On Mixing and Edge Expansion Properties in Randomized Broadcasting

Abstract

In this paper, we consider a popular randomized broadcasting algorithm called push-algorithm defined as follows. Initially, one vertex of a graph G=(V,E) owns a piece of information which is spread iteratively to all other vertices: in each timestep t=1,2,… every informed vertex chooses a neighbor uniformly at random and informs it. The question is how many time steps are required until all vertices become informed (with high probability). For various graph classes, involved methods have been developed in order to show an upper bound of $\mathcal{O}(\log N+\mathop{\mathrm{diam}}(G))$on the runtime of the push-algorithm, where N is the number of vertices and $\mathop{\mathrm{diam}}(G)$denotes the diameter of G. However, no asymptotically tight bound on the runtime based on the mixing time of random walks has been established. In this work we fill this gap by deriving an upper bound of $\mathcal{O}(\mathsf {T}_{\mathop{\mathrm{mix}}}+\log N)$, where $\mathsf{T}_{\mathop{\mathrm{mix}}}$denotes the mixing time of a certain random walk on G. After that we prove upper bounds that are based on certain edge expansion properties of G. However, for hypercubes neither the bound based on the mixing time nor the bounds based on edge expansion properties are tight. That is why we develop a general way to combine these two approaches by which we can deduce that the runtime of the push-algorithm is Θ(log N) on every Hamming graph.