Efficient information exchange in the random phone-call model

We consider the gossiping problem in the classical random phone-call model introduced by Demers et. al. ([6]). We are given a complete graph, in which every node has an initial message to be disseminated to all other nodes. In each step every node is allowed to establish a communication channel with a randomly chosen neighbour. Karp et al. [15] proved that it is possible to design a randomized procedure performing O(nloglogn) transmissions that accomplishes broadcasting in time O(logn), with probability 1-n ^-1. In this paper we provide a lower bound argument that proves Ω(nlogn) message complexity for any O(logn)-time randomized gossiping algorithm, with probability 1-o(1). This should be seen as a separation result between broadcasting and gossiping in the random phone-call model. We study gossiping at the two opposite points of the time and message complexity trade-off. We show that one can perform gossiping based on exchange of O(n•logn/loglogn) messages in time O(log² n/loglogn), and based on exchange of O(nloglogn) messages with the time complexity Both results hold wit probability 1-n ^-1. Finally, we consider a model in which each node is allowed to store a small set of neighbours participating in its earlier transmissions. We show that in this model randomized gossiping based on exchange of O(nloglogn) messages can be obtained in time O(logn), with probability 1-n ^-1.