Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model

• Jurek Czyzowicz
• Konstantinos Georgiou
• Ryan Killick
• Evangelos Kranakis
• Danny Krizanc
• Manuel Lafond
• Lata Narayanan
• Jaroslav Opatrny
• Sunil Shende
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)

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.

Keywords

Energy Evacuation Linear Robot Speed Time Trade-offs Wireless communication

References

1. 1.
Ahlswede, R., Wegener, I.: Search Problems. Wiley-Interscience, Chichester (1987)
2. 2.
Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Springer, Boston (2003).
3. 3.
Baeza Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)
4. 4.
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000)
5. 5.
Beck, A.: On the linear search problem. Isr. J. Math. 2(4), 221–228 (1964)
6. 6.
Bellman, R.: An optimal search. SIAM Rev. 5(3), 274 (1963)
7. 7.
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).
8. 8.
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).
9. 9.
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).
10. 10.
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).
11. 11.
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)Google Scholar
12. 12.
Czyzowicz, J.: Time-energy tradeoffs for evacuation by two robots in the wireless model. CoRR, abs/1905.06783 (2019)Google Scholar
13. 13.
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)Google Scholar
14. 14.
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).
15. 15.
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).
16. 16.
Czyzowicz, J., et al.: Search on a line by byzantine robots. In: Proceedings of 27th ISAAC, pp. 27:1–27:12 (2016)Google Scholar
17. 17.
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).
18. 18.
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)Google Scholar
19. 19.
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).
20. 20.
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).
21. 21.
Demaine, E.D., Fekete, S.P., Gal, S.: Online searching with turn cost. Theor. Comput. Sci. 361(2), 342–355 (2006)
22. 22.
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)Google Scholar
23. 23.
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)Google Scholar
24. 24.
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)
25. 25.
Stone, L.: Theory of Optimal Search. Academic Press, New York (1975)

© Springer Nature Switzerland AG 2019

Authors and Affiliations

• Jurek Czyzowicz
• 1
• Konstantinos Georgiou
• 2
• Ryan Killick
• 3
• Evangelos Kranakis
• 3
Email author
• Danny Krizanc
• 4
• Manuel Lafond
• 5
• Lata Narayanan
• 6
• Jaroslav Opatrny
• 6
• Sunil Shende
• 7
1. 1.Départemant d’informatiqueUniversité du Québec en OutaouaisGatineauCanada