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Many Random Walks Are Faster Than One†

Published online by Cambridge University Press:  07 April 2011

NOGA ALON ,
CHEN AVIN ,
MICHAL KOUCKÝ ,
GADY KOZMA ,
ZVI LOTKER  and
MARK R. TUTTLE
NOGA ALON
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel (e-mail: nogaa@tau.ac.il)
CHEN AVIN
Affiliation:
Department of Communication Systems Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel (e-mail: avin@cse.bgu.ac.il, zvilo@cse.bgu.ac.il)
MICHAL KOUCKÝ
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic (e-mail: koucky@math.cas.cz)
GADY KOZMA
Affiliation:
Department of Mathematics, The Weizmann Institute of Science, Rehovot, Israel (e-mail: gady.kozma@weizmann.ac.il)
ZVI LOTKER
Affiliation:
Department of Communication Systems Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel (e-mail: avin@cse.bgu.ac.il, zvilo@cse.bgu.ac.il)
MARK R. TUTTLE
Affiliation:
Intel Corporation, Hudson, Massachusetts, USA (e-mail: tuttle@acm.org)
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Abstract

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 4 , July 2011 , pp. 481 - 502
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Aldous, D. J. (1983) On the time taken by random walks on finite groups to visit every state. Z. Wahrsch. Verw. Gebiete 62 361–374.CrossRefGoogle Scholar
[2]Aldous, D. J. (1989) Lower bounds for covering times for reversible Markov chains and random walks on graphs. J. Theoret. Probab. 2 91–100.CrossRefGoogle Scholar
[3]Aldous, D. J. (1991) Threshold limits for cover times. J. Theoret. Probab. 4 197–211.CrossRefGoogle Scholar
[4]Aldous, D. and Fill, J. (1999) Reversible Markov chains and random walks on graphs. Unpublished. http://stat-www.berkeley.edu/users/aldous/RWG/book.html.Google Scholar
[5]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
[6]Alon, N. (1986) Eigenvalues and expanders. Combinatorica 6 83–96.CrossRefGoogle Scholar
[7]Alon, N., Avin, C., Koucký, M., Kozma, G., Lotker, Z. and Tuttle, M. R. (2008) Many random walks are faster than one. In SPAA 2008: Proc. 20th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 119–128.Google Scholar
[8]Armoni, R., Ta-Shma, A., Wigderson, A. and Zhou, S. (2000) A (log n)4/3 space algorithm for (s,t) connectivity in undirected graphs. J. Assoc. Comput. Mach. 47 294–311.CrossRefGoogle Scholar
[9]Avin, C. and Brito, C. (2004) Efficient and robust query processing in dynamic environments using random walk techniques. In Proc. Third International Symposium on Information Processing in Sensor Networks, pp. 277–286.CrossRefGoogle Scholar
[10]Avin, C. and Ercal, G. (2005) On the cover time of random geometric graphs. In Proc. Automata, Languages and Programming, 32nd International Colloquium, ICALP05, pp. 677–689.Google Scholar
[11]Bar-Yossef, Z., Friedman, R. and Kliot, G. (2006) RaWMS: Random walk based lightweight membership service for wireless ad hoc network. In MobiHoc '06: Proc. Seventh ACM International Symposium on Mobile ad hoc Networking and Computing (New York 2006), ACM Press, pp. 238–249.CrossRefGoogle Scholar
[12]Barnes, G. and Feige, U. (1997) A spectrum of time–space tradeoffs for undirected s–t connectivity. J. Comput. System Sci. 54 305–316.Google Scholar
[13]Braginsky, D. and Estrin, D. (2002) Rumor routing algorithm for sensor networks. In Proc. First ACM Internat. Workshop on Wireless Sensor Networks and Applications, ACM Press, pp. 22–31.CrossRefGoogle Scholar
[14]Broder, A. and Karlin, A. (1989) Bounds on the cover time. J. Theoret. Probab. 2 101–120.CrossRefGoogle Scholar
[15]Broder, A., Karlin, A., Raghavan, P. and Upfal, E. (1989) Trading space for time in undirected s–t connectivity. In Proc. ACM Symp. Theory of Computing, pp. 543–549.CrossRefGoogle Scholar
[16]Chandra, A. K., Raghavan, P., Ruzzo, W. L. and Smolensky, R. (1989) The electrical resistance of a graph captures its commute and cover times. In Proc. 21st Annual ACM Symposium on Theory of Computing, ACM Press, pp. 574–586.Google Scholar
