Almost Optimal Asynchronous Rendezvous in Infinite Multidimensional Grids

  • Evangelos Bampas
  • Jurek Czyzowicz
  • Leszek Gąsieniec
  • David Ilcinkas
  • Arnaud Labourel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6343)


Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylogd). This bound for the case of 2d -grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ ) lower bound is straightforward.

Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylogd) length trajectory.

We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r 1 and r 2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of \(O((\frac{d}{r})^\delta {\rm polylog}(\frac{d}{r}))\), where r =  min (r 1, r 2) and for r ≥ 1.


Hamiltonian Path Network Design Problem Port Number Graph Exploration Rendezvous Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abraham, I., Dolev, D., Malkhi, D.: LLS: a locality aware location service for mobile ad hoc networks. In: Proc. DIALM-POMC 2004, pp. 75–84 (2004)Google Scholar
  2. 2.
    Alpern, S.: The rendezvous search problem. SIAM J. on Control and Optimization 33, 673–683 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alpern, S., Gal, S.: The theory of search games and rendezvous. Int. Series in Operations research and Management Science, vol. 55. Kluwer Academic Publisher, Dordrecht (2002)Google Scholar
  4. 4.
    Alpern, J., Baston, V., Essegaier, S.: Rendezvous search on a graph. Journal of Applied Probability 36, 223–231 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Anderson, E., Fekete, S.: Asymmetric rendezvous on the plane. In: Proc. 14th Annual ACM Symp. on Computational Geometry, pp. 365–373 (1998)Google Scholar
  6. 6.
    Anderson, E., Fekete, S.: Two-dimensional rendezvous search. Operations Research 49, 107–118 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Baston, V., Gal, S.: Rendezvous on the line when the players’ initial distance is given by an unknown probability distribution. SIAM J. on Control and Optimization 36, 1880–1889 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Baston, V., Gal, S.: Rendezvous search when marks are left at the starting points. Naval Res. Log. 48, 722–731 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bose, P., Morin, P., Stojmenovic, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wireless Networks 7(6), 609–616 (2001)zbMATHCrossRefGoogle Scholar
  10. 10.
    Buchin, K.: Constructing Delaunay Triangulations along Space-Filling Curves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 119–130. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Transactions on Algorithms 4(4), 1–18 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Collins, A., Czyzowicz, J., Gasieniec, L., Labourel, A.: Tell me where I am so I can meet you sooner: Asynchronous rendezvous with location information. In: Proc. of ICALP 2010 (2010)Google Scholar
  13. 13.
    Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Solving the Robots Gathering Problem. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1181–1196. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Czyzowicz, J., Dobrev, S., Gasieniec, L., Ilcinkas, D., Jansson, J., Klasing, R., Lignos, I., Martin, R.A., Sadakane, K., Sung, W.-K.: More Efficient Periodic Traversal in Anonymous Undirected Graphs. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 167–181. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: Proc. of SODA 2010, pp. 22–30 (2010)Google Scholar
  16. 16.
    De Marco, G., Gargano, L., Kranakis, E., Krizanc, D., Pelc, A., Vaccaro, U.: Asynchronous deterministic rendezvous in graphs. Theoretical Computer Science 355, 315–326 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dobrev, S., Jansson, J., Sadakane, K., Sung, W.-K.: Finding short right-hand-on-the-wall walks in graphs. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 127–139. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Emek, Y., Gasieniec, L., Kantor, E., Pelc, A., Peleg, D., Su, C.: Broadcasting in UDG radio networks with unknown topology. Distributed Computing 21(5), 331–351 (2009)CrossRefGoogle Scholar
  20. 20.
    Emek, Y., Kantor, E., Peleg, D.: On the effect of the deployment setting on broadcasting in Euclidean radio networks. In: Proc. PODC 2008, pp. 223–232 (2008)Google Scholar
