Coalescing Walks on Rotor-Router Systems

  • Colin Cooper
  • Tomasz Radzik
  • Nicolás Rivera
  • Takeharu Shiraga
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9439)


We consider the rotor-router mechanism for distributing particles in an undirected graph. If the last particle passing through a vertex v took an edge (v,u), then the next time a particle is at v, it will leave v along the next edge (v,w) according to a fixed cyclic order of edges adjacent to v. The system works in synchronized steps and when two or more particles meet at the same vertex, they coalesce into one particle. A k-particle configuration of such a system is stable, if it does not lead to any coalescing. For 2 ≤ k ≤ n, we give the full characterization of stable k-particle configurations for cycles. We also show sufficient conditions for regular graphs with n vertices to admit n-particle stable configurations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous, D., Fill, J.A.: Reversible markov chains and random walks on graphs 2002. Unfinished monograph, recompiled (2014).
  2. 2.
    Aldous, D.J.: Meeting times for independent markov chains. Stochastic Processes and their Applications 38(2), 185–193 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alon, N., Avin, C., Koucky, M., Kozma, G., Lotker, Z., Tuttle, M.R.: Many random walks are faster than one. In: Proc. 20th Annual Symposium on Parallelism in Algorithms and Architectures, SPAA 2008, pp. 119–128. ACM (2008)Google Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Bampas, E., Gasieniec, L., Klasing, R., Kosowski, A., Radzik, T.: Robustness of the rotor-router mechanism. In: Abdelzaher, T., Raynal, M., Santoro, N. (eds.) OPODIS 2009. LNCS, vol. 5923, pp. 345–358. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Bhatt, S.N., Even, S., Greenberg, D.S., Tayar, R.: Traversing directed eulerian mazes. J. Graph Algorithms Appl. 6(2), 157–173 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chalopin, J., Das, S., Gawrychowski, P., Kosowski, A., Labourel, A., Uznanski, P.: Lock-in problem for parallel rotor-router walks. CoRR, abs/1407.3200 (2014)Google Scholar
  8. 8.
    Cooper, C., Elsässer, R., Ono, H., Radzik, T.: Coalescing random walks and voting on connected graphs. SIAM J. Discrete Math. 27(4), 1748–1758 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cooper, C., Frieze, A.M., Radzik, T.: Multiple random walks in random regular graphs. SIAM J. Discrete Math. 23(4), 1738–1761 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dereniowski, D., Kosowski, A., Pajak, D., Uznanski, P.: Bounds on the cover time of parallel rotor walks. In: 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014, pp. 263–275 (2014)Google Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Elsässer, R., Sauerwald, T.: Tight bounds for the cover time of multiple random walks. Theor. Comput. Sci. 412(24), 2623–2641 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Israeli, A., Jalfon, M.: Token management schemes and random walks yield self-stabilizing mutual exclusion. In: Proceedings of the Ninth Annual ACM Symposium on Principles of Distributed Computing, PODC 1990, pp. 119–131. ACM (1990)Google Scholar
  14. 14.
    Kosowski, A., Pająk, D.: 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.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 544–555. Springer, Heidelberg (2014)Google Scholar
  15. 15.
    Lovász, L., Plummer, D.: Matching Theory. AMS Chelsea Publishing Series. American Mathematical Soc. (2009)Google Scholar
  16. 16.
    Oliveira, R.: On the coalescence time of reversible random walks. Trans. Amer. Math. Soc. 364, 2109–2128 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Priezzhev, V.B., Dhar, D., Dhar, A., Krishnamurthy, S.: Eulerian walkers as a model of self-organized criticality. Phys. Rev. Lett. 77, 5079–5082 (1996)CrossRefGoogle Scholar
  18. 18.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Smell as a computational resource - A lesson we can learn from the ant. In: ISTCS, pp. 219–230 (1996)Google Scholar
  19. 19.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Distributed covering by ant-robots using evaporating traces. IEEE T. Robotics and Automation 15(5), 918–933 (1999)CrossRefGoogle Scholar
  20. 20.
    Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165–186 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Colin Cooper
    • 1
  • Tomasz Radzik
    • 1
  • Nicolás Rivera
    • 1
  • Takeharu Shiraga
    • 2
  1. 1.Department of InformaticsKing’s College LondonLondonUnited Kingdom
  2. 2.Theoretical Computer Science Group, Department of InformaticsKyushu UniversityFukuokaJapan

Personalised recommendations