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Randomized range-maxima in nearly-constant parallel time

Extended abstract
  • Omer Berkman
  • Yossi Matias
  • Uzi Vishkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)

Abstract

Given an array of n input numbers, the range-maxima problem is to preprocess the data so that queries of the type “what is the maximum value in subarray [i..j]” can be answered quickly using one processor. We present a randomized preprocessing algorithm that runs in O(log*n) time with high probability, using an optimal number of processors on a CRCW PRAM; each query can be processed in constant time by one processor. We also present a randomized algorithm for a parallel comparison model. Using an optimal number of processors, the preprocessing algorithm runs in O(α(n)) time with high probability; each query can be processed in O(α(n)) time by one processor. (α(n) is the inverse of Ackermann function.) A constant time query can be achieved by some slowdown in the performance of the preprocessing stage.

Keywords

Constant Time Maximum Computation Complete Binary Tree Lower Common Ancestor Preprocessing Algorithm 
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.

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References

  1. 1.
    N. Alon and Y. Azar. The average complexity of deterministic and randomized parallel comparison-sorting algorithms. SIAM J. Comput., 17:1178–1192, 1988.CrossRefGoogle Scholar
  2. 2.
    N. Alon and B. Schieber. Optimal preprocessing for answering on-line product queries. Technical Report 71/87, Eskenasy Inst. of Comp. Sc., Tel Aviv Univ., 1987.Google Scholar
  3. 3.
    A. Amir, G. M. Landau, and U. Vishkin. Efficient pattern matching with scaling. In SODA '90, pages 344–357, 1990.Google Scholar
  4. 4.
    D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian paths and matchings. J. Comput. Syst. Sci., 18:155–193, 1979.CrossRefGoogle Scholar
  5. 5.
    Y. Azar and U. Vishkin. Tight comparison bounds on the complexity of parallel sorting. SIAM J. Comput., 16:458–464, 1987.CrossRefGoogle Scholar
  6. 6.
    O. Berkman, Y. Matias, and U. Vishkin. Randomized range-maxima in nearlyconstant parallel time. Technical Report UMIACS-TR-91-161, Institute for Advanced Computer Studies, Univ. of Maryland, 1991. To appear in j. computational complexity.Google Scholar
  7. 7.
    O. Berkman, B. Schieber, and U. Vishkin. Some doubly logarithmic parallel algorithms based on finding all nearest smaller values. Technical Report UMIACS-TR-88-79, Univ. of Maryland Inst. for Advanced Computer Studies, Oct. 1988. To appear in J. Algorithms as ‘Optimal doubly logarithmic parallel algorithms based on finding all nearest smaller values'.Google Scholar
  8. 8.
    O. Berkman and U. Vishkin. Recursive *-tree parallel data-structure. In FOCS '89, pages 196–202, 1989. Also in UMIACS-TR-90-40, Institute for Advanced Computer Studies, Uniy. of Maryland, March 1991. To appear in SIAM J. Comput. Google Scholar
  9. 9.
    R. B. Boppana. The average-case parallel complexity of sorting. Inf. Process. Lett., 33:145–146, 1989.CrossRefGoogle Scholar
  10. 10.
    A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. J. Comput. Syst. Sci., 30:130–145, 1985.CrossRefGoogle Scholar
  11. 11.
    H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Math. Statistics, 23:493–507, 1952.Google Scholar
  12. 12.
    P. F. Dietz. Heap construction in the parallel comparison tree model. In SWAT '92, pages 140–150, July 1992.Google Scholar
  13. 13.
    H. N. Gabow, J. L. Bentley, and R. E. Tarjan. Scaling and related techniques for geometry problems. In STOC '84, pages 135–143, 1984.Google Scholar
  14. 14.
    J. Gil and Y. Matias. Fast hashing on a PRAM—designing by expectation. In SODA '91, pages 271–280, 1991.Google Scholar
  15. 15.
    J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In FOCS '91, pages 698–710, Oct. 1991.Google Scholar
  16. 16.
    J. Gil and L. Rudolph. Counting and packing in parallel. In ICPP '86, pages 1000–1002, 1986.Google Scholar
  17. 17.
    D. Harel and R. E. Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338–355, 1984.CrossRefGoogle Scholar
  18. 18.
    P. D. MacKenzie. Load balancing requires Ω(log* n) expected time. In SODA '92, pages 94–99, Jan. 1992.Google Scholar
  19. 19.
    Y. Matias and U. Vishkin. Converting high probability into nearly-constant time—with applications to parallel hashing. In STOC '91, pages 307–316, 1991. Also in UMIACS-TR-91-65, Institute for Advanced Computer Studies, Univ. of Maryland, April 1991.Google Scholar
  20. 20.
    V. Ramachandran and U. Vishkin. Efficient parallel triconnectivity in logarithmic parallel time. In AWOC '88, pages 33–42, 1988.Google Scholar
  21. 21.
    R. Reischuk. Probabilistic parallel algorithms for sorting and selection. SIAM J. Comput., 14(2):396–409, May 1985.Google Scholar
  22. 22.
    B. Schieber. Design and analysis of some parallel algorithms. PhD thesis, Dept. of Computer Science, Tel Aviv Univ., 1987.Google Scholar
  23. 23.
    Y. Shiloach and U. Vishkin. Finding the maximum, merging, and sorting in a parallel computation model. J. of Alg., 2:88–102, 1981.CrossRefGoogle Scholar
  24. 24.
    R. E. Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22(2):215–225, Apr. 1975.CrossRefGoogle Scholar
  25. 25.
    L. G. Valiant. Parallelism in comparison problems. SIAM J. Comput., 4:348–355, 1975.CrossRefGoogle Scholar
  26. 26.
    U. Vishkin. Structural parallel algorithmics. In ICALP '91, pages 363–380, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Omer Berkman
    • 1
  • Yossi Matias
    • 2
  • Uzi Vishkin
    • 1
  1. 1.Dept. of Computing, School of Physical Sciences and EngineeringKing's College LondonStrand, LondonEngland
  2. 2.Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA

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