Abstract
Consider a set of mobile robots placed on distinct nodes of a discrete, anonymous, and bidirectional ring. Asynchronously, each robot takes a snapshot of the ring, determining the size of the ring and which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or not. In the first case, it performs the computed move, eventually. This model of computation is known as Look-Compute-Move. The computation depends on the required task. In this paper, we solve both the well-known Gathering and Exclusive Searching tasks. In the former problem, all robots must simultaneously occupy the same node, eventually. In the latter problem, the aim is to clear all edges of the graph. An edge is cleared if it is traversed by a robot or if both its endpoints are occupied. We consider the exclusive searching where it must be ensured that two robots never occupy the same node. Moreover, since the robots are oblivious, the clearing is perpetual, i.e., the ring is cleared infinitely often. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a characterization of the initial configurations that permit the resolution of the problems under very weak assumptions. More precisely, we provide a full characterization (except for few pathological cases) of the initial configurations for which gathering can be solved. The algorithm relies on the necessary assumption of the local-weak multiplicity detection. This means that during the Look phase a robot detects also whether the node it occupies is occupied by other robots, without acquiring the exact number. For the exclusive searching, we characterize all (except for few pathological cases) aperiodic configurations from which the problem is feasible. We also provide some impossibility results for the case of periodic configurations.
Notes
Configuration \({{\mathcal {C}}} = (0,1,0,1,1,0,1,1)\) is the only exception, see Sect. 6.3.
Except for the case of \({{\mathcal {C}}}^{\min } = (0^{i},1,0^x,1,(0^{i},1,0^{x-1},1)^{\ell })\) for \(\ell =1\) and \(x=1\) (i.e. \({{\mathcal {C}}}^{\min } = (0^i,1,0,1,0^i,1,1)\)) where \({\textsc {reduce}} _1\) is performed. This case will be discussed in Sect. 6.3, along with the case \({{\mathcal {C}}}^{\min } = (0^{i},1,0^x,1,0^{i},1,0^{y},1)\), \(y<x\).
References
Agmon, N., Peleg, D.: Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM J. Comput. 36(1), 58–82 (2006)
Bampas, E., Czyzowicz, J., Gąsieniec, L., Ilcinkas, D., Labourel, A.: Almost optimal asynchronous rendezvous in infinite multidimensional grids. In: Proceedings of the 24th International Symposium on Distributed Computing (DISC), Lecture Notes in Computer Science, vol. 6343, pp. 297–311 (2010)
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. Springer (2013)
Blin, L., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Exclusive perpetual ring exploration without chirality. In: Proceedings of the 24th International Symposium on Distributed Computing (DISC), Lecture Notes in Computer Science, vol. 6343, pp. 312–327 (2010)
Bonnet, F., Milani, A., Potop-Butucaru, M., Tixeuil, S.: Asynchronous exclusive perpetual grid exploration without sense of direction. In: 15th International Conference on Principles of Distributed Systems (OPODIS), Lecture Notes in Computer Science, vol. 7109, pp. 251–265. Springer (2011)
Chalopin, J., Das, S.: Rendezvous of mobile agents without agreement on local orientation. In: Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), vol. 6199, pp. 515–526 (2010)
Chalopin, J., Flocchini, P., Mans, B., Santoro, N.: Network exploration by silent and oblivious robots. In: 36th International Workshop on Graph Theoretic Concepts in Computer Science (WG), Lecture Notes in Computer Science, vol. 6410, pp. 208–219. Springer (2010)
Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: gathering. SIAM J. Comput. 41(4), 829–879 (2012)
Czyzowicz, J., Gąsieniec, L., Pelc, A.: Gathering few fat mobile robots in the plane. Theor. Comput. Sci. 410(6–7), 481–499 (2009)
D’Angelo, G., Di Stefano, G., Klasing, R., Navarra, A.: Gathering of robots on anonymous grids and trees without multiplicity detection. Theor. Comput. Sci. 610, 158–168 (2016)
