Parallel comparison algorithms for approximation problems

The authors consider that they have n elements from a totally ordered domain and are allowed to perform p parallel comparisons in each time unit (round). They determine, up to a constant factor, the time complexity of several approximation problems in the common parallel comparison tree model of L.G. Valiant, for all admissible values of n, p, and epsilon , where epsilon is an accuracy parameter determining the quality of the required approximation. The problems considered include the approximate maximum problem, approximate sorting, and approximate merging. The results imply, as special cases, all the known results about the time complexity of parallel sorting, parallel merging, and parallel selection of the maximum (in the comparison model). They highlight one very special but representative result concerning the approximate maximum problem. They wish to find, among the given n elements, one which belongs to the biggest n/2, where in each round they are allowed to ask n binary comparisons. They show that log/sup */n+ Theta (1) rounds are both necessary and sufficient in the best algorithm for this problem.<<ETX>>

[1]  Noga Alon,et al.  Tight complexity bounds for parallel comparison sorting , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[2]  Noga Alon,et al.  Expanders, sorting in rounds and superconcentrators of limited depth , 1985, STOC '85.

[3]  Noga Alon,et al.  The average complexity of deterministic and randomized parallel comparison sorting algorithms , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[4]  B. Bollobás,et al.  Sorting in one round , 1981 .

[5]  Noga Alon,et al.  Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory , 1986, Comb..

[6]  Uzi Vishkin,et al.  Finding the Maximum, Merging, and Sorting in a Parallel Computation Model , 1981, J. Algorithms.

[7]  R. Häggkvist,et al.  Sorting and Merging in Rounds , 1982 .

[8]  Pavol Hell,et al.  Parallel Sorting with Constant Time for Comparisons , 1981, SIAM J. Comput..

[9]  Frank Harary,et al.  Graph Theory , 2016 .

[10]  Allan Borodin,et al.  Routing, Merging, and Sorting on Parallel Models of Computation , 1985, J. Comput. Syst. Sci..

[11]  János Komlós,et al.  Deterministic selection in O(loglog N) parallel time , 1986, STOC '86.

[12]  Noga Alon,et al.  Finding an Approximate Maximum , 1989, SIAM J. Comput..

[13]  E. Szemerédi,et al.  Sorting inc logn parallel steps , 1983 .

[14]  Frank Thomson Leighton,et al.  Tight Bounds on the Complexity of Parallel Sorting , 1984, IEEE Transactions on Computers.

[15]  Noga Alon,et al.  Sorting, Approximate Sorting, and Searching in Rounds , 1988, SIAM J. Discret. Math..

[16]  János Komlós,et al.  An 0(n log n) sorting network , 1983, STOC.

[17]  Selim G. Akl,et al.  Parallel Sorting Algorithms , 1985 .

[18]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[19]  Clyde P. Kruskal,et al.  Searching, Merging, and Sorting in Parallel Computation , 1983, IEEE Transactions on Computers.

[20]  János Komlós,et al.  Almost Sorting in one Round , 1989, Adv. Comput. Res..

[21]  Nicholas Pippenger,et al.  Sorting and Selecting in Rounds , 1987, SIAM J. Comput..

[22]  Noga Alon,et al.  The Average Complexity of Deterministic and Randomized Parallel Comparison-Sorting Algorithms , 1988, SIAM J. Comput..

[23]  Béla Bollobás,et al.  Sorting and Graphs , 1985 .

[24]  Yossi Azar,et al.  Tight Comparison Bounds on the Complexity of Parallel Sorting , 2018, SIAM J. Comput..

[25]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[26]  B. Bollobás,et al.  Graphs whose every transitive orientation contains almost every relation , 1987 .