Abstract
We analyse the cover time of a random walk on a random graph of a given degree sequence. Weights are assigned to the edges of the graph using a certain type of scheme that uses only local degree knowledge. This biases the transitions of the walk towards lower degree vertices. We demonstrate that, with high probability, the cover time is at most \((1+o(1))\frac{d-1}{d-2}8n\log n\), where d is the minimum degree. This is in contrast to the precise cover time of \((1+o(1))\frac{d-1}{d-2}\frac{\theta }{d} n\log n\) (with high probability) given in [1] for a simple (i.e., unbiased) random walk on the same graph model. Here \(\theta \) is the average degree and since the ratio \(\theta /d\) can be arbitrarily large, or go to infinity with n, we see that the scheme can give an unbounded speed up for sparse graphs.
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References
Abdullah, M., Cooper, C., Frieze, A.M.: Cover time of a random graph with given degree sequence. Discrete Math. 312(21), 3146–3163 (2012)
Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs. http://stat-www.berkeley.edu/pub/users/aldous/RWG/book.html
Aleliunas, R., Karp, R.M., Lipton, R.J., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, pp.218–223 (1979)
Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R., Tiwari, P.: The electrical resistance of a graph captures its commute and cover times. Comput. Complex. 6, 312–340 (1997)
Doyle, P.G., Snell, J.L.: Random Walks and Electrical Networks (2006)
Feige, U.: A tight upper bound for the cover time of random walks on graphs. Random Struct. Algorithms 6, 51–54 (1995)
Feige, U.: A tight lower bound for the cover time of random walks on graphs. Random Struct. Algorithms 6, 433–438 (1995)
Ikeda, S., Kubo, I., Yamashita, M.: The hitting and the cover times of random walks on finite graphs using local degree information. Theor. Comput. Sci. 410, 94–100 (2009)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18, 1149–1178 (1989)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS Press (2008)
Lovász, L.: Random walks on graphs: a survey. In: Miklós, D., Sós, V.T., Szonyi, T. (eds.) Combinatorics, Paul Erdős is Eighty, vol. 2, János Bolyai Mathematical Society, Budapest, pp. 353–398 (1996)
Lawler, G.F., Sokal, A.D.: Bounds on the \(L^2\) spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Tran. Amer. Math. Soc. 309, 557–580 (1988)
Matthews, P.: Covering problems for Markov chains. Ann. Prob. 16, 1215–1228 (1988)
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Abdullah, M.A., Cooper, C., Draief, M. (2016). Speeding Up Cover Time of Sparse Graphs Using Local Knowledge. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_1
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