Fault-tolerant Routing Methods in Crossed Cubes

In this study, we propose two methods for fault-tolerant routing in the crossed cube. The first method, Method 1, at each node that has the message to the destination node, its neighbor nodes into the set of the forward neighbor nodes, the set of the sideward neighbor nodes, and the set of the backward neighbor nodes by calculating the distances from the neighbor nodes to the destination node. Then, Method 1 chooses one of the neighbor nodes and forwards the message to the node. Based on the observation of the execution of Method 1, we found that the routing tends to fail at the nodes with 3 hops to the destination node. Hence, we have introduced another method, Method 2, that executes the depth-first search at the nodes that are at 3 hops to the destination node. By adopting a simple fault-tolerant routing method, Simple, as the baseline, we conducted a computer experiment in 11-, 12-, and 13-dimensional crossed cubes, CQ11, CQ12, and CQ13. As a result, Method 1 showed better ratios of successful routings than a baseline method by at most 0.0973 in CQ11, 0.2082 in CQ12, and 0.139 in CQ13, respectively. Also, Method 2 showed better ratios than a baseline method by at most 0.1195 in CQ11, 0.2366 in CQ12, and 0.674 in CQ13, respectively. The average path lengths were also improved by Method 1 compared to the baseline method by at most 2.35 in CQ11, 2.07 in CQ12, and 3.63 in CQ13, respectively. Moreover, Method 2 improved them by 1.89 in CQ11, 1.61 in CQ12, and 3.12 in CQ13, respectively.