Advertisement

A Topological Perspective on Distributed Network Algorithms

  • Armando Castañeda
  • Pierre FraigniaudEmail author
  • Ami Paz
  • Sergio Rajsbaum
  • Matthieu Roy
  • Corentin Travers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11639)

Abstract

More than two decades ago, combinatorial topology was shown to be useful for analyzing distributed fault-tolerant algorithms in shared memory systems and in message passing systems. In this work, we show that combinatorial topology can also be useful for analyzing distributed algorithms in