Abstract
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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N. Alon, andY. Azar: Sorting, approximate sorting and searching in rounds,SIAM J. Discrete Math. 1 (1988), 269–280.
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.
N. Alon, andY. Azar: Finding an approximate maximum,SIAM J. Comput. 18 (1989), 258–267.
N. Alon, andY. Azar: Parallel comparison algorithms for approximation problems, Proc 29th IEEE FOCS, Yorktown Heights, NY 1988, IEEE Press, 194–203.
N. Alon, Y. Azar, andU. Vishkin: Tight complexity bounds for parallel comparison sorting, Proc. 27th IEEE FOCS, Toronto, 1986, 502–510.
S. Akl: Parallel Sorting Algorithm, Academic Press, 1985.
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.
M. Ajtai, J. Komlós, W.L. Steiger, andE. Szemerédi: Almost sorting in one round, Advances in Computing Research, to appear.
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.
N. Alon: Expanders, sorting in rounds and superconcentrators of limited depth, Proc. 17th ACM STOC (1985), 98–102.
N. Alon: Eigenvalues, geometric expanders, sorting in rounds and Ramsey Theory,Combinatorica 6 (1986), 207–219.
Y. Azar, andN. Pippenger: Parallel selection,Discrete Applied Math. 27 (1990), 49–58.
Y. Azar, andU. Vishkin: Tight comparison bounds on the complexity of parallel sorting,SIAM J. Comput. 3 (1987), 458–464.
B. Bollobás, andG. Brightwell: Graphs whose every transitive orientation contains almost every relation,Israel J. Math.,59 (1987), 112–128.
B. Bollobás:Extremal Graph Theory, Academic Press, London and New York, 1978.
B. Bollobás, andM. Rosenfeld: Sorting in one round,Israel J. Math. 38 (1981), 154–160.
B. Bollobás, andA. Thomason: Parallel sorting,Discrete Applied Math. 6 (1983), 1–11.
B. Bollobás, andP. Hell: Sorting and Graphs, inGraphs and Orders, I. Rival ed., D. Reidel (1985), 169–184.
B. Bollobás:Random Graphs, Academic Press (1986), Chapter 15 (Sorting algorithms).
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.
R. Häggkvist, andP. Hell: Graphs and parallel comparison algorithms,Congr. Num. 29 (1980), 497–509.
R. Häggkvist, andP. Hell: Parallel sorting with constant time for comparisons,SIAM J. Comput. 10 (1981), 465–472.
R. Häggkvist, andP. Hell: Sorting and merging in rounds,SIAM J. Algeb. and Disc. Math. 3 (1982), 465–473.
D.E. Knuth:The Art of Computer Programming,3 Sorting and Searching, Addison Wesley 1973.
C.P. Kruskal: Searching, merging and sorting in parallel computation,IEEE Trans. Comput. 32 (1983), 942–946.
F.T. Leighton: Tight bounds on the complexity of parallel sorting, Proc. 16th ACM STOC (1984), 71–80.
N. Pippenger: Sorting and selecting in rounds,SIAM J. Comput. 6 (1986), 1032–1038.
S. Scheele: Final report to office of environmental education, Dept, of Health, Education and Welfare, Social Engineering Technology, Los Angeles, CA, 1977.
Y. Shiloach, andU. Vishkin: Finding the maximum, merging and sorting in a parallel model of computation,J. Algorithms 2 (1981), 88–102.
L.G. Valiant: Parallelism in comparison problems,SIAM J. Comp. 4 (1975), 348–355.
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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). https://doi.org/10.1007/BF01206355
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DOI: https://doi.org/10.1007/BF01206355