A Distributed Algorithm to Find Hamiltonian Cycles in \(\mathcal{G}(n, p)\) Random Graphs

  • Eythan Levy
  • Guy Louchard
  • Jordi Petit
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3405)


In this paper, we present a distributed algorithm to find Hamiltonian cycles in \(\mathcal{G}(n, p)\) graphs. The algorithm works in a synchronous distributed setting. It finds a Hamiltonian cycle in \(\mathcal{G}(n, p)\) with high probability when \(p=\omega(\sqrt{log n}/n^{1/4})\), and terminates in linear worst-case number of pulses, and in expected O(n 3/4 + ε ) pulses. The algorithm requires, in each node of the network, only O(n) space and O(n) internal instructions.


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  1. 1.
    Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for hamiltonian circuits and matchings. Journal of Computer and System Sciences 18, 155–193 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Awerbuch, B.: Optimal distributed algorithms for minimum-weight spanning tree, counting, leader election and related problems. In: Proc. 19th Symp. on Theory of Computing, pp. 230–240 (1987)Google Scholar
  3. 3.
    Bollobás, B.: Random graphs, 2nd edn. Academic Press, London (2001)MATHGoogle Scholar
  4. 4.
    Bollobás, B., Fenner, T.I., Frieze, A.M.: An algorithm for finding Hamilton paths and cycles in random graphs. Combinatorica 7(4), 327–341 (1987)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Díaz, J., Petit, J., Serna, M.: Faulty random geometric networks. Parallel Processing Letters 10(4), 343–357 (2001)CrossRefGoogle Scholar
  6. 6.
    Díaz, J., Petit, J., Serna, M.: A random graph model for optical smart dust networks. IEEE Transactions on Mobile Computing 2(3), 186–196 (2003)CrossRefGoogle Scholar
  7. 7.
    Faloutsos, M., Molle, M.: Optimal distributed algorithm for minimum spanning trees revisited. In: Symposium on Principles of Distributed Computing, pp. 231–237 (1995)Google Scholar
  8. 8.
    Flajolet, P., Sedgewick, R.: The average case analysis of algorithms: Saddle point asymptotics. Technical Report RR-2376, INRIA (1994)Google Scholar
  9. 9.
    Frieze, A.: Parallel algorithms for finding hamilton cycles in random graphs. Information Processing Letters 25, 111–117 (1987)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gurevich, Y., Shelah, S.: Expected computation time for hamiltonian path problem. SIAM Journal on Computing 16(3), 486–502 (1987)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hitczenko, P., Louchard, G.: Distinctness of compositions of an integer: A probabilistic analysis. Random Structures & Algorithms 19(3-4), 407–437 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Janson, S., Łuczak, T., Rucinski, A.: Random graphs. Wiley, New York (2000)MATHGoogle Scholar
  13. 13.
    Kleinberg, J.M., Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.S.: The Web as a graph: Measurements, models and methods. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, p. 1. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Kumar, S.R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Extracting large-scale knowledge bases from the web. VLDB Journal, 639–650 (1999)Google Scholar
  15. 15.
    Levy, E.: Distributed algorithms for finding hamilton cycles in faulty random geometric graphs. Mémoire de licence (master’s thesis), Université Libre de Bruxelles (2002), http://www.ulb.ac.be/di/scsi/elevy/
  16. 16.
    Levy, E.: Analyse et conception d’un algorithme de cycle hamiltonien pour graphes aléatoires du type g(n,p). Mémoire de DEA, Ecole Polytechnique, Paris (2003), http://www.ulb.ac.be/di/scsi/elevy/
  17. 17.
    MacKenzie, P.D., Stout, Q.F.: Optimal parallel construction of hamiltonian cycles and spanning trees in random graphs. In: ACM Symposium on Parallel Algorithms and Architectures, pp. 224–229 (1993)Google Scholar
  18. 18.
    Nikoletseas, S., Spirakis, P.: Efficient communication establishment in adverse communication environments. In: Rolim, J. (ed.) ICALP Workshops. Proceedings in Informatics, vol. 8, pp. 215–226. Carleton Scientific (2000)Google Scholar
  19. 19.
    Tel, G.: Introduction to Distributed Algorithms, 2nd edn. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  20. 20.
    Thomason, A.G.: A simple linear expected time algorithm for finding a hamilton path. Discrete Mathematics 75, 373–379 (1989)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eythan Levy
    • 1
  • Guy Louchard
    • 1
  • Jordi Petit
    • 2
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelona, Catalonia

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