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)


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.


Disk Exit Robot Search and Fetch Treasure 


  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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Springer, Heidelberg (2003)MATHGoogle Scholar
  4. 4.
    Baeza Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comp. 106(2), 234–252 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baeza-Yates, R., Schott, R.: Parallel searching in the plane. Comput. Geom. 5(3), 143–154 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beck, A.: On the linear search problem. Israel J. Math. 2(4), 221–228 (1964)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bellman, R.: An optimal search. SIAM Rev. 5(3), 274–274 (1963)CrossRefGoogle Scholar
  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 CrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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 CrossRefGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  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 CrossRefGoogle Scholar
  17. 17.
    Gluss, B.: An alternative solution to the lost at sea problem. Naval Res, Logistics Q. 8(1), 117–122 (1961)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Isbell, J.R.: Pursuit around a hole. Naval Res. Logistics Q. 14(4), 569–571 (1967)CrossRefMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  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 CrossRefGoogle Scholar
  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)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Konstantinos Georgiou
    • 1
  • George Karakostas
    • 2
  • Evangelos Kranakis
    • 3
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada
  2. 2.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations