Path establishment of permutations in a rearrangeable interconnection network

One of the most widely used parallel systems is the one which uses a multistage interconnection network to connect processors and memory modules, such as Omega and inverse Omega (denoted as Omega^-1) networks. Appending an Omega network to an Omega^-1 network becomes an Omega^-1.Omega network. This paper aims at establishing paths for routing an arbitrary permutation in a (2log₂ N-1) stage Omega^-1.Omega network. First, we present the set of permutations which can be realized in an Omega network. The path establishment for such an Omega-passable permutation can be easily achieved by using top or bottom control. Secondly, an approach is developed to rearrange an arbitrary input permutation in an Omega^-1 network such that the output permutation turns out to be an Omega-passable permutation. It is also shown that removing the last stage in Omega^-1 network has no influence on this rearrangement result. Henceforth, an arbitrary permutation can be realized in the reduced (2log₂ N-1) stage Omega^-1.Omega network. Based on the results presented in this paper, it will be shown that several networks are also rearrangeable by using the lemmas and theorems presented by Abdennadher and Feng¹.