Rendezvous of Distance-Aware Mobile Agents in Unknown Graphs

  • Shantanu Das
  • Dariusz Dereniowski
  • Adrian Kosowski
  • Przemysław Uznański
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8576)

Abstract

We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent’s label. It is well known that even for simple graphs of degree Δ, the rendezvous time can be exponential in Δ in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these distance-aware agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance D and the size of the smaller of the two agent labels l =  min (l1, l2). Our algorithm has a time complexity of O(Δ(D + logl)) and we show an almost matching lower bound of Ω(Δ(D + logl/logΔ)) on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness.

Keywords

Mobile Agent Rendezvous Synchronous Anonymous Networks Distance Oracle Lower Bounds 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer (2003)Google Scholar
  2. 2.
    Baston, V., Gal, S.: Rendezvous search when marks are left at the starting points. Naval Research Logistics 48(8), 722–731 (2001)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bampas, E., Czyzowicz, J., Gąsieniec, L., Ilcinkas, D., Labourel, A.: Almost optimal asynchronous rendezvous in infinite multidimensional grids. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 297–311. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Chalopin, J., Das, S., Santoro, N.: Rendezvous of Mobile Agents in Unknown Graphs with Faulty Links. In: Pelc, A. (ed.) DISC 2007. LNCS, vol. 4731, pp. 108–122. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Chalopin, J., Das, S., Widmayer, P.: Rendezvous of Mobile Agents in Directed Graphs. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 282–296. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Czyżowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 22–30 (2010)Google Scholar
  7. 7.
    Czyżowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: Log-space rendezvous in arbitrary graphs. Distributed Computing 25, 165–178 (2012)CrossRefMATHGoogle Scholar
  8. 8.
    Czyżowicz, J., Kosowski, A., Pelc, A.: Time vs. space trade-offs for rendezvous in trees. Distributed Computing 27(2), 95–109 (2014)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dieudonné, Y., Pelc, A.: Vincent Villain. How to meet asynchronously at polynomial cost. In: Proc 32nd Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 92–99 (2013)Google Scholar
  11. 11.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theoretical Computer Science 337(1-3), 147–168 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    De Marco, G., Gargano, L., Kranakis, E., Krizanc, D., Pelc, A., Vaccaro, U.: Asynchronous deterministic rendezvous in graphs. Theoretical Computer Science 355(3), 315–326 (2006)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fraigniaud, P., Ilcinkas, D., Pelc, A.: Oracle size: A new measure of difficulty for communication problems. In: Proc. 25th Ann ACM Symposium on Principles of Distributed Computing (PODC), pp. 179–187 (2006)Google Scholar
  14. 14.
    Fraigniaud, P., Pelc, A.: Deterministic Rendezvous in Trees with Little Memory. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 242–256. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Klasing, R., Markou, E., Pelc, A.: Gathering asynchronous oblivious mobile robots in a ring. Theoretical Computer Science 390(1), 27–39 (2008)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Kranakis, E., Krizanc, D., Markou, E.: The mobile agent rendezvous problem in the ring. Morgan and Claypool Publishers (2010)Google Scholar
  17. 17.
    Kowalski, D.R., Malinowski, A.: How to meet in anonymous network. Theoretical Computer Science 399(1-2), 141–156 (2008)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Pelc, A.: Deterministic rendezvous in networks: A comprehensive survey. Networks 59, 331–347 (2012)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Stachowiak, G.: Asynchronous deterministic rendezvous on the line. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 497–508. Springer, Heidelberg (2009)Google Scholar
  20. 20.
    Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proc 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 599–608 (2007)Google Scholar
  21. 21.
    Yamashita, M., Kameda, T.: Computing on anonymous networks: Part I–Characterizing the solvable cases. IEEE Transactions on Parallel and Distributed Systems 7(1), 69–89 (1996)CrossRefGoogle Scholar
  22. 22.
    Yu, X., Yung, M.: Agent rendezvous: A dynamic symmetry-breaking problem. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 610–621. Springer, Heidelberg (1996)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Shantanu Das
    • 1
  • Dariusz Dereniowski
    • 2
  • Adrian Kosowski
    • 3
    • 4
  • Przemysław Uznański
    • 1
  1. 1.LIFAix-Marseille University and CNRSMarseilleFrance
  2. 2.Dept. of Algorithms and System ModelingGdańsk University of TechnologyGdańskPoland
  3. 3.GANG ProjectInria ParisFrance
  4. 4.LIAFAParis Diderot University and CNRSFrance

Personalised recommendations