Node-to-set and set-to-set cluster fault tolerant routing in hypercubes

We study node-to-set and set-to-set fault tolerant routing problems in n-dimensional hypercubes Hn. Node-to-set routing problem is that given a node s and a set of nodes T = {t1,...,tk}, finds k node-disjoint paths s&rarrti, 1&lei&lek. Set-to-set routing problem is that given two sets of nodes S = {S1,...,Sk} and T = {t1,...,tk}, finds k node-disjoint paths, each path connects a node of S and a node of T. From Menger's theorem, it is known that these two problems in Hn can tolerate at most n-k arbitrary faulty nodes. In this paper, we prove that both routing problems can tolerate n-k arbitrary faulty subgraphs (called cluster) of diameter 1. For 2&lek&len, we show that, in the presence of at most n-k faulty clusters of diameter at most 1, the k paths of length at most n+2 for node-to-set routing in Hn can be found in O(kn) optimal time and the k paths of length at most n+k+2 for set-to-set routing in Hn can be found in O(kn log k) time. The upper bound n+2 on the length of the paths for node-to-set routing in Hn is optimal.