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Parallel Algorithms for the Classes of +or-2^b DESCEND and ASCEND Computations on a SIMD Hypercube
December 1993 (vol. 4 no. 12)
pp. 1372-1381

Derives a simple lower bound for performing a 2/sup b/ permutation on an N-PE SIMDhypercube, proving that log N-b routing steps are needed even if one allows an arbitrarymapping of elements to processors. An algorithm for performing a 2/sup b/ permutationusing exactly log N-b full-duplex routing steps that is slightly more efficient thanpreviously known O(log N-b) algorithms, which perform the permutation as an Omega orOmega /sup -1/ mapping, is presented. The author has also identified a general class ofparallel computations called +or-2/sup b/ descend, which includes Batcher's odd-evenmerge and many other algorithms. An efficient algorithm for performing any computation in this class in O(log N) steps on an N-PE SIMD hypercube is given. A related class of parallel computations called +or-2/sup b/ ascend is also defined. This class appears to be more difficult than +or-2/sup b/ descend. A simple O(log/sup 2/ N/log log) N algorithm for this class on a SIMD hypercube, requiring Theta (log log N) space per processor is developed.

[1] K. E. Batcher, "Sorting networks and their applications," inProc. AFIPS 1968 Spring Joint Comput., vol. 32, 1968, pp. 307-314.
[2] T. Y. Feng, "A survey of interconnection networks,"Computer, vol. 14, no. 12, pp. 12-27, 1981.
[3] S. L. Johnsson, "Communication efficient basic linear algebra computations on hypercube architectures,"J. Parallel Distributed Comput., pp. 133-172, 1987.
[4] D. E. Knuth,The Art of Computer Programming, Vol. 3, Reading, MA: Addison-Wesley, 1973.
[5] M. Kumar and D. S. Hirschberg, "An efficient implementation of batcher's odd-even merge algorithm and its application in parallel sorting schemes,"IEEE Trans. Comput., vol. C-32, pp. 254-264, 1983.
[6] D. Lawrie, "Access and alignment of data in an array processor,"IEEE Trans. Comput., vol. C-24, no. 12, pp. 1145-1155, 1975.
[7] D. Nassimi and S. Sahni, "Parallel permutation and sorting algorithms and a new generalized connection network,"J. ACM, vol. 29, no. 3, pp. 642-667, 1982.
[8] D. Nassimi and Y. D. Tsai, "Efficient implementations of a class of±2bparallel computations on a SIMD hypercube," inProc. 5th Int. Parallel Processing Symp., Apr. 1991, pp. 2-9.
[9] D. Nassimi, "Nearly logarithmic-time parallel algorithms for the class of±2bASCEND computations on a SIMD hypercube," inProc. 6th Int. Parallel Processing Symp., Mar. 1992, pp. 122-129.
[10] D. Nassimi, "Parallel algorithms for PM2B-ASCEND computations on a SIMD hypercube using multiple levels of iteration-grouping," inProc. 1992 Int. Conf. Parallel Processing, Aug. 1992.
[11] F. P. Preparata and J. Vuillemin, "The cube-connected cycle: A versatile network for parallel computation,"Commun. ACM, vol. 24, pp. 300-309, May 1981.
[12] Y. Saad and M. H. Schultz, "Topological properties of hypercubes,"IEEE Trans. Comput., pp. 907-913, 1988.
[13] H. J. Siegel, "Analysis techniques for SIMD machine interconnection networks and the effects of processor address masks,"IEEE Trans. Comput., vol. 26, no. 2, pp. 153-161, 1977.
[14] H. Stone, "Parallel processing with the perfect shuffle,IEEE Trans. Comput., vol. C-20, pp. 153-161, 1971.

Index Terms:
Index Terms2/sup b/ permutation; SIMD hypercube; routing steps; full-duplex routing; parallelcomputations; efficient algorithm; +or-2/sup b/ ascend; +or-2/sup b/ descend; cyclicshift; odd-even merge; parallel prefix; PM2I interconnection; computational complexity;hypercube networks; parallel algorithms
D. Nassimi, "Parallel Algorithms for the Classes of +or-2^b DESCEND and ASCEND Computations on a SIMD Hypercube," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 12, pp. 1372-1381, Dec. 1993, doi:10.1109/71.250118
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