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cs.DS updates on arXiv.org

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
Tight bounds for undirected graph exploration with pebble...
Yann Disser, Jan Hackfeld, Max Klimm · 2018-05-09 · via cs.DS updates on arXiv.org

We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $Θ(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles, or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory $Θ(\log \log n)$ pebbles are necessary and sufficient for exploration. We further prove that the same bound holds for the number of collaborating agents needed for exploration. For the upper bound, we devise an algorithm for a single agent with constant memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$ pebbles, even when $n$ is unknown. The algorithm terminates after polynomial time and returns to the starting vertex. Since an additional agent is at least as powerful as a pebble, this implies that $\mathcal{O}(\log \log n)$ agents with constant memory can explore any $n$-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph of size $n$ is already $Ω(\log \log n)$ when we allow each agent to have at most $\mathcal{O}( \log n ^{1-\varepsilon})$ bits of memory for any $\varepsilon>0$. This also implies that a single agent with sublogarithmic memory needs $Θ(\log \log n)$ pebbles to explore any $n$-vertex graph.