Simulating permutation networks on hypercubes

Abstract

Various permutation interconnection networks have recently been suggested as an alternative to the hypercube. We investigate embeddings of these permutation networks on hypercubes. Our embeddings exhibit a marked trade-off between dilation and expansion and for the n-dimensional star network have the following dilation and expansion bounds:

• 1. dilation 2 and expansion O($2^{n^2 }$ / n ⁿ),

• 2. dilation O(log n) and expansion O((2e) ⁿ),

• 3. dilation O(log n log log n) and expansion O(e ⁿ /n),

• 4. dilation O(log ² n) and expansion O((e/2) ⁿ/$\sqrt n$), and

• 5. dilation n(log n − 2) and expansion n/2.

The embeddings are, in fact, optimum or nearly optimum in both dilation and expansion for small values of n, which are the important cases in practice. We also show that any star network with dimension at least 4 is not a subgraph of a hypercube and that any embedding with O(1) expansion must have dilation Ω(log n).

Partial support from the Louisiana Education and Quality Support Fund (LEQSF) is gratefully acknowledged.

Partial support from a Texas Advanced Research Project Grant is gratefully acknowledged.

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