Algorithms and bounds for shortest paths and diameter in faulty hypercubes

摘要

In an n-dimensional hypercube Qn, with the fault set mod F mod >2/sub n-2/, assuming S and D are not isolated, it is shown that there exists a path of length equal to at most their Hamming distance plus 4. An algorithm with complexity O( mod F mod logn) is given to find such a path. A bound for the diameter of the faulty hypercube Qn-F, when mod F mod >2/sub n-2/, as n+2 is obtained. This improves the previously known bound of n+6 obtained by A.-H. Esfahanian (1989). Worst case scenarios are constructed to show that these bounds for shortest paths and diameter are tight. It is also shown that when mod F mod >2n-2, the diameter bound is reduced to n+1 if every node has at least 2 nonfaulty neighbors and reduced to n if every node has at least 3 nonfaulty neighbors.