An optimal message routing algorithm for circulant networks

A k-circulant network G(n; h1, h2, … , hk) is an undirected graph where the node set is Zn={0,1,…,n-1}Zn={0,1,…,n-1} and the edge set is the union of sets of unordered pairs Ei={(u,u+sign(i)∗h|i|(modn))|u∈Zn}, for i ∈ {−k, … , −1, 1, … , k}. We present an optimal (i.e. using shortest paths) dynamic two-terminal message routing algorithm for k-circulant networks, k ⩾ 2. Instead of computing the shortest paths in advance or using routing tables, our algorithm uses only the address of the final destination to determine the next node to which the message must be sent in order to stay on one of the shortest paths to its destination. We introduce the restricted shortest paths, which are used by our routing algorithm, and present an efficient algorithm for their construction in 2-circulant graphs.