Aggarwal, A. and Huang, M-D.A. "Network complexity of sorting and graph problems and simulating CRCW PRAMS by interconnection networks." In Proceedings of the 3rd Aegean Workshop on Computing: VLSI Algorithms and Architectures, volume 319 of Lecture Notes in Computer Science, (Reif, J., ed.), Springer-Verlag, pp. 339-350, July 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....permitted is combining two partial results to get one. 87 B BSP Sorting, Merging, and Bucketing The problems of sorting, merging, and bucketing have been studied extensively in the literature because of their many important applications and because of their intrinsic theoretical significance [4, 5, 10, 22, 48, 39, 74, 91, 105, 112, 141]. Many solution strategies have been studied in some depth and their actual performances reported. In this section, we restrict ourselves to the modest goal of providing optimal BSP algorithms for the case n 1 Gammaffl = p for some 0 ffl 1. B.1 Merging Merging is a fundamental operation on ....

Aggarwal, A. and Huang, M-D.A. "Network complexity of sorting and graph problems and simulating CRCW PRAMS by interconnection networks." In Proceedings of the 3rd Aegean Workshop on Computing: VLSI Algorithms and Architectures, volume 319 of Lecture Notes in Computer Science, (Reif, J., ed.), Springer-Verlag, pp. 339-350, July 1988.

....runs in O(log n log p= log(p=n) time [14] The problem of sorting on cube type computers when n p has been addressed by a number of researchers. Aggarwal and Huang created an algorithm that sorts in O( n=p) log n(log n= log(n=p) ff ) time, where ff = 3 Gamma log 3) log 3 Gamma 1) 2:419 [1]. This was improved by Cypher and Sanz, who created an algorithm called Cubesort that runs in O( n=p) log 2 n= log(n=p) time, provided p log (x) p n for some positive integer x (see Section 2.2 for the definition of log (x) p) A further improvement was given by Plaxton s Smoothsort ....

Alok Aggarwal and Ming-Deh A. Huang. Network complexity of sorting and graph problems and simulating CRCWPRAMS by interconnection networks (preliminary version). In Proc. 3rd Aegean Workshop on Computing, pages 339--350, 1988.

.... based on other assumptions, we refer the reader to [15] 16] and [18] One may verify that BitonicSort provides optimal speedup over sequential sorting only if p = O(2 p log n ) Two recent algorithms, which we refer to as CubeSort (Cypher and Sanz, 5] and ColumnSort (Aggarwal and Huang, [1]) have improved this result significantly. Both of these algorithms are optimal if n exceeds p by a polynomial factor, that is, if n = p 1 ffl for any constant ffl 0. ColumnSort is based on Leighton s technique for sorting n values by performing a constant number of smaller sorts [12] Note ....

....located at the fringe of S, those in F(H d ; S) can send tokens out of S, and these can only transmit one token per time step. Therefore, 2 Algorithm Running Time Range BitonicSort [2] 3] 9] O( n=p) log 2 p) n = Omega Gamma p) MergeSort [13] O(log 2 p= log(p=n) n = O(p) ColumnSort [1] O( n log n) p) n = Omega Gamma p 1 ffl ) ffl 0 CubeSort [5] O( n=p) log 2 p= log(n=p) essentially n = Omega Gamma p) Table 1: Previous sorting algorithms for the weak model. the running time of Balance must be at least (n Gamma dn=medn=pe) f(H d ; dn=me) Given Lemma 2.1, we can ....

A. Aggarwal and M.-D. A. Huang. Network complexity of sorting and graph problems and simulating CRCW PRAMs by interconnection networks. In J. H. Reif, editor, Lecture Notes in Computer Science: VLSI Algorithms and Architectures (AWOC 88), vol. 319, pages 339--350. Springer-Verlag, 1988.

....be bn=pc or dn=pe values at any processor, and that the set of values within any particular processor be sorted. There has been a great deal of previous research related to the problem of sorting on the hypercube. For the 1 port model of the hypercube that we have been considering thus far, see [1], 4] 7] 9] 10] and [12] For examples of results based on other assumptions, we refer the reader to [13] 15] 17] and [18] The time bounds for the merging and sorting algorithms described in this section do not apply to the 1 port model of computation that we have been considering up ....

A. Aggarwal and M.-D. A. Huang. Network complexity of sorting and graph problems and simulating CRCW PRAMs by interconnection networks. In J. H. Reif, editor, Lecture Notes in Computer Science: VLSI Algorithms and Architectures (AWOC 88), vol. 319, pages 339--350. Springer-Verlag, 1988. 13

....such as the tree, multi dimensional mesh, hypercube, butterfly and shuffle exchange. The lower bound is proven in Sections 4 and 5. Note that this lower bound disproves a claim of Aggarwal and Huang stating that optimal speedup is possible for selection on the hypercube and shuffle exchange [1]. When n=p is sufficiently large (for example, greater than log 2 p on the hypercube and shuffle exchange) the lower bound is tight to within a multiplicative constant. The matching upper bound is provided by the algorithm Select presented in [14] Section 3 contains a brief description of this ....

