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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
Path and Ancestor Queries on Trees with Multidimensional ...
Meng He, Serikzhan Kazi · 2019-10-03 · via cs.DS updates on arXiv.org

We consider an ordinal tree $T$ on $n$ nodes, with each node assigned a $d$-dimensional weight vector $\pnt{w} \in \{1,2,\ldots,n\}^d,$ where $d \in \mathbb{N}$ is a constant. We study path queries as generalizations of well-known {\textit{orthogonal range queries}}, with one of the dimensions being tree topology rather than a linear order. Since in our definitions $d$ only represents the number of dimensions of the weight vector without taking the tree topology into account, a path query in a tree with $d$-dimensional weight vectors generalize the corresponding $(d+1)$-dimensional orthogonal range query. We solve {\textit{ancestor dominance reporting}} problem as a direct generalization of dominance reporting problem, %in time $Ø((\lg^{d-1} n)/(\lg\lg n)^{d-2}+k)$ in time $Ø(\lg^{d-1}{n}+k)$ %and space of $Ø(n(\lg n)^{d-1}/(\lg \lg n)^{d-2})$ words, and space of $Ø(n\lg^{d-2}n)$ words, where $k$ is the size of the output, for $d \geq 2.$ We also achieve a tradeoff of $Ø(n\lg^{d-2+\eps}{n})$ words of space, with query time of $Ø((\lg^{d-1} n)/(\lg\lg n)^{d-2}+k),$ for the same problem, when $d \geq 3.$ We solve {\textit{path successor problem}} in $Ø(n\lg^{d-1}{n})$ words of space and time $Ø(\lg^{d-1+\eps}{n})$ for $d \geq 1$ and an arbitrary constant $\eps > 0.$ We propose a solution to {\textit{path counting problem}}, with $Ø(n(\lg{n}/\lg\lg{n})^{d-1})$ words of space and $Ø((\lg{n}/\lg\lg{n})^{d})$ query time, for $d \geq 1.$ Finally, we solve {\textit{path reporting problem}} in $Ø(n\lg^{d-1+\eps}{n})$ words of space and $Ø((\lg^{d-1}{n})/(\lg\lg{n})^{d-2}+k)$ query time, for $d \geq 2.$ These results match or nearly match the best tradeoffs of the respective range queries. We are also the first to solve path successor even for $d = 1$.