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

The fast Fourier transform (FFT) is undoubtedly an essential primitive that has been applied in various fields of science and engineering. In this paper, we present a decomposition method for parallelization of multi-dimensional FFTs with smallest communication amount for all ranges of the number of processes compared to previously proposed methods. This is achieved by two distinguishing features: adaptive decomposition and transpose order awareness. In the proposed method, the FFT data are decomposed based on a row-wise basis that maps the multi-dimensional data into one-dimensional data, and translates the corresponding coordinates from multi-dimensions into one-dimension so that the resultant one-dimensional data can be divided and allocated equally to the processes. As a result, differently from previous works that have the dimensions of decomposition pre-defined, our method can adaptively decompose the FFT data on the lowest possible dimensions depending on the number of processes. In addition, this row-wise decomposition provides plenty of alternatives in data transpose, and different transpose order results in different amount of communication. We identify the best transpose orders with smallest communication amounts for the 3-D, 4-D, and 5-D FFTs by analyzing all possible cases. Given both communication efficiency and scalability, our method is promising in development of highly efficient parallel packages for the FFT. ∗Corresponding author. Tel.: +81 761 51 1987 Email addresses: duytvt@{jaist.ac.jp,issp.u-tokyo.ac.jp} (Truong Vinh Truong Duy), [email protected] (Taisuke Ozaki) Preprint submitted to Computer Physics Communications February 26, 2013 ar X iv :1 30 2. 61 89 v1 [ cs .N A ] 6 F eb 2 01 3