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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
Are We Still Missing an Item?
Roey Magen · 2024-01-12 · via cs.DS updates on arXiv.org

The missing item problem, as introduced by Stoeckl in his work at SODA 23, focuses on continually identifying a missing element $e$ in a stream of elements ${e_1, ..., e_{\ell}}$ from the set $\{1,2,...,n\}$, such that $e \neq e_i$ for any $i \in \{1,...,\ell\}$. Stoeckl's investigation primarily delves into scenarios with $\ell<n$, providing bounds for the (i) deterministic case, (ii) the static case -- where the algorithm might be randomized but the stream is fixed in advanced and (iii) the adversarially robust case -- where the algorithm is randomized and each stream element can be chosen depending on earlier algorithm outputs. Building upon this foundation, our paper addresses previously unexplored aspects of the missing item problem. In the first segment, we examine the static setting with a long stream, where the length of the steam $\ell$ is close to or even exceeds the size of the universe $n$. We present an algorithm demonstrating that even when $\ell$ is very close to $n$ (say $\ell=n-1$), polylog($n$) bits of memory suffice to identify the missing item. When the stream's length $\ell$ exceeds the size of the universe $n$ i.e. $\ell = n +k$, we show a tight bound of roughly $Θ(k)$. The second segment focuses on the adversarially robust setting. We show a lower bound for a pseudo-deterministic error-zero (where the algorithm reports its errors) algorithm of approximating $Ω(\ell)$, up to polylog factors. Based on Stoeckl's work and the previous result, we establish a tight bound for a random-start (only use randomness at initialization) error-zero streaming algorithm of roughly $Θ(\sqrt{\ell})$.