Optimal fault-tolerant routing algorithm and fault-tolerant diameter in directed double-loop networks

This paper addresses the reliability of directed double-loop networks G(N;r,s), and studies the problems about optimal fault-tolerant routing, fault-tolerant diameter, etc., in G(N;r,s). Firstly, we study the shapes of the L-shaped tiles which are the minimum distance diagram of directed double-loop networks; we divide them into four types. There are different parameters for each type. According to the solutions to the congruence equation, then we study the distributions of the optimal equivalent nodes for different destination nodes, and present an optimal fault-tolerant routing algorithm and a formula computing the fault-tolerant diameter. Finally, we prove that there is a lower bound for fault-tolerant diameter, and show that there may be lots of double optimal directed double-loop networks in some infinite clusters of G(N;r,s). According to the proposed fault-tolerant routing algorithm, the reliability and transmission performance will be optimal when some faults occur in G(N;r,s).