TY - GEN

T1 - How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees

AU - Nonaka, Yoshiaki

AU - Ono, Hirotaka

AU - Kijima, Shuji

AU - Yamashita, Masafumi

PY - 2011/12/1

Y1 - 2011/12/1

N2 - 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(n 3) 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(n 2) hitting time and O(n 2 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.

AB - 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(n 3) 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(n 2) hitting time and O(n 2 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.

UR - http://www.scopus.com/inward/record.url?scp=84864631238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864631238&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84864631238

SN - 9781920682989

T3 - Conferences in Research and Practice in Information Technology Series

SP - 63

EP - 68

BT - Theory of Computing 2011 - Proceedings of the 17th Computing

T2 - Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011

Y2 - 17 January 2011 through 20 January 2011

ER -