Shawn M. Larson

The Mobius cubes

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. >

On generalized twisted cubes

Abstract We briefly survey some of the results on twisted cubes which have been published in the last decade. We point out that a number of n-dimensional twisted cubes with diameter about n 2 are known. These twisted cubes are better than the recent ⌈ 2n 3 ⌉ diameter twisted cube given by F. Chedid and R. Chedid (1993). We discuss embedding other networks within twisted cubes and point out an error in one supposed embedding given by Chedid and Chedid.

The 'Mobius cubes': improved cubelike networks for parallel computation

The Mobius cubes are created, by rearranging in a systematic manner, some of the edges of a hypercube. This rearrangement results in smaller distances between processors; where distance is the number of communication links which must be traversed. The authors show that the n-dimensional Mobius cubes have a diameter of about n/2 and expected distance of about n/3. These distances are a considerable savings over the diameter of n and expected distance of n/2 for the n-dimensional hypercube. The authors show that the Mobius cubes have a slightly more complicated algorithm than the hypercube. While the asymmetry of the Mobius cubes may give rise to communications bottlenecks, they report preliminary experiments showing the bottle necks are not significant. They compare their Mobius cubes to other variants and indicate some advantages for the Mobius cubes.<<ETX>>

Static and dynamic performance of the Möbius cubes (short version)

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 1/2 the diameter of the equivalent hypercube, and that the average number of steps between processors for a Mobius cube is about 2/3 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 report results of simulation studies on the dynamic message-passing performance of the hypercube, the Twisted Cube of Hilbers et. al. [8], and the Mobius cubes. Our results agree with those of Abraham [2], showing that the Twisted Cube has worse dynamic performance than the hypercube. But our results show that the Mobius cubes and in particular the 1-Mobius cube have better dynamic performance than the hypercube.

Smaller diameters in hypercube-variant networks

A number of hypercube-variant networks attempt to improve the hypercube by adding extra connections and thus reducing the diameter of the constructed network. We briefly outline a model which describes these variant networks. Further, we show that by restricting this model, we can describe hypercube variants with exactly the same number of edges as the hypercube. We mention several such networks which all have diameter about n/2. We describe a new network within this class that has diameter about 2n/5, thus improving the best known previous bound by a constant factor. We show that within a limited construction paradigm our network is best possible.

A linear equation model for twisted cube networks

The Twisted 3-cube is an interconnection network that twists the edges of the 3-dimensional hypercube to produce a network with diameter 2 and expected distance 11/8. A number of papers have shown that the Twisted 3-cube can be generalized into higher dimensional cube-like networks. We show that many of these networks can be described using a simple model. We place bounds on the diameter and expected distances of networks in this model, and show that the dynamic performance of these networks can match or improve upon the hypercubes performance in most conditions.