### Abstract

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.

Original language | English |
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Title of host publication | Theory of Computing 2011 - Proceedings of the 17th Computing |

Subtitle of host publication | The Australasian Theory Symposium, CATS 2011 |

Pages | 63-68 |

Number of pages | 6 |

Publication status | Published - Dec 1 2011 |

Event | Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011 - Perth, WA, Australia Duration: Jan 17 2011 → Jan 20 2011 |

### Publication series

Name | Conferences in Research and Practice in Information Technology Series |
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Volume | 119 |

ISSN (Print) | 1445-1336 |

### Other

Other | Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011 |
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Country | Australia |

City | Perth, WA |

Period | 1/17/11 → 1/20/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computer Science Applications
- Hardware and Architecture
- Information Systems
- Software

### Cite this

*Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011*(pp. 63-68). (Conferences in Research and Practice in Information Technology Series; Vol. 119).

**How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees.** / Nonaka, Yoshiaki; Ono, Hirotaka; Kijima, Shuji; Yamashita, Masafumi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011.*Conferences in Research and Practice in Information Technology Series, vol. 119, pp. 63-68, Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011, Perth, WA, Australia, 1/17/11.

}

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.

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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

ER -