Different Speeds Suffice for Rendezvous of Two Agents on Arbitrary Graphs

  • Evangelos Kranakis
  • Danny Krizanc
  • Euripides Markou
  • Aris Pagourtzis
  • Felipe Ramírez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10139)

Abstract

We consider the rendezvous problem for two robots on an arbitrary connected graph with n vertices and all its edges of length one. Two robots are initially located on two different vertices of the graph and can traverse its edges with different but constant speeds. The robots do not know their own speed. During their movement they are allowed to meet on either vertices or edges of the graph. Depending on certain conditions reflecting the knowledge of the robots we show that a rendezvous algorithm is always possible on a general connected graph.

More specifically, we give new rendezvous algorithms for two robots as follows. (1) In unknown topologies. We provide a polynomial time rendezvous algorithm based on universal exploration sequences, assuming that n is known to the robots. (2) In known topologies. In this case we prove the existence of more efficient rendezvous algorithms by considering the special case of the two-dimensional torus.

Keywords

Graph Mobile agents Rendezvous Speeds Universal exploration sequence 

References

  1. 1.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovasz, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: FOCS, pp. 218–223. IEEE (1979)Google Scholar
  2. 2.
    Alpern, S.: The rendezvous search problem. SIAM J. Control Optim. 33(3), 673–683 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alpern, S.: Rendezvous search: a personal perspective. Oper. Res. 50(5), 772–795 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer Academic Publishers, New York (2002). International Series in Operations Research and Management ScienceMATHGoogle Scholar
  5. 5.
    Czyzowicz, J., Ilcinkas, D., Labourel, A., Pelc, A.: Asynchronous deterministic rendezvous in bounded terrains. TCS 412(50), 6926–6937 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: log-space rendezvous in arbitrary graphs. Distrib. Comput. 25(2), 165–178 (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Czyzowicz, J., Kosowski, A., Pelc, A.: Deterministic rendezvous of asynchronous bounded-memory agents in polygonal terrains. Theor. Comput. Syst. 52(2), 179–199 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Marco, G., Gargano, L., Kranakis, E., Krizanc, D., Pelc, A., Vaccaro, U.: Asynchronous deterministic rendezvous in graphs. Theoret. Comput. Sci. 355(3), 315–326 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2013, pp. 92–99 (2013)Google Scholar
  11. 11.
    Feinerman, O., Korman, A., Kutten, S., Rodeh, Y.: Fast rendezvous on a cycle by agents with different speeds. In: Chatterjee, M., Cao, J., Kothapalli, K., Rajsbaum, S. (eds.) ICDCN 2014. LNCS, vol. 8314, pp. 1–13. Springer, Heidelberg (2014). doi:10.1007/978-3-642-45249-9_1 CrossRefGoogle Scholar
  12. 12.
    Huus, E., Kranakis, E.: Rendezvous of many agents with different speeds in a cycle. In: Papavassiliou, S., Ruehrup, S. (eds.) ADHOC-NOW 2015. LNCS, vol. 9143, pp. 195–209. Springer, Heidelberg (2015). doi:10.1007/978-3-319-19662-6_14 CrossRefGoogle Scholar
  13. 13.
    Koucky, M.: Universal traversal sequences with backtracking. J. Comput. Syst. Sci. 65, 717–726 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kranakis, E., Krizanc, D., MacQuarrie, F., Shende, S.: Randomized rendezvous on a ring for agents with different speeds. In: Proceedings of the 15th International Conference on Distributed Computing and Networking (ICDCN) (2015)Google Scholar
  15. 15.
    Kranakis, E., Krizanc, D., Markou, E.: The mobile agent rendezvous problem in the ring: an introduction. Synthesis Lectures on Distributed Computing Theory Series. Morgan and Claypool Publishers, San Rafael (2010)Google Scholar
  16. 16.
    Pelc, A.: Deterministic rendezvous in networks: a comprehensive survey. Networks 59, 331–347 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sawchuk, C.: Mobile Agent Rendezvous in the Ring. Ph.D. thesis, Carleton University (2004)Google Scholar
  19. 19.
    Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts, strongly universal exploration sequences. ACM Trans. Algorithms 10(3), 12 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Yu, X., Yung, M.: Agent rendezvous: a dynamic symmetry-breaking problem. In: Meyer, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 610–621. Springer, Heidelberg (1996). doi:10.1007/3-540-61440-0_163 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Evangelos Kranakis
    • 1
  • Danny Krizanc
    • 2
  • Euripides Markou
    • 3
  • Aris Pagourtzis
    • 4
  • Felipe Ramírez
    • 2
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA
  3. 3.Department of Computer Science and Biomedical InformaticsUniversity of ThessalyVolosGreece
  4. 4.School of Electronic and Computer EngineeringNational Technical University of AthensZografouGreece

Personalised recommendations