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.

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

[2]  Donald E. Knuth,et al.  The Art of Computer Programming, Vol. 3: Sorting and Searching , 1974 .

[3]  Allan Borodin,et al.  Routing, merging and sorting on parallel models of computation , 1982, STOC '82.

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

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

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

[7]  Yossi Azar,et al.  Parallel selection , 1990, Discret. Appl. Math..

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

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

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

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

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

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

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

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

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

[17]  Noga Alon,et al.  Parallel comparison algorithms for approximation problems , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[18]  Uzi Vishkin,et al.  Finding the maximum, merging and sorting in a parallel computation model , 1981, CONPAR.

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

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

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

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

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

[24]  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).

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

[26]  Béla Bollobás,et al.  Parallel sorting , 1983, Discret. Appl. Math..

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

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