Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Randomized algorithms
Random walks, universal traversal sequences, and the complexity of maze problems
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
The hitting and cover times of random walks on finite graphs using local degree information
Theoretical Computer Science
Impact of local topological information on random walks on finite graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Maximum hitting time for random walks on graphs
Random Structures & Algorithms
A tight upper bound on the cover time for random walks on graphs
Random Structures & Algorithms
A tight lower bound on the cover time for random walks on graphs
Random Structures & Algorithms
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Random walk is a powerful tool, not only for modeling, but also for practical use such as the Internet crawlers. Standard random walks on graphs have been well studied; It is well-known that both hitting time and cover time of a standard random walk are bounded by O(n3) for any graph with n vertices, besides the bound is tight for some graphs. Ikeda et al. (2003) provided "β-random walk," which realizes O(n2) hitting time and O(n2 log n) cover times for any graph, thus it archives, in a sense, "n-times improvement" compared to the standard random walk. This paper is concerned with optimizations of hitting and cover times, by drawing a comparison between the standard random walk and the fastest random walk. We show for any tree that the hitting time of the standard random walk is at most O(n)-times longer than one of the fastest random walk. Similarly, the cover time of the standard random walk is at most O(√n log n)-times longer than the fastest one, for any tree. We also show that our bound for the hitting time is tight by giving examples, while we only give a lower bound Ω(√n/log n) for the cover time.