Broadcasting in hypercubes with randomly distributed Byzantine faults
We study all-to-all broadcasting in hypercubes with randomly distributed Byzantine faults. We construct an efficient broadcasting scheme BC1-n-cube running on the n-dimensional hypercube (n-cube for short) in 2n rounds, where for communication by each node of the n-cube, only one of its links is used in each round. The scheme BC1-n-cube can tolerate ⌊(n −1)/2⌋ Byzantine node and/or link faults in the worst case. If there are at most f Byzantine faulty nodes randomly distributed in the n-cube, BC1-n-cube succeeds with a probability higher than 1-(64nf/2n)⌈n/2⌉. In other words, if 1/(64nk) of all the nodes (i.e., 2n/(64nk) nodes) fail in Byzantine manner randomly in the n-cube, then the scheme succeeds with a probability higher than 1 −k−⌈ n/2 ⌉. We also consider the case where all nodes are faultless but links may fail randomly in the n-cube. Scheme BC1-n-cube succeeds with a probability higher than 1 −k−⌈ n/2 ⌉ provided that not more than l/(64(n + 1)k) of all the links in the n-cube fail in Byzantine manner randomly. For the case where only links may fail, we give another broadcasting scheme BC2-n-cube which runs in 2n2 rounds. Broadcasting by BC2-n-cube is successful with a high probability if the number of Byzantine faulty links randomly distributed in the n-cube is not more than a constant fraction of the total number of links. That is, it succeeds with a probability higher than 1−n·k−⌈ n/2 ⌉ if l/(48k) of all the links in the n-cube fail in Byzantine manner randomly.
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