Methods for distributed unicast in hypercubes

Abstract

Unicast algorithms in off-line routing have been used for one-to-one communication between a source node and a destination node in an n -dimensional hypercube, denoted as H n . A node is called k -safe, where 0⩽ k ⩽ n , if it has at least k healthy neighbors, and H n is called k -safe if every node in it is k -safe. A k -safe H n is connected if the number of faulty nodes, | F |, does not exceed 2 k ( n − k )−1. In this paper, we propose two methods for distributed routing. The first method has been presented in [Proc. 7th Int. IEEE Conf. Electron., Circ. Syst., Jounieh, Lebanon, December, 2000, p. 194]. The second method that has not been addressed before, can be used for off-line routing. In the case of off-line routing we avoid the cost of collecting global information about the faulty nodes, and the cost of getting information about H n , whether it is k -safe or not. The minimum requirements of the proposed methods is to have the path between the source and the destination connected. Hence, they may work when H n is disconnected, which is an important advantage. The time cost of the first method may be O( mn 4 ), and the expected length of the routing path between source and destination may be O( mn 5 ), where 1⩽ m ⩽ n . The time cost of the second method may be O( en n /2 ), the space cost may be O( en n /2 ), and the expected length of the routing path between source and destination may be d ( s , t ).