Many random walks are faster than one

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 probablistic 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.

[1]  Thomas Sauerwald,et al.  Tight bounds for the cover time of multiple random walks , 2009, Theor. Comput. Sci..

[2]  C. Avin,et al.  Efficient and robust query processing in dynamic environments using random walk techniques , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[3]  Eli Upfal,et al.  Trading space for time in undirected s-t connectivity , 1989, STOC '89.

[4]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[5]  Christos Gkantsidis,et al.  Random walks in peer-to-peer networks: Algorithms and evaluation , 2006, Perform. Evaluation.

[6]  David Zuckerman,et al.  A technique for lower bounding the cover time , 1990, STOC '90.

[7]  Israel A. Wagner,et al.  Robotic Exploration, Brownian Motion and Electrical Resistance , 1998, RANDOM.

[8]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[9]  Anna R. Karlin,et al.  Bounds on the cover time , 1989 .

[10]  Johan Jonasson.  On the Cover Time for Random Walks on Random Graphs , 1998, Comb. Probab. Comput..

[11]  Prabhakar Raghavan,et al.  The electrical resistance of a graph captures its commute and cover times , 2005, computational complexity.

[12]  Uriel Feige,et al.  A Tight Lower Bound on the Cover Time for Random Walks on Graphs , 1995, Random Struct. Algorithms.

[13]  D. Aldous.  Threshold limits for cover times , 1991 .

[14]  Endre Szemerédi,et al.  Undirected Connectivity in O(l~gl*~ n) Space* , 1992 .

[15]  Alan M. Frieze,et al.  Multiple Random Walks and Interacting Particle Systems , 2009, ICALP.

[16]  Sergio D. Servetto,et al.  Constrained random walks on random graphs: routing algorithms for large scale wireless sensor networks , 2002, WSNA '02.

[17]  Deborah Estrin,et al.  Rumor routing algorthim for sensor networks , 2002, WSNA '02.

[18]  Omer Reingold,et al.  How Well Do Random Walks Parallelize? , 2009, APPROX-RANDOM.

[19]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[20]  Uriel Feige.  A Spectrum of Time-Space Trade-Offs for Undirected s-t Connectivity , 1997, J. Comput. Syst. Sci..

[21]  Jennifer L. Welch,et al.  Random walk for self-stabilizing group communication in ad hoc networks , 2002, IEEE Transactions on Mobile Computing.

[22]  Roy Friedman,et al.  RaWMS -: random walk based lightweight membership service for wireless ad hoc network , 2006, MobiHoc '06.

[23]  Christos Gkantsidis,et al.  Random walks in peer-to-peer networks , 2004, IEEE INFOCOM 2004.

[24]  Endre Szemerédi,et al.  Undirected Connectivity in O(log ^1.5 n) Space , 1992, FOCS.

[25]  E. Szemerédi,et al.  Undirected connectivity in O(log/sup 1.5/n) space , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[26]  Uriel Feige,et al.  Short random walks on graphs , 1993, SIAM J. Discret. Math..

[27]  Uri Zwick,et al.  An Optimal Randomised Logarithmic Time Connectivity Algorithm for the EREW PRAM , 1996, J. Comput. Syst. Sci..

[28]  Venkatesh Saligrama,et al.  A Random-Walk Model for Distributed Computation in Energy-Limited Networks , 2006 .

[29]  P. Matthews.  Covering Problems for Brownian Motion on Spheres , 1988 .

[30]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[31]  D. Zuckerman,et al.  Covering times of random walks on bounded degree trees and other graphs , 1989 .

[32]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[33]  Chen Avin,et al.  On the Cover Time of Random Geometric Graphs , 2005, ICALP.

[34]  Uriel Feige,et al.  Short Random Walks on Graphs , 1996, SIAM J. Discret. Math..

[35]  B. Krishnamachari,et al.  Active Query Forwarding in Sensor Networks ( ACQUIRE ) , 2002 .

[36]  Alan M. Frieze,et al.  The cover time of sparse random graphs. , 2003, SODA '03.

[37]  Uriel Feige,et al.  A Tight Upper Bound on the Cover Time for Random Walks on Graphs , 1995, Random Struct. Algorithms.

[38]  Noam Nisan,et al.  Fast connected components algorithms for the EREW PRAM , 1992, SPAA '92.

[39]  O. Schramm,et al.  On the Cover Time of Planar Graphs , 2000, math/0002034.

[40]  Avi Wigderson,et al.  An O(log(n)4/3) space algorithm for (s, t) connectivity in undirected graphs , 2000, JACM.