Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Randomized 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
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
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Arandom walk on a finite graph G = (V,E) is random token circulation on vertices of G. A token on a vertex in V moves to one of its adjacent vertices according to a transition probability matrix P. It is known that both of the hitting time and the cover time of the standard random walk are bounded by O(|V|3), in which the token randomly moves to an adjacent vertex with the uniform probability. This estimation is tight in a sense, that is, there exist graphs for which the hitting time and cover times of the standard random walk are Ω(|V|3). Thus the following questions naturally arise: is it possible to speed up a random walk, that is, to design a transition probability for G that achieves a faster cover time? Or, how large (or small) is the lower bound on the cover time of random walks on G? In this paper, we investigate how we can/cannot design a faster random walk in terms of the cover time. We give necessary conditions for a graph G to have a linear cover time random walk, i,e., the cover time of the random walk on G is O(|V|). We also present a class of graphs that have a linear cover time. As a byproduct, we obtain the lower bound Ω(|V| log |V|) of the cover time of any random walk on trees.