DISC 2015: Distributed Computing pp 123-139

# Limit Behavior of the Multi-agent Rotor-Router System

• Jérémie Chalopin
• Shantanu Das
• Paweł Gawrychowski
• Arnaud Labourel
• Przemysław Uznański
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9363)

## Abstract

The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al. (2003) proved that a single rotor-router walk on any graph with m edges and diameter D stabilizes to a traversal of an Eulerian circuit on the set of all 2m directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most 2mD steps.

The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the experimental observation that a system of $$k>1$$ parallel walks also stabilizes with a period of length at most 2m steps. In this work we disprove this observation, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of $$\mathcal {O}(m^4D^2 + mD\log k)$$ and $$\mathcal {O}(m^5k^2)$$ on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of $$k=2$$ walks.

## Keywords

Load Balance Stabilization Time Exit Pointer Limit Behavior Graph Exploration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Afek, Y., Gafni, E.: Distributed algorithms for unidirectional networks. SIAM J. Comput. 23(6), 1152–1178 (1994)
2. 2.
Akbari, H., Berenbrink, P.: Parallel rotor walks on finite graphs and applications in discrete load balancing. In: SPAA, pp. 186–195. ACM (2013)Google Scholar
3. 3.
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)
4. 4.
Bampas, E., Gąsieniec, 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)
5. 5.
Berenbrink, P., Klasing, R., Kosowski, A., Mallmann-Trenn, F., Uznański, P.: Improved analysis of deterministic load-balancing schemes. In: PODC, pp. 301–310 (2015)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)
7. 7.
Cooper, J.N., Spencer, J.: Simulating a random walk with constant error. Combinatorics, Probability & Computing 15(6), 815–822 (2006)
8. 8.
Dereniowski, D., Kosowski, A., Pająk, D., Uznański, P.: Bounds on the cover time of parallel rotor walks. In: STACS. LIPIcs, vol. 25, pp. 263–275 (2014)Google Scholar
9. 9.
Doerr, B., Friedrich, T.: Deterministic random walks on the two-dimensional grid. Combinatorics, Probability & Computing 18(1–2), 123–144 (2009)
10. 10.
Fraenkel, A.S.: Economic traversal of labyrinths. Mathematics Magazine 43, 125–130 (1970)
11. 11.
Friedrich, T., Sauerwald, T.: The cover time of deterministic random walks. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 130–139. Springer, Heidelberg (2010)
12. 12.
Gąsieniec, L., Radzik, T.: Memory efficient anonymous graph exploration. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 14–29. Springer, Heidelberg (2008)
13. 13.
Kiwi, M., Ndoundam, R., Tchuente, M., Goles, E.: No polynomial bound for the period of the parallel chip firing game on graphs. Theoretical Computer Science 136, 527–532 (1994)
14. 14.
Klasing, R., Kosowski, A., Pająk, D., Sauerwald, T.: The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks. In: PODC, pp. 365–374 (2013)Google Scholar
15. 15.
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
16. 16.
Landau, E.: Uber die maximalordnung der permutationen gegebenen grades. Arch. Math. Phys. 5, 92–103 (1903)
17. 17.
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)
18. 18.
Rabani, Y., Sinclair, A., Wanka, R.: Local divergence of Markov chains and the analysis of iterative load-balancing schemes. In: FOCS, pp. 694–703, November 1998Google Scholar
19. 19.
Sauerwald, T., Sun, H.: Tight bounds for randomized load balancing on arbitrary network topologies. In: FOCS, pp. 341–350 (2012)Google Scholar
20. 20.
Shiraga, T., Yamauchi, Y., Kijima, S., Yamashita, M.: $$\mathit{L}_{\infty }$$-discrepancy analysis of polynomial-time deterministic samplers emulating rapidly mixing chains. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds.) COCOON 2014. LNCS, vol. 8591, pp. 25–36. Springer, Heidelberg (2014) Google Scholar
21. 21.
Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Distributed covering by ant-robots using evaporating traces. IEEE Trans. Robotics and Automation 15, 918–933 (1999)
22. 22.
Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165–186 (2003)

## Authors and Affiliations

• Jérémie Chalopin
• 1
• Shantanu Das
• 1
• Paweł Gawrychowski
• 2