Multinode Broadcast in Hypercubes and Rings with Randomly Distributed Length of Packets

We consider a multinode broadcast (MNB) in a hypercube and in a ring network of processors. This is the communication task where we want each node of the network to broadcast a packet to all the other nodes. The communication model that we use is different than those considered in the literature so far. In particular, we assume that the lengths of the packets that are broadcast are not fixed, but are distributed according to some probabilistic rule, and we compare the optimal times required to execute the MNB for variable and for fixed packet lengths. For large hypercubes we show under very general probabilistic assumptions on the packet lengths, that the MNB is completed in essentially the same time as when the packet lengths are fixed. In particular, we show that the MNB is completed by time (1 + δ)T_s with probability at least 1 - ∊, for any positive ∊ and δ, where T_s is the optimal time required to execute the MNB when the packet lengths are fixed at their mean, provided that the size of the hypercube is large enough. In the case of the ring we prove that the average time required to execute a MNB when the packet lengths are exponentially distributed exceeds by a factor of In n the corresponding time for the case where the packet lengths are fixed at their mean, where n is the number of nodes of the ring.