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<p><b>Abstract</b>—To embed a ring in a hypercube is to find a <it>Hamiltonian cycle</it> through every node of the hypercube. It is obvious that no 2<super><it>n</it></super>-node Hamiltonian cycle exists in an <it>n</it>-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 <it>n</it>− 2, at least one 2<super><it>n</it></super>-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 <it>n</it>-dimensional hypercube (<it>Q</it><sub><it>n</it></sub>), and the complexity is <it>O</it>(<it>n</it>) parallel steps. The algorithm is based on the recursion property of the hypercube and the <it>free-link dimension</it> concept. In some cases, even when the number of faulty links is larger than <it>n</it>− 2, Hamiltonian cycles may still exist. We will show that the largest possible number of faulty links that can be tolerated is 2<super><it>n</it>−1</super>− 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.</p>
Hamiltonian cycle, faulty link, hypercube, free-link dimension, reconfiguration.

Y. Leu and S. Kuo, "Distributed Fault-Tolerant Ring Embedding and Reconfiguration in Hypercubes," in IEEE Transactions on Computers, vol. 48, no. , pp. 81-88, 1999.
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