ALGOSENSORS 2016: Algorithms for Sensor Systems pp 80-94

# Search-and-Fetch with One Robot on a Disk

(Track: Wireless and Geometry)
• Konstantinos Georgiou
• George Karakostas
• Evangelos Kranakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10050)

## Abstract

A robot is located at a point in the plane. A treasure and an exit, both stationary, are located at unknown (to the robot) positions both at distance one from the robot. Starting from its initial position, the robot aims to fetch the treasure to the exit. At any time the robot can move anywhere on the disk with constant speed. The robot detects an interesting point (treasure or exit) only if it passes over the exact location of that point. Given that an adversary controls the locations of both the treasure and the exit on the perimeter, we are interested in designing algorithms that minimize the treasure-evacuation time, i.e. the time it takes for the treasure to be discovered and brought to the exit by the robot.

In this paper we differentiate how the robot’s knowledge of the distance between the two interesting points affects the overall evacuation time. We demonstrate sthe difference between knowing the exact value of that distance versus knowing only a lower bound and provide search algorithms for both cases. In the former case we give an algorithm which is off from the optimal algorithm (that does not know the locations of the treasure and the exit) by no more than $$\frac{4 \sqrt{2}+3 \pi +2}{6 \sqrt{2}+2 \pi +2}\le 1.019$$ multiplicatively, or $$\frac{\pi }{2}-\sqrt{2}\le 0.157$$ additively. In the latter case we provide an algorithm which is shown to be optimal.

### Keywords

Disk Exit Robot Search and Fetch Treasure

### References

1. 1.
Ahlswede, R., Wegener, I.: Search Problems. Wiley-Interscience (1987)Google Scholar
2. 2.
Alpern, S.: Find-and-fetch search on a tree. Oper. Res. 59(5), 1258–1268 (2011)
3. 3.
Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Springer, Heidelberg (2003)
4. 4.
Baeza Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comp. 106(2), 234–252 (1993)
5. 5.
Baeza-Yates, R., Schott, R.: Parallel searching in the plane. Comput. Geom. 5(3), 143–154 (1995)
6. 6.
Beck, A.: On the linear search problem. Israel J. Math. 2(4), 221–228 (1964)
7. 7.
Bellman, R.: An optimal search. SIAM Rev. 5(3), 274–274 (1963)
8. 8.
Berman, P.: On-line searching and navigation. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms. LNCS, vol. 1442, pp. 232–241. Springer, Heidelberg (1998). doi:10.1007/BFb0029571
9. 9.
Bonato, A., Nowakowski, R.: The game of cops and robbers on graphs. In: AMS (2011)Google Scholar
10. 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). doi:10.1007/978-3-662-46078-8_14 Google Scholar
11. 11.
Chung, T.H., Hollinger, G.A., Isler, V.: Search and pursuit-evasion in mobile robotics. Auton. Robots 31(4), 299–316 (2011)
12. 12.
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). doi:10.1007/978-3-662-45174-8_9 Google Scholar
13. 13.
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, Heidelberg (2015). doi:10.1007/978-3-319-18173-8_10
14. 14.
Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S.: Wireless autonomous robot evacuation from equilateral triangles and squares. In: ADHOC-NOW 2015, Athens, Greece, June 29–July 1, 2015, Proceedings, pp. 181–194 (2015)Google Scholar
15. 15.
Demaine, E.D., Fekete, S.P., Gal, S.: Online searching with turn cost. Theor. Comput. Sci. 361(2), 342–355 (2006)
16. 16.
Fekete, S., Gray, C., Kröller, A.: Evacuation of rectilinear polygons. In: Wu, W., Daescu, O. (eds.) COCOA 2010. LNCS, vol. 6508, pp. 21–30. Springer, Heidelberg (2010). doi:10.1007/978-3-642-17458-2_3
17. 17.
Gluss, B.: An alternative solution to the lost at sea problem. Naval Res, Logistics Q. 8(1), 117–122 (1961)
18. 18.
Isbell, J.R.: Pursuit around a hole. Naval Res. Logistics Q. 14(4), 569–571 (1967)
19. 19.
Jennings, J.S., Whelan, G., Evans, W.F.: Cooperative search and rescue with a team of mobile robots. In: ICAR, pp. 193–200. IEEE (1997)Google Scholar
20. 20.
Kao, M.-Y., Ma, Y., Sipser, M., Yin, Y.: Optimal constructions of hybrid algorithms. J. Algorithms 29(1), 142–164 (1998)
21. 21.
Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. Inf. Comp. 131(1), 63–79 (1996)
22. 22.
Koutsoupias, E., Papadimitriou, C., Yannakakis, M.: Searching a fixed graph. In: Meyer, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 280–289. Springer, Heidelberg (1996). doi:10.1007/3-540-61440-0_135
23. 23.
Nahin, P., Chases, E.: The Mathematics of Pursuit and Evasion. Princeton University Press (2012)Google Scholar
24. 24.
Stone, L.: Theory of optimal search. Academic Press, New York (1975)

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## Authors and Affiliations

• Konstantinos Georgiou
• 1
• George Karakostas
• 2
• Evangelos Kranakis
• 3