A. Aggarwal and M.-D. A. Huang. Network complexity of sorting and graph problems and simulating CRCW PRAMs by interconnection networks. In J. H. Reif, editor, VLSI Algorithms and Architectures: Proceedings of the 3rd Aegean Workshop on Computing, Lecture Notes in Computer Science, volume 319, pages 339--350. Springer-Verlag, 1988.

....(1 Gamma fi) Gamma2 )25 log n Gammalog (n=m) for sorting on the ER PRAM(m) where m = O(n fi ) However, this algorithm has substantial overhead and is considerably more involved then the one presented in this paper. A recursive version of Columnsort is also used by Aggarwal and Huang in [AH88] to obtain an algorithm for sorting in fixed connection networks. Subsequent to a preliminary version of this paper in [ABK95] Adler [Adl96] provides an algorithm for sorting in the CR PRAM(m) that is considerably faster than the lower bound for the ER PRAM(m) presented in this paper. Thus, that ....

A. Aggarwal and M. Huang. Network Complexity of Sorting and Graph Problems and Simulating CRCW PRAMs by Interconnection Networks. In Proceedings of 3rd Aegean Workshop on Computing, LNCS , pp. 339-350, 1988.

....2, kksort can be implemented to run with maximal queue size k. 3.3 Generalizations. The ideas underlying our k k sorting algorithm are very general, and can in fact be applied to large classes of networks. Interestingly, the resulting algorithm is a variation of Leighton s Columnsort algorithm [2, 13]. In this subsection, we briefly describe this generalized algorithm, and show how it can be efficiently implemented on multi dimensional meshes. This leads to an algorithm for k k sorting on meshes of arbitrary dimension whose running time matches the bisection lower bound to within a lower order ....

Aggarwal, A., M.D. Huang, `Network Complexity of Sorting and Graph Problems and Simulating CRCW PRAMs by Interconnection Networks,' VLSI Algorithms and Architectures (AWOC 88), LNCS 319, pp. 50-59, ACM, 1992.

....The analysis for our algorithms is the deterministic, worst case analysis. The weighted selection problem can be solved sequentially in linear time by using the selection algorithm [4] as a subroutine. On coarse grain hypercubes, this problem can certainly be solved by sorting algorithms [1, 7] in O( n p log n) computation and communication time when n = O(p 1 ) but such solutions use O(n log n) total operations and have high communication complexity. Plaxton [11, 12] presents three coarse grain hypercube algorithms for the unweighted case; the best one [12] achieves a worst case ....

A. Aggarwal and M.-D. Huang. Network complexity of sorting and graph problems and simulating CRCW PRAMs by interconnection networks, pages 339 -- 350. Lecture Notes in Computer Science Vol. 319: VLSI Algorithms and Architectures, Springer Verlag, 1988.

....2, kksort can be implemented to run with maximal queue size k. 3.3 Generalizations. The ideas underlying our k k sorting algorithm are very general, and can in fact be applied to large classes of networks. Interestingly, the resulting algorithm is a variation of Leighton s Columnsort algorithm [2, 13]. In this subsection, we briefly describe this generalized algorithm, and show how it can be efficiently implemented on multi dimensional meshes. This leads to an algorithm for k k sorting on meshes of arbitrary dimension whose running time matches the bisection lower bound to within a lower order ....

Aggarwal, A., M.D. Huang, `Network Complexity of Sorting and Graph Problems and Simulating CRCW PRAMs by Interconnection Networks,' VLSI Algorithms and Architectures (AWOC 88), LNCS 319, pp. 50-59, ACM, 1992.

....integer addition. The only operator employed by our algorithms is addition. As output, the scan returns a vector in which each position has the sum, according to the operator, of those input values in lesser positions. For example, a plus scan (integer addition as the operator) of the vector [ 4 7 1 0 5 2 6 4 8 1 9 5 ] yields [ 0 4 11 12 12 17 19 25 29 37 38 47 ] as the result of the scan. Operation Symbolic Time Actual Time Arithmetic A Delta (n=p) 1 Delta (n=2048) Cube Swap Q Delta (n=p) 40 Delta (n=2048) Send (routing) R Delta (n=p) 130 Delta (n=2048) Scan (parallel prefix) 3A Delta (n=p) S 3 ....

A. Aggarwal and M.-D. A. Huang. Network complexity of sorting and graph problems and simulating CRCW PRAMs by interconnection networks. In J. H. Reif, editor, Lecture Notes in Computer Science: VLSI Algorithms and Architectures (AWOC 88), vol. 319, pages 339--350. Springer-Verlag, 1988.

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A. Aggarwal and M.-D. A. Huang, Network Complexity of Sorting and Graph Problems and Simulating CRCW PRAMs by Interconnection Networks; Lecture Notes in Computer Science VLSI Algorithms and Architectures (AWOC 88) (ed. by John Reif), vol. 319, pp. 339-350, Springer-Verlag, 1988.

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