Abstract
We consider the evacuation problem on a circle for three robots, at most one of which is faulty. The three robots starting from the center of a unit circle search for an exit placed at an unknown location on the perimeter (of the circle). During the search, robots can communicate wirelessly at any distance. The goal is to minimize the time that the latest non-faulty robot reaches the exit.
Our main contributions are two intuitive evacuation protocols for the non-faulty robots to reach the exit in two well-studied fault-models, Crash and Byzantine. Moreover, we complement our positive results by lower bounds in both models. A summary of our results reads as follows:
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Case of Crash Faults:
Lower Bound \({\approx }5.188\); Upper Bound \({\approx }6.309\),
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Case of Byzantine Faults:
Lower Bound \({\approx }5.948\); Upper Bound \({\approx }6.921\),
For comparison, it is known (see [11]) that in the case of three non-faulty robots with wireless communication we have a lower bound of 4.159, and an upper bound of 4.219 for evacuation, while for two non-faulty robots \(1 + 2\pi /3 + \sqrt{3} \approx 4.779\) is a tight upper and lower bound for evacuation.
J. Czyzowicz, K. Georgiou, and E. Kranakis—Research supported in part by NSERC Discovery grant.
J. Czyzowicz and W. Rytter—Supported in part by grant NCN2014/13/B/ST6/00770 of the Polish Science Centre.
M. Włodarczyk—Supported in part by the National Science Centre of Poland Grant UMO-2016/21/N/ST6/00968.
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References
Agmon, N., Peleg, D.: Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM J. Comput. 36(1), 56–82 (2006)
Baeza-Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)
Baeza-Yates, R., Schott, R.: Parallel searching in the plane. Comput. Geom. 5(3), 143–154 (1995)
Beck, A.: On the linear search problem. Israel J. Math. 2(4), 221–228 (1964)
Bellman, R.: An optimal search. SIAM Rev. 5(3), 274–274 (1963)
Bouzid, Z., Potop-Butucaru, M.G., Tixeuil, S.: Optimal byzantine-rezilient convergence in uni-dimensional robot network. Theoret. Comput. Sci. 411(34–36), 3154–3168 (2010)
Brandt, S., Laufenberg, F., Lv, Y., Stolz, D., Wattenhofer, R.: Collaboration without communication: evacuating two robots from a disk. In: Fotakis, D., Pagourtzis, A., Paschos, V.T. (eds.) CIAC 2017. LNCS, vol. 10236, pp. 104–115. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57586-5_10
Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. In: Frey, H., Li, X., Ruehrup, S. (eds.) ADHOC-NOW 2011. LNCS, vol. 6811, pp. 346–359. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22450-8_27
Cohen, R., Peleg, D.: Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput. 41(1), 1516–1528 (2005)
Cohen, R., Peleg, D.: Convergence of autonomous mobile robots with inaccurate sensors and movements. SIAM J. Comput. 38(1), 276–302 (2008)
Czyzowicz, J., Gąsieniec, L., Gorry, T., Kranakis, E., Martin, R., Pajak, D.: Evacuating robots via unknown exit in a disk. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 122–136. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45174-8_9
Czyzowicz, J., Gasieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Taleb, N.: When patrolmen become corrupted: monitoring a graph using faulty mobile robots. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 343–354. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48971-0_30
Czyzowicz, J., Georgiou, K., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S.: Search on a line by byzantine robots. In: 27th International Symposium on Algorithms and Computation, ISAAC 2016, 12–14 December 2016, Sydney, Australia, pp. 27:1–27:12 (2016)
Czyzowicz, J., Georgiou, K., Kranakis, E., Narayanan, L., Opatrny, J., Vogtenhuber, B.: Evacuating robots from a disk using face-to-face communication (extended abstract). In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 140–152. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18173-8_10
Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J.: Search on a line with faulty robots. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25–28, pp. 405–414 (2016)
Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S.: Wireless autonomous robot evacuation from equilateral triangles and squares. In: Papavassiliou, S., Ruehrup, S. (eds.) ADHOC-NOW 2015. LNCS, vol. 9143, pp. 181–194. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19662-6_13
Défago, X., Gradinariu, M., Messika, S., Raipin-Parvédy, P.: Fault-tolerant and self-stabilizing mobile robots gathering. In: Dolev, S. (ed.) DISC 2006. LNCS, vol. 4167, pp. 46–60. Springer, Heidelberg (2006). https://doi.org/10.1007/11864219_4
Dieudonné, Y., Pelc, A., Peleg, D.: Gathering despite mischief. ACM Trans. Algorithms (TALG) 11(1), 1 (2014)
Hromkovič, J., Klasing, R., Monien, B., Peine, R.: Dissemination of information in interconnection networks (broadcasting & gossiping). In: Du, D.Z., Hsu, D.F. (eds.) Combinatorial Network Theory, pp. 125–212. Springer, Boston (1996). https://doi.org/10.1007/978-1-4757-2491-2_5
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)
Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: Proceedings of the Forty-second ACM Symposium on Theory of Computing, pp. 513–522. ACM (2010)
Lamport, L., Shostak, R., Pease, M.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. (TOPLAS) 4(3), 382–401 (1982)
Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, Burlington (1996)
Souissi, S., Défago, X., Yamashita, M.: Gathering asynchronous mobile robots with inaccurate compasses. In: Shvartsman, M.M.A.A. (ed.) OPODIS 2006. LNCS, vol. 4305, pp. 333–349. Springer, Heidelberg (2006). https://doi.org/10.1007/11945529_24
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)
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Czyzowicz, J. et al. (2017). Evacuation from a Disc in the Presence of a Faulty Robot. In: Das, S., Tixeuil, S. (eds) Structural Information and Communication Complexity. SIROCCO 2017. Lecture Notes in Computer Science(), vol 10641. Springer, Cham. https://doi.org/10.1007/978-3-319-72050-0_10
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