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. © Springer-Verlag 2009.
CITATION STYLE
Nonaka, Y., Ono, H., Sadakane, K., & Yamashita, M. (2009). How to design a linear cover time random walk on a finite graph. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5792 LNCS, pp. 104–116). https://doi.org/10.1007/978-3-642-04944-6_9
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