Abstract
We consider the problem of evacuating two robots from a bounded area, through an unknown exit located on the boundary. Initially, the robots are in the center of the area and throughout the evacuation process they can only communicate with each other when they are at the same point at the same time. Having a visibility range of 0, the robots can only identify the location of the exit if they are already at the exit position. The task is to minimize the time it takes until both robots reach the exit, for a worst-case placement of the exit. For unit disks, an upper bound of 5.628 for the evacuation time is presented in [8]. Using the insight that, perhaps surprisingly, a forced meeting of the two robots as performed in the respective algorithm does not provide an exchange of any non-trivial information, we design a simpler algorithm that achieves an upper bound of 5.625. Our numerical simulations suggest that this bound is optimal for the considered natural class of algorithms. For dealing with the technical difficulties in analyzing the algorithm, we formulate a powerful new criterion that, for a given algorithm, reduces the number of possible worst-case exits radically. This criterion is of independent interest and can be applied to any area shape. Due to space restrictions, this version of the paper contains no proofs or illustrating figures; the full version can be found at http://disco.ethz.ch/publications/ciac2017-robotevac.pdf.
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- 1.
Recall that upon finding the exit, a robot immediately takes the shortest possible tour to meet the other robot and communicate the location of the exit. We call the point where this meeting happens the pick-up point.
- 2.
Note that a robot can also infer information from the fact that the other robot is not at the same point as it is. For instance, it may conclude that the other robot has not already found the exit in some specific segment of the perimeter, since otherwise the other robot would have picked him up at the latest at the current position. This indirect information transfer plays an important role in our arguments that the robots cannot infer any non-trivial information from a forced meeting.
- 3.
We emphasize that \(R_1\) does not calculate a shortest route to the point where \(R_2\) is when \(R_1\) finds the exit, but rather the shortest route for picking \(R_2\) up, knowing that and how \(R_2\) will move until being picked up.
- 4.
These parameters are chosen in a way that for the (only) three possible global worst-case exit positions (determined in the following), the evacuation times are the same up to numerical precision. While the parameter values were determined numerically, we give a rigorous proof for the correctness of the claimed evacuation time.
- 5.
The same holds for the shape of the cut, by the same reason.
References
Alpern, S.: The rendezvous search problem. SIAM J. Control Optim. 33(3), 673–683 (1995)
Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching with uncertainty extended abstract. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 176–189. Springer, Heidelberg (1988). doi:10.1007/3-540-19487-8_20
Beck, A., Newman, D.J.: Yet more on the linear search problem. Isr. J. Math. 8(4), 419–429 (1970)
Borowiecki, P., Das, S., Dereniowski, D., Kuszner, Ł.: Distributed evacuation in graphs with multiple exits. In: Suomela, J. (ed.) SIROCCO 2016. LNCS, vol. 9988, pp. 228–241. Springer, Cham (2016). doi:10.1007/978-3-319-48314-6_15
Chrobak, M., Gasieniec, L., Gorry, T., Martin, R.: Group search on the line. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015. LNCS, vol. 8939, pp. 164–176. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46078-8_14
Czyzowicz, J., Dobrev, S., Georgiou, K., Kranakis, E., MacQuarrie, F.: Evacuating two robots from multiple unknown exits in a circle. In: ICDCN (2016). doi:10.1145/2833312.2833318
Czyzowicz, J., Gasieniec, 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). doi:10.1007/978-3-662-45174-8_9
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). doi:10.1007/978-3-319-18173-8_10
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). doi:10.1007/978-3-319-19662-6_13
Dessmark, A., Fraigniaud, P., Pelc, A.: Deterministic rendezvous in graphs. In: Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 184–195. Springer, Heidelberg (2003). doi:10.1007/978-3-540-39658-1_19
Emek, Y., Langner, T., Stolz, D., Uitto, J., Wattenhofer, R.: How many ants does it take to find the food? In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 263–278. Springer, Cham (2014). doi:10.1007/978-3-319-09620-9_21
Feinerman, O., Korman, A.: Memory lower bounds for randomized collaborative search and implications for biology. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 61–75. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33651-5_5
Feinerman, O., Korman, A., Lotker, Z., Sereni, J.-S.: Collaborative search on the plane without communication. In: PODC (2012). doi:10.1145/2332432.2332444
Förster, K.-T., Nuridini, R., Uitto, J., Wattenhofer, R.: Lower bounds for the capture time: linear, quadratic, and beyond. In: Scheideler, C. (ed.) Structural Information and Communication Complexity. LNCS, vol. 9439, pp. 342–356. Springer, Cham (2015). doi:10.1007/978-3-319-25258-2_24
Förster, K.-T., Wattenhofer, R.: Directed graph exploration. In: Baldoni, R., Flocchini, P., Binoy, R. (eds.) OPODIS 2012. LNCS, vol. 7702, pp. 151–165. Springer, Heidelberg (2012). doi:10.1007/978-3-642-35476-2_11
Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 451–462. Springer, Heidelberg (2004). doi:10.1007/978-3-540-28629-5_34
Kranakis, E., Krizanc, D., Rajsbaum, S.: Mobile agent rendezvous: a survey. In: Flocchini, P., Gasieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 1–9. Springer, Heidelberg (2006). doi:10.1007/11780823_1
Lamprou, I., Martin, R., Schewe, S.: Fast two-robot disk evacuation with wireless communication. In: Gavoille, C., Ilcinkas, D. (eds.) DISC 2016. LNCS, vol. 9888, pp. 1–15. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53426-7_1
Megow, N., Mehlhorn, K., Schweitzer, P.: Online graph exploration: new results on old and new algorithms. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6756, pp. 478–489. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22012-8_38
Nowakowski, R.J., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discret. Math. 43(2–3), 235–239 (1983). doi:10.1016/0012-365X(83)90160-7
Parsons, T.D.: Pursuit-evasion in a graph. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 426–441. Springer, Heidelberg (1978)
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Brandt, S., Laufenberg, F., Lv, Y., Stolz, D., Wattenhofer, R. (2017). Collaboration Without Communication: Evacuating Two Robots from a Disk. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_10
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