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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
Simple and Faster algorithm for Reachability in a Decreme...
Manoj Gupta · 2015-05-01 · via cs.DS updates on arXiv.org

Consider the problem of maintaining source sink reachability($st$-Reachability), single source reachability(SSR) and strongly connected component(SCC) in an edge decremental directed graph. In particular, we design a randomized algorithm that maintains with high probability: 1) $st$-Reachability in $\tilde{O}(mn^{4/5})$ total update time. 2) $st$-Reachability in a total update time of $\tilde{O}(n^{8/3})$ in a dense graph. 3) SSR in a total update time of $\tilde{O}(m n^{9/10})$. 4) SCC in a total update time of $\tilde{O}(m n^{9/10})$. For all the above problems, we improve upon the previous best algorithm (by Henzinger et. al. (STOC 2014)). Our main focus is maintaining $st$-Reachability in an edge decremental directed graph (other problems can be reduced to $st$-Reachability). The classical algorithm of Even and Shiloach (JACM 81) solved this problem in $O(1)$ query time and $O(mn)$ total update time. Recently, Henzinger, Krinninger and Nanongkai (STOC 2014) designed a randomized algorithm which achieves an update time of $\tilde{O}(m n^{0.98})$ and broke the long-standing $O(mn)$ bound of Even and Shiloach. However, they designed four algorithms $A_i (1\le i \le 4)$ such that for graphs having total number of edges between $m_i$ and $m_{i+1}$ ($m_{i+1} > m_i$), $A_i$ outperforms other three algorithms. That is, one of the four algorithms may be faster for a particular density range of edges, but it may be too slow asymptotically for the other ranges. Our main contribution is that we design a {\it single} algorithm which works for all types of graphs. Not only is our algorithm faster, it is much simpler than the algorithm designed by Henzinger et.al. (STOC 2014).