K-pairwise cluster fault tolerant routing in hypercubes

In this paper, we introduce a general fault tolerant routing problem, cluster fault tolerant routing, which is a natural extension of the well studied node fault tolerant routing problem. A cluster is a connected subgraph of a graph G, and a cluster is faulty if all nodes in it are faulty. In cluster fault tolerant routing (abbreviated as CFT routing), we are interested in the number of faulty clusters and the size of the clusters that an interconnection network can tolerate for certain routing problems. As a case study, we investigate the following k-pairwise CFT routing in n-dimensional hypercubes H_n: Given a set of faulty clusters and k distinct nonfaulty node pairs (s₁ , t₁), ..., (s_k, t_k) in H_n , find k fault-free node-disjoint paths s_i→t_i , 1⩽i⩽k. We show that H_n can tolerate n-2 faulty clusters of diameter one, plus one faulty node for the k-pairwise CFT routing with k=1. For n⩽4 and 2⩽k⩽[n/2], we prove that H _n can tolerate n-2k+1 faulty clusters of diameter one for the k-pairwise CFT routing. We also give an O(kn log n) time algorithm which finds the k paths for the mentioned problem. Our algorithm implies an O(n² log n) time algorithm for the k-pairwise node-disjoint paths problem in H_n, which improves the previous result of O(n³ log n)