Brief Announcement: Ultra-Fast Asynchronous Randomized Rumor Spreading

Standard randomized rumor spreading algorithms propagate a piece of information, so-called the rumor, in a given network that proceed in synchronized rounds. Starting with a single informed node, in each subsequent round, every node calls a random neighbor in order to exchange the rumor (by sending the rumor to the neighbor (push algorithm) or asking it from the neighbor (pull algorithm)). Panagiotou et al. [ISAAC'13] considered a multiple-call version of the algorithms where each node is enabled to make more than one call in each round. The number of calls of a node is independently chosen from a probability distribution R. Seeking for a more realistic model, we propose an asynchronous version of the multiple-call algorithms on fully connected networks. In our model, each node has an independent Poisson clock whose rate may differ from others. Basically, the clock rate of each node is independently drawn from a probability distribution R at the beginning of the process. The push algorithm starts with a single informed node, when the clock of an informed node rings, the node contacts a random neighbor and sends (pushes) the rumor to the neighbor. Similarly, in the push-pull, if the clock of a node rings, then the node contacts a random neighbor in order to exchange the rumor. We study the effect of R on the spreading time of the algorithms, which is the time that the algorithm needs to inform all nodes with high probability. In this work, we show that if R is a power law distribution with exponent β \in(2,3)$ and $\varepsilon\in[1/n, 1-1/n]$ be an arbitrary number. Then in expectation, after $O(1+łog(1/\varepsilon))$ time the push-pull algorithm informs at least $(1-\varepsilon)n$ nodes. Moreover, if R is an arbitrary distribution with bounded mean and variance, we show that the push algorithm spreads the rumor in a complete network with n nodes in $\fracłog n \matbbE R \pm O(łogłog n)$ time, with high probability.