Distributed fault-tolerant ring embedding and reconfiguration in hypercubes

To embed a ring in a hypercube is to find a Hamiltonian cycle through every node of the hypercube. It is obvious that no 2 ⁿ-node Hamiltonian cycle exists in an n-dimensional faulty hypercube which has at least one faulty node. However, if a hypercube has faulty links only and the number of faulty links is at most n • •2, at least one 2 ⁿ-node Hamiltonian cycle can be found. In this paper, we propose a distributed ring-embedding algorithm that can find a Hamiltonian cycle in a fault-free or faulty n-dimensional hypercube (Q _n), and the complexity is O(n) parallel steps. The algorithm is based on the recursion property of the hypercube and the free-link dimension concept. In some cases, even when the number of faulty links is larger than n • •2, Hamiltonian cycles may still exist. We will show that the largest possible number of faulty links that can be tolerated is 2 ^{n• 1} • •1. The performance and the constraints of the fault-tolerant algorithm is also analyzed in detail in this paper. Furthermore, a dynamic reconfiguration algorithm for an embedded ring is proposed and discussed. Due to the distributed nature of the algorithms, they are useful for the simulation of ring-based multiprocessors on MIMD hypercube multiprocessors.