Social networks spread rumors in sublogarithmic time
Proceedings of the forty-third annual ACM symposium on Theory of computing
Ultra-fast rumor spreading in social networks
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Rumor spreading and vertex expansion on regular graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Diameter and broadcast time of random geometric graphs in arbitrary dimensions
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Coalescing random walks and voting on graphs
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
Randomized information dissemination in dynamic environments
IEEE/ACM Transactions on Networking (TON)
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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.