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Multiphase Complete Exchange: A Theoretical Analysis
February 1996 (vol. 45 no. 2)
pp. 220-229

Abstract-Complete Exchange requires each of N processors to send a unique message to each of the remaining N− 1 processors. For a circuit switched hypercube with N = 2d processors, the Direct and Standard algorithms for Complete Exchange are time optimal for very large and very small message sizes, respectively. For intermediate sizes, a hybrid Multiphase algorithm is better. This carries out Direct exchanges on a set of subcubes whose dimensions are a partition of the integer d. The best such algorithm for a given message size m could hitherto only be found by enumerating all partitions of d.

The Multiphase algorithm is analyzed assuming a high performance communication network. It is proved that only algorithms corresponding to equipartitions of d (partitions in which the maximum and minimum elements differ by at most one) can possibly be optimal. The run times of these algorithms plotted against m form a hull of optimality. It is proved that, although there is an exponential number of partitions, 1) the number of faces on this hull is $\Theta \left( {\sqrt d} \right)$, 2) the hull can be found in $\Theta \left( {\sqrt d} \right)$ time, and 3) once it has been found, the optimal algorithm for any given m can be found in Θ(log d) time.

These results provide a very fast technique for minimizing communication overhead in many important applications, such as matrix transpose, fast Fourier transform, and alternating directions implicit (ADI).

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Index Terms:
Circuit switching, complete exchange, communication overhead, hypercube, multiphase algorithm, parallel computing.
Shahid H. Bokhari, "Multiphase Complete Exchange: A Theoretical Analysis," IEEE Transactions on Computers, vol. 45, no. 2, pp. 220-229, Feb. 1996, doi:10.1109/12.485374
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