Adaptive Fault-Tolerant Routing in Hypercube Multicomputers

A connected hypercube with faulty links and/or nodes is called an injured hypercube. To enable any non-faulty node to communicate with any other non-faulty node in an injured hypcrcube, the information on component failures has to be made available to non-faulty nodes so as to route messages around the faulty components. We propose first a distributed adaptive fault-tolerant routing scheme for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probability and the expected length of a resulting path is very close to that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, we also introduce a routing scheme based on depth-first search which works in the presence of an arbitrary number of faulty components. Due to the insufficient information on faulty components, however, the paths chosen by the above scheme may not always be the shortest. To guarantee all messages to be routed via shortest paths, we propose to equip every node with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are also given to illustrate bur results.