# Gathering on rings under the Look–Compute–Move model

- 274 Downloads
- 17 Citations

## Abstract

A set of robots arbitrarily placed on different nodes of an anonymous ring have to meet at one common node and there remain. This problem is known in the literature as the *gathering*. Anonymous and oblivious robots operate in Look–Compute–Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), decides whether to stay idle or to move to one of its neighbors (Compute), and in the latter case makes the computed move instantaneously (Move). Cycles are asynchronous among robots. Moreover, each robot is empowered by the so called *multiplicity detection* capability, that is, it is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. The described problem has been extensively studied during the last years. However, the known solutions work only for specific initial configurations and leave some open cases. In this paper, we provide an algorithm which solves the general problem but for few marginal and specific cases, and is able to detect all the ungatherable configurations. It is worth noting that our new algorithm makes use of some previous techniques and unifies them with new strategies in order to deal with any initial configuration, even those left open by previous works.

## Keywords

Distributed algorithm Asynchronous system Gathering Oblivious robots## References

- 1.Alpern, S.: The rendezvous search problem. SIAM J. Control Optim.
**33**, 673–683 (1995)CrossRefzbMATHMathSciNetGoogle Scholar - 2.Bampas, E., Czyzowicz, J., Gasieniec, L., Ilcinkas, D., Labourel, A.: Almost optimal asynchronous rendezvous in infinite multidimensional grids. In: Proceedings of the 24th Internatioanal Symposium on Distributed Computing (DISC), Lecture Notes in Computer Science, vol. 6343, pp. 297–311 (2010)Google Scholar
- 3.Blin, L., Burman, J., Nisse, N.: Exclusive graph searching. In: Proceedings of the 21st Annual European Symposium on Algorithms (ESA), Lecture Notes in Computer Science, vol. 8125, pp. 181–192 (2013)Google Scholar
- 4.Blin, L., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Exclusive perpetual ring exploration without chirality. In: Procedings of the 24th International Symposium on Distributed Computing (DISC), Lecture Notes in Computer Science, vol. 6343, pp. 312–327. Springer (2010)Google Scholar
- 5.Chalopin, J., Das, S.: Rendezvous of mobile agents without agreement on local orientation. In: Proceedings of the 37th International Conference on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 6199, pp. 515–526 (2010)Google Scholar
- 6.Cord-Landwehr, A., Degener, B., Fischer, M., Hüllmann, M., Kempkes, B., Klaas, A., Kling, P., Kurras, S., Märtens, M., Der Heide, F.M.A., Raupach, C., Swierkot, K., Warner, D., Weddemann, C., Wonisch, D.: A new approach for analyzing convergence algorithms for mobile robots. In: Proceedings of the 38th International Conference on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 6756, pp. 650–661 (2011)Google Scholar
- 7.Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. In: Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 22–30 (2010)Google Scholar
- 8.D’Angelo, G., Di Stefano, G., Klasing, R., Navarra, A.: Gathering of robots on anonymous grids without multiplicity detection. In: Proceedings of the 19th International Colloquium on Structural Information and Communication Complexity (SIROCCO), Lecture Notes in Computer Science, vol. 7355, pp. 327–338 (2012)Google Scholar
- 9.D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering of six robots on anonymous symmetric rings. In: Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), Lecture Notes in Computer Science, vol. 6796, pp. 174–185 (2011)Google Scholar
- 10.D’Angelo, G., Di Stefano, G., Navarra, A.: How to gather asynchronous oblivious robots on anonymous rings. In: Proceedings of the 26th International Symposium on Distributed Computing (DISC), Lecture Notes in Computer Science, vol. 7611, pp. 330–344 (2012)Google Scholar
- 11.D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering asynchronous and oblivious robots on basic graph topologies under the look–compute–move model. In: S. Alpern, R. Fokkink, L. Gasieniec, R. Lindelauf, V. Subrahmanian (eds.) Search Theory: A Game Theoretic Perspective, pp. 197–222. Springer, Berlin (2013)Google Scholar
- 12.D’Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., Suchan, K.: A unified approach for different tasks on rings in robot-based computing systems. In: Proceedings of the 15th IEEE IPDPS Workshop on Advances in Parallel and Distributed Computational Models (APDCM), pp. 667–676 (2013)Google Scholar
- 13.Degener, B., Kempkes, B., Langner, T., Meyer auf der Heide, F., Pietrzyk, P., Wattenhofer, R.: A tight runtime bound for synchronous gathering of autonomous robots with limited visibility. In: Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 139–148 (2011)Google Scholar
- 14.Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica
**46**, 69–96 (2006)CrossRefzbMATHMathSciNetGoogle Scholar - 15.Di Stefano, G., Navarra, A.: Optimal gathering of oblivious robots in anonymous graphs. In: Proceedings of the 20th International Colloquium on Structural Information and Communication Complexity (SIROCCO), Lecture Notes in Computer Science, vol. 8179, pp. 213–224 (2013)Google Scholar
- 16.Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Remembering without memory: tree exploration by asynchronous oblivious robots. Theor. Comput. Sci.
**411**(14–15), 1583–1598 (2010)CrossRefzbMATHMathSciNetGoogle Scholar - 17.Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Computing without communicating: ring exploration by asynchronous oblivious robots. Algorithmica
**65**(3), 562–583 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 18.Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci.
**337**, 147–168 (2005) Google Scholar - 19.Izumi, T., Izumi, T., Kamei, S., Ooshita, F.: Randomized gathering of mobile robots with local-multiplicity detection. In: Proceedings of the 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), Lecture Notes in Computer Science, vol. 5873, pp. 384–398 (2009)Google Scholar
- 20.Izumi, T., Izumi, T., Kamei, S., Ooshita, F.: Mobile robots gathering algorithm with local weak multiplicity in rings. In: Proceedings of the 17th International Colloquium on Structural Information and Communication Complexity (SIROCCO), Lecture Notes in Computer Science, vol. 6058, pp. 101–113 (2010)Google Scholar
- 21.Kamei, S., Lamani, A., Ooshita, F., Tixeuil, S.: Asynchronous mobile robot gathering from symmetric configurations. In: Proceedings of the 18th International Colloquium on Structural Information and Communication Complexity (SIROCCO), Lecture Notes in Computer Science, vol. 6796, pp. 150–161 (2011)Google Scholar
- 22.Kamei, S., Lamani, A., Ooshita, F., Tixeuil, S.: Gathering an even number of robots in an odd ring without global multiplicity detection. In: Proceedings of 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), LNCS, vol. 7464, pp. 542–553 (2012)Google Scholar
- 23.Klasing, R., Kosowski, A., Navarra, A.: Taking advantage of symmetries: gathering of many asynchronous oblivious robots on a ring. Theor. Comput. Sci.
**411**, 3235–3246 (2010)CrossRefzbMATHMathSciNetGoogle Scholar - 24.Klasing, R., Markou, E., Pelc, A.: Gathering asynchronous oblivious mobile robots in a ring. Theor. Comput. Sci.
**390**, 27–39 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 25.Koren, M.: Gathering small number of mobile asynchronous robots on ring. Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej. Technologie Informacyjne
**18**, 325–331 (2010)Google Scholar - 26.Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: formation of geometric patterns. SIAM J. Comput.
**28**(4), 1347–1363 (1999)CrossRefzbMATHMathSciNetGoogle Scholar - 27.Yamashita, M., Souissi, S., Défago, X.: Gathering two stateless mobile robots using very inaccurate compasses in finite time. In: Proceedings of the 1st International Conference on Robot Communication and Coordination (RoboComm), pp. 48:1–48:4 (2007)Google Scholar