The gossip problem involves communicating a unique item from each node in a graph to every other node. We study the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receiver. We study a number of topologies including the complete graph, grids, hypercubes and rings. De nitive new optimal time algorithms are derived for complete graphs, rings, regular grids and toroidal grids that signi cantly extend existing results. In particular, we settle an open problem about minimum time gossiping in complete graphs. Speci cally, for a graph with N nodes, at least log N communication steps, where the logarithm is in the base of the golden ratio , are required by any algorithm under the weakest model of communication. This bound, which is approximately 1:44 log 2 N , can be realized for some networks and so the result is optimal.
We consider a model of multicast communication in a network whereby multiple sources have messages to disseminate among all sites of a network. We propose that the messages from all sources are disseminated along the same spanning tree of the network and consider the problem of constructing an optimal such tree. One measure for suitability of the construction is the sum of distances from all sources to all other vertices. We show that finding the exact solution in this case in [Formula: see text]-hard (in the strong sense). We then investigate solutions for some restricted classes of graphs and give efficient algorithms for those. We also consider an alternative measure of goodness for the spanning tree, being the maximum eccentricity of a source. We show that the problem of finding such a minimum eccentricity spanning tree is somewhat easier to solve and give a pseudo-polynomial solution algorithm.
The gossip problem involves communicating a unique item from every node in a graph to every other node. We study the minimum time required to do this for the binary hypercube under two models of communication. In the rst model, all communication links may be used concurrently but each may only carry information in one direction at a time. In the second, weaker model each node may be involved in only one communication at a time either as sender or receiver. In both cases, simple algorithms exist which are close to optimal. This paper shows that neither of these algorithms is optimal by exhibiting faster algorithms. In the rst case an optimal algorithm is obtained.
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