2-D SIMD algorithms for perfect shuffle networks

In this paper we study a set of basic algorithms for SIMD Perfect Shuffle networks. These algorithms were studied in the works of Schwartz (ACM TOPLAS 2 (1980), 484-521) and Nassimi and Sahni (IEEE Trans. Comput. C 30, 2 (Feb. 1981), 101-107), for the 1-D case, where the size of the problem N is the same as the number of processors P. For the 2-D case of N = L * P, studied also by Gottlieb and Kruskal (Ultracomputer Note No. 11) and Kruskal (Ph.D. dissertation, Courant Institute, New York University, 1981), we improve the run time of several algorithms. In particular, the run time of Row Reduction and Parallel Prefix is improved from O(L * P) to O(L + log P). An adaptive method for gathering global information is introduced, implying a fast algorithm for Smoothing (off-line load balancing). Optimal algorithms for Cartesian Product and for the Transpose operations are devised. Other problems, important variants of these problems, and lowerbounds can be found in the work of Ben-Asher, Egozi, Schuster (Hebrew University Tech. Rep. 88-14, Oct. 1988).