Fault-tolerant multicasting in hypercube multicomputers based on local safety information

A fully unsafe hypercube can be split into a unique set of maximal safe subcubes. Multicasting in a maximal safe subcube can be completed reliably based on information related to the maximal safe subcube. It is found that there always exists a time-optimal multicasting if (1) the multicast source is locally safe in the minimum subcube called a multicast subcube that contains the source and destinations, or (2) the spanning subcube between each destination and the source is safe. We show a sufficient condition for the existence of a multicasting: The multicast subcube is safe or the spanning subcube between the source and each destination is either safe or is contained in a safe subcube. The proposed fault-tolerant multicast algorithm can always find a time-optimal multicast if the multicast subcube is contained in a maximal safe subcube, in which the source is locally safe. Similarly, a time-optimal multicast can always be found by the algorithm if each destination and the source are contained in a maximal safe subcube, in which the source is locally safe. A multicast can always be carried out successfully if the multicast subcube is contained in a maximal safe subcube, or each destination and the source or a neighbor of the source are contained in a maximal safe subcube.