Transposing Arrays on Multicomputers Using de Bruijn Sequences

Transposing an N × N array that is distributed rowor column-wise across P = N processors is a fundamental communication task that requires time-consuming interprocessor communication. It is the underlying communication task for the fast Fourier transform of long sequences and multi-dimensional arrays. It is also the key communication task for certain weather and climate models. A parallel transposition algorithm is presented for hypercube and mesh connected multicomputers with programmable networks. The optimal scheduling of network transmissions is not unique and known to be non-trivial. Here, scheduling is determined by a single de Bruijn sequence of N bits. The elements in each processor are first preordered and then, in groups of log2P adjacent elements, either transmitted or not transmitted, depending on the corresponding bit in the de Bruijn sequence. The algorithm is optimal both in overall time and the time that any individual element is in the network. The results are extended to other communication tasks including shuffles, bit reversal, index reversal, and general index-digit permutation. The case P ≠ N and rectangular arrays with nonpower-of-two dimensions are also discussed. Algorithms for mesh connected multicomputers are developed by embedding the hypercube in the mesh. The optimal implementation of the algorithms requires certain architectural features that are not currently available in the marketplace. 1 National Center for Atmospheric Research, P.O. Box 3000, Boulder, Colorado 80307, which is sponsored by the National Science Foundation; email: [email protected]; fax: 303 497 1286.