Unicast in hypercubes with large number of faulty nodes

Unicast in computer/communication networks is a one-to-one communication between a source node s and a destination node t. We propose three algorithms which find a nonfaulty routing path between s and t for unicast in the hypercube with a large number of faulty nodes. Given the n-dimensional hypercube H-n and a set F of faulty nodes, node u epsilon H-n is called k-safe if u has at least k nonfaulty neighbors. The H-n is called k-safe if every node of H-n is k-safe. it has been known that for 0 less than or equal to k less than or equal to n/2, a k-safe H-n is connected if \ F \ less than or equal to 2(k)(n - k) - 1. Our first algorithm finds a nonfaulty path of length at most d(s, t) + 4 in O(n) time for unicast between 1-safe s and t in the H-n with \ F \ less than or equal to 2n - 3, where d(s, t) is the distance between s and t. The second algorithm finds a nonfaulty path of length at most d(s, t) i 6 in O(n) time for unicast in the 2-safe H-n with \ F \ less than or equal to 4n - 9. The third algorithm finds a nonfaulty path of length at most d(s, t)+ O(k(2)) in time O(\ F \ + n) for unicast in the 2-safe H-n with \ F \ less than or equal to 2(k)(n - k) - 1 (0 less than or equal to k less than or equal to n/2). The time complexities of the algorithms are optimal. We show that in the worst case, the length of the nonfaulty path between s and t in a k-safe H-n with \ F \ less than or equal to 2(k)(n - k) - 1 is at least d(s, t) + 2(k + 1) for 0 less than or equal to k less than or equal to n/2. This implies that the path lengths found by the algorithms for unicast in the 1-safe and 2-safe hypercubes are optimal.