Fast and Dependable Communication 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}. The following results are shown. We prove that the node-connectivity ? of an N-HR is equal to its degree, say ?, by presenting an algorithm for the explicit construction of ? node-disjoint paths connecting nodes s and t. The length of these paths is bounded by ?logD? + 3, where D is the positional distance between s and t. Finally, we show a node-to-node communication scheme for HRs that requires only ?logD? + 3 rounds, even in the presence of up to ?-1 node failures.