How to Design a Linear Cover Time Random Walk on a Finite Graph

• Yoshiaki Nonaka
• Hirotaka Ono
• Masafumi Yamashita
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5792)

Abstract

A random 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 $${\it \Omega}(|V|^3)$$. 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 $${\it \Omega}(|V|\log |V|)$$ of the cover time of any random walk on trees.

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Authors and Affiliations

• Yoshiaki Nonaka
• 1
• Hirotaka Ono
• 1
• 2