Rendezvous of Heterogeneous Mobile Agents in Edge-Weighted Networks

  • Dariusz Dereniowski
  • Ralf Klasing
  • Adrian Kosowski
  • Łukasz Kuszner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8576)

Abstract

We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time \(T_{\textup{OPT}}\) in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time \(O(n T_{\textup{OPT}})\) in a n-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to \(\Theta (T_{\textup{OPT}})\) when the agents are allowed to exchange Θ(n) bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time \(\Theta (T_{\textup{OPT}})\).

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. International Series in Operations Research and Managment Science. Kluwer Academic Publishers, Boston (2003)Google Scholar
  2. 2.
    Anderson, E., Fekete, S.: Asymmetric rendezvous on the plane. In: Proceedings of 14th Annual ACM Symposium on Computational Geometry (SoCG), pp. 365–373 (1998)Google Scholar
  3. 3.
    Anderson, E., Fekete, S.: Two-dimensional rendezvous search. Operations Research 49(1), 107–118 (2001)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    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
  5. 5.
    Baston, V., Gal, S.: Rendezvous on the line when the players’ initial distance is given by an unknown probability distribution. SIAM Journal on Control and Optimization 36(6), 1880–1889 (1998)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Baston, V., Gal, S.: Rendezvous search when marks are left at the starting points. Naval Research Logistics 48(8), 722–731 (2001)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chalopin, J., Das, S., Mihalák, M., Penna, P., Widmayer, P.: Data delivery by energy-constrained mobile agents. In: Flocchini, P., Gao, J., Kranakis, E., Meyer auf der Heide, F. (eds.) ALGOSENSORS 2013. LNCS, vol. 8243, pp. 111–122. Springer, Heidelberg (2013)Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Collins, A., Czyzowicz, J., Gąsieniec, L., Labourel, A.: Tell me where I am so I can meet you sooner. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part II. LNCS, vol. 6199, pp. 502–514. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Czyzowicz, J., Gasieniec, L., Georgiou, K., Kranakis, E., MacQuarrie, F.: The beachcombers’ problem: Walking and searching with mobile robots. CoRR, abs/1304.7693 (2013)Google Scholar
  11. 11.
    Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E.: Boundary patrolling by mobile agents with distinct maximal speeds. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 701–712. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: log-space rendezvous in arbitrary graphs. Distributed Computing 25(2), 165–178 (2012)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Czyzowicz, J., Kranakis, E., Pacheco, E.: Localization for a system of colliding robots. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 508–519. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Dessmark, A., Fraigniaud, P., Kowalski, D.R., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46(1), 69–96 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. CoRR, abs/1301.7119 (2013)Google Scholar
  16. 16.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci. 337(1-3), 147–168 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    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
  18. 18.
    Gal, S.: Rendezvous search on the line. Operations Research 47(6), 974–976 (1999)CrossRefMATHGoogle Scholar
  19. 19.
    Guilbault, S., Pelc, A.: Asynchronous rendezvous of anonymous agents in arbitrary graphs. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 421–434. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Israeli, A., Jalfon, M.: Token management schemes and random walks yield self-stabilizing mutual exclusion. In: Dwork, C. (ed.) PODC, pp. 119–131. ACM (1990)Google Scholar
  21. 21.
    Kalyanasundaram, B., Schintger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discret. Math. 5(4), 545–557 (1992)CrossRefMATHGoogle Scholar
  22. 22.
    Kawamura, A., Kobayashi, Y.: Fence patrolling by mobile agents with distinct speeds. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 598–608. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Kowalski, D.R., Malinowski, A.: How to meet in anonymous network. Theoretical Computer Science 399(1-2), 141–156 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Lim, W., Alpern, S.: Minimax rendezvous on the line. SIAM Journal on Control and Optimization 34(5), 1650–1665 (1996)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Marco, G.D., 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
  26. 26.
    Schelling, T.: The strategy of conflict. Oxford University Press, Oxford (1960)Google Scholar
  27. 27.
    Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proceedings of 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 599–608 (2007)Google Scholar
  28. 28.
    Thomas, L.: Finding your kids when they are lost. Journal of the Operational Research Society 43(6), 637–639 (1992)CrossRefMATHGoogle Scholar
  29. 29.
    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

  • Dariusz Dereniowski
    • 1
  • Ralf Klasing
    • 2
  • Adrian Kosowski
    • 3
    • 4
  • Łukasz Kuszner
    • 1
  1. 1.Department of Algorithms and System ModelingGdańsk University of TechnologyPoland
  2. 2.LaBRICNRS and University of BordeauxFrance
  3. 3.GANG ProjectInria ParisFrance
  4. 4.LIAFACNRS and Paris Diderot UniversityFrance

Personalised recommendations