The Mobius cubes

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Abstract
The Mobius cubes are hypercube variants that give better performance with the same number of links and processors. We show that the diameter of the Mobius cubes is about one half the diameter of the equivalent hypercube, and that the average number of steps between processors for a Mobius cube is about two-thirds of the average for a hypercube. We give an efficient routing algorithm for the Mobius cubes. This routing algorithm finds a shortest path and operates in time proportional to the dimension of the cube. We also give efficient broadcast algorithms for the Mobius cubes. We show that the Mobius cubes contain ring networks and other networks. We report results of simulation studies on the dynamic message-passing performance of the hypercube, the Twisted Cube of P.A.J. Hilbers et al. (1987), and the Mobius cubes. Our results are in agreement with S. Abraham (1990), showing that the Twisted Cube has worse dynamic performance than the hypercube, but our results show that the 1-Mobius cube has dynamic performance superior to that of the hypercube. This contradicts current literature, which implies that twisted cube variants will have worse dynamic performance.

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