Derandomizing Random Walks in Undirected Graphs Using Locally Fair Exploration Strategies

  • Colin Cooper
  • David Ilcinkas
  • Ralf Klasing
  • Adrian Kosowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5556)


We consider the problem of exploring an anonymous undirected graph using an oblivious robot. The studied exploration strategies are designed so that the next edge in the robot’s walk is chosen using only local information, and so that some local equity (fairness) criterion is satisfied for the adjacent undirected edges. Such strategies can be seen as an attempt to derandomize random walks, and are natural undirected counterparts of the rotor-router model for symmetric directed graphs.

The first of the studied strategies, known as Oldest-First (OF), always chooses the neighboring edge for which the most time has elapsed since its last traversal. Unlike in the case of symmetric directed graphs, we show that such a strategy in some cases leads to exponential cover time. We then consider another strategy called Least-Used-First (LUF) which always uses adjacent edges which have been traversed the smallest number of times. We show that any Least-Used-First exploration covers a graph G = (V,E) of diameter \(\mathit{D}\) within time \(O(\mathit{D}|E|)\), and in the long run traverses all edges of G with the same frequency.


Random Walk Undirected Graph Time Moment Directed Edge Exploration Strategy 
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.


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  1. 1.
    Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs (2001),
  2. 2.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal sequences and the complexity of maze problems. In: Proceedings of the 20th Annual IEEE Symposium on the Foundations of Computer Science (FOCS 1979), pp. 218–223 (1979)Google Scholar
  3. 3.
    Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: Exploring and mapping directed graphs. Information and Computation 176(1), 1–21 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhatt, S.N., Even, S., Greenberg, D.S., Tayar, R.: Traversing directed eulerian mazes. Journal of Graph Algorithms and Applications 6(2), 157–173 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cooper, J., Doerr, B., Friedrich, T., Spencer, J.: Deterministic Random Walks on Regular Trees. In: Proceedings of 19th ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 766–772 (2008)Google Scholar
  6. 6.
    Deng, X., Papadimitriou, C.H.: Exploring an Unknown Graph. Journal of Graph Theory 32(3), 265–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Doerr, B., Friedrich, T.: Deterministic Random Walks on the Two-Dimensional Grid. Combinatorics, Probability and Computing (to appear, 2009), doi:10.1017/S0963548308009589Google 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)MathSciNetCrossRefzbMATHGoogle 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.
    Hemmerling, A.: Labyrinth Problems: Labyrinth-Searching Abilities of Automata. Teubner-Texte zur Mathematik 114 (1989)Google Scholar
  11. 11.
    Ikeda, S., Kubo, I., Yamashita, M.: The hitting and cover times of random walks on finite graphs using local degree information. Theoretical Computer Science 410(1), 94–100 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kahn, J., Kim, J.H., Lovász, L., Vu, V.H.: The cover time, the blanket time, and the Matthews bound. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 467–475. IEEE, Los Alamitos (2000)CrossRefGoogle Scholar
  13. 13.
    Koenig, S.: Complexity of Edge Counting. In: Goal-Directed Acting with Incomplete Information. Technical Report CMU-CS-97-199, Carnegie Mellon University (1997)Google Scholar
  14. 14.
    Koenig, S., Simmons, R.G.: Easy and Hard Testbeds for Real-Time Search Algorithms. In: Proceedings of the National Conference on Artificial Intelligence, pp. 279–285 (1996)Google Scholar
  15. 15.
    Koucký, M.: Universal traversal sequences with backtracking. Journal of Computer and System Sciences 65(4), 717–726 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lovász, L.: Random walks on graphs: A survey. Bolyai Society Mathematical Studies 2, 353–397 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Reingold, O.: Undirected ST-connectivity in log-space. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 376–385 (2005)Google Scholar
  18. 18.
    Saks, M.E.: Randomization and derandomization in space-bounded computation. In: Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pp. 128–149 (1996)Google Scholar
  19. 19.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Smell as a Computational Resource — a Lesson We Can Learn from the Ants. In: Proceedings of Fourth Israeli Symposium on Theory of Computing and Systems (ISTCS 1996), pp. 219–230 (1996)Google Scholar
  20. 20.
    Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Distributed Covering by Ant-Robots Using Evaporating Traces. IEEE Transactions on Robotics and Automation 15(5), 918–933 (1999)CrossRefGoogle Scholar
  21. 21.
    Winkler, P., Zuckerman, D.: Multiple cover time. Random Structures and Algorithms 9(4), 403–411 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yanovski, V., Wagner, I.A., Bruckstein, A.M.: A Distributed Ant Algorithm for Efficiently Patrolling a Network. Algorithmica 37, 165–186 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Colin Cooper
    • 1
  • David Ilcinkas
    • 2
  • Ralf Klasing
    • 2
  • Adrian Kosowski
    • 2
    • 3
  1. 1.Dept of Computer ScienceKing’s College LondonUK
  2. 2.LaBRI, CNRS and Université de BordeauxFrance
  3. 3.Dept of Algorithms and System ModelingGdańsk University of TechnologyPoland

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