Abstract
We consider (n, f)-search on a circle, a search problem of a hidden exit on a circle of unit radius for \(n > 1\) robots, f of which are faulty. All the robots start at the centre of the circle and can move anywhere with maximum speed 1. During the search, robots may communicate wirelessly. All messages transmitted by all robots are tagged with the robots’ unique identifiers which cannot be corrupted. The search is considered complete when the exit is found by a non-faulty robot (which must visit its location) and the remaining non-faulty robots know the correct location of the exit.
We study two models of faulty robots. First, crash-faulty robots may stop operating as instructed, and thereafter they remain nonfunctional. Second, Byzantine-faulty robots may transmit untrue messages at any time during the search so as to mislead the non-faulty robots, e.g., lie about the location of the exit.
When there are only crash fault robots, we provide optimal algorithms for the (n, f)-search problem, with optimal worst-case search completion time \(1+\frac{(f+1)2\pi }{n}\). Our main technical contribution pertains to optimal algorithms for (n, 1)-search with a Byzantine-faulty robot, minimizing the worst-case search completion time, which equals \(1+\frac{4\pi }{n}\).
K. Georgiou and E. Kranakis—Research supported in part by NSERC Discovery grant.
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Notes
- 1.
Figures in this paper depict robot trajectories during the execution of our search algorithm. They restrict to cases where the first announcement is made while robots search their first sector of length \(\theta =\frac{2\pi }{n}\), and no other announcement is made until time \(1+\theta \). It is assumed that agent \(a_0\) makes the first announcement. A black square shows the location of the announcement; a white square shows the locations of other agents at that time. A solid dot shows the starting positions of the robots on the unit circle (starting from the center of the circle, they move directly, in time 1, to their starting positions). Recall that the arc length between the starting position of \(a_0\) and the point of the announcement is denoted by t (hence, the announcement takes place in time \(1+t\)).
References
Ahlswede, R., Wegener, I.: Search Problems. Wiley, Hoboken (1987)
Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous, vol. 55. Springer, Heidelberg (2003). https://doi.org/10.1007/b100809
Baeza-Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)
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)
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., et al.: Evacuation from a disc in the presence of a faulty robot. In: Das, S., Tixeuil, S. (eds.) SIROCCO 2017. LNCS, vol. 10641, pp. 158–173. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72050-0_10
Czyzowicz, J., Georgiou, K., Kranakis, E.: Group search and evacuation. In: Flocchini, P., Prencipe, G., Santoro, N. (eds.) Distributed Computing by Mobile Entities. LNCS, vol. 11340, pp. 335–370. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11072-7_14
Czyzowicz, J., et al.: Search on a line by byzantine robots. In: ISAAC, pp. 27:1–27:12 (2016)
Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J.: Search on a line with faulty robots. In: PODC, pp. 405–414. ACM (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
Demaine, E.D., Fekete, S.P., Gal, S.: Online searching with turn cost. Theor. Comput. Sci. 361(2), 342–355 (2006)
Czyzowicz, J., Georgiou, K., Kranakis, E., Narayanan, L., Opatrny, J., Vogtenhuber, B.: Evacuating using face-to-face communication. CoRR, abs/1501.04985 (2015)
Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. Inf. Comput. 131(1), 63–79 (1996)
Stone, L.: Theory of Optimal Search. Academic Press, New York (1975)
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Georgiou, K., Kranakis, E., Leonardos, N., Pagourtzis, A., Papaioannou, I. (2019). Optimal Circle Search Despite the Presence of Faulty Robots. In: Dressler, F., Scheideler, C. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2019. Lecture Notes in Computer Science(), vol 11931. Springer, Cham. https://doi.org/10.1007/978-3-030-34405-4_11
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