D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering asynchronous and oblivious robots on basic graph topologies under the look-compute-move model. In: Alpern, S., Fokkink, R., Gąsieniec, L., Lindelauf, R., Subrahmanian, V. (eds.) Search Theory: A Game Theoretic Perspective, pp. 197–222. Springer, Berlin (2013)
D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering on rings under the look-compute-move model. Distrib. Comput. 27(4), 255–285 (2014)
D’Angelo, G., Di Stefano, G., Navarra, A.: Gathering six oblivious robots on anonymous symmetric rings. J. Discrete Algorithms 26, 16–27 (2014)
D’Angelo, G., Di Stefano, G., Navarra, A., Nisse, N., Suchan, K.: Computing on rings by oblivious robots: a unified approach for different tasks. Algorithmica 4(72), 1055–1096 (2015)
D’Angelo, G., Navarra, A., Nisse, N.: Gathering and exclusive searching on rings under minimal assumptions. In: Proceedings of the 15th International Conference on Distributed Computing and Networking (ICDCN), Lecture Notes in Computer Science, vol. 8314, pp. 149–164. Springer (2014)
Di Stafano, G., Navarra, A.: Gathering of oblivious robots on infinite grids with minimum traveled distance. Inf. Comput. (to appear)
Di Stefano, G., Montanari, P., Navarra, A.: About ungatherability of oblivious and asynchronous robots on anonymous rings. In: Proceedings of the 26th International Workshop on Combinatorial Algorithms (IWOCA’15), Lecture Notes in Computer Science, vol. 9538, pp. 136–147. Springer (2016)
Dieudonne, Y., Pelc, A., Peleg, D.: Gathering despite mischief. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 527–540 (2012)
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)
Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: How many oblivious robots can explore a line. Inf. Process. Lett. 111(20), 1027–1031 (2011)
Flocchini, P., Ilcinkas, D., Pelc, A., Santoro, N.: Computing without communicating: ring exploration by asynchronous oblivious robots. Algorithmica 65(3), 562–583 (2013)
Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Synthesis Lectures on Distributed Computing Theory, vol. 3. Morgan & Claypool (2012). doi:10.2200/S00440ED1V01Y201208DCT010
Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Hard tasks for weak robots: the role of common knowledge in pattern formation by autonomous mobile robots. In: 10th International Symposium on Algorithms and Computation (ISAAC), Lecture Notes in Computer Science, vol. 1741, pp. 93–102. Springer (1999)
Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)
Ilcinkas, D., Nisse, N., Soguet, D.: The cost of monotonicity in distributed graph searching. Distrib. Comput. 22(2), 117–127 (2009)
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)
Izumi, T., Souissi, S., Katayama, Y., Inuzuka, N., Défago, X., Wada, K., Yamashita, M.: The gathering problem for two oblivious robots with unreliable compasses. SIAM J. Comput. 41(1), 26–46 (2012)
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)
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 the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), Lecture Notes in Computer Science, vol. 7464, pp. 542–553 (2012)
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)
Klasing, R., Markou, E., Pelc, A.: Gathering asynchronous oblivious mobile robots in a ring. Theor. Comput. Sci. 390, 27–39 (2008)
Kranakis, E., Krizanc, D., Markou, E.: The Mobile Agent Rendezvous Problem in the Ring. Synthesis Lectures on Distributed Computing Theory, vol. 1. Morgan & Claypool (2010). doi:10.2200/S00278ED1V01Y201004DCT001
Prencipe, G.: Instantaneous actions vs. full asynchronicity: controlling and coordinating a set of autonomous mobile robots. In: Proceedings of the 7th Italian Conference on Theoretical Computer Science (ICTCS), pp. 154–171 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Preliminary results concerning this work were presented in the 15th International Conference on Distributed Computing and Networking (ICDCN’14) [15].
Rights and permissions
About this article
Cite this article
D’Angelo, G., Navarra, A. & Nisse, N. A unified approach for gathering and exclusive searching on rings under weak assumptions. Distrib. Comput. 30, 17–48 (2017). https://doi.org/10.1007/s00446-016-0274-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00446-016-0274-y