On Mixing and Edge Expansion Properties in Randomized Broadcasting

Abstract

A very simple and natural broadcasting algorithm is the so-called push algorithm which has several applications in the area of distributed computing. Initially, only one vertex of a graph G = (V,E) owns a piece of information which is spread iteratively to all other vertices: in each time step t = 1,2,... every informed vertex chooses some neighbor uniformly at random which then becomes informed and may itself inform other vertices in the succeeding time steps. The crucial question is how many time steps are required such that all vertices become informed (with high probability).

For various graph classes, involved methods have been developed in order to show an upper bound of ${\mathcal{O}}(\log N+{\operatorname{diam}}(G))$, where N is the number of vertices and ${\operatorname{diam}}(G)$ denotes the diameter of G. However, currently no asymptotically tight bound on the runtime of the push algorithm based on the mixing time exists. In this work we fill this gap by deriving an upper bound of ${\mathcal{O}}(\log N + {\mathsf{T}}_{\operatorname{mix}})$, where ${\mathsf{T}}_{{\operatorname{mix}}}$ denotes the mixing time of a certain random walk on G. After that we give a simple but useful upper bound which is based on a certain average value of the edge expansion of G. Unfortunately, both approaches do not give the right bound for Hypercubes. Therefore, we develop a general way to combine them and prove that the runtime of the push algorithm is Θ(logN) on every Hamming graph.

This work was partially supported by German Science Foundation (DFG) Research Training Group GK-693 of the Paderborn Institute for Scientific Computation (PaSCo) and by the Integrated Project IST-15964 “Algorithmic Principles for Building Efficient Overlay Networks” (AEOLUS) of the European Union.