Abstract
We consider the problem of graph exploration by a team of k agents, which follow the so-called rotor router mechanism. Agents move in synchronous rounds, and each node successively propagates agents which visit it along its outgoing arcs in round-robin fashion. It has recently been established by Dereniowski et al. (STACS 2014) that the rotor-router cover time of a graph G, i.e., the number of steps required by the team of agents to visit all of the nodes of G, satisfies a lower bound of Ω(mD/k) and an upper bound of O(mD/logk) for any graph with m edges and diameter D. In this paper, we consider the question of how the cover time of the rotor-router depends on k for many important graph classes. We determine the precise asymptotic value of the rotor-router cover time for all values of k for degree-restricted expanders, random graphs, and constant-dimensional tori. For hypercubes, we also resolve the question precisely, except for values of k much larger than n. Our results can be compared to those obtained by Elsässer and Sauerwald (ICALP 2009) in an analogous study of the cover time of k independent parallel random walks in a graph; for the rotor-router, we obtain tight bounds in a slightly broader spectrum of cases. Our proofs take advantage of a relation which we develop, linking the cover time of the rotor-router to the mixing time of the random walk and the local divergence of a discrete diffusion process on the considered graph.
Research partially supported by ANR project DISPLEXITY and by NCN under contract DEC-2011/02/A/ST6/00201.
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References
Akbari, H., Berenbrink, P.: Parallel rotor walks on finite graphs and applications in discrete load balancing. In: SPAA, pp. 186–195 (2013)
Aldous, D., Fill, J.: Reversible markov chains and random walks on graphs (2001), http://stat-www.berkeley.edu/users/aldous/RWG/book.html
Alon, N., Avin, C., Koucký, M., Kozma, G., Lotker, Z., Tuttle, M.R.: Many random walks are faster than one. Combinatorics, Probability & Computing 20(4), 481–502 (2011)
Bampas, E., Gąsieniec, L., Hanusse, N., Ilcinkas, D., Klasing, R., Kosowski, A.: Euler tour lock-in problem in the rotor-router model. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 423–435. Springer, Heidelberg (2009)
Berenbrink, P., Cooper, C., Friedetzky, T., Friedrich, T., Sauerwald, T.: Randomized diffusion for indivisible loads. In: Randall, D. (ed.) SODA, pp. 429–439. SIAM (2011)
Cooper, J.N., Doerr, B., Spencer, J.H., Tardos, G.: Deterministic random walks on the integers. Eur. J. Comb. 28(8), 2072–2090 (2007)
Dereniowski, D., Kosowski, A., Pajak, D., Uznanski, P.: Bounds on the cover time of parallel rotor walks. In: Mayr, E.W., Portier, N. (eds.) STACS. LIPIcs, vol. 25, pp. 263–275. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2014)
Doerr, B., Friedrich, T.: Deterministic random walks on the two-dimensional grid. Combinatorics, Probability, and Computing 18(1-2), 123–144 (2009)
Efremenko, K., Reingold, O.: How well do random walks parallelize? In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX and RANDOM 2009. LNCS, vol. 5687, pp. 476–489. Springer, Heidelberg (2009)
Elsässer, R., Sauerwald, T.: Tight bounds for the cover time of multiple random walks. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 415–426. Springer, Heidelberg (2009)
Elsässer, R., Sauerwald, T.: Tight bounds for the cover time of multiple random walks. Theor. Comput. Sci. 412(24), 2623–2641 (2011)
Friedrich, T., Gairing, M., Sauerwald, T.: Quasirandom load balancing. SIAM J. Comput. 41(4), 747–771 (2012)
Kijima, S., Koga, K., Makino, K.: Deterministic random walks on finite graphs. In: Martinez, C., Hwang, H.-K. (eds.) ANALCO, pp. 18–27. SIAM (2012)
Klasing, R., Kosowski, A., Pajak, D., Sauerwald, T.: The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks. In: PODC, pp. 365–374 (2013)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society (2006)
Priezzhev, V., Dhar, D., Dhar, A., Krishnamurthy, S.: Eulerian walkers as a model of self-organized criticality. Phys. Rev. Lett. 77(25), 5079–5082 (1996)
Rabani, Y., Sinclair, A., Wanka, R.: Local divergence of markov chains and the analysis of iterative load balancing schemes. In: FOCS, pp. 694–705. IEEE Computer Society (1998)
Uznanski, P.: Personal communication (2014)
Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165–186 (2003)
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Kosowski, A., Pająk, D. (2014). Does Adding More Agents Make a Difference? A Case Study of Cover Time for the Rotor-Router. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43951-7_46
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