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The cover time, the blanket time, and the Matthews bound

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4 Author(s)
Kahn, J. ; Dept. of Math., Rutgers Univ., New Brunswick, NJ, USA ; Kim, J.H. ; Lovasz, L. ; Vu, V.H.

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 (P. Matthews, 1988) on the cover time, C, and prove that M/2<C= O(M(lnlnn)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((lnlnn)2), improving the previous bound O(lnn) due to Matthews. The blanket time B was introduced by P. Winkler and D. Zuckerman (1996): 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(Clnn). Our bounds above are also valid for the blanket time, and so it follows that B=O(C(lnlnn)2)

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Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on

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