The set of \((N/2^i)\)-distance graphs of \(C_N\) and its application to efficient information broadcast among \(N\) nodes over \(K_N\)

The set \(S\) of \(\left(\frac{N}{2^i}\right)\)-distance graphs of order \(N\) cycle graphs \(\mathbb{C}_N\) is defined here as \[S = \left\{ s_i= \bigcup_1^{\frac{N}{2^i}} \mathbb{C}_{2^i} \quad\Bigg\vert\quad i=1,2, \dots, \log N\right\},\] where w.o.l.o.g. \(N=2^k\), \(\forall k=2, 3, \dots\), and \(\log\) is base two. The utility of the computation of \(S\) is demonstrated by a \(\mathcal{O}(|S| = \log N)\)-step implementation of various information broadcasts and their corresponding duals (i.e., reduction) among \(N\) completely-connected nodes \(\mathbb{K}_N\) exchanging messages under a realistic \((1, 1, 2)\) communication model (i.e., concurrent one in-port and one out-port over a duplex connection). Information broadcast over \(\mathbb{K}_N\) under \((1,1,2)\) is currently implemented with \(\mathcal{O}(N)\) steps using a series of \(N\) circular 1-shift operations (or one circular \(N\)--shift). Algorithmically, the \(i\)th element \(s_i\in S\) partially coincides with the \(i\)th dimension of a \(\mathbb{K}_N\)-embedded \((\log N)\)-cube.