An Algorithm to Find \(k\)-Tight Optimal Double-Loop Networks

Abstract

A double-loop network (DLN) \(G(N;1,s)\) with \(1 < s < N\), is a digraph with the vertex set \(V = \{0,1,\ldots,N – 1\}\) and the edge set \(E=\{u\to v\mid v-u\equiv 1,s \pmod{N}, u,v \in V\}\). Let \(D(N;1,s)\) be the diameter of \(G\) and let us define \(D(N) = \min\{D(N;1,s)\mid 1 < s < N\}\) and \(lb(N) = \lceil\sqrt{3N}\rceil – 2\). A given DLN \(G(N;1,s)\) is called \(k\)-tight if \(D(N;1,s) = lb(N) + k\) (\(k \geq 0\)). A \(k\)-tight DLN is called optimal if \(D(N) = lb(N) + k\) (\(k \geq 0\)). 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 (\(k \geq 0\)), 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.