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Tree Decontamination with Temporary Immunity

  • Paola Flocchini
  • Bernard Mans
  • Nicola Santoro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

Consider a tree network that has been contaminated by a persistent and active virus: when infected, a network site will continuously attempt to spread the virus to all its neighbours. The decontamination problem is that of disinfecting the entire network using a team of mobile antiviral system agents, called cleaners, avoiding any recontamination of decontaminated areas. A cleaner is able to decontaminate any infected node it visits; once the cleaner departs, the decontaminated node is immune for t ≥ 0 time units to viral attacks from infected neighbours. After the immunity time t is elapsed, re-contamination can occur. The primary research objective is to determine the minimum team size, as well as the solution strategy, that is the protocol that would enable such a minimal team of cleaners to perform the task. The network decontamination problem has been extensively investigated in the literature, and a very large number of studies exist on the subject. However, all the existing work is limited to the special case t = 0. In this paper we examine the tree decontamination problem for any value t ≥ 0. We determine the minimum team size necessary to disinfect any given tree with immunity time t. Further we show how to compute for all nodes of the tree the minimum team size and implicitly the solution strategy starting from each starting node; these computations use a total of Θ(n) time (serially) or Θ(n) messages (distributively). We then provide a complete structural characterization of the class of trees that can be decontaminated with k agents and immunity time t; we do so by identifying the forbidden subgraphs and analyzing their properties. Finally, we consider generic decontamination algorithms, i.e. protocols that work unchanged in a large class of trees, with little knowledge of their topological structure. We prove that, for each immunity time t ≥ 0, all trees of height at most h can be decontaminated by a team of \(k=\lfloor {{2 h} \over t+2 }\rfloor\) agents whose only knowledge of the tree is the bound h. The proof is constructive.

Keywords

Network decontamination tree networks mobile agents antiviral agents distributed algorithm 

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References

  1. 1.
    Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: Proc. 14th Symp. Parallel Algorithms and Architectures (SPAA 2002), pp. 200–209 (2002)Google Scholar
  2. 2.
    Barrière, L., Fraignaud, P., Santoro, N., Thilikos, D.: Searching is not jumping. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 34–45. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Bienstock, D.: Graph searching, path-width, tree-width and related problems. DIMACS Series in Disc. Maths. and Theo. Comp. Sci. 5, 33–49 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bienstock, D., Seymour, P.: Monotonicity in graph searching. J. Algorithms 12, 239–245 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blin, L., Fraignaud, P., Nisse, N., Vial, S.: Distributed chasing of network intruders. Theoretical Computer Science 399(1-2), 12–37 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Breisch, R.: An intuitive approach to speleotopology. S.W. Cavers  VI(5), 72–78 (1967)Google Scholar
  7. 7.
    Ellis, J., Sudborough, H., Turner, J.: The vertex separation and search number of a graph. Information and Computation 113(1), 50–79 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Flocchini, P., Huang, M.J., Luccio, F.L.: Decontamination of hypercubes by mobile agents. Networks (to appear, 2008)Google Scholar
  9. 9.
    Flocchini, P., Huang, M.J., Luccio, F.L.: Decontamination of chordal rings and tori using mobile agents. Int. J. of Foundation of Computer Science 18(3), 547–564 (2007)CrossRefMATHGoogle Scholar
  10. 10.
    Flocchini, P., Luccio, F.L., Song, L.X.: Size optimal strategies for capturing an intruder in mesh networks. In: Proc. Int. Conf. on Comm. in Computing (CIC 2005), pp. 200–206 (2005)Google Scholar
  11. 11.
    Flocchini, P., Nayak, A., Shulz, A.: Cleaning an arbitrary regular network with mobile agents. In: Chakraborty, G. (ed.) ICDCIT 2005. LNCS, vol. 3816, pp. 132–142. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Flocchini, P., Santoro, N.: Distributed Security Algorithms For Mobile Agents. In: Cao, J., Das, S. (eds.) Mobile Agents in Networking and Distributed Computing. Wiley, Chichester (2008)Google Scholar
  13. 13.
    Fomin, F., Golovach, P.: Graph searching and interval completion. SIAM J. on Discrete Mathematics 13(4), 454–464 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fomin, F., Thilikos, D., Todineau, I.: Connected graph searching in outerplanar graphs. In: Proc. 7th Int. Conf. on Graph Theory (ICGT 2005) (2005)Google Scholar
  15. 15.
    Fraigniaud, P., Nisse, N.: Connected treewidth and connected graph searching. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 479–490. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Fraigniaud, P., Nisse, N.: Monotony properties of connected visible graph searching. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 229–240. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Kirousis, L., Papadimitriou, C.: Interval graphs and searching. Discrete Mathematics 55, 181–184 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kirousis, L., Papadimitriou, C.: Searching and pebbling. Theoretical Computer Science 47(2), 205–218 (1986)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lapaugh, A.: Recontamination does not help to search a graph. J. of the ACM 40(2), 224–245 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Luccio, F.L.: Intruder capture in Sierpiński graphs. In: Proc. 4th Int. Conf. on Fun with Algorithms, pp. 249–261 (2007)Google Scholar
  21. 21.
    Luccio, F., Pagli, L., Santoro, N.: Network decontamination with local immunization. Int. J. of Foundation of Computer Science 18(3), 457–474 (2007)CrossRefMATHGoogle Scholar
  22. 22.
    Megiddo, N., Hakimi, S., Garey, M., Johnson, D., Papadimitriou, C.: The complexity of searching a graph. J. of the ACM 35(1), 18–44 (1988)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Parson, T.: Pursuit-evasion in a graph. In: Theory and Applications of Graphs. Lecture Notes in Mathematics, pp. 426–441. Springer, Heidelberg (1976)Google Scholar
  24. 24.
    Santoro, N.: Design and Analysis of Distributed Algorithms. Wiley, Chichester (2007)MATHGoogle Scholar
  25. 25.
    Takahashi, A., Ueno, S., Kajitani, Y.: Mixed searching and proper-path-width. Theoretical Computer Science 137(2), 253–268 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yamamoto, M., Takahashi, K., Hagiya, M., Nishizaki, S.-Y.: Formalization of graph search algorithms and its applications. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 479–496. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Paola Flocchini
    • 1
  • Bernard Mans
    • 2
  • Nicola Santoro
    • 3
  1. 1.University of OttawaOttawaCanada
  2. 2.Macquarie UniversitySydneyAustralia
  3. 3.Carleton UniversityOttawaCanada

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