More Efficient Periodic Traversal in Anonymous Undirected Graphs

  • Jurek Czyzowicz
  • Stefan Dobrev
  • Leszek Gąsieniec
  • David Ilcinkas
  • Jesper Jansson
  • Ralf Klasing
  • Ioannis Lignos
  • Russell Martin
  • Kunihiko Sadakane
  • Wing-Kin Sung
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)


We consider the problem of periodic graph exploration in which a mobile entity with (at most) constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in the minimisation of the length of the exploration period.

This problem is unsolvable if the local port numbers are set arbitrarily, see [1]. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. [2] described an algorithm for assigning port numbers, and an oblivious agent (i.e., an agent with no persistent memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was proved in [3] to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most \(4\frac{1}{3}n\) for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Finally, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case.


Span Tree Edge Incident Single Edge Port Number Span Subgraph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Budach, L.: Automata and labyrinths. Mathematische Nachrichten, 195–282 (1978)Google Scholar
  2. 2.
    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
  3. 3.
    Gąsieniec, L., Klasing, R., Martin, R., Navarra, A., Zhang, X.: Fast periodic graph exploration with constant memory. J. Computer and System Science 74(5), 808–822 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Transactions on Algorithms 4(4), 331–344 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rollik, H.: Automaten in planaren graphen. Acta Informatica 13, 287–298 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Computing 29, 1164–1188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bender, M., Slonim, D.K.: The power of team exploration: two robots can learn unlabeled directed graphs. In: Proc. 35th Annual Symposium on Foundations of Computer Science (FOCS), pp. 75–85 (1994)Google Scholar
  9. 9.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. J. Graph Theory 32(3), 265–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fleischer, R., Trippen, G.: Exploring an unknown graph efficiently. In: Proc. 13th Annual European Symposium on Algorithms (ESA), pp. 11–22 (2005)Google Scholar
  11. 11.
    Awerbuch, B., Betke, M., Rivest, R., Singh, M.: Piecemeal graph exploration by a mobile robot. Information and Computation 152(2), 155–172 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Betke, M., Rivest, R., Singh, M.: Piecemeal learning of an unknown environment. Machine Learning 18(2-3), 231–254 (1995)CrossRefGoogle Scholar
  13. 13.
    Duncan, C., Kobourov, S., Kumar, V.: Optimal constrained graph exploration. ACM Transaction on Algorithms 2(3), 380–402 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Fraigniaud, P., Ilcinkas, D., Rajsbaum, S., Tixeuil, S.: The reduced automata technique for graph exploration space lower bounds. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 1–26. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33, 281–295 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cook, S.A., Rackoff, C.: Space lower bounds for maze threadability on restricted machines. SIAM J. Computing 9(3), 636–652 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garey, M., Johnson, D., Tarjan, R.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Computing 5(4), 704–714 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ilcinkas, D.: Setting port numbers for fast graph exploration. Theoretical Computer Science 401, 236–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jurek Czyzowicz
    • 1
  • Stefan Dobrev
    • 2
  • Leszek Gąsieniec
    • 3
  • David Ilcinkas
    • 4
  • Jesper Jansson
    • 5
  • Ralf Klasing
    • 4
  • Ioannis Lignos
    • 6
  • Russell Martin
    • 3
  • Kunihiko Sadakane
    • 7
  • Wing-Kin Sung
    • 8
  1. 1.Département d’InformatiqueUniversité du Québec en OutaouaisGatineauCanada
  2. 2.Institute of MathematicsSlovak Academy of SciencesBratislavaSlovak Republic
  3. 3.Department of Computer ScienceUniversity of LiverpoolLiverpoolU.K.
  4. 4.LaBRICNRS and Université de BordeauxTalenceFrance
  5. 5.Ochanomizu UniversityTokyoJapan
  6. 6.Department of Computer ScienceDurham UniversityDurhamUK
  7. 7.Principles of Informatics Research DivisionNational Institute of InformaticsTokyoJapan
  8. 8.Department of Computer ScienceNational University of SingaporeSingapore

Personalised recommendations