Tree Exploration by a Swarm of Mobile Agents

  • Jurek Czyzowicz
  • Andrzej Pelc
  • Mélanie Roy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7702)


A swarm of mobile agents starting at the root of a tree has to explore it: every node of the tree has to be visited by at least one agent. In every round, each agent can remain idle or move to an adjacent node. In any round all agents have to be at distance at most d, where d is a parameter called the range of the swarm. The goal is to explore the tree as fast as possible.

If the topology of the tree is known to the agents, we establish optimal exploration time for any range d and give an optimal exploration algorithm. The formula for the optimal exploration time of a tree by a swarm of agents depends on the range of the swarm and on the characteristics of the tree. If the tree is unknown, the quality of an exploration algorithm \(\mathcal{A}\) is measured by comparing its time to that of the optimal algorithm having full knowledge of the tree. The ratio between these times, maximized over all starting nodes and over all trees, is called the overhead of algorithm \(\mathcal{A}\). Overhead 2 is achieved when the swarm executes a DFS, remaining together all the time. We show that this overhead cannot be improved, for any range d.


algorithm exploration swarm of mobile agents tree 


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  1. 1.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29, 1164–1188 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Averbakh, I., Berman, O.: A heuristic with worst-case analysis for minimax routing of two traveling salesmen on a tree. Discr. Appl. Math. 68, 17–32 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bender, M.A., Fernandez, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: exploring and mapping directed graphs. In: Proc. 30th Ann. Symp. on Theory of Computing, STOC 1998, pp. 269–278 (1998)Google Scholar
  4. 4.
    Bender, M.A., Slonim, D.: The power of team exploration: Two robots can learn unlabeled directed graphs. In: Proc. 35th Ann. Symp. on Foundations of Computer Science, FOCS 1994, pp. 75–85 (1994)Google Scholar
  5. 5.
    Betke, M., Rivest, R., Singh, M.: Piecemeal learning of an unknown environment. Machine Learning 18, 231–254 (1995)Google Scholar
  6. 6.
    Chalopin, J., Flocchini, P., Mans, B., Santoro, N.: Network Exploration by Silent and Oblivious Robots. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 208–219. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Das, S., Flocchini, P., Kutten, S., Nayak, A., Santoro, N.: Map construction of unknown graphs by multiple agents. Theoretical Computer Science 385, 34–48 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. J. of Graph Theory 32, 265–297 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dessmark, A., Pelc, A.: Optimal graph exploration without good maps. Theoretical Computer Science 326, 343–362 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Devismes, S.: Optimal exploration of small rings. In: Proc. 3rd Int. Workshop on Reliability, Availability, and Security, WRAS 2010 (2010)Google Scholar
  11. 11.
    Devismes, S., Petit, F., Tixeuil, S.: Optimal Probabilistic Ring Exploration by Semi-synchronous Oblivious Robots. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 195–208. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree exploration with little memory. Journal of Algorithms 51, 38–63 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Duncan, C.A., Kobourov, S.G., Kumar, V.S.A.: Optimal constrained graph exploration. In: Proc. 12th Ann. ACM-SIAM Symp. on Discrete Algorithms, SODA 2001, pp. 807–814 (2001)Google Scholar
  14. 14.
    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
  15. 15.
    Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Remembering without memory: tree exploration by asynchronous oblivious robots. Theoretical Computer Science 411, 1544–1557 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Computing Without Communicating: Ring Exploration by Asynchronous Oblivious Robots. In: Tovar, E., Tsigas, P., Fouchal, H. (eds.) OPODIS 2007. LNCS, vol. 4878, pp. 105–118. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Fraigniaud, P., Gasieniec, L., Kowalski, D., Pelc, A.: Collective tree exploration. Networks 48, 166–177 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Frederickson, G.N., Hecht, M.S., Kim, C.E.: Approximation algorithms for some routing problems. SIAM J. Comput. 7, 178–193 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lamani, A., Potop-Butucaru, M.G., Tixeuil, S.: Optimal Deterministic Ring Exploration with Oblivious Asynchronous Robots. In: Patt-Shamir, B., Ekim, T. (eds.) SIROCCO 2010. LNCS, vol. 6058, pp. 183–196. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. Journal of Algorithms 33, 281–295 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jurek Czyzowicz
    • 1
  • Andrzej Pelc
    • 1
  • Mélanie Roy
    • 1
  1. 1.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada

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