### Abstract

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

Original language | English |
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Title of host publication | Stochastic Algorithms |

Subtitle of host publication | Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings |

Pages | 104-116 |

Number of pages | 13 |

DOIs | |

Publication status | Published - Dec 1 2009 |

Event | 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009 - Sapporo, Japan Duration: Oct 26 2009 → Oct 28 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5792 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009 |
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Country | Japan |

City | Sapporo |

Period | 10/26/09 → 10/28/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Stochastic Algorithms: Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings*(pp. 104-116). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5792 LNCS). https://doi.org/10.1007/978-3-642-04944-6_9

**How to design a linear cover time random walk on a finite graph.** / Nonaka, Yoshiaki; Ono, Hirotaka; Sadakane, Kunihiko; Yamashita, Masafumi.

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

*Stochastic Algorithms: Foundations and Applications - 5th International Symposium, SAGA 2009, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5792 LNCS, pp. 104-116, 5th Symposium on Stochastic Algorithms, Foundations and Applications, SAGA 2009, Sapporo, Japan, 10/26/09. https://doi.org/10.1007/978-3-642-04944-6_9

}

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 - 2009/12/1

Y1 - 2009/12/1

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

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

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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

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

ER -