An algorithm to find k-tight optimal double-loop networks

A double-loop network(DLN) G(N;1, s) with 1 < s < N, is a digraph with the vertex set V = {0, 1, ..., N - 1} and the edge set E = {u -> v vertical bar v - u equivalent to 1, s( mod N), u, v is an element of V}. Let D(N;1, s) be the diameter of G and let us define D(N) = min{D(N; 1, s)vertical bar 1 < s < N} and lb(N) = [root 3N] - 2. A given DLN G(N;1, s) is called k-tight if D(N; 1, s) = lb(N) + k(k >= 0). A k-tight DLN is called optimal if D(N) = lb(N) + k(k >= 0). It is known that finding k-tight optimal DLN is a difficult task as the value k increases. In this work, a practical algorithm is derived for finding k-tight optimal double-loop networks(k >= 0), and it is proved that the average complexity to judge whether there exists a k-tight L-shaped tile with N nodes is O(k(2)). As application examples, we give some 9-tight optimal DLN and their infinite families.