A new approach to fast control of r2×r2 3-stage Benes networks of r×r crossbar switches

The authors introduce an approach to fast control of N×N three-stage Benes networks of r×r crossbar switches as building blocks. The approach consists of setting the leftmost column of switches to an appropriately chosen configuration so that the network becomes self-routed while still able to realize a given family of permutations. This approach requires that, for any given family of permutations, a configuration for the leftmost column be found. Such a family is called compatibles; and the configuration of the leftmost column is called the compatibility factor. Compatibility is characterized, and a technique to determine compatibility and the compatibility factor is developed and applied to Ω-realizable permutations, the permutations needed to emulate a hypercube, and the families of permutations required by FFT, bitonic sorting, tree computations, multidimensional mesh and torus computations, and multigrid computations. An O(log² N) time routing algorithm for the three-stage Benes is also developed. Finally, since only three compatibility factors are required by the preceding families of permutations, it is proposed that the first column be replaced by three multiplexed connections yielding a self-routing network with strong communication capabilities