How robust is the n-cube?

AbstractThe n-cube network is called faulty if it contains any faulty processor or any faulty link. For any number k we want to compute the minimum number f(n, k) of faults which is necessary for an adversary to make every (n − k)-dimensional subcube faulty. Reversely formulated: The existence of an (n − k)-dimensional non-faulty subcube can be guaranteed, if there are less than f(n, k) faults in the n-cube. In this paper several lower and upper bounds for f(n, k) are derived such that the resulting gaps are “small.” For instance if k ≥ 2 is constant, then f(n, k) = θ(logn). Especially for k = 2 and large n: f(n, 2) ∈ [[αn⌉]: [αn]⌉ + 2], where αn =logn + ½ log log n + ½. Or if k = ω(log log n) then 2k < f(n, k) < 2(1 + ɛ)k, with ɛ chosen arbitrarily small. The aforementioned upper bounds are obtained by analysing the behaviour of an adversary who makes “worst-case” distributions of a given number of faulty processors. For k = 2 the “worst-case” distribution is obtained constructively. In the general case the constructive methods presented in this paper lead to a (rather “bad”) upper bound which can be significantly improved by probabilistic arguments. The bounds mentioned above change if the notions are relativized with respect to some given parallel fault-checking procedure P. In this case only subcubes which are possible outputs of P must be made faulty by the adversary. The notion of directed chromatic index is defined in order to analyse the case k = 2. Relations between the directed chromatic index and the chromatic number are derived, which are of interest in their own right