, pp. 139–149 (1995)Google Scholar
  8. 8.
    Fischer, J., Heun, V.: Space efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput. 40(2), 465–492 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fischer, J., Mäkinen, V., Välimäki, N.: Space efficient string mining under frequency constraints. In: Proc. ICDM, pp. 193–202. IEEE Computer Society (2008)Google Scholar
  10. 10.
    Franceschini, G., Grossi, R.: A general technique for managing strings in comparison-driven data structures. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 606–617. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Kärkkäinen, J., Sanders, P., Burkhardt, S.: Linear work suffix array construction. J. ACM 53(6), 1–19 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Raman, R.: Range extremum queries. In: Smyth, B. (ed.) IWOCA 2012. LNCS, vol. 7643, pp. 280–287. Springer, Heidelberg (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Lars Arge
    • 1
  • Johannes Fischer
    • 2
  • Peter Sanders
    • 2
  • Nodari Sitchinava
    • 2
  1. 1.MADALGOAarhus UniversityAarhusDenmark
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations