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The fast Fourier transform (FFT) is undoubtedly an essential primitive thathas been applied in various fields of science and engineering. In this paper,we present a decomposition method for parallelization of multi-dimensional FFTswith smallest communication amount for all ranges of the number of processescompared to previously proposed methods. This is achieved by two distinguishingfeatures: adaptive decomposition and transpose order awareness. In the proposedmethod, the FFT data are decomposed based on a row-wise basis that maps themulti-dimensional data into one-dimensional data, and translates thecorresponding coordinates from multi-dimensions into one-dimension so that theresultant one-dimensional data can be divided and allocated equally to theprocesses. As a result, differently from previous works that have thedimensions of decomposition pre-defined, our method can adaptively decomposethe FFT data on the lowest possible dimensions depending on the number ofprocesses. In addition, this row-wise decomposition provides plenty ofalternatives in data transpose, and different transpose order results indifferent amount of communication. We identify the best transpose orders withsmallest communication amounts for the 3-D, 4-D, and 5-D FFTs by analyzing allpossible cases. Given both communication efficiency and scalability, our methodis promising in development of highly efficient parallel packages for the FFT.

A decomposition method with minimum communication amount for parallelization of multi-dimensional FFTs