Multiprocessor FFTs

Several multiprocessor FFTs are developed in this paper for both vector multiprocessors with shared memory and the hypercube. Two FFTs for vector multiprocessors are given that compute an ordered transform and have a stride of one except for a single ‘link’ step. Since multiple FFTs provide additional options for both vectoriation and distribution we show that a single FFT can be performed in terms of two multiple FFTs and development of algorithms that minimize interprocessor communication. On a hypercube of dimension d the unordered FFT requires d + 1 parallel transmissions. The ordered FFT requires from 1.5 d + 2 to 2 d + 1 parallel transmissions depending on the length of the sequence. It is also shown that a class of orderings called index-digit permutations which includes matrix transposition, the perfect shuffle, and digit reversal can be performed with less than or equal to 1.5 d parallel transmissions.