Linear Search by a Pair of Distinct-Speed Robots

  • Evangelos Bampas
  • Jurek Czyzowicz
  • Leszek Gąsieniec
  • David Ilcinkas
  • Ralf Klasing
  • Tomasz Kociumaka
  • Dominik Pająk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9988)

Abstract

Two mobile robots are initially placed at the same point on an infinite line. Each robot may move on the line in either direction not exceeding its maximal speed. The robots need to find a stationary target placed at an unknown location on the line. The search is completed when both robots arrive at the target point. The target is discovered at the moment when either robot arrives at its position. The robot knowing the placement of the target may communicate it to the other robot. We look for the algorithm with the shortest possible search time (i.e. the worst-case time at which both robots meet at the target) measured as a function of the target distance from the origin (i.e. the time required to travel directly from the starting point to the target at unit velocity).

We consider two standard models of communication between the robots, namely wireless communication and communication by meeting. In the case of communication by meeting, a robot learns about the target while sharing the same location with the robot possessing this knowledge. We propose here an optimal search strategy for two robots including the respective lower bound argument, for the full spectrum of their maximal speeds. This extends the main result of Chrobak et al. (SOFSEM 2015) referring to the exact complexity of the problem for the case when the speed of the slower robot is at least one third of the faster one. In addition, we consider also the wireless communication model, in which a message sent by one robot is instantly received by the other robot, regardless of their current positions on the line. In this model, we design an optimal strategy whenever the faster robot is at most 6 times faster than the slower one.

References

  1. 1.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. International Series in Operations Research & Management Science, vol. 55. Kluwer Academic Publishers, New York (2002)MATHGoogle Scholar
  2. 2.
    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 CrossRefGoogle Scholar
  3. 3.
    Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baeza-Yates, R.A., Schott, R.: Parallel searching in the plane. Comput. Geom. 5, 143–154 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barajas, J., Serra, O.: The lonely runner with seven runners. Electr. J. Comb. 15(1), 1–18 (2008). http://www.combinatorics.org/Volume_15/Abstracts/v15i1r48.html MathSciNetMATHGoogle 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 (1963)CrossRefGoogle Scholar
  8. 8.
    Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: exploring and mapping directed graphs. Inf. Comput. 176(1), 1–21 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bender, M.A., Slonim, D.K.: The power of team exploration: two robots can learn unlabeled directed graphs. In: 35th IEEE Annual Symposium on Foundations of Computer Science, FOCS 1994, pp. 75–85. IEEE Computer Society (1994)Google Scholar
  10. 10.
    Bose, P., Carufel, J.D., Durocher, S.: Searching on a line: a complete characterization of the optimal solution. Theor. Comput. Sci. 569, 24–42 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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-Testing. LNCS, vol. 8939, pp. 164–176. Springer, Heidelberg (2015)Google 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)Google Scholar
  13. 13.
    Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E.: Boundary patrolling by mobile agents with distinct maximal speeds. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 701–712. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Dereniowski, D., Disser, Y., Kosowski, A., Pająk, D., Uznański, P.: Fast collaborative graph exploration. Inf. Comput. 243, 37–49 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dudek, G., Romanik, K., Whitesides, S.: Localizing a robot with minimum travel. SIAM J. Comput. 27(2), 583–604 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Feinerman, O., Korman, A., Lotker, Z., Sereni, J.: Collaborative search on the plane without communication. In: Kowalski, D., Panconesi, A. (eds.) ACM Symposium on Principles of Distributed Computing, PODC 2012, pp. 77–86. ACM (2012)Google Scholar
  17. 17.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fraigniaud, P., Gąsieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jaillet, P., Stafford, M.: Online searching. Oper. Res. 49(4), 501–515 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kawamura, A., Kobayashi, Y.: Fence patrolling by mobile agents with distinct speeds. Distrib. Comput. 28(2), 147–154 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nahin, P.J.: Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton Puzzlers, Princeton University Press, Princeton (2012)MATHGoogle Scholar
  22. 22.
    Pelc, A.: Deterministic rendezvous in networks: a comprehensive survey. Networks 59(3), 331–347 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wills, J.M.: Zwei Sätze über inhomogene diophantische Approximation von Irrationalzahlen. Monatsh. Math. 71(3), 263–269 (1967)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Evangelos Bampas
    • 1
  • Jurek Czyzowicz
    • 2
  • Leszek Gąsieniec
    • 3
  • David Ilcinkas
    • 4
  • Ralf Klasing
    • 4
  • Tomasz Kociumaka
    • 5
  • Dominik Pająk
    • 6
  1. 1.LIF, CNRS, Aix-Marseille UniversityMarseilleFrance
  2. 2.Département d’informatiqueUniversité du Québec en OutaouaisGatineauCanada
  3. 3.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  4. 4.LaBRI, CNRS, University of BordeauxTalenceFrance
  5. 5.Institute of InformaticsUniversity of WarsawWarsawPoland
  6. 6.Institute of InformaticsWrocław University of TechnologyWrocławPoland

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