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Capture on Grids and Tori with Different Numbers of Cops

  • Fabrizio LuccioEmail author
  • Linda Pagli
Conference paper
  • 272 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11657)

Abstract

This paper is a contribution to the classical cops and robber problem on a graph, directed to two-dimensional grids and tori. We apply some new concepts for solving the problem on grids and apply these concepts to give a new algorithm for the capture on tori. Then we consider using any number k of cops, give efficient algorithms for this case yielding a capture time \(t_k\), and compute the minimum value of k needed for any given capture time. We introduce the concept of work \(w_k=k\cdot t_k\) of an algorithm and study a possible speed-up using larger teams of cops.

Keywords

Cops Robber Capture time Grid Tori Work Speed-up 

References

  1. 1.
    Alspach, B.: Searching and sweeping graphs a brief survey. Le Matematiche 59(I), 5–37 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bhattacharya, S., Paul, G., Sanyal, S.: A cops and robber game in multidimensional grids. Discrete Appl. Math. 158, 1745–1751 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhattacharya, S., Banerjee, A., Badyopadhay, S.: CORBA-based analysis of multi-agent behavior. J. Comput. Sci. Technol. 20(1), 118–124 (2005).  https://doi.org/10.1007/s11390-005-0013-5CrossRefGoogle Scholar
  4. 4.
    Blin, L., Fraignaud, P., Nisse, N., Vial, S.: Distributed chasing of network intruders. Theoret. Comput. Sci. 399, 12–37 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonato, A., Nowakovski, R.: The Game of Cops and Robbers on Graphs. American Mathematical Society, Rhode Island (2011)CrossRefGoogle Scholar
  6. 6.
    Cohen, N., Hilaire, M., Martins, N.A., Nisse, N., Perennes, S.: Spy-game on graphs: complexity and simple topologies. Theoret. Comput. Sci. 725, 1–15 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dawes, R.: Some pursuit-evasion problems on grids. Inf. Process. Lett. 43, 241–247 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dumitrescu, A., Kok, H., Suzuki, I., Żyliński, P.: Vision-based pursuit-evasion in a grid. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 53–64. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-69903-3_7CrossRefGoogle Scholar
  9. 9.
    Ellis, J., Warren, R.: Lower bounds on the pathwidth of some grid-like graphs. Discrete Appl. Math. 156, 545–555 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fomin, F., Golovach, P., Kratochvil, J., Nisse, N., Suchan, K.: Pursuing a fast robber on a graph. Theoret. Comput. Sci. 411, 1167–1181 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Goldstein, F., Reingold, E.: The complexity of pursuing a graph. Theoret. Comput. Sci. 143, 93–112 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kinnersley, W.B.: Cops and Robbers is EXPTIME-complete. J. Comb. Theory, Series B 111, 201–220 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Luccio, F., Pagli, L.: Cops and robber on grids and tori. arXiv:1708.08255v2 (2019). https://arxiv.org/abs/1708.08255
  14. 14.
    Maamoun, M., Meyniel, H.: On a game of policemen and robber. Discrete Appl. Math. 17, 18–44 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mehrabian, A.: The capture time of grids. Discrete Math. 311, 102–105 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Neufeld, S.: A pursuit-evasion problem on a grid. Inf. Proc. Let. 58, 5–9 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Neufeld, S., Nowakovsky, R.J.: A game on cops and robbers played on products of graphs. Discrete Math. 186, 253–268 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nowakowski, R.J., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Math. 43, 253–259 (1983)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Quillot, A.: These di \(3^{\circ }\) cycle. Universit de Paris VI, 131–145 (1978)Google Scholar
  20. 20.
    Sugihara, K., Suzuki, I.: Optimal algorithm for a pursuit-evasion problem. SIAM J. Discrete Math. 2, 126–143 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of PisaPisaItaly

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