Parallel comparison algorithms for approximation problems


Suppose we haven elements from a totally ordered domain, and we are allowed to performp parallel comparisons in each time unit (=round). In this paper we determine, up to a constant factor, the time complexity of several approximation problems in the common parallel comparison tree model of Valiant, for all admissible values ofn, p and ɛ, where ɛ is an accuracy parameter determining the quality of the required approximation. The problems considered include the approximate maximum problem, approximate sorting and approximate merging. Our results imply as special cases, all the known results about the time complexity for parallel sorting, parallel merging and parallel selection of the maximum (in the comparison model), up to a constant factor. We mention one very special but representative result concerning the approximate maximum problem; suppose we wish to find, among the givenn elements, one which belongs to the biggestn/2, where in each round we are allowed to askn binary comparisons. We show that log* n+O(1) rounds are both necessary and sufficient in the best algorithm for this problem.

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  1. [1]

    N. Alon, andY. Azar: Sorting, approximate sorting and searching in rounds,SIAM J. Discrete Math. 1 (1988), 269–280.

    Google Scholar 

  2. [2]

    N. Alon, andY. Azar: The average complexity of deterministic and randomized parallel comparison sorting algorithms, Proc. 28th IEEE FOCS, Los Angeles, CA 1987, IEEE Press, 489–498; Also:SIAM J. Comput. 17 (1988), 1178–1192.

  3. [3]

    N. Alon, andY. Azar: Finding an approximate maximum,SIAM J. Comput. 18 (1989), 258–267.

    Google Scholar 

  4. [4]

    N. Alon, andY. Azar: Parallel comparison algorithms for approximation problems, Proc 29th IEEE FOCS, Yorktown Heights, NY 1988, IEEE Press, 194–203.

  5. [5]

    N. Alon, Y. Azar, andU. Vishkin: Tight complexity bounds for parallel comparison sorting, Proc. 27th IEEE FOCS, Toronto, 1986, 502–510.

  6. [6]

    S. Akl: Parallel Sorting Algorithm, Academic Press, 1985.

  7. [7]

    M. Ajtai, J. Komlós, W.L. Steiger, andE. Szemerédi: Deterministic selection inO(log logn) parallel time, Proc. 18th ACM STOC, Berkeley, California, 1986, 188–195.

  8. [8]

    M. Ajtai, J. Komlós, W.L. Steiger, andE. Szemerédi: Almost sorting in one round, Advances in Computing Research, to appear.

  9. [9]

    M. Ajtai, J. Komlós, andE. Szemerédi: AnO(nlogn) sorting network, Proc. 15th ACM STOC (1983), 1–9; Also,M. Ajtai, J. Komlós, andE. Szemerédi: Sorting inc logn parallel steps, Combinatorica3 (1983), 1–19.

  10. [10]

    N. Alon: Expanders, sorting in rounds and superconcentrators of limited depth, Proc. 17th ACM STOC (1985), 98–102.

  11. [11]

    N. Alon: Eigenvalues, geometric expanders, sorting in rounds and Ramsey Theory,Combinatorica 6 (1986), 207–219.

    Google Scholar 

  12. [12]

    Y. Azar, andN. Pippenger: Parallel selection,Discrete Applied Math. 27 (1990), 49–58.

    Google Scholar 

  13. [13]

    Y. Azar, andU. Vishkin: Tight comparison bounds on the complexity of parallel sorting,SIAM J. Comput. 3 (1987), 458–464.

    Google Scholar 

  14. [14]

    B. Bollobás, andG. Brightwell: Graphs whose every transitive orientation contains almost every relation,Israel J. Math.,59 (1987), 112–128.

    Google Scholar 

  15. [15]

    B. Bollobás:Extremal Graph Theory, Academic Press, London and New York, 1978.

    Google Scholar 

  16. [16]

    B. Bollobás, andM. Rosenfeld: Sorting in one round,Israel J. Math. 38 (1981), 154–160.

    Google Scholar 

  17. [17]

    B. Bollobás, andA. Thomason: Parallel sorting,Discrete Applied Math. 6 (1983), 1–11.

    Google Scholar 

  18. [18]

    B. Bollobás, andP. Hell: Sorting and Graphs, inGraphs and Orders, I. Rival ed., D. Reidel (1985), 169–184.

  19. [19]

    B. Bollobás:Random Graphs, Academic Press (1986), Chapter 15 (Sorting algorithms).

  20. [20]

    A. Borodin, andJ.E. Hopcroft: Routing, merging and sorting on parallel models of computation,J. Comput. System Sci. 30 (1985), 130–145. Also: Proc. 14th ACM STOC (1982), 338–344.

    Google Scholar 

  21. [21]

    R. Häggkvist, andP. Hell: Graphs and parallel comparison algorithms,Congr. Num. 29 (1980), 497–509.

    Google Scholar 

  22. [22]

    R. Häggkvist, andP. Hell: Parallel sorting with constant time for comparisons,SIAM J. Comput. 10 (1981), 465–472.

    Google Scholar 

  23. [23]

    R. Häggkvist, andP. Hell: Sorting and merging in rounds,SIAM J. Algeb. and Disc. Math. 3 (1982), 465–473.

    Google Scholar 

  24. [24]

    D.E. Knuth:The Art of Computer Programming,3 Sorting and Searching, Addison Wesley 1973.

  25. [25]

    C.P. Kruskal: Searching, merging and sorting in parallel computation,IEEE Trans. Comput. 32 (1983), 942–946.

    Google Scholar 

  26. [26]

    F.T. Leighton: Tight bounds on the complexity of parallel sorting, Proc. 16th ACM STOC (1984), 71–80.

  27. [27]

    N. Pippenger: Sorting and selecting in rounds,SIAM J. Comput. 6 (1986), 1032–1038.

    Google Scholar 

  28. [28]

    S. Scheele: Final report to office of environmental education, Dept, of Health, Education and Welfare, Social Engineering Technology, Los Angeles, CA, 1977.

    Google Scholar 

  29. [29]

    Y. Shiloach, andU. Vishkin: Finding the maximum, merging and sorting in a parallel model of computation,J. Algorithms 2 (1981), 88–102.

    Google Scholar 

  30. [30]

    L.G. Valiant: Parallelism in comparison problems,SIAM J. Comp. 4 (1975), 348–355.

    Google Scholar 

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Research supported in part by Allon Fellowship, by a Bat Sheva de Rothschild grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.

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Alon, N., Azar, Y. Parallel comparison algorithms for approximation problems. Combinatorica 11, 97–122 (1991).

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