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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Pull and Push&Pull in Random Evolving Graphs
Rami Daknama · 2018-01-01 · via cs.DS updates on arXiv.org

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 Erdős-Rényi 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 $γ:= 2(1-e^{-a})-(1-e^{-a})^2/a$; we prove that the expected number of rounds needed by Push&Pull is $\log_{1+γ}(n)+1/a\ln(n)+\mathcal{O}(1)$; as a byproduct, we obtain large deviation bounds, too.