Randomized Rendez-Vous with Limited Memory

  • Evangelos Kranakis
  • Danny Krizanc
  • Pat Morin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

We present a tradeoff between the expected time for two identical agents to rendez-vous on a synchronous, anonymous, oriented ring and the memory requirements of the agents. In particular, we show that there exists a 2t state agent, which can achieve rendez-vous on an n node ring in expected time O( n2/2t + 2t ) and that any t/2 state agent requires expected time Ω( n2/2t ). As a corollary we observe that Θ(loglogn) bits of memory are necessary and sufficient to achieve rendez-vous in linear time.

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References

  1. 1.
    Alpern, S.: The rendezvous search problem. SIAM Journal of Control and Optimization 33, 673–683 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer Academic Publishers, Norwell, Massachusetts (2003)MATHGoogle Scholar
  3. 3.
    Coppersmith, D., Tetali, P., Winkler, P.: Collisions among random walks on a graph. SIAM Journal of Discrete Mathematics 6, 363–374 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dessmark, A., Fraigniaud, P., Pelc, A.: Deterministic rendezvous in graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 184–195. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Dobrev, S., Flocchini, P., Prencipe, G., Santoro, N.: Multiple agents rendezvous in a ring in spite of a black hole. In: Papatriantafilou, M., Hunel, P. (eds.) OPODIS 2003. LNCS, vol. 3144, pp. 34–46. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Flocchini, P., Kranakis, E., Krizanc, D., Luccio, F., Santoro, N., Sawchuk, C.: Mobile agent rendezvous when tokens fail. In: Kralovic, R., Sýkora, O. (eds.) SIROCCO 2004. LNCS, vol. 3104, pp. 161–172. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Flocchini, P., Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Multiple mobile agent rendezvous in the ring. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 599–608. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Gasieniec, L., Kranakis, E., Krizanc, D., Zhang, X.: Optimal memory rendezvous of anonymous mobile agents in a uni-directional ring. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 282–292. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Kowalski, D., Malinowski, A.: How to meet in an anonymous network. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 44–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Kowalski, D., Pelc, A.: Polynomial deterministic rendezvous in arbitrary graphs. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 644–656. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Kranakis, E., Krizanc, D.: An algorithmic theory of mobile agents. In: Symposium on Trustworthy Global Computing (2006)Google Scholar
  12. 12.
    Kranakis, E., Krizanc, D., Markou, E.: Mobile agent rendezvous in a synchronous torus. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 653–664. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Kranakis, E., Krizanc, D., Rajasbaum, S.: Mobile agent rendezvous: A survey. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 1–9. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Mobile agent rendezvous search problem in the ring. In: Proc. International Conference on Distributed Computing Systems (ICDCS), pp. 592–599 (2003)Google Scholar
  15. 15.
    De Marco, G., Gargano, L., Kranakis, E., Krizanc, D., Pelc, A., Vacaro, U.: Asynchronous deterministic rendezvous in graphs. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 271–282. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  17. 17.
    Ross, S.M.: Probability Models for Computer Science. Harcourt Academic Press, London (2002)Google Scholar
  18. 18.
    Roy, N., Dudek, G.: Collaborative robot exploration and rendezvous: Algorithms, performance bounds and observations. Autonomous Robots 11, 117–136 (2001)MATHCrossRefGoogle Scholar
  19. 19.
    Santoro, N.: Design and Analysis of Distributed Algorithms. Wiley, Hoboken (2006)Google Scholar
  20. 20.
    Sawchuk, C.: Mobile Agent Rendezvous in the Ring. PhD thesis, Carleton University, School of Computer Science, Ottawa, Canada (2004)Google Scholar
  21. 21.
    Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal of Computing 28, 1347–1363 (1999)MATHCrossRefMathSciNetGoogle 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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Evangelos Kranakis
    • 1
  • Danny Krizanc
    • 2
  • Pat Morin
    • 1
  1. 1.School of Computer ScienceCarleton University 
  2. 2.Department of Mathematics and Computer ScienceWesleyan University 

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