Journal of Computer and System Sciences
Introduction to parallel algorithms and architectures: array, trees, hypercubes
Introduction to parallel algorithms and architectures: array, trees, hypercubes
Randomized algorithms
A stochastic process on the hypercube with applications to peer-to-peer networks
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables
Journal of Automated Reasoning
Tight Bounds for the Cover Time of Multiple Random Walks
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Speeding up random walks with neighborhood exploration
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Tight bounds for the cover time of multiple random walks
Theoretical Computer Science
Proceedings of the twenty-fourth annual ACM symposium on Parallelism in algorithms and architectures
Coalescing-branching random walks on graphs
Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
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In this paper we study the following covering process defined over an arbitrary directed graph. Each node is initially uncovered and is assigned a random integer rank drawn from a suitable range. The process then proceeds in rounds. In each round, a uniformly random node is selected and its lowest-ranked uncovered outgoing neighbor, if any, is covered. We prove that if each node has in-degree $\theta({\it d})$ and out-degree O(d), then with high probability, every node is covered within $O(n+ \frac{n \ {\rm log}\ n}{d})$ rounds, matching a lower bound due to Alon. Alon has also shown that, for a certain class of d-regular expander graphs, the upper bound holds no matter what method is used to choose the uncovered neighbor. In contrast, we show that for arbitrary d-regular graphs, the method used to choose the uncovered neighbor can affect the cover time by more than a constant factor.