### Abstract

We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds on the cover time, C, and prove that M/2≤C = O(M(ln ln n)^{2}). We give a deterministic polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((ln ln n)^{2}), improving the previous bound O(ln n) due to Matthews. The blanket time B was introduced by Winkler and Zuckerman: it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C≤B. Winkler and Zuckerman conjectured B = O(C) and proved B = O(C ln n). Our bounds above are also valid for the blanket time, and so it follows that B = O(C(ln ln n)^{2}).

Original language | English |
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Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Publisher | IEEE |

Pages | 467-475 |

Number of pages | 9 |

Publication status | Published - 2000 |

Event | 41st Annual Symposium on Foundations of Computer Science (FOCS 2000) - Redondo Beach, CA, USA Duration: Nov 12 2000 → Nov 14 2000 |

### Other

Other | 41st Annual Symposium on Foundations of Computer Science (FOCS 2000) |
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City | Redondo Beach, CA, USA |

Period | 11/12/00 → 11/14/00 |

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### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 467-475). IEEE.

**Cover time, the blanket time, and the Matthews bound.** / Kahn, J.; Kim, J. H.; Lovász, L.; Vu, V. H.

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

*Annual Symposium on Foundations of Computer Science - Proceedings.*IEEE, pp. 467-475, 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), Redondo Beach, CA, USA, 11/12/00.

}

TY - GEN

T1 - Cover time, the blanket time, and the Matthews bound

AU - Kahn, J.

AU - Kim, J. H.

AU - Lovász, L.

AU - Vu, V. H.

PY - 2000

Y1 - 2000

N2 - We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds on the cover time, C, and prove that M/2≤C = O(M(ln ln n)2). We give a deterministic polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((ln ln n)2), improving the previous bound O(ln n) due to Matthews. The blanket time B was introduced by Winkler and Zuckerman: it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C≤B. Winkler and Zuckerman conjectured B = O(C) and proved B = O(C ln n). Our bounds above are also valid for the blanket time, and so it follows that B = O(C(ln ln n)2).

AB - We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds on the cover time, C, and prove that M/2≤C = O(M(ln ln n)2). We give a deterministic polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((ln ln n)2), improving the previous bound O(ln n) due to Matthews. The blanket time B was introduced by Winkler and Zuckerman: it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C≤B. Winkler and Zuckerman conjectured B = O(C) and proved B = O(C ln n). Our bounds above are also valid for the blanket time, and so it follows that B = O(C(ln ln n)2).

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M3 - Conference contribution

SP - 467

EP - 475

BT - Annual Symposium on Foundations of Computer Science - Proceedings

PB - IEEE

ER -