Linear time algorithms for fault tolerant routing in hypercubes and star graphs

In this paper, we study the following node-to-node fault tolerant routing problem: In the presence of up to n-1 faulty nodes, find a fault-free path which connects any two non-faulty nodes s and t in an n-connected graph. For node-to-node fault tolerant routing in n-dimensional hypercubes Hn, we give an algorithm which finds a fault-free path s&rarrt of length at most d(s, t)+2[log n/d(s, t)] in O(n) time, where d(s, t) is the distance between s and t. We also show that a fault-free path s&rarrt in Hn of length at most d(s, t)+2i, 1[ii-1+n) time. For node-to-node fault tolerant routing in n-dimensional star graphs Gn, we give an algorithm which finds a fault-free path s&rarrt of length at most min{d(Gn)+3, d(s, t)+6} in O(n) time, where d(G$-$/) = [3(n-1)/2] is the diameter of GN. It is previously known that, in Hn, a fault-free path s&rarrt of length at most d(s, t) for d(s, t) = n and at most d(s, t)+2 for d(s, t)n, a fault-free path s&rarrt of length at most min{d(Gn)+1, d(s, t)+4} can be found in O(d(s, t)n) time. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.