Setting Port Numbers for Fast Graph Exploration

  • David Ilcinkas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4056)


We consider the problem of periodic graph exploration by a finite automaton in which an automaton with a constant number of states has to explore all unknown anonymous graphs of arbitrary size and arbitrary maximum degree. In anonymous graphs, nodes are not labeled but edges are labeled in a local manner (called local orientation) so that the automaton is able to distinguish them. Precisely, the edges incident to a node v are given port numbers from 1 to d v , where d v is the degree of v.

Periodic graph exploration means visiting every node infinitely often. We are interested in the length of the period, i.e., the maximum number of edge traversals between two consecutive visits of any node by the automaton in the same state and entering the node by the same port. This problem is unsolvable if local orientations are set arbitrarily. Given this impossibility result, we address the following problem: what is the mimimum function π(n) such that there exist an algorithm for setting the local orientation, and a finite automaton using it, such that the automaton explores all graphs of size n within the period π(n)?

The best result so far is the upper bound π(n) ≤10n, by Dobrev et al. [SIROCCO 2005], using an automaton with no memory (i.e. only one state). In this paper we prove a better upper bound π(n) ≤4n. Our automaton uses three states but performs periodic exploration independently of its starting position and initial state.


Span Tree Local Orientation Finite Automaton Port Number Impossibility Result 
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.
    Afek, Y., Gafni, E.: Distributed Algorithms for Unidirectional Networks. SIAM J. Computing 23(6), 1152–1178 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Computing 29, 1164–1188 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bender, M., Fernandez, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: Exploring and mapping directed graphs. Information and Computation 176(1), 1–21 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Budach, L.: Automata and labyrinths. Math. Nachrichten, 195–282 (1978)Google Scholar
  5. 5.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-Guided Graph Exploration by a Finite Automaton. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 335–346. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Cook, S., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM J. on Computing 9(3), 636–652 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Das, S., Flocchini, P., Nayak, A., Santoro, N.: Distributed Exploration of an Unknown Graph. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 99–114. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. J. Graph Theory 32(3), 265–297 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree Exploration with Little Memory. J. Algorithms 51(1), 38–63 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dobrev, S., Jansson, J., Sadakane, K., Sung, W.-K.: Finding Short Right-Hand-on-the-Wall Walks in Graphs. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 127–139. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Dudek, G., Jenkin, M., Milios, E., Wilkes, D.: Robotic Exploration as Graph Construction. IEEE Transaction on Robotics and Automation 7(6), 859–865 (1991)CrossRefGoogle Scholar
  12. 12.
    Duncan, C., Kobourov, S., Kumar, V.: Optimal constrained graph exploration. In: 12th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 807–814 (2001)Google Scholar
  13. 13.
    Fleischer, R., Trippen, G.: Exploring an unknown graph efficiently. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 11–22. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Flocchini, P., Mans, B., Santoro, N.: Sense of direction in distributed computing. Theoretical Computer Science 291(1), 29–53 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fraigniaud, P., Gąsieniec, L., Kowalski, D.R., Pelc, A.: Collective Tree Exploration. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 141–151. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Fraigniaud, P., Gavoille, C., Mans, B.: Interval routing schemes allow broadcasting with linear message-complexity. Distributed Computing 14(4), 217–229 (2001)CrossRefGoogle Scholar
  17. 17.
    Fraigniaud, P., Ilcinkas, D.: Digraphs Exploration with Little Memory. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 246–257. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph Exploration by a Finite Automaton. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 451–462. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Hemmerling, A.: Labyrinth Problems: Labyrinth-Searching Abilities of Automata. Teubner-Texte zur Mathematik, vol. 114. B. G. Teubner Verlagsgesellschaft, Leipzig (1989)zbMATHGoogle Scholar
  20. 20.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33(2), 281–295 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rao, N., Kareti, S., Shi, W., Iyengar, S.: Robot navigation in unknown terrains: Introductory survey of non-heuristic algorithms. Tech. Report ORNL/TM-12410, Oak Ridge National Lab (1993)Google Scholar
  22. 22.
    Reingold, O.: Undirected ST-Connectivity in Log-Space. In: 37th ACM Symp. on Theory of Computing (STOC), pp. 376–385 (2005)Google Scholar
  23. 23.
    Rollik, H.: Automaten in planaren Graphen. Acta Informatica 13, 287–298 (1980) (also in LNCS 67, pp. 266–275 (1979))Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Ilcinkas
    • 1
  1. 1.LRIUniversité Paris-SudFrance

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