Pull and Push&Pull in Random Evolving Graphs

The Push, the Pull and the Push&Pull algorithms are well-studied rumor spreading protocols. In all three, in the beginning one node of a graph is informed. In the Push setting, every round every informed node chooses a neighbor uniformly at random and, if it is not already informed anyway, informs it. In the Pull setting, each round each uninformed node chooses a neighbor uniformly at random and asks it for the rumor; if the asked neighbor is informed, now also the asking node is informed. Push&Pull is a combination of Push and Pull: In each round, each node picks a neighbor uniformly at random. If at least one of both knows the rumor, after this round, both know the rumor. Clementi et al. have considered Push in settings where the underlying graph changes each round. In one setting they investigated, in each round the underlying graph is a newly sampled Erd\H{o}s-R\'enyi random graph $G(n,p)$. They show that if $p\geq 1/n$ then with probability $1-o(1)$ (as $n\rightarrow \infty$) the number of rounds needed until all nodes are informed is $\mathcal{O}(\ln(n))$. Doerr and Kostrygin introduced a general framework to analyze rumor spreading algorithms; using this framework, for $a>0$ and $p=a/n$ they improved the previous results in the described setting: The expected number of rounds needed by Push was determined to be $\log_{2-e^{-a}}(n)+1/(1-e^{-a})\ln(n)+\mathcal{O}(1)$; also large deviation bounds were obtained. Using their framework, we investigate Pull and Push&Pull in that setting: We prove that the expected number of rounds needed by Pull to inform all nodes is $\log_{2-e^{-a}}(n)+1/a \ln(n)+\mathcal{O}(1)$. Let $\gamma := 2(1-e^{-a})-(1-e^{-a})^2/a$; we prove that the expected number of rounds needed by Push&Pull is $\log_{1+\gamma}(n)+1/a\ln(n)+\mathcal{O}(1)$; as a byproduct, we obtain large deviation bounds, too.