Gossiping in Minimal Time

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. Deenitive new optimal time algorithms are derived for complete graphs, rings, regular grids and toroidal grids that signiicantly extend existing results. In particular, we settle an open problem about minimum time gossiping in complete graphs. Speciically, 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 .