Advertisement

Explorable Families of Graphs

  • Andrzej PelcEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11085)

Abstract

Graph exploration is one of the fundamental tasks performed by a mobile agent in a graph. An n-node graph has unlabeled nodes, and all ports at any node of degree d are arbitrarily numbered \(0,\dots , d-1\). A mobile agent, initially situated at some starting node v, has to visit all nodes of the graph and stop. In the absence of any initial knowledge of the graph the task of deterministic exploration is often impossible. On the other hand, for some families of graphs it is possible to design deterministic exploration algorithms working for any graph of the family. We call such families of graphs explorable. Examples of explorable families are all finite families of graphs, as well as the family of all trees.

In this paper we study the problem of which families of graphs are explorable. We characterize all such families, and then ask the question whether there exists a universal deterministic algorithm that, given an explorable family of graphs, explores any graph of this family, without knowing which graph of the family is being explored. The answer to this question turns out to depend on how the explorable family is given to the hypothetical universal algorithm. If the algorithm can get the answer to any yes/no question about the family, then such a universal algorithm can be constructed. If, on the other hand, the algorithm can be only given an algorithmic description of the input explorable family, then such a universal deterministic algorithm does not exist.

Keywords

Algorithm Graph Exploration Mobile agent Explorable family of graphs 

References

  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29, 1164–1188 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aleliunas, R., Karp, R., Lipton, R., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: Proceedings of 20th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1979), pp. 218–223 (1979)Google Scholar
  3. 3.
    Awerbuch, B., Betke, M., Rivest, R.L., Singh, M.: Piecemeal graph exploration by a mobile robot. Inf. Comput. 152, 155–172 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bar-Eli, E., Berman, P., Fiat, A., Yan, R.: On-line navigation in a room. J. Algorithms 17, 319–341 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bender, M.A., Fernandez, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: exploring and mapping directed graphs. Inf. Comput. 176, 1–21 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bender, M.A., Slonim, D.: The power of team exploration: two robots can learn unlabeled directed graphs. In: Proceedings of 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), pp. 75–85 (1994)Google Scholar
  7. 7.
    Betke, M., Rivest, R., Singh, M.: Piecemeal learning of an unknown environment. Mach. Learn. 18, 231–254 (1995)Google Scholar
  8. 8.
    Blum, A., Raghavan, P., Schieber, B.: Navigating in unfamiliar geometric terrain. SIAM J. Comput. 26, 110–137 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chalopin, J., Das, S., Kosowski, A.: Constructing a map of an anonymous graph: applications of universal sequences. In: Lu, C., Masuzawa, T., Mosbah, M. (eds.) OPODIS 2010. LNCS, vol. 6490, pp. 119–134. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-17653-1_10CrossRefGoogle Scholar
  10. 10.
    Deng, X., Kameda, T., Papadimitriou, C.H.: How to learn an unknown environment I: the rectilinear case. J. ACM 45, 215–245 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree exploration with little memory. J. Algorithms 51, 38–63 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Duncan, C.A., Kobourov, S.G., Anil Kumar, V.S.: Optimal constrained graph exploration. ACM Trans. Algorithms 2, 380–402 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput. 206, 1276–1287 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fraigniaud, P., Ilcinkas, D.: Directed graphs exploration with little memory. In: Proceedings of 21st Symposium on Theoretical Aspects of Computer Science (STACS 2004), pp. 246–257 (2004)Google Scholar
  15. 15.
    Gorain, B., Pelc, A.: Deterministic graph exploration with advice. In: Proceedings of 44th International Colloquium on Automata, Languages and Programming (ICALP 2017), pp. 132:1–132:14 (2017)Google Scholar
  16. 16.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33, 281–295 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pelc, A., Tiane, A.: Efficient grid exploration with a stationary token. Int. J. Found. Comput. Sci. 25, 247–262 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rao, N.S.V., Kareti, S., Shi, W., Iyengar, S.S.: Robot navigation in unknown terrains: introductory survey of non-heuristic algorithms, Technical report ORNL/TM-12410, Oak Ridge National Laboratory, July 1993Google Scholar
  19. 19.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55, 17:1–17:24 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: Part I - characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7, 69–89 (1996)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

Personalised recommendations