Characterizations of Generalized Butter y Networks

A multistage network is a Generalized Butterry Network (GBN) if there are 2 n end-to-end paths, (P 0 ; P 1 ; ; P n?1), that cover all the vertices and are vertex disjoint and for any two consecutive stages k and k + 1 there is an integer`s.t. the k-th vertex of P i is adjacent to the (k + 1)-th vertex of P j ii the binary presentations of i and j diier at most at thè-th bit. The GBN family contains most of the important interconnection networks including the Butterry networks, the Bene s networks and the Batcher bitonic sorting networks. This work establishes several characterizations and properties of the GBN family that were previously unknown even for the special case of the Butterry. One characterization is based on automorphism as follows: A multistage network G is a GBN ii it is 2-regular and for any two end-to-end paths, P 0 and P 00 , there is an automorphism of G that swaps P 0 and P 00. Another one is based on the existence of paths that shortcut or bypass a certain path and yet anther is based on a certain symmetry of the distance function. These results, and all the other results of this work, are founded on the Layered Cross Product (LCP) introduced by Even and Litman. It is sometimes useful to assign labels to the vertices or edges of a network. A notable example is the min/max labeling of the edges of a comparator network. To facilitate the study of labeled multistage networks we introduce the Labeled LCP which is an associative and commutative variant of the LCP that operates on vertex-labeled or edge-labeled multistage networks. We show that, under certain conditions, this operator commutes with the concatenation operator. Building on that, we show that the Batcher bitonic sorting network with its min/max labeling is a product of very simple labeled networks { bamboos or rosaries. Moreover, this network, as well as any network which is a product of labeled rosaries under a certain algebra of labels, satisses a variant of the above path swapping property. Namely, for any two end-to-end paths having identical labels at each stage, there is a label-preserving automorphism that swaps these paths.