Group Search on the Line

  • Marek Chrobak
  • Leszek Gąsieniec
  • Thomas Gorry
  • Russell Martin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8939)

Abstract

In this paper we consider the group search problem, or evacu- ation problem, in which k mobile entities (\({\cal M}{\cal E}\)s) located on the line perform search for a specific destination. The \({\cal M}{\cal E}\)s are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All \({\cal M}{\cal E}\)s must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d − o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k ≥ 1 unit-speed \({\cal M}{\cal E}\)s. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of \({\cal M}{\cal E}\)s, group search cannot be performed faster than in time 9d − o(d). We also examine the case of k = 2 \({\cal M}{\cal E}\)s with different speeds, showing a surprising result that the bound of 9d can be achieved when one \({\cal M}{\cal E}\) has unit speed, and the other \({\cal M}{\cal E}\) moves with speed at least 1/3.

Keywords

evacuation group search mobile entity 

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References

  1. 1.
    Alpern, S., Baston, V., Essegaier, S.: Rendezvous search on a graph. J. Applied Probability 36(1), 223–231 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Kluwer Academic Publishing, Dordrecht (2003)MATHGoogle Scholar
  3. 3.
    Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching with uncertainty. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 176–189. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  4. 4.
    Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching in the plane. Information and Computation 106(2), 234–252 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Baeza-Yates, R.A., Schott, R.: Parallel searching in the plane. Computational Geometric Theory and Applications 5(3), 143–154 (1995)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bellman, R.: Minimization problem. Bull. AMS 62(3), 270 (1956)CrossRefGoogle Scholar
  7. 7.
    Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: Exploring and mapping directed graphs. In: STOC 1998, pp. 269–278 (1998)Google Scholar
  8. 8.
    Bose, P., De Carufel, J.-L., Durocher, S.: Revisiting the problem of searching on a line. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 205–216. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Collins, A., Czyzowicz, J., Gąsieniec, L., Labourel, A.: Tell me where I am so I can meet you sooner. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 502–514. Springer, Heidelberg (2010)Google Scholar
  10. 10.
    Dieudonné, Y., Pelc, A.: Anonymous meeting in networks. In: SODA 2013, pp. 737–747 (2013)Google Scholar
  11. 11.
    Ghosh, S.K., Klein, R.: Online algorithms for searching and exploration in the plane. Computer Science Review 4(4), 189–201 (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Hammar, M., Nilsson, B.J., Schuierer, S.: Parallel searching on m rays. Comput. Geom. 18(3), 125–139 (2001)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Jeż, A., Łopuzański, J.: On the two-dimensional cow search problem. Information Processing Letters 131(11), 543–547 (2009)Google Scholar
  14. 14.
    Kao, M.Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Information and Computation 109(1), 63–79 (1996)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Koutsoupias, E., Papadimitriou, C.H., Yannakakis, M.: Searching a fixed graph. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 280–289. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  16. 16.
    Li, H., Chong, K.P.: Search on lines and graphs. In: Proc. 48th IEEE Conference on Decision and Control, 2009 held Jointly with the 2009 28th Chinese Control Conference (CDC/CCC 2009), vol. 109(11), pp. 5780–5785 (2009)Google Scholar
  17. 17.
    Temple, T., Frazzoli, E.: Whittle-indexability of the cow path problem. In: American Control Conference (ACC), pp. 4152–4158 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Marek Chrobak
    • 1
  • Leszek Gąsieniec
    • 2
  • Thomas Gorry
    • 2
  • Russell Martin
    • 2
  1. 1.Dept of Computer Science and EngineeringUniversity of CaliforniaRiversideUSA
  2. 2.Dept of Computer ScienceUniversity of LiverpoolLiverpoolUnited Kingdom

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