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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
Election vs. Selection: Two Ways of Finding the Largest N...
Avery Miller, Andrzej Pelc · 2014-11-06 · via cs.DS updates on arXiv.org

Finding the node with the largest label in a network, modeled as an undirected connected graph, is one of the fundamental problems in distributed computing. This is the way in which $\textit{leader election}$ is usually solved. We consider two distinct tasks in which the largest-labeled node is found deterministically. In $\textit{selection}$, this node must output 1 and all other nodes must output 0. In $\textit{election}$, the other nodes must additionally learn the largest label. Our aim is to compare the difficulty of these two tasks executed under stringent running time constraints. The measure of difficulty is the amount of information that nodes of the network must initially possess in order to solve the given task in an imposed amount of time. Following the standard framework of $\textit{algorithms with advice}$, this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire graph. The length of this string is called the $\textit{size of advice}$. Consider the class of $n$-node graphs with any diameter $diam \leq D$. If time is larger than $diam$, then both tasks can be solved without advice. For the task of $\textit{election}$, we show that if time is smaller than $diam$, then the optimal size of advice is $Θ(\log n)$, and if time is exactly $diam$, then the optimal size of advice is $Θ(\log D)$. For the task of $\textit{selection}$, the situation changes dramatically, even within the class of rings. Indeed, for the class of rings, we show that, if time is $O(diam^ε)$, for any $ε<1$, then the optimal size of advice is $Θ(\log D)$, and, if time is $Θ(diam)$ (and at most $diam$) then this optimal size is $Θ(\log \log D)$.