Abstract
Two robots stand at the origin of the infinite line and are tasked with searching collaboratively for an exit at an unknown location on the line. They can travel at maximum speed b and can change speed or direction at any time. The two robots can communicate with each other at any distance and at any time. The task is completed when the last robot arrives at the exit and evacuates. We study time-energy tradeoffs for the above evacuation problem. The evacuation time is the time it takes the last robot to reach the exit. The energy it takes for a robot to travel a distance x at speed s is measured as \(xs^2\). The total and makespan evacuation energies are respectively the sum and maximum of the energy consumption of the two robots while executing the evacuation algorithm.
Assuming that the maximum speed is b, and the evacuation time is at most cd, where d is the distance of the exit from the origin, we study the problem of minimizing the total energy consumption of the robots. We prove that the problem is solvable only for \(bc \ge 3\). For the case \(bc=3\), we give an optimal algorithm, and give upper bounds on the energy for the case \(bc>3\).
We also consider the problem of minimizing the evacuation time when the available energy is bounded by \(\varDelta \). Surprisingly, when \(\varDelta \) is a constant, independent of the distance d of the exit from the origin, we prove that evacuation is possible in time \(O(d^{3/2}\log d)\), and this is optimal up to a logarithmic factor. When \(\varDelta \) is linear in d, we give upper bounds on the evacuation time.
A full version of this work is available on the Computing Research Repository [12].
J. Czyzowicz, K. Georgiou, E. Kranakis, M. Lafond, L. Narayanan and J. Opatrny—Research supported in part by NSERC Discovery grant.
R. Killick—Research supported by the Ontario Graduate Scholarship.
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Notes
- 1.
The constant of proportionality has (SI) units kg / m and depends, among other things, on the shape of the object and the density of the fluid through which it moves.
References
Ahlswede, R., Wegener, I.: Search Problems. Wiley-Interscience, Chichester (1987)
Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Springer, Boston (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)
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000)
Beck, A.: On the linear search problem. Isr. J. Math. 2(4), 221–228 (1964)
Bellman, R.: An optimal search. SIAM Rev. 5(3), 274 (1963)
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
Chrobak, M., Gąsieniec, 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). https://doi.org/10.1007/978-3-662-46078-8_14
Chuangpishit, H., Georgiou, K., Sharma, P.: Average case - worst case tradeoffs for evacuating 2 robots from the disk in the face-to-face model. In: Gilbert, S., Hughes, D., Krishnamachari, B. (eds.) ALGOSENSORS 2018. LNCS, vol. 11410, pp. 62–82. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-14094-6_5
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.: Energy/time trade-offs for linear-search. In: The 46th International Colloquium on Automata, Languages and Programming (ICALP 2019) (2019, to appear)
Czyzowicz, J.: Time-energy tradeoffs for evacuation by two robots in the wireless model. CoRR, abs/1905.06783 (2019)
Czyzowicz, J., et al.: God save the queen. In: 9th International Conference on Fun with Algorithms (FUN 2018). LIPIcs, vol. 100, pp. 16:1–16:20 (2018)
Czyzowicz, J., et al.: Priority evacuation from a disk using mobile robots. In: Lotker, Z., Patt-Shamir, B. (eds.) SIROCCO 2018. LNCS, vol. 11085, pp. 392–407. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01325-7_32
Czyzowicz, J., Georgiou, K., Kranakis, E.: Group search and evacuation. In: Flocchini, P., Prencipe, G., Santoro, N. (eds.) Distributed Computing by Mobile Entities: Current Research in Moving and Computing, Chap. 14. 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: Proceedings of 27th ISAAC, 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: Proceeding of PODC, pp. 405–413. ACM (2016)
Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S.: Linear search with terrain-dependent speeds. In: Fotakis, D., Pagourtzis, A., Paschos, V.T. (eds.) CIAC 2017. LNCS, vol. 10236, pp. 430–441. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57586-5_36
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)
Georgiou, K., Karakostas, G., Kranakis, E.: Search-and-fetch with one robot on a disk - (track: wireless and geometry). In: Proceedings of 12th ALGOSENSORS 2016, pp. 80–94 (2016)
Georgiou, K., Karakostas, G., Kranakis, E.: Search-and-fetch with 2 robots on a disk - wireless and face-to-face communication models. In: Liberatore, F., Parlier, G.H., Demange, M. (eds.) Proceedings of the 6th International Conference on Operations Research and Enterprise Systems, ICORES 2017, Porto, Portugal, 23–25 February 2017, pp. 15–26. SciTePress (2017)
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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Czyzowicz, J. et al. (2019). Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model. In: Censor-Hillel, K., Flammini, M. (eds) Structural Information and Communication Complexity. SIROCCO 2019. Lecture Notes in Computer Science(), vol 11639. Springer, Cham. https://doi.org/10.1007/978-3-030-24922-9_13
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