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Setting Ports in an Anonymous Network: How to Reduce the Level of Symmetry?

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Book cover Structural Information and Communication Complexity (SIROCCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9988))

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Abstract

A fundamental question in the setting of anonymous graphs concerns the ability of nodes to spontaneously break symmetries, based on their local perception of the network. In contrast to previous work, which focuses on symmetry breaking under arbitrary port labelings, in this paper, we study the following design question: Given an anonymous graph G without port labels, how to assign labels to the ports of G, in interval form at each vertex, so that symmetry breaking can be achieved using a message-passing protocol requiring as few rounds of synchronous communication as possible?

More formally, for an integer \(l>0\), the truncated view \(\mathcal {V}_l(v)\) of a node v of a port-labeled graph is defined as a tree encoding labels encountered along all walks in the network which originate from node v and have length at most l, and we ask about an assignment of labels to the ports of G so that the views \({\mathcal {V}_{l}}(v)\) are distinct for all nodes \(v\in V\), with the goal being to minimize l.

We present such efficient port labelings for any graph G, and we exhibit examples of graphs showing that the derived bounds are asymptotically optimal in general. More precisely, our results imply the following statements.

  1. 1.

    For any graph G with n nodes and diameter D, a uniformly random port labeling achieves \(l = O(\min (D,\log n))\), w.h.p.

  2. 2.

    For any graph G with n nodes and diameter D, it is possible to construct in polynomial time a labeling that satisfies \(l = O(\min (D,\log n))\).

  3. 3.

    For any integers \(n\ge 2\) and \(D \le \log _2 n-\log _2\log _2 n\), there exists a graph G with n nodes and diameter D which satisfies \(l \ge \frac{1}{2} D - \frac{5}{2}\).

Research partially supported by the ANR project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of the “Investments for the future” Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02). Research partially supported by the National Science Centre, Poland - grant number 2015/17/B/ST6/01897.

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Notes

  1. 1.

    With high probability means here with probability at least \(1-O(\text {polylog} n/n)\).

References

  1. Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243(1–3), 21–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boldi, P., Vigna, S.: Universal dynamic synchronous self-stabilization. Distrib. Comput. 15(3), 137–153 (2002)

    Article  Google Scholar 

  3. Chalopin, J., Das, S., Kosowski, A.: Constructing a map of an anonymous graph: applications of universal sequences. In: Lu, C., Masuzawa, T., Mosbah, M. (eds.) OPODIS 2010. LNCS, vol. 6490, pp. 119–134. Springer, Heidelberg (2010). doi:10.1007/978-3-642-17653-1_10

    Chapter  Google Scholar 

  4. Chalopin, J., Métivier, Y.: An efficient message passing election algorithm based on Mazurkiewicz’s algorithm. Fundamenta Informaticae 80(1–3), 221–246 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Czyzowicz, J., Dobrev, S., Gasieniec, L., Ilcinkas, D., Jansson, J., Klasing, R., Lignos, I., Martin, R., Sadakane, K., Sung, W.: More efficient periodic traversal in anonymous undirected graphs. Theor. Comput. Sci. 444, 60–76 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: log-space rendezvous in arbitrary graphs. Distrib. Comput. 25(2), 165–178 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Das, S., Flocchini, P., Nayak, A., Santoro, N.: Effective elections for anonymous mobile agents. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 732–743. Springer, Heidelberg (2006). doi:10.1007/11940128_73

    Chapter  Google Scholar 

  8. Das, S., Mihalák, M., Šrámek, R., Vicari, E., Widmayer, P.: Rendezvous of mobile agents when tokens fail anytime. In: Baker, T.P., Bui, A., Tixeuil, S. (eds.) OPODIS 2008. LNCS, vol. 5401, pp. 463–480. Springer, Heidelberg (2008). doi:10.1007/978-3-540-92221-6_29

    Chapter  Google Scholar 

  9. Dereniowski, D., Kosowski, A., Pajak, D.: Distinguishing views in symmetric networks: a tight lower bound. Theor. Comput. Sci. 582, 27–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput. 72(2), 132–143 (2012)

    Article  MATH  Google Scholar 

  11. Dereniowski, D., Pelc, A.: Leader election for anonymous asynchronous agents in arbitrary networks. Distrib. Comput. 27(1), 21–38 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dereniowski, D., Pelc, A.: Topology recognition and leader election in colored networks. Theor. Comput. Sci. 621, 92–102 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dobrev, S., Jansson, J., Sadakane, K., Sung, W.-K.: Finding short right-hand-on-the-wall walks in graphs. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 127–139. Springer, Heidelberg (2005). doi:10.1007/11429647_12

    Chapter  Google Scholar 

  14. Flocchini, P., Roncato, A., Santoro, N.: Computing on anonymous networks with sense of direction. Theor. Comput. Sci. 1–3(301), 355–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fraigniaud, P., Pelc, A.: Decidability classes for mobile agents computing. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 362–374. Springer, Heidelberg (2012). doi:10.1007/978-3-642-29344-3_31

    Chapter  Google Scholar 

  16. Fusco, E.G., Pelc, A.: Knowledge, level of symmetry, and time of leader election. Distrib. Comput. 28(4), 221–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gasieniec, L., Klasing, R., Martin, R.A., Navarra, A., Zhang, X.: Fast periodic graph exploration with constant memory. J. Comput. Syst. Sci. 74(5), 808–822 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gąsieniec, L., Radzik, T.: Memory efficient anonymous graph exploration. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 14–29. Springer, Heidelberg (2008). doi:10.1007/978-3-540-92248-3_2

    Chapter  Google Scholar 

  19. Guilbault, S., Pelc, A.: Asynchronous rendezvous of anonymous agents in arbitrary graphs. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 421–434. Springer, Heidelberg (2011). doi:10.1007/978-3-642-25873-2_29

    Chapter  Google Scholar 

  20. Hendrickx, J.M.: Views in a graph: to which depth must equality be checked? IEEE Trans. Parallel Distrib. Syst. 25(7), 1907–1912 (2014)

    Article  Google Scholar 

  21. Ilcinkas, D.: Setting port numbers for fast graph exploration. Theor. Comput. Sci. 401(1–3), 236–242 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kosowski, A., Navarra, A.: Graph decomposition for memoryless periodic exploration. Algorithmica 63(1–2), 26–38 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Krebs, A., Verbitsky, O.: Universal covers, color refinement, two-variable counting logic: lower bounds for the depth. In: Proceedings of 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2015), pp. 689–700. IEEE (2015)

    Google Scholar 

  24. Norris, N.: Universal covers of graphs: isomorphism to depth \({N}-1\) implies isomorphism to all depths. Discrete Appl. Math. 56(1), 61–74 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Steinová, M.: On the power of local orientations. In: Shvartsman, A.A., Felber, P. (eds.) SIROCCO 2008. LNCS, vol. 5058, pp. 156–169. Springer, Heidelberg (2008). doi:10.1007/978-3-540-69355-0_14

    Chapter  Google Scholar 

  26. Tani, S.: Compression of view on anonymous networks – folded view. IEEE Trans. Parallel Distrib. Syst. 23(2), 255–262 (2012)

    Article  Google Scholar 

  27. Tani, S., Kobayashi, H., Matsumoto, K.: Exact quantum algorithms for the leader election problem. ACM Trans. Comput. Theor. 4(1), 1 (2012)

    Article  MATH  Google Scholar 

  28. Yamashita, M., Kameda, T.: Computing functions on asynchronous anonymous networks. Math. Syst. Theor. 29(4), 331–356 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yamashita, M., Kameda, T.: Computing on anonymous networks: part I-characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7(1), 69–89 (1996)

    Article  Google Scholar 

  30. Yamashita, M., Kameda, T.: Leader election problem on networks in which processor identity numbers are not distinct. IEEE Trans. Parallel Distrib. Syst. 10(9), 878–887 (1999)

    Article  Google Scholar 

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Acknowlegdements

The authors would like to thank Philippe Duchon and David Ilcinkas for proposing the problem, and for some initial discussions.

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Authors and Affiliations

  1. LaBRI, CNRS — Université de Bordeaux, 33400, Talence, France

    Ralf Klasing

  2. Inria and IRIF, CNRS — Université Paris Diderot, 75013, Paris, France

    Adrian Kosowski

  3. Faculty of Fundamental Problems of Technology, Institute of Informatics, Wroclaw University of Technology, 50-370, Wroclaw, Poland

    Dominik Pajak

Authors
  1. Ralf Klasing
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  2. Adrian Kosowski
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  3. Dominik Pajak
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Corresponding author

Correspondence to Dominik Pajak .

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Editors and Affiliations

  1. Aalto University, Espoo, Finland

    Jukka Suomela

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Klasing, R., Kosowski, A., Pajak, D. (2016). Setting Ports in an Anonymous Network: How to Reduce the Level of Symmetry? . In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-48314-6_3

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