Ordered Fast Fourier Transforms on a Massively Parallel Hypercube Multiprocessor

We examine design alternatives for ordered FFT algorithms on massively parallel hypercube multiprocessors such as the Connection Machine. Particular emphasis is placed on reducing communication which is known to dominate the overall computing time. To this end we combine the order and computational phases of the FFT and also use sequence to processor maps that reduce communication. The class of ordered transforms is expanded to include any FFT in which the order of the transform is the same as that of the input sequence. Two such orderings are examined, namely, "standard-order" and "A-order" which can be implemented with equal ease on the Connection Machine where orderings are determined by geometries and priorities. If the sequence has N = 2 elements and the hypercube has P = 2 processors then a standard-order FFT can be implemented with d +r ⁄2+1 parallel transmissions. An Aorder sequence can be transformed with 2d −r ⁄2 parallel transmissions which is r −d +1 fewer than the standard order. A parallel method for computing the trigonometric coefficients is presented that does not use trigonometric functions or interprocessor communication. A performance of 0.9 GFLOPS was obtained for an A-order transform on the Connection Machine. 1 Department of Computer Science, University of California at Los Angles, Los Angles, California 90024-1596. 2 National Center for Atmospheric Research, Boulder, Colorado 80307, which is sponsored by the National Science Foundation. 3 This work was supported by the NAS Systems Division via Cooperative Agreement NCC 2-387 between NASA and the University Space Research Association (USRA). It was performed while the authors were visiting the Research Institute for Advanced Computer Science (RIACS), NASA Ames Research Center, Moffett Field, CA 94035.