A Survey of Randomness and Parallism in Comparison Problems

  • Danny Krizanc
Part of the Combinatorial Optimization book series (COOP, volume 5)

Abstract

A survey of results for the problems of selection, merging and sorting in the Randomized Parallel Comparison Tree (RPCT) model is given. The results indicate that while randomization “helps” in the case of selection, it does not provide any advantage for the cases of merging and sorting, in this model.

Keywords

Sorting Paral Dian 

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References

  1. [1]
    A. Aho, J. Hopcroft And J. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974.MATHGoogle Scholar
  2. [2]
    M. Ajtai, J. KomlÓs and E. SzemerÉdi, An O(N log N) Sorting Network, Proc. of 15th ACM Symp. on Theory of Computing, 1983, pp. 1–9.Google Scholar
  3. [3]
    M. Ajtai, J. KomlÓs, W. L. Steiger, and E. SzemerÉdi, Optimal Parallel Selection Has Complexity O(log log N), J. of Computer and System Sciences, 38 (1989), pp. 125–133.MATHGoogle Scholar
  4. [4]
    N. Alon And Y. Azar, Sorting, Approximate Sorting and Searching in Rounds, SIAM J. of Discrete Mathematics, 1 (1988), pp. 269–280.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    N. Alon And Y. Azar, The Average Complexity of Deterministic and Randomized Parallel Comparison Sorting Algorithms, SIAM J of Computing, 17 (1988), pp. 1178–1192.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    N. Alon, Y. Azar And U. Vishkin, Tight Complexity Bounds for Parallel Comparison Sorting, Proc. of 29th IEEE Symp. on Foundations of Computer Science, 1986, pp. 502–510.Google Scholar
  7. [7]
    Y. Azar, Parallel Comparison Merging of Many–Ordered Lists, Theoretical Computer Science, 83 (1991), pp. 275–285.Google Scholar
  8. [8]
    Y. Azar And N. Pippenger, Parallel Selection, Discrete Applied Mathematics, 27 (1990), pp. 49–58.MathSciNetMATHGoogle Scholar
  9. [9]
    Y. Azar And U. Vishkin, Tight Comparison Bounds on the Complexity of Parallel Sorting, SIAM J. of Computing, 16 (1987), pp. 458–464.MathSciNetGoogle Scholar
  10. [10]
    R. Boppana, The Average-Case Parallel Complexity of Sorting, Information Processing Letters, 33 (1989), pp. 145–146.Google Scholar
  11. [11]
    A. Borodin And J. E. Hopcroft, Routing, Merging and Sorting on Parallel Models of Computation, J. Computer and SystemJSciences, 30 (1985), pp. 130–145.Google Scholar
  12. [12]
    V. Chvatal, The Tail of the Hypergeometric Distribution, Discrete Mathematics, 25 (1979), pp. 285–287.Google Scholar
  13. [13]
    A. Condon And L. Narayanan, Upper and Lower Bounds for Selection on the Mesh, Proc. of Symp. on Parallel and Distributed Processing, 1994, pp. 497–504.Google Scholar
  14. [14]
    R. Cypher And G. Plaxton, Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers, Proc. of 22nd ACM Symp. on Theory of Computing, 1990, pp. 193–203.Google Scholar
  15. [15]
    R. Floyd And R. Rivest, Expected Time Bounds for Selection, Communications of the ACM, 18 (1975), pp. 165–172.Google Scholar
  16. [16]
    M. GerÉB-Graus And D. Krizanc, The Average Complexity of Parallel Comparison Merging, SIAM J. of Computing, 21 (1992), pp. 43–47.Google Scholar
  17. [17]
    R. HÄGgkvist And P. Hell, Graphs and Parallel Comparison Algorithms, Congr. Numer., 29 (1980), pp. 497–509.Google Scholar
  18. [18]
    H. J. Karloff And P. Raghavan, Randomized Algorithms and Pseudorandom Numbers, Proc. of 20th ACM Symp. on Theory of Computing, 1988, pp. 310–321.Google Scholar
  19. [19]
    C. Kaklamanis And D. Krizanc, Optimal Sorting on Mesh-Connected Processor Arrays, Proc. of the 4th ACM Symp. on Parallel Algorithms and Architectures, 1992, pp. 50–59.Google Scholar
  20. [20]
    C. Kaklamanis, D. Krizanc, L. Narayanan And T. Tsantilas, Randomized Sorting and Selection on Mesh-Connected Processor Arrays, Proc. of the 3rd ACM Symp. on Parallel Algorithms and Architectures, 1991, pp. 17–28.Google Scholar
  21. [21]
    D. E. Knuth, The Art of Computer Programming, vol. 3, Addison-Wesley, Reading, Mass., 1973.Google Scholar
  22. [22]
    D. Krizanc, Time-Randomness Tradeoffs in Parallel Computation, Journal of Algorithms, 20 (1996), pp. 1–19.Google Scholar
  23. [23]
    D. Krizanc And L. Narayanan, Optimal Algorithms for Selection on a Mesh-Connected Processor Array, Proc. of IEEE Symp. on Parallel and Distributed Processing, 1992, pp. 70–76.Google Scholar
  24. [24]
    D. Krizanc, L. Narayanan And R. Raman, Fast Deterministic Selection on a Mesh-Connected Processor Array, Algorithmica, 15 (1996), pp. 319–332.Google Scholar
  25. [25]
    D. Krizanc, D. Peleg And E. Upfal, A Time-Randomness Tradeoff for Oblivious Routing, Proc. of 20th ACM Symp. on Theory of Computing, 1988, pp. 93–102.Google Scholar
  26. [26]
    C. P. Kruskal, Searching, Merging and Sorting in Parallel Computation, IEEE Trans, on Computers, C–32 (1983), pp. 942–946.Google Scholar
  27. [27]
    T. Leighton, C. Leiserson, B. Maggs, And M. Klugerman, Advanced Parallel and VLSI Computation Lecture Notes, MIT/LCS/RSS–24, July 1994.Google Scholar
  28. [28]
    U. Manber And M. Tompa, The Effect of Number of Hamiltonian Paths on the Complexity of a Vertex-Coloring Problem, SIAM J. of Computing, 13 (1984), pp. 109–115.Google Scholar
  29. [29]
    N. Meggido, Parallel Algorithms for Finding the Maximum and the Median Almost Surely in Constant-time, Carnegie-Mellon University Technical Report, Oct. 1982.Google Scholar
  30. [30]
    D. Peleg And E. Upfal, A Time-Randomness Tradeoff for Oblivious Routing, SIAM J. of Computing, 20 (1989), pp. 396–409.Google Scholar
  31. [31]
    N. Pippenger, Sorting and Selecting in Rounds, SIAM J. of Computing, 16 (1987), pp. 1032–1038.Google Scholar
  32. [32]
    S. Rajasekaran, Randomized Parallel Selection, Proc. of Foundations of Software Technology and Theoretical Computer Science Conf., 1990, pp. 215–224.Google Scholar
  33. [33]
    S. Rajasekaran, Sorting and Selection on Interconnection Networks, DI-MACS Series on Discrete Mathematics and Theoretical Computer Science, 21 (1995), pp. 275–296.Google Scholar
  34. [34]
    S. Rajasekaran And J. Reif, Derivation of Randomized Sorting and Selection Algorithms, in Parallel Algorithm Derivation and Program Transformation, Kluwer Academic Publishers, 1993, pp. 187–205.Google Scholar
  35. [35]
    J. Reif And L. Valiant, A Logarithmic Time Sort for Linear Size Networks, J. of the ACM, 34 (1987), pp. 60–76.Google Scholar
  36. [36]
    R. Reischuk, Probabilistic Parallel Algorithms for Sorting and Selection, SIAM J. of Computing, 14 (1985), pp. 396–409.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    C. E. Shannon, A Mathematical Theory of Communication, Bell Systems Technical Journal, 27 (1948), pp. 379–423 and 623–656.MathSciNetGoogle Scholar
  38. [38]
    P. Turan, On the Theory of Graphs, Colloq. Math., 3 (1954), pp. 19–34.MathSciNetGoogle Scholar
  39. [39]
    L. G. Valiant, Parallelism in Comparison Problems, SIAM J. of Computing, 4 (1975), pp. 348–355.MathSciNetMATHGoogle Scholar
  40. [40]
    A. C-C. Yao, Probabilistic Computations: Towards a Unified Measure of Complexity, Proc. of 18th Symp. on Foundations of Computer Science, 1977, pp. 222–227.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Danny Krizanc
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada

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