TY - GEN

T1 - How to design a linear cover time random walk on a finite graph

AU - Nonaka, Yoshiaki

AU - Ono, Hirotaka

AU - Sadakane, Kunihiko

AU - Yamashita, Masafumi

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-642-04944-6_9

DO - 10.1007/978-3-642-04944-6_9

M3 - Conference contribution

AN - SCOPUS:78650636517

SN - 3642049435

SN - 9783642049439

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 104

EP - 116

BT - Stochastic Algorithms

T2 - 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009

Y2 - 26 October 2009 through 28 October 2009

ER -