Construction of Dual Optimal Bidirectional Double-Loop Networks for Optimal Routing

Bidirectional double-loop networks (BDLNs) are widely used in computer networks for their simplicity, symmetry and scalability. One common way to improve their performance is to decrease the diameter and average distance. Attempts have been made to find BDLNs with minimal diameters; however, such BDLNs will not necessarily have the minimum average distance. In this paper, we construct dual optimal BDLNs with minimum diameters and average distances using an efficient method based on coordinate embedding and transforming. First, we get the lower bounds of both the diameter and average distance by embedding a BDLN into Cartesian coordinates. Then, we construct tight optimal BDLNs that provide the aforementioned lower bounds based on an embedding graph. On the basis of node distribution regularity in tight optimal BDLNs, we construct dual optimal BDLNs with minimum diameters and average distances for any number of nodes. Finally, we present on-demand optimal message routing algorithms for the dual optimal BDLNs that we have constructed. The presented algorithms do not require routing tables and are efficient, requiring little computation.