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Distributedly Testing Cycle-Freeness

  • Heger Arfaoui
  • Pierre FraigniaudEmail author
  • David Ilcinkas
  • Fabien Mathieu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

We tackle local distributed testing of graph properties. This framework is well suited to contexts in which data dispersed among the nodes of a network can be collected by some central authority (like in, e.g., sensor networks). In local distributed testing, each node can provide the central authority with just a few information about what it perceives from its neighboring environment, and, based on the collected information, the central authority is aiming at deciding whether or not the network satisfies some property. We analyze in depth the prominent example of checking cycle-freeness, and establish tight bounds on the amount of information to be transferred by each node to the central authority for deciding cycle-freeness. In particular, we show that distributedly testing cycle-freeness requires at least \(\lceil \log d \rceil -1\) bits of information per node in graphs with maximum degree \(d\), even for connected graphs. Our proof is based on a novel version of the seminal result by Naor and Stockmeyer (1995) enabling to reduce the study of certain kinds of algorithms to order-invariant algorithms, and on an appropriate use of the known fact that every free group can be linearly ordered.

References

  1. 1.
    Alon, N.: Subdivided graphs have linear ramsey numbers. J. Graph Theor. 18(4), 343–347 (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Shapira, A.: A characterization of easily testable induced subgraphs. Comb. Probab. Comput. 15(6), 791–805 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arfaoui, H., Fraigniaud, P., Pelc, A.: Local decision and verification with bounded-size outputs. In: Higashino, T., Katayama, Y., Masuzawa, T., Potop-Butucaru, M., Yamashita, M. (eds.) SSS 2013. LNCS, vol. 8255, pp. 133–147. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Becker, F., Kosowski, A., Nisse, N., Rapaport, I., Suchan, K.: Allowing each node to communicate only once in a distributed system: shared whiteboard models. In: Proceediings 24th ACM Sympsium on Parallelism in Algorithms and Architectures (SPAA), pp. 11–17 (2012)Google Scholar
  5. 5.
    Becker, F., Matamala, M., Nisse, N., Rapaport, I., Suchan, K., Todinca, I.: Adding a referee to an interconnection network: what can(not) be computed in one round. In: Proceedings 25th IEEE International Sympusim on Parallel and Distributed Processing (IPDPS), pp. 508–514 (2011)Google Scholar
  6. 6.
    Corneil, D., Lerchs, H., Burlingham, L.: Complement reducible graphs. Discrete Appl. Math. 3(3), 163–174 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(13), 77–144 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Czumaj, A., Goldreich, O., Ron, D., Seshadhri, C., Shapira, A., Sohler, C.: Finding cycles and trees in sublinear time. Electron. Colloq. Comput. Complex. (ECCC) 19, 35 (2012)Google Scholar
  9. 9.
    Dolev, S., Gouda, M.G., Schneider, M.: Memory requirements for silent stabilization. Acta Inf. 36(6), 447–462 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fraigniaud, P., Göös, M., Korman, A., Suomela, J.: What can be decided locally without identifiers? In: Proceedings of the 32nd ACM Symposium on Principles of Distributed Computing (PODC), pp. 157–165 (2013)Google Scholar
  11. 11.
    Fraigniaud, P., Korman, A., Parter, M., Peleg, D.: Randomized distributed decision. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 371–385. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Fraigniaud, P., Korman, A., Peleg, D.: Local distributed decision. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 708–717 (2011)Google Scholar
  13. 13.
    Fraigniaud, P., Korman, A., Peleg, D.: Towards a complexity theory for local distributed computing. J. ACM 60(5), 35 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goldreich, O. (ed.): Property Testing. LNCS, vol. 6390. Springer, Heidelberg (2010)zbMATHGoogle Scholar
  15. 15.
    Goldreich, O., Ron, D.: Property testing in bounded degree graphs. Algorithmica 32(2), 302–343 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Göös, M., Hirvonen, J., Suomela, J.: Lower bounds for local approximation. J. ACM 60(5), 39 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Göös, M., Suomela, J.: Locally checkable proofs. In: Proceedings of the 30th ACM Sympusim on Principles of Distributed Computing (PODC), pp. 159–168 (2011)Google Scholar
  18. 18.
    Itkis, G., Levin, L.A.: Fast and lean self-stabilizing asynchronous protocols. In: 35th IEEE Sympsium on Foundations of Computer Science (FOCS), pp. 226–239 (1994)Google Scholar
  19. 19.
    Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22(4), 215–233 (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Heger Arfaoui
    • 1
  • Pierre Fraigniaud
    • 1
    Email author
  • David Ilcinkas
    • 2
  • Fabien Mathieu
    • 3
  1. 1.CNRS, University Paris DiderotParisFrance
  2. 2.CNRS, University of BordeauxTalenceFrance
  3. 3.Alcatel-Lucent Bell LabsMarcoussisFrance

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