Optimal mappings of m dimensional FFT communication to k dimensional mesh for arbitrary m and k
The FFT communication patterns are important to not only FFT algorithms, but also many other algorithms over one or higher dimensional. The mapping of m dimensional FFT communication to k dimensional mesh has previously been considered only for the following special cases (a) m=1 or 2, k=1 or 2, (b) m=1 or 2, k=log(n) where n is the size of the machine. In this paper, we present the optimal mappings of m dimensional FFT communication onto k dimensional mesh for arbitrary m and k. The mappings are optimal since the communication distances in the logarithmic steps sum to exactly the diameter of the mesh regardless of the dimension cr the shape of the mesh. An m-k shuffle permutation, which subsumes perfect shuffle, is introduced and used to derive some of the optimal mappings. As a by-product, an optimal broadcast algorithm over any dimensional mesh, including binary hypercube as a special case, is presented.
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