FAULT-TOLERANT NETWORKS BASED ON THE DE BRUIJN GRAPH

We introduce a new class of networks based on the de Bruijn graph. These directed graphs are regular of degree k, have N = k(n) vertices for some n, and can tolerate up to k - 2 node faults. Their fault-free diameter is n = log(k)N, and this increases by at most 1 hop in the presence of k - 2 faults. This class is very rich: for any given N = k(n), we can construct at least 2N different such graphs. This is in sharp contrast to most other such constructions (including the de Bruijn graph), in which only one graph exists for each N. We also show how to implement certain algorithms on these networks. First, we exhibit an optimal algorithm for performing fault-tolerant routing in these networks. Given a source vertex S, a destination vertex D, and a set F of faulty vertices, our algorithm executes in time O(n\F\), and this is asymptotically optimal. It is interesting that this time bound is independent of the value of k. The techniques that we employ draw from the known theory of string overlaps, and are therefore likely to be applicable to other similar problems. We also show how to compute strongly-connected components on these networks in the presence of arbitrarily many faults; this uses some constructions from the theory of finite-state automata. Finally, we show how to handle time-varying faults. These algorithms run in time polynomial in n = log(k)N and the number of faults.