The set of \((N/2^i)\)-distance graphs of \(C_N\) and its application to efficient information broadcast among \(N\) nodes over \(K_N\)
- Published
- Accepted
- Subject Areas
- Computer Networks and Communications, Distributed and Parallel Computing, Optimization Theory and Computation
- Keywords
- graph theory
- Copyright
- © 2015 Pabico
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ PrePrints) and either DOI or URL of the article must be cited.
- Cite this article
- 2015. The set of \((N/2^i)\)-distance graphs of \(C_N\) and its application to efficient information broadcast among \(N\) nodes over \(K_N\) PeerJ PrePrints 3:e1372v1 https://doi.org/10.7287/peerj.preprints.1372v1
Abstract
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.
Author Comment
Contributed abstract to the 2015 Mathematical Society of the Philippines CALABARZON Annual Convention and General Assembly (MSPC 2015) held in Los Baños, Laguna, on 18 July 2015.