[17]Cooper, C. and Frieze, A. (2003) The cover time of sparse random graphs. In Proc. 14th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA-03 (Baltimore 2003), ACM Press, pp. 140–147.Google Scholar
[18]Cooper, C., Frieze, A. M. and Radzik, T. (2009) Multiple random walks and interacting particle systems. In ICALP 2 (Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S. E. and Thomas, W., eds), Vol. 5556 of Lecture Notes in Computer Science, Springer, pp. 399–410.Google Scholar
[19]Dolev, S., Schiller, E. and Welch, J. L. (2006) Random walk for self-stabilizing group communication in ad hoc networks. IEEE Trans. Mob. Comput. 5 893–905.CrossRefGoogle Scholar
[20]Efremenko, K. and Reingold, O. (2009) How well do random walks parallelize? In APPROX–RANDOM 2009 (Dinur, I., Jansen, K., Naor, J. and Rolim, J. D. P., eds), Vol. 5687 of Lecture Notes in Computer Science, Springer, pp. 476–489.Google Scholar
[21]Elsässer, R. and Sauerwald, T. (2009) Tight bounds for the cover time of multiple random walks. In ICALP 1 (Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S. E. and Thomas, W., eds), Vol. 5555 of Lecture Notes in Computer Science, Springer, pp. 415–426.Google Scholar
[22]Feige, U. (1995) A tight upper bound on the cover time for random walks on graphs. Random Struct. Alg. 6 51–54.CrossRefGoogle Scholar
[23]Feige, U. (1995) A tight lower bound on the cover time for random walks on graphs. Random Struct. Alg. 6 433–438.CrossRefGoogle Scholar
[24]Feige, U. (1996) Short random walks on graphs. SIAM J. Discrete Math. 1 19–28.Google Scholar
[25]Gkantsidis, C., Mihail, M. and Saberi, A. (2006) Random walks in peer-to-peer networks: Algorithms and evaluation. Perform. Eval. 63 241–263.CrossRefGoogle Scholar
[26]Halperin, S. and Zwick, U. (1996) An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. J. Comput. System Sci. 53 395–416.CrossRefGoogle Scholar
[27]Jerrum, M. and Sinclair, A. (1997) The Markov chain Monte Carlo method: An approach to approximate counting and integration. In Approximations for NP-hard Problems (Hochbaum, D., ed.), PWS Publishing, Boston, pp. 482–520.Google Scholar
[28]Jonasson, J. (1998) On the cover time for random walks on random graphs. Combin. Probab. Comput. 7 265–279.CrossRefGoogle Scholar
[29]Jonasson, J. and Schramm, O. (2000) On the cover time of planar graphs. Electron. Commun. Probab. 5 85–90.CrossRefGoogle Scholar
[30]Karger, D. R., Nisan, N. and Parnas, M. (1999) Fast connected components algorithms for the {EREW} {PRAM}. SIAM J. Comput. 28 1021–1034.CrossRefGoogle Scholar
[31]Lovász, L. (1996) Random walks on graphs: A survey. In Combinatorics: Paul Erdös is Eighty, Vol. 2 (Keszthely 1993), Vol. 2 of Bolyai Soc. Math. Stud., János Bolyai Mathematical Society, Budapest, pp. 353–397.Google Scholar
[32]Matthews, P. (1988) Covering problems for Brownian motion on spheres. Ann. Probab. 16 189–199.CrossRefGoogle Scholar
[33]Nisan, N., Szemerédi, E. and Wigderson, A. (1992) Undirected connectivity in O(log1.5n) space. In Proc. 33rd Annual Symposium on Foundations of Computer Science, pp. 24–29.Google Scholar
[34]Sadagopan, N., Krishnamachari, B. and Helmy, A. (2005) ACQUIRE: Active query forwarding in sensor networks. J. Ad Hoc Networks 3 91–113.CrossRefGoogle Scholar
[35]Servetto, S. D. and Barrenechea, G. (2002) Constrained random walks on random graphs: Routing algorithms for large scale wireless sensor networks. In Proc. First ACM Internat. Workshop on Wireless Sensor Networks and Applications, ACM Press, pp. 12–21.CrossRefGoogle Scholar
[36]Wagner, I. A., Lindenbaum, M. and Bruckstein, A. M. (1998) Robotic exploration, Brownian motion and electrical resistance. In RANDOM'98, Vol. 1518 of Lecture Notes in Computer Science, Springer, pp. 116–130.Google Scholar
[37]Zuckerman, D. (1989) Covering times of random walks on bounded degree trees and other graphs. J. Theoret. Probab. 2 147–157.CrossRefGoogle Scholar
[38]Zuckerman, D. (1990) A technique for lower bounding the cover time. In Proc. 22nd Annual ACM Symposium on Theory of Computing, ACM Press, pp. 254–259.Google Scholar