An efficient algorithm to find optimal double loop networks Original Research Article

The problem of finding optimal diameter double loop networks with a fixed number of vertices has been widely studied. In this work, we give an algorithmic solution of the problem by using a geometrical approach. Given a fixed number of vertices n, the general problem is to find “steps” s1, s2 ∈ Zn, such that the digraph G(n; s1, s2) with set of vertices V = Zn and adjacencies given by i → i + s1 (mod n) and i → i + s2 (mod n) has minimum diameter d(n). A lower bound of this diameter is known to be be lb(n)=⌈√3n⌉−2. So, given n, the algorithm has as outputs s1,s2 and the minimum integer κ = κ(n) such that d(n;s1,s2)=d(n)=lb(n)+κ The running time complexity of the algorithm is O(κ3)O(logn)-O(κ) is unknown but it is upper-bounded by O(4√n). Moreover, in most of the cases the algorithm also gives (as a by-product) an infinite family of digraphs with increasing order and diameter as above, to which the obtained digraph G(n; S1,S2) belongs.