Class-congruence property and two-phase routing of Borel Cayley graphs

Dense, symmetric graphs are useful interconnection models for multicomputer systems. Borel Cayley graphs, the densest degree-4 graphs for a range of diameters, are attractive candidates. However, the group-theoretic representation of these graphs makes the development of efficient routing algorithms difficult. In earlier reports, we showed that all degree-4 Borel Cayley graphs have generalized chordal ring (GCR) and chordal ring (CR) representations. We present the class-congruence property and use this property to develop the two-phase routing algorithm for Borel Cayley graphs in a special GCR representation. The algorithm requires a small space complexity of O(p+k) for n=p×k nodes. Although suboptimal, the algorithm finds paths with length bounded by 2D, where D is the diameter. Furthermore, our computer implementation of the algorithm on networks with 1,081 and 15,657 nodes shows that the average path length is on the order of the diameter. The performance of the algorithm is compared with that of existing optimal and suboptimal algorithms