On a class of concatenated (2log₂N-1)-stage interconnection networks

This paper investigates topological properties of a class of concatenated (2log₂N-1)-stage interconnection networks and introduces their interstage correlations. A concatenated (2log₂ N-1)-stage interconnection network is constructed by merging two (log₂N)-stage cube-type networks with overlapped center stages. In constructing concatenated (2log₂N-1)-stage interconnection networks, we consider all self-routable (log₂N)-stage cube-type networks which can establish a path to an output port with a self-routing tag. Each routing tag for a cube-type network can be generated by reordering destination address bits using a bit permutation mapping function. Based on two bit-reordering mapping functions for a given (2log₂N-1)-stage interconnection network, we formulate its interstage correlations into an algebraic function. Then, the topological equivalences among concatenated (2log₂N-1)-stage interconnection networks are introduced. The proposed classification scheme implies that any two concatenated networks in the same class are topologically equivalent and have isomorphic interstage correlation functions. This means that all the concatenated networks in the same class can use the same routing and application algorithms. In addition, several related important characteristics of each class are discussed. We expect that the results of this paper provide us with an important background for developing a graphical representation model and a bidirectional tag scheme