Finding and Displaying the Shortest Paths in Hyper-Rings

A graph G= (V, E) is called a hyper-ring with N nodes (N-HR for short) if V = {0, ..., N-1} and E = {(u,v)| v-u modulo N is a power of 2}. In other words, a hyper-ring with N nodes is a graph in which all of the edges connect nodes that are at distances of powers of 2 from each other. Hyper-rings are a multi-machine organization that is, in a sense, a generalization of the standard hyper-cube architecture. One of the most powerful properties of the hyper-rings is that the node-connectivity of an N-HR has been proven to be equal to its degree. This paper discusses hyper-rings, their construction and properties, and an interactive software that allows the user to generate and display hyper-rings of arbitrary size and dimension. The algorithm also finds and displays, under 3D rotations, all node-disjoint shortest paths between any two user-specified nodes.