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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.

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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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References

[1]Aldous, D. (1989) An introduction to covering problems for random walks on graphs. J. Theoret. Probab. 2 87–89.CrossRefGoogle Scholar
[2]Aldous, D. J. (1991) Random walk covering of some special trees. J. Math. Anal. Appl. 157 271–283.CrossRefGoogle Scholar
[3]Aldous, D. and Fill, J. A.Reversible Markov Chains and Random Walks on Graphs. In preparation. http://www.stat.berkeley.edu/~aldous/RWG/book.html.Google Scholar
[4]Aleliunas, R., Karp, R. M., Lipton, R. J., Lovász, L. and Rackoff, C. (1979) Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979), IEEE, pp. 218–223.Google Scholar
[5]Barlow, M. T. (1985) Continuity of local times for Lévy processes. Z. Wahrsch. Verw. Gebiete 69 23–35.CrossRefGoogle Scholar
[6]Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.CrossRefGoogle Scholar
[7]Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005) Random subgraphs of finite graphs I: The scaling window under the triangle condition. Random Struct. Alg. 27 137–184.Google Scholar
[8]Bridgland, M. F. (1987) Universal traversal sequences for paths and cycles. J. Algorithms 8 395–404.CrossRefGoogle Scholar
[9]Broder, A. (1990) Universal sequences and graph cover times: A short survey. In Sequences (Naples/Positano 1988), Springer, pp. 109–122.Google Scholar
[10]Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R. and Tiwari, P. (1996/97) The electrical resistance of a graph captures its commute and cover times. Comput. Complexity 6 312–340.Google Scholar
[11]Cooper, C. and Frieze, A. (2008) The cover time of the giant component of a random graph. Random Struct. Alg. 32 401–439.Google Scholar
[12]Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. Anatomy of a young giant component in the random graph. Random Struct. Alg., to appear. Available at: http://arxiv.org/abs/0906.1839.Google Scholar
[13]Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2010) Diameters in supercritical random graphs via first passage percolation. Combin. Probab. Comput. 19 729–751.CrossRefGoogle Scholar
[14]Ding, J., Lee, J. R. and Peres, Y. Cover times, blanket times, and majorizing measures. Preprint. Available at: http://arxiv.org/abs/1004.4371.Google Scholar
[15]Dudley, R. M. (1967) The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290–330.CrossRefGoogle Scholar
[16]Erdős, P. and Rényi, A. (1959) On random graphs I. Publ. Math. Debrecen 6 290–297.CrossRefGoogle Scholar
[17]Erdős, P. and Rényi, A. (1961) On the evolution of random graphs. Bull. Inst. Internat. Statist. 38 343–347.Google Scholar
[18]Feige, U. (1995) A tight upper bound on the cover time for random walks on graphs. Random Struct. Alg. 6 51–54.Google Scholar
[19]Feige, U. (1995) A tight lower bound on the cover time for random walks on graphs. Random Struct. Alg. 6 433–438.CrossRefGoogle Scholar
[20]Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edition, Wiley.Google Scholar
[21]Heydenreich, M. and van der Hofstad, R. (2007) Random graph asymptotics on high-dimensional tori. Comm. Math. Phys. 270 335–358.Google Scholar
[22]Heydenreich, M. and van der Hofstad, R. Random graph asymptotics on high-dimensional tori II: Volume, diameter and mixing time. Probability Theory and Related Fields, to appear.Google Scholar
[23]Janson, S., Łuczak, T. and Rucinski, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience.CrossRefGoogle Scholar
[24]Jonasson, J. (2000) Lollipop graphs are extremal for commute times. Random Struct. Alg. 16 131–142.3.0.CO;2-3>CrossRefGoogle Scholar
[25]Kahn, J., Kim, J. H., Lovász, L. and Vu, V. H. (2000) The cover time, the blanket time, and the Matthews bound. In 41st Annual Symposium on Foundations of Computer Science (Redondo Beach 2000), IEEE Comput. Soc. Press, pp. 467–475.CrossRefGoogle Scholar
[26]Karlin, Anna R. and Raghavan, P. (1995) Random walks and undirected graph connectivity: A survey. In Discrete Probability and Algorithms (Minneapolis 1993), Vol. 72 of IMA Volumes in Mathematics and its Applications, Springer, pp. 95–101.CrossRefGoogle Scholar
[27]Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton–Watson process with mean one and finite variance. Teor. Verojatnost. i Primenen. 11 579–611.Google Scholar
[28]Kolchin, V. F. (1986) Random Mappings, Translation Series in Mathematics and Engineering, Optimization Software Inc.Google Scholar
[29]Kozma, G. and Nachmias, A. (2009) The Alexander–Orbach conjecture holds in high dimensions. Inventiones Math. 178 635–654.CrossRefGoogle Scholar
[30]Kozma, G. and Nachmias, A. A note about critical percolation on finite graphs. J. Theoret. Probab., to appear.Google Scholar
[31]Levin, D. A., Peres, Y. and Wilmer, E. L. (2009) Markov Chains and Mixing Times, AMS.Google Scholar
[32]Łuczak, T. (1990) Component behavior near the critical point of the random graph process. Random Struct. Alg. 1 287–310.CrossRefGoogle Scholar
[33]Luczak, M. and Winkler, P. (2004) Building uniformly random subtrees. Random Struct. Alg. 24 420–443.Google Scholar
[34]Lyons, R. and Peres, Y. (2008) Probability on Trees and Networks, Cambridge University Press, in preparation. Available at: http://mypage.iu.edu/~rdlyons/prbtree/book.pdf.Google Scholar
[35]Matthews, P. (1988) Covering problems for Markov chains. Ann. Probab. 16 1215–1228.CrossRefGoogle Scholar
[36]Mihail, M. and Papadimitriou, C. H. (1994) On the random walk method for protocol testing. In Computer Aided Verification (Stanford 1994), Vol. 818 of Lecture Notes in Computer Science, Springer, pp. 132–141.CrossRefGoogle Scholar
[37]Nachmias, A. (2009) Mean-field conditions for percolation on finite graphs. Geometric Funct. Anal. 19 1171–1194.Google Scholar
[38]Nachmias, A. and Peres, Y. (2008) Critical random graphs: Diameter and mixing time. Ann. Probab. 36 1267–1286.Google Scholar
[39]Nachmias, A. and Peres, Y. Critical percolation on random regular graphs. Random Struct. Alg., to appear.Google Scholar
[40]Nash-Williams, C.St, J. A. (1959) Random walk and electric currents in networks. Proc. Cambridge Philos. Soc. 55 181–194.Google Scholar
[41]Pittel, B. (2008) Edge percolation on a random regular graph of low degree. Ann. Probab. 36 1359–1389.Google Scholar
[42]Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes, Vol. 233 of Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
[43]Talagrand, M. (2005) The Generic Chaining: Upper and Lower Bounds of Stochastic Processes, Springer Monographs in Mathematics, Springer.Google Scholar
[44]Tetali, P. (1991) Random walks and the effective resistance of networks. J. Theoret. Probab. 4 101–109.Google Scholar
[45]Winkler, P. and Zuckerman, D. (1996) Multiple cover time. Random Struct. Alg. 9 403–411.3.0.CO;2-0>CrossRefGoogle Scholar