Symmetric Rendezvous with Advice: How to Rendezvous in a Disk

  • Konstantinos GeorgiouEmail author
  • Jay Griffiths
  • Yuval Yakubov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11085)


In the classic Symmetric Rendezvous problem on a Line (SRL), two robots at known distance 2 but unknown direction execute the same randomized algorithm trying to minimize the expected rendezvous time. A long standing conjecture is that the best possible rendezvous time is 4.25 with known upper and lower bounds being very close to that value. We introduce and study a geometric variation of SRL that we call Symmetric Rendezvous in a Disk (SRD) where two robots at distance 2 have a common reference point at distance \(\rho \). We show that even when \(\rho \) is not too small, the two robots can meet in expected time that is less than 4.25. Part of our contribution is that we demonstrate how to adjust known, even simple and provably non-optimal, algorithms for SRL, effectively improving their performance in the presence of a reference point. Special to our algorithms for SRD is that, unlike in SRL, for every fixed \(\rho \) the worst case distance traveled, i.e. energy that is used, in our algorithms is finite. In particular, we show that the energy of our algorithms is \(O\left( \rho ^2\right) \), while we also explore time-energy tradeoffs, concluding that one may be efficient both with respect to time and energy, with only a minor compromise on the optimal termination time.


  1. 1.
    Alpern, S.: Hide and seek games. Seminar (1976)Google Scholar
  2. 2.
    Alpern, S.: The rendezvous search problem. SIAM J. Control Optim. 33(3), 673–711 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alpern, S.: Rendezvous search: a personal perspective. Oper. Res. 50(5), 772–795 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alpern, S.: Ten Open Problems in Rendezvous Search, pp. 223–230. Springer, New York (2013). Scholar
  5. 5.
    Alpern, S., Baston, V.: Rendezvous on a planar lattice. Oper. Res. 53(6), 996–1006 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alpern, S., Baston, V.: A common notion of clockwise can help in planar rendezvous. Eur. J. Oper. Res. 175(2), 688–706 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alpern, S., Baston, V.: Rendezvous in higher dimensions. SIAM J. Control Optim. 44(6), 2233–2252 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Alpern, S., Gal, S.: Rendezvous search on the line with distinguishable players. SIAM J. Control Optim. 33(4), 1270–1276 (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Alpern, S., Gal, S.: The theory of search games and rendezvous. In: International Series in Operations Research & Management Science, vol. 55. Springer, Heidelberg (2003).
  10. 10.
    Steve Alpern and Wei Shi Lim: Rendezvous of three agents on the line. Naval Res. Logist. (NRL) 49(3), 244–255 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Anaya, J., Chalopin, J., Czyzowicz, J., Labourel, A., Pelc, A., Vaxès, Y.: Collecting information by power-aware mobile agents. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 46–60. Springer, Heidelberg (2012). Scholar
  12. 12.
    Anderson, E.J., Weber, R.R.: The rendezvous problem on discrete locations. J. Appl. Probab. 27(4), 839–851 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Anderson, E.J., Essegaier, S.: Rendezvous search on the line with indistinguishable players. SIAM J. Control Optim. 33(6), 1637–1642 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Rendezvous and election of mobile agents: impact of sense of direction. Theory Comput. Syst. 40(2), 143–162 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Baston, V.J.: Two rendezvous search problems on the line. Naval Res. Logist. 46, 335–340 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Baston, V., Gal, S.: Rendezvous on the line when the players’ initial distance is given by an unknown probability distribution. SIAM J. Control Optim. 36(6), 1880–1889 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Beveridge, A., Ozsoyeller, D., Isler, V.: Symmetric rendezvous on the line with an unknown initial distance. Technical report (2011)Google Scholar
  18. 18.
    Chester, E.J., Tütüncü, R.H.: Rendezvous search on the labeled line. Oper. Res. 52(2), 330–334 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Collins, A., Czyzowicz, J., Gąsieniec, L., Kosowski, A., Martin, R.: Synchronous rendezvous for location-aware agents. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 447–459. Springer, Heidelberg (2011). Scholar
  20. 20.
    Cooper, C., Frieze, A., Radzik, T.: Multiple random walks and interacting particle systems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 399–410. Springer, Heidelberg (2009). Scholar
  21. 21.
    Czyzowicz, J., Dobrev, S., Kranakis, E., Krizanc, D.: The power of tokens: rendezvous and symmetry detection for two mobile agents in a ring. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 234–246. Springer, Heidelberg (2008). Scholar
  22. 22.
    Czyzowicz, J., Pelc, A., Labourel, A.: How to meet asynchronously (almost) everywhere. ACM Trans. Algorithms 8(4), 37:1–37:14 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Das, S.: Distributed computing with mobile agents: solving rendezvous and related problems. Ph.D. thesis, University of Ottawa, Canada (2007)Google Scholar
  24. 24.
    Das, S.: Mobile agent rendezvous in a ring using faulty tokens. In: Rao, S., Chatterjee, M., Jayanti, P., Murthy, C.S.R., Saha, S.K. (eds.) ICDCN 2008. LNCS, vol. 4904, pp. 292–297. Springer, Heidelberg (2007). Scholar
  25. 25.
    Das, S., Luccio, F.L., Markou, E.: Mobile agents rendezvous in spite of a malicious agent. In: Bose, P., Gąsieniec, L.A., Römer, K., Wattenhofer, R. (eds.) ALGOSENSORS 2015. LNCS, vol. 9536, pp. 211–224. Springer, Cham (2015). Scholar
  26. 26.
    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). Scholar
  27. 27.
    Flocchini, P., Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Multiple mobile agent rendezvous in a ring. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 599–608. Springer, Heidelberg (2004). Scholar
  28. 28.
    Georgiou, K., Griffiths, J., Yakubov, Y.: Symmetric rendezvous with advice: How to rendezvous in a disk. CoRR, abs/1805.03351 (2018)Google Scholar
  29. 29.
    Han, Q., Donglei, D., Vera, J., Zuluaga, L.F.: Improved bounds for the symmetric rendezvous value on the line. Oper. Res. 56(3), 772–782 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    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). Scholar
  31. 31.
    Kranakis, E., Krizanc, D., Markou, E.: The Mobile Agent Rendezvous Problem in the Ring. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers, San Rafael (2010)CrossRefGoogle Scholar
  32. 32.
    Kranakis, E., Santoro, N., Sawchuk, C., Krizanc, D.: Mobile agent rendezvous in a ring. In: Distributed Computing Systems, pp. 592–599. IEEE (2003)Google Scholar
  33. 33.
    Pelc, A.: Deterministic rendezvous in networks: a comprehensive survey. Networks 59(3), 331–347 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Prencipe, G.: Impossibility of gathering by a set of autonomous mobile robots. Theor. Comput. Sci 384(2–3), 222–231 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences. ACM Trans. Algorithms 10(3), 12:1–12:15 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Patchrawat Patch Uthaisombut: Symmetric rendezvous search on the line using move patterns with different lengths. Working paper (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Konstantinos Georgiou
    • 1
    Email author
  • Jay Griffiths
    • 1
  • Yuval Yakubov
    • 1
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada

Personalised recommendations