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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
Compressing Dynamic Fully Indexable Dictionaries in Word-RAM
Gabriel Marques Domingues · 2026-03-24 · via cs.DS updates on arXiv.org

We study the problem of constructing a dynamic fully indexable dictionary (FID) in the Word-RAM model using space close to the information-theoretic lower bound. A FID is a data-structure that encodes a bit-vector $B$ of length $u$ and answers, for $b\in\{0,1\}$, $\texttt{rank}_b(B, x)=|{\{y\leq x~|~B[y]=b\}}|$ and $\texttt{select}_b(B, r)=\min\{0\leq x<u~|~\texttt{rank}_b(B, x)=r\}$ ($-1$ if empty). A dynamic FID supports updates that modify a single bit of $B$, i.e., $B[i]\gets b$. We work in the Word-RAM model with $w$-bit words, assuming $w\geq \operatorname{lg} u$. Integer multiplication takes $\mathcal{O}(1)$ time. Our memory model is $\mathcal{M}_B$, allowing access to a fixed precomputed table of $τ=\operatorname{polylog}(w)$ words, which can be computed in $\mathcal{O}(wτ)$ time. In this paper, we show a dynamic FID based on the famous fusion-tree data-structure of P{ă}tra{ş}cu and Thorup [FOCS 2014], modified to use fewer bits and to support $\texttt{select}_0$. Let $n$ denote the number of ones in $B$. We describe a parametric construction: for every $ε\leq 1/2$, there is a dynamic FID using $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(nw^ε/ε)\text{ bits}$$ taking $\mathcal{O}({1/ε+\log_w(n)})$ time for $\texttt{rank}_0/\texttt{rank}_1/\texttt{select}_0$ and updates, and $\mathcal{O}({\log_w(n)})$ time for $\texttt{select}_1$. All time bounds are worst-case. For $ε={1/\sqrt{\operatorname{lg} w}}$, we reduce the space to $\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log w)$ bits. For $ε=Θ(1)$, the running time matches the lower bound of Fredman and Saks [STOC 1989]. This is the first deterministic dynamic FID in the standard Word-RAM model that achieves $o(n\sqrt{w})$ bits of redundancy in $\mathcal{M}_B$ (e.g., $ε=1/4$), and optimal worst-case time.