Robustness of the Rotor-router Mechanism

  • Evangelos Bampas
  • Leszek Gąsieniec
  • Ralf Klasing
  • Adrian Kosowski
  • Tomasz Radzik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5923)

Abstract

We consider the model of exploration of an undirected graph G by a single agent which is called the rotor-router mechanism or the Propp machine (among other names). Let πv indicate the edge adjacent to a node v which the agent took on its last exit from v. The next time when the agent enters node v, first a “rotor” at node v advances pointer πv to the edge \({\it next}(\pi_v)\) which is next after the edge πv in a fixed cyclic order of the edges adjacent to v. Then the agent is directed onto edge πv to move to the next node. It was shown before that after initial O(mD) steps, the agent periodically follows one established Eulerian cycle, that is, in each period of 2m consecutive steps the agent traverses each edge exactly twice, once in each direction. The parameters m and D are the number of edges in G and the diameter of G. We investigate robustness of such exploration in presence of faults in the pointers πv or dynamic changes in the graph. We show that after the exploration establishes an Eulerian cycle,

  • if at some step the values of k pointers πv are arbitrarily changed, then a new Eulerian cycle is established within O(km) steps;

  • if at some step k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps;

  • if at some step an edge is deleted from the graph, then a new Eulerian cycle is established within O(γm) steps, where γ is the smallest number of edges in a cycle in graph G containing the deleted edge.

Our proofs are based on the relation between Eulerian cycles and spanning trees known as the “BEST” Theorem (after de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte).

Keywords

Graph exploration Rotor-router mechanism Propp machine Network faults Dynamic graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: Exploring and mapping directed graphs. Inf. Comput. 176(1), 1–21 (2002)MATHCrossRefGoogle Scholar
  2. 2.
    Bhatt, S.N., Even, S., Greenberg, D.S., Tayar, R.: Traversing directed Eulerian mazes. J. Graph Algorithms Appl. 6(2), 157–173 (2002)MATHMathSciNetGoogle Scholar
  3. 3.
    de Bruijn, N.G., Aardenne-Ehrenfest, T.: Circuits and trees in oriented linear graphs. Simon Stevin (Bull. Belgian Math. Soc.) 28, 203–217 (1951)MATHGoogle Scholar
  4. 4.
    Cooper, C., Ilcinkas, D., Klasing, R., Kosowski, A.: Derandomizing random walks in undirected graphs using locally fair exploration strategies. In: Albers, S., et al. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 411–422. Springer, Heidelberg (2009)Google Scholar
  5. 5.
    Cooper, J.N., Spencer, J.: Simulating a random walk with constant error. Combinatorics, Probability & Computing 15(6), 815–822 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. Journal of Graph Theory 32(3), 265–297 (1999)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Doerr, B., Friedrich, T.: Deterministic random walks on the two-dimensional grid. Combinatorics, Probability & Computing 18(1-2), 123–144 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoretical Computer Science 345(2-3), 331–344 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gąsieniec, L., Pelc, A., Radzik, T., Zhang, X.: Tree exploration with logarithmic memory. In: Proceedings 19th ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 585–594 (2007)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4) (2008)Google Scholar
  13. 13.
    Tutte, W.T., Smith, C.A.B.: On unicursal paths in a network of degree 4. The American Mathematical Monthly 48(4), 233–237 (1941)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Smell as a computational resource - a lesson we can learn from the ant. In: Proc. 4th Israel Symposium on Theory of Computing and Systems, ISTCS 1996, pp. 219–230 (1996)Google Scholar
  15. 15.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Distributed covering by ant-robots using evaporating traces. IEEE Transactions on Robotics and Automation 15, 918–933 (1999)CrossRefGoogle Scholar
  16. 16.
    Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A distributed ant algorithm for efficiently patrolling a network. Algorithmica 37(3), 165–186 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Evangelos Bampas
    • 1
    • 3
  • Leszek Gąsieniec
    • 2
  • Ralf Klasing
    • 3
  • Adrian Kosowski
    • 4
    • 3
  • Tomasz Radzik
    • 5
  1. 1.School of Elec. & Comp. Eng.National Technical University of AthensGreece
  2. 2.Dept of Computer ScienceUniv. of LiverpoolUK
  3. 3.LaBRICNRS / INRIA / Univ. of BordeauxFrance
  4. 4.Dept of Algorithms and System ModelingGdańsk Univ. of TechnologyPoland
  5. 5.Dept of Computer ScienceKing’s College LondonUK

Personalised recommendations