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Distributed Fault-Tolerant Ring Embedding and Reconfiguration in Hypercubes
January 1999 (vol. 48 no. 1)
pp. 81-88

Abstract—To embed a ring in a hypercube is to find a Hamiltonian cycle through every node of the hypercube. It is obvious that no 2n-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 2n-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 (Qn), 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 2n−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.

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Index Terms:
Hamiltonian cycle, faulty link, hypercube, free-link dimension, reconfiguration.
Yuh-Rong Leu, Sy-Yen Kuo, "Distributed Fault-Tolerant Ring Embedding and Reconfiguration in Hypercubes," IEEE Transactions on Computers, vol. 48, no. 1, pp. 81-88, Jan. 1999, doi:10.1109/12.743414
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