Optimal embeddings of butterfly-like graphs in the hypercube

Abstract

We present optimal embeddings of three genres of butterfly-like graphs in the (boolean) hypercube; each embedding is specified via a linear-time algorithm. Our first embedding finds an instance of the FFT graph as a subgraph of the smallest hypercube that is big enough to hold it; thus, we embed then-level FFT graph, which has (n+1)2 n vertices, in the (n+⌈log2(n+1)⌉)-dimensional hypercube, with unit dilation. This embedding yields a mapping of the pipelined FFT algorithm on the hypercube architecture, which is optimal in all resources (time, processor utilization, load balancing, etc.) and which is on-line in the sense that inputs can be added to the transform even during the computation. Second, we find optimal embeddings of then-level butterfly graph and then-level cube-connected cycles graph, each of which hasn2 n vertices, in the (n+⌈log2n⌉)-dimensional hypercube. These embeddings, too, have optimal dilation, congestion, and expansion. The dilation is 1+(n mod 2), which is best possible. Our embeddings indicate that these two bounded-degree approximations to the hypercube do not have any communication power that is not already present in the hypercube.