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The Evolution of the Cover Time

Published online by Cambridge University Press:  15 February 2011

MARTIN T. BARLOW ,
JIAN DING ,
ASAF NACHMIAS  and
YUVAL PERES
MARTIN T. BARLOW
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z4 (e-mail: barlow@math.ubc.ca)
JIAN DING
Affiliation:
Department of Statistics, University of California at Berkeley, Berkeley, CA 94720, USA (e-mail: jding@berkeley.edu)
ASAF NACHMIAS
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: asafnach@math.mit.edu)
YUVAL PERES
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: peres@microsoft.com)
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Abstract

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovász and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Rényi random graph G(n, c/n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Rényi random graph G(n, 1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0, 1}n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3/2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 3 , May 2011 , pp. 331 - 345
Copyright
Copyright © Cambridge University Press 2011

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