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 κ of an N -HR is equal to its degree, say δ, by presenting an algorithm for the explicit construction of δ 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 δ−1 node failures.