Performing BMMC permutations in two passes through the expanded delta network and MasPar MP-2

This paper examines routing of BMMC (bit-matrix-multiply/complement) permutations on two types of multistage interconnection networks: the expanded delta network and the global router of the MasPar MP-2. BMMC permutations are an important class of permutations that has been well-studied on various multistage networks. The class of BMMC permutations includes as subclasses Gray-code and inverse Gray-code permutations and the entire subclass of bit-permute/complement (BPC) permutations, which in turn includes matrix transpose (with power-of-2 dimensions), bit reversal, vector reversal, hypercube, and matrix reblocking permutations. There are four results in this paper. First, we use linear-algebraic techniques to derive an algorithm to perform any BMMC permutation in at most two passes on the expanded delta network. Second, we use linear-algebraic techniques to derive an algorithm to perform any BMMC permutation in at most two passes on the global router of the MasPar MP-2. Third, we use linear-algebraic and combinatorial analysis to determine the distribution of all BMMC permutations when routed naively through the MP-2 global router and show that most, but not all, BMMC permutations require only one or two passes anyway. We can apply our two-pass algorithms in those cases when naive routing requires more than two passes. Fourth, we present experimental evidence to support our analysis.