  21. 21.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous oblivious robots with limited visibility. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 247–258. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoretical Computer Science 345(2-3), 331–344 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gal, S.: Rendezvous search on the line. Operations Research 47, 974–976 (1999)zbMATHCrossRefGoogle Scholar
  24. 24.
    Gasieniec, L., Klasing, R., Martin, R.A., Navarra, A., Zhang, X.: Fast periodic graph exploration with constant memory. J. on Computer Systems and Sciences 74(5), 808–822 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gotsman, C., Lindenbaum, M.: On the metric properties of discrete space-filling curves. IEEE Transactions on Image Processing 5(5), 794–797 (1996)CrossRefGoogle Scholar
  26. 26.
    Ilcinkas, D.: Setting Port Numbers for Fast Graph Exploration. Theor. Comput. Sci. 401(1-3), 236–242 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Israeli, A., Jalfon, M.: Token management schemes and random walks yield self stabilizing mutual exclusion. In: Proc. PODC’90, pp. 119–131 (1990)Google Scholar
  28. 28.
    Klasing, R., Kosowski, A., Navarra, A.: Taking advantage of symmetries: gathering of asynchronous oblivious robots on a ring. In: Baker, T.P., Bui, A., Tixeuil, S. (eds.) OPODIS 2008. LNCS, vol. 5401, pp. 446–462. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    Klasing, R., Markou, E., Pelc, A.: Gathering asynchronous oblivious mobile robots in a ring. Theoretical Computer Science 390, 27–39 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Kosowski, A., Navarra, A.: Graph Decomposition for Improving Memoryless Periodic Exploration. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 501–512. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  31. 31.
    Kowalski, D., Malinowski, A.: How to meet in anonymous network. Theoretical Computer Science 399, 141–156 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Kozma, G., Lotker, Z., Sharir, M., Stupp, G.: Geometrically aware communication in random wireless networks. In: Proc. PODC 2004, pp. 310–319 (2004)Google Scholar
  33. 33.
    Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Mobile agent rendezvous in a ring. In: Proc. 23rd International Conference on Distributed Computing Systems (ICDCS 2003), pp. 592–599 (2003)Google Scholar
  34. 34.
    Kuhn, F., Wattenhofer, R., Zhang, Y., Zollinger, A.: Geometric ad-hoc routing: theory and practice. In: Proc. PODC 2003, pp. 63–72 (2003)Google Scholar
  35. 35.
    Lim, W., Alpern, S.: Minimax rendezvous on the line. SIAM J. on Control and Optimization 34, 1650–1665 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Moon, B., Jagadish, H.V., Faloutsos, C., Saltz, J.H.: Analysis of the Clustering Properties of the Hilbert Space-Filling Curve. IEEE Transactions on Knowledge Data Engineering 14(1), 124–141 (2001)CrossRefGoogle Scholar
  37. 37.
    Prencipe, G.: Impossibility of gathering by a set of autonomous mobile robots. Theoretical Computer Science 384, 222–231 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Schelling, T.: The strategy of conflict. Oxford University Press, Oxford (1960)Google Scholar
  39. 39.
    Stachowiak, G.: Asynchronous Deterministic Rendezvous on the Line. In: Nielsen, M., Kucera, A., Miltersen, P.B., Palamidessi, C., Tuma, P., Valencia, F.D. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 497–508. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  40. 40.
    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 2007), pp. 599–608 (2007)Google Scholar
  41. 41.
    Thomas, L.: Finding your kids when they are lost. Journal on Operational Res. Soc. 43, 637–639 (1992)zbMATHGoogle Scholar
  42. 42.
    Xu, B., Chen, D.Z.: Density-Based Data Clustering Algorithms for Lower Dimensions Using Space-Filling Curves. In: Zhou, Z.-H., Li, H., Yang, Q. (eds.) PAKDD 2007. LNCS (LNAI), vol. 4426, pp. 997–1005. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  43. 43.
    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 2010

Authors and Affiliations

  • Evangelos Bampas
    • 1
  • Jurek Czyzowicz
    • 2
  • Leszek Gąsieniec
    • 3
  • David Ilcinkas
    • 1
  • Arnaud Labourel
    • 1
  1. 1.LaBRICNRS / INRIA / Université de Bordeaux 
  2. 2.Université du Québec 
  3. 3.University of Liverpool 

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