On the Runtime and Robustness of Randomized Broadcasting

In this paper, we study the following randomized broadcasting protocol. At some time t an information r is placed at one of the nodes of a graph. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. We begin by developing tight lower and upper bounds on the runtime of the algorithm described above. First, it is shown that on ∆-regular graphs this algorithm requires at least log2− 1 ∆ n + log( ∆ ∆−1 ) ∆ n − o(log n) ≈ 1.69 log2 n rounds to inform all n nodes. Together with a result of Pittel (On Spreading a Rumor, SIAM Journal on Applied Mathematics, 47(1):213-223) this bound implies that the algorithm has best performance on complete graphs among all regular graphs. For general graphs, we prove a slightly weaker lower bound of log2− 1 ∆ n + log4 n − o(log n) ≈ 1.5 log2 n, where ∆ denotes the maximum degree of G. We also prove two general upper bounds, (1 + o(1))n lnn and O(n∆δ ), respectively, where δ denotes the minimum degree. The second part of this paper is devoted to the analysis of fault-tolerance. We show that if the informed nodes are allowed to fail in some step with probability 1 − p, then the broadcasting time increases by at most a factor 6/p. As a by-product, we determine the performance of agent based broadcasting in certain graphs and obtain bounds for the runtime of randomized broadcasting on Cartesian products of graphs.