Optimal Simulations by Butterfly Networks: Extended Abstract,
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We investigate the power of the Butterfly network which is the FFT network with inputs and outputs identified relative to other proposed multicomputer interconnection networks, by considering how efficiently the Butterfly can simulate the other networks Formally we ask, How efficiently can one embed the graph underlying the other network in the graph underlying the Butterfly We measure the efficiency of an embedding of a graph G in a graph H in terms of the dilation, or, the maximum amount that any edge of G is stretched by the embedding the expansion, or, the ratio of the number of vertices of H to the number of vertices of G. We present three simulations that are optimal, to within constant factors 1 Any complete binary tree can be embedded in a Butterfly graph, with simultaneous dilation O1 and expansion O1. 2 The n-vertex X-tree can be embedded in a Butterfly graph with simultaneous dilation Olog log n and expansion O1 no embedding has better dilation, independent of expansion. 3 Any embedding of the n x n mesh in the Butterfly graph must have dilation log n, independent of expansion any embedding of the mesh in the Butterfly graph achieves this dilation. Thus, we have simulations of complete-binary-tree machines, X-tree machines, and mesh computers on Butterfly machines, that are optimal in resource utilization expansion and delay dilation, to within constant factors.
- Command, Control and Communications Systems