Infinite Grid Exploration by Disoriented Robots

  • Quentin BramasEmail author
  • Stéphane Devismes
  • Pascal Lafourcade
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)


We deal with a set of autonomous robots moving on an infinite grid. Those robots are opaque, have limited visibility capabilities, and run using synchronous Look-Compute-Move cycles. They all agree on a common chirality, but have no global compass. Finally, they may use lights of different colors, but except from that, robots have neither persistent memories, nor communication mean. We consider the infinite grid exploration (IGE) problem. We first show that two robots are not sufficient in our settings to solve the problem, even when robots have a common coordinate system. We then show that if the robots’ coordinate systems are not self-consistent, three or four robots are not sufficient to solve the problem neither. Finally, we present three algorithms that solve the IGE problem in various settings. The first algorithm uses six robots with constant colors and a visibility range of one. The second one uses the minimum number of robots, i.e., five, as well as five modifiable colors, still under visibility one. The last algorithm requires seven oblivious anonymous robots, yet assuming visibility two. Notice that the two last algorithms also achieve exclusiveness.


  1. 1.
    Adhikary, R., Bose, K., Kundu, M.K., Sau, B.: Mutual visibility by asynchronous robots on infinite grid. In: Gilbert, S., Hughes, D., Krishnamachari, B. (eds.) ALGOSENSORS 2018. LNCS, vol. 11410, pp. 83–101. Springer, Cham (2019). Scholar
  2. 2.
    Baldoni, R., Bonnet, F., Milani, A., Raynal, M.: Anonymous graph exploration without collision by mobile robots. Inf. Process. Lett. 109(2), 98–103 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonnet, F., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Asynchronous exclusive perpetual grid exploration without sense of direction. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 251–265. Springer, Heidelberg (2011). Scholar
  4. 4.
    Bose, K., Adhikary, R., Kundu, M.K., Sau, B.: Arbitrary pattern formation on infinite grid by asynchronous oblivious robots. In: Das, G.K., Mandal, P.S., Mukhopadhyaya, K., Nakano, S. (eds.) WALCOM 2019. LNCS, vol. 11355, pp. 354–366. Springer, Cham (2019). Scholar
  5. 5.
    Bramas, Q., Devismes, S., Lafourcade, P.: Infinite grid exploration by disoriented robots. Technical report, arXiv, May 2019.
  6. 6.
    Bramas, Q., Devismes, S., Lafourcade, P.: Infinite grid exploration by disoriented robots: animations, May 2019.
  7. 7.
    Brandt, S., Uitto, J., Wattenhofer, R.: A tight bound for semi-synchronous collaborative grid exploration. In: 32nd International Symposium on Distributed Computing (DISC), New Orleans, Louisiana, October 2018Google Scholar
  8. 8.
    Das, S., Flocchini, P., Prencipe, G., Santoro, N., Yamashita, M.: Autonomous mobile robots with lights. Theor. Comput. Sci. 609(P1), 171–184 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Devismes, S., Lamani, A., Petit, F., Raymond, P., Tixeuil, S.: Optimal grid exploration by asynchronous oblivious robots. In: Richa, A.W., Scheideler, C. (eds.) SSS 2012. LNCS, vol. 7596, pp. 64–76. Springer, Heidelberg (2012). Scholar
  10. 10.
    Di Stefano, G., Navarra, A.: Gathering of oblivious robots on infinite grids with minimum traveled distance. Inf. Comput. 254, 377–391 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dieudonné, Y., Petit, F.: Circle formation of weak robots and Lyndon words. Inf. Process. Lett. 101(4), 156–162 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dutta, D., Dey, T., Chaudhuri, S.G.: Gathering multiple robots in a ring and an infinite grid. In: Krishnan, P., Radha Krishna, P., Parida, L. (eds.) ICDCIT 2017. LNCS, vol. 10109, pp. 15–26. Springer, Cham (2017). Scholar
  13. 13.
    Efrima, A., Peleg, D.: Distributed models and algorithms for mobile robot systems. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 70–87. Springer, Heidelberg (2007). Scholar
  14. 14.
    Emek, Y., Langner, T., Stolz, D., Uitto, J., Wattenhofer, R.: How many ants does it take to find the food? Theor. Comput. Sci. 608(P3), 255–267 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci. 337(1), 147–168 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Luna, G.A.D., Flocchini, P., Chaudhuri, S.G., Poloni, F., Santoro, N., Viglietta, G.: Mutual visibility by luminous robots without collisions. Inf. Comput. 254, 392–418 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ooshita, F., Datta, A.K.: Brief announcement: feasibility of weak gathering in connected-over-time dynamic rings. In: Izumi, T., Kuznetsov, P. (eds.) SSS 2018. LNCS, vol. 11201, pp. 393–397. Springer, Cham (2018). Scholar
  18. 18.
    Peleg, D.: Distributed coordination algorithms for mobile robot swarms: new directions and challenges. In: Pal, A., Kshemkalyani, A.D., Kumar, R., Gupta, A. (eds.) IWDC 2005. LNCS, vol. 3741, pp. 1–12. Springer, Heidelberg (2005). Scholar
  19. 19.
    Yang, Y., Souissi, S., Défago, X., Takizawa, M.: Fault-tolerant flocking for a group of autonomous mobile robots. J. Syst. Softw. 84(1), 29–36 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Quentin Bramas
    • 1
    Email author
  • Stéphane Devismes
    • 2
  • Pascal Lafourcade
    • 3
  1. 1.University of Strasbourg, ICUBE, CNRSStrasbourgFrance
  2. 2.Université Grenoble Alpes, VERIMAGGrenobleFrance
  3. 3.University Clermont Auvergne, CNRS, UMR 6158, LIMOSClermont-FerrandFrance

Personalised recommendations