On the Complexity of Fixed-Schedule Neighbourhood Learning in Wireless Ad Hoc Radio Networks

  • Avery MillerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8243)


Consider a synchronous static radio network of \(n\) nodes represented by an undirected graph with maximum degree \(\varDelta \). Suppose that each node has a unique ID from \(\{1,\ldots ,N\}\), where \(N \gg n\). In the complete neighbourhood learning task, each node \(p\) must produce a set \(L_p\) of IDs such that ID \(i \in L_p\) if and only if \(p\) has a neighbour with ID \(i\). We study the complexity of this task when it is assumed that each node fixes its entire transmission schedule at the start of the algorithm. We prove a \(\varOmega (\frac{\varDelta ^2}{\log \varDelta }\log {N})\)-slot lower bound on schedule length that holds in very general models, e.g., when nodes possess collision detectors, messages can be of arbitrary size, and nodes know the schedules being followed by all other nodes. We also prove a similar result for the SINR model of radio networks. To prove these results, we introduce a new generalization of cover-free families of sets, which may be of independent interest. We also show a separation between the class of fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out some transmissions from their schedule.


  1. 1.
    Chaudhuri, S., Radhakrishnan, J.: Deterministic restrictions in circuit complexity. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC ’96), pp. 30–36 (1996)Google Scholar
  2. 2.
    Chlebus, B., Gasieniec, L., Gibbons, A., Pelc, A., Rytter, W.: Deterministic broadcasting in unknown radio networks. In: SODA, pp. 861–870 (2000)Google Scholar
  3. 3.
    Clementi, A., Monti, A., Silvestri, R.: Selective families, superimposed codes, and broadcasting on unknown radio networks. In: SODA, pp. 709–718 (2001)Google Scholar
  4. 4.
    De Bonis, A., Gasieniec, L., Vaccaro, U.: Generalized framework for selectors with applications in optimal group testing. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    De Bonis, A., Vaccaro, U.: Constructions of generalized superimposed codes with applications to group testing and conflict resolution in multiple access channels. Theor. Comput. Sci. 306(1–3), 223–243 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Du, D., Hwang, F.: Combinatorial Group Testing and Its Applications. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  8. 8.
    D’yachkov, A.G., Vilenkin, P., Torney, D., Macula, A.: Families of finite sets in which no intersection of sets is covered by the union of s others. J. Comb. Theor. Ser. A 99(2), 195–218 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dyachkov, A.G., Rykov, V.V.: Bounds on the length of disjunctive codes. Probl. Peredachi Informatsii 18(3), 7–13 (1982)MathSciNetGoogle Scholar
  10. 10.
    Dyachkov, A.G., Rykov, V.V., Rashad, A.M.: Superimposed distance codes. Probl. Control. Inf. Theor. 18, 237–250 (1989)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Engel, K.: Sperner Theory. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Erdös, P., Frankl, P., Füredi, Z.: Families of finite sets in which no set is covered by the union of \(r\) others. Israel J. Math. 51, 79–89 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Füredi, Z.: On \(r\)-cover-free families. J. Comb. Theor. Ser. A 73(1), 172–173 (1996)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gasieniec, L., Pagourtzis, A., Potapov, I., Radzik, T.: Deterministic communication in radio networks with large labels. Algorithmica 47(1), 97–117 (2007)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Trans. Inf. Theor. 46(2), 388–404 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Jurdzinski, T., Kowalski, D.R.: Distributed backbone structure for algorithms in the SINR model of wireless networks. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 106–120. Springer, Heidelberg (2012)Google Scholar
  17. 17.
    Kautz, W., Singleton, R.: Nonrandom binary superimposed codes. IEEE Trans. Inf. Theor. 10(4), 363–377 (1964)CrossRefzbMATHGoogle Scholar
  18. 18.
    Komlós, J., Greenberg, A.: An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels. IEEE Trans. Inf. Theor. 31(2), 302–306 (1985)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krishnamurthy, S., Thoppian, M., Kuppa, S., Chandrasekaran, R., Mittal, N., Venkatesan, S., Prakash, R.: Time-efficient distributed layer-2 auto-configuration for cognitive radio networks. Comput. Netw. 52(4), 831–849 (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mittal, N., Krishnamurthy, S., Chandrasekaran, R., Venkatesan, S., Zeng, Y.: On neighbor discovery in cognitive radio networks. J. Parallel Distrib. Comput. 69, 623–637 (2009)CrossRefGoogle Scholar
  21. 21.
    Porat, E., Rothschild, A.: Explicit non-adaptive combinatorial group testing schemes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 748–759. Springer, Heidelberg (2008)Google Scholar
  22. 22.
    Ruszinkó, M.: On the upper bound of the size of the r-cover-free families. J. Comb. Theor. Ser. A 66, 302–310 (1994)CrossRefzbMATHGoogle Scholar
  23. 23.
    Sperner, E.: Ein satz über untermengen einer endlichen menge. Math. Z. 27(1), 544–548 (1928)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Stinson, D., Wei, R.: Generalized cover-free families. Discrete Math. 279(1–3), 463–477 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Yu, D., Wang, Y., Hua, Q., Lau, F.: Distributed local broadcasting algorithms in the physical interference model. In: DCOSS, pp. 1–8 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations