, Volume 63, Issue 1–2, pp 158–190 | Cite as

A Distributed Algorithm for Computing the Node Search Number in Trees

  • David Coudert
  • Florian Huc
  • Dorian MazauricEmail author


We present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis et al. (Inf. Comput. 113(1):50–79, 1994). It can be executed in an asynchronous environment, requires an overall computation time of O(nlog n), and n messages of log 3 n+4 bits each.

The main contribution of this work lies in the data structure proposed to design our algorithm, called hierarchical decomposition. This simple and flexible data structure is used for four operations: updating the node search number after addition or deletion of any tree-edges in a distributed fashion; computing it in a tree whose edges are added sequentially and in any order; computing other graph invariants such as the process number and the edge search number, by changing only initialization rules; extending our algorithms for trees and forests of unknown size (using messages of up to 2log 3 n+5 bits).


Distributed algorithm Edge search number Graph searching Node search number Pathwidth Process number 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barrière, L., Flocchini, P., Fraigniaud, P., Santoro, N.: Capture of an intruder by mobile agents. In: 14th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), San Diego, California, USA (part of FCRC 2003), June 2002, pp. 200–209. ACM, New York (2002) Google Scholar
  2. 2.
    Barrière, L., Fraigniaud, P., Santoro, N., Thilikos, D.M.: Searching is not jumping. In: Bodlaender, H.L. (ed.) 29th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), Elspeet, The Netherlands, June 2003. Lecture Notes in Computer Science, vol. 2880, pp. 34–45. Springer, Berlin (2003) CrossRefGoogle Scholar
  3. 3.
    Breisch, R.L.: An intuitive approach to speleotopology. Southwest. Cavers VI(5), 72–78 (1967) Google Scholar
  4. 4.
    Coudert, D., Huc, F., Mazauric, D., Nisse, N., Sereni, J.-S.: Reconfiguration of the routing in WDM networks with two classes of services. In: 13th Conference on Optical Network Design and Modeling (ONDM), Braunschweig, Germany, February 2009, pp. 1–6. IEEE, Los Alamitos (2009) Google Scholar
  5. 5.
    Coudert, D., Perennes, S., Pham, Q.-C., Sereni, J.-S.: Rerouting requests in WDM networks. In: 7èmes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications (AlgoTel’05), Presqu’île de Giens, France, May 2005, pp. 17–20 (2005) Google Scholar
  6. 6.
    Coudert, D., Sereni, J.-S.: Characterization of graphs and digraphs with small process number. Discrete Appl. Math. 159, 1094–1109 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Díaz, J., Petit, J., Serna, M.: A survey on graph layout problems. ACM Comput. Surv. 34(3), 313–356 (2002) CrossRefGoogle Scholar
  8. 8.
    Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Inf. Comput. 113(1), 50–79 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fomin, F.V., Fraigniaud, P., Nisse, N.: Nondeterministic graph searching: from pathwidth to treewidth. Algorithmica 53(3), 358–373 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399(3), 236–245 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Golovach, P.A.: Search number, node search number, and vertex separator of a graph. Vestn. Leningr. Univ., Math. 24(1), 88–90 (1991) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kinnersley, N.G.: The vertex separation number of a graph equals its pathwidth. Inf. Process. Lett. 42(6), 345–350 (1992). doi: 10.1016/0020-0190(92)90234-M MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kirousis, M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47(2), 205–218 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    LaPaugh, A.S.: Recontamination does not help to search a graph. J. Assoc. Comput. Mach. 40(2), 224–245 (1993). doi: 10.1145/151261.151263 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Megiddo, N., Hakimi, S.L., Garey, M.R., Johnson, D.S., Papadimitriou, C.H.: The complexity of searching a graph. J. Assoc. Comput. Mach. 35(1), 18–44 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mihai, R., Todinca, I.: Pathwidth is NP-hard for weighted trees. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) 3rd International Workshop on Frontiers in Algorithmics (FAW), Hefei, China, June 2009, Lecture Notes in Computer Science, vol. 5598, pp. 181–195. Springer, Berlin (2009) CrossRefGoogle Scholar
  17. 17.
    Parsons, T.D.: Pursuit-evasion in a graph. In: Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 426–441. Springer, Berlin (1978) CrossRefGoogle Scholar
  18. 18.
    Peng, S.-L., Hob, C.-W., Hsu, T.-S., Ko, M.-T., Tanga, C.Y.: Edge and node searching problems on trees. Theor. Comput. Sci. 240(2), 429–446 (2000) zbMATHCrossRefGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory, Ser. B 35(1), 39–61 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Scheffler, P.: A linear algorithm for the pathwidth of trees. In: R. Bodendiek, R.H. (ed.) Topics in Combinatorics and Graph Theory, pp. 613–620. Physica-Verlag, Heidelberg (1990) Google Scholar
  21. 21.
    Skodinis, K.: Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time. J. Algorithms 47(1), 40–59 (2003) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.MASCOTTE, INRIA, I3S, CNRSUniv. Nice SophiaSophia Antipolis CedexFrance
  2. 2.EPFLLaboratory of Distributed ProgrammingLausanneSwitzerland

Personalised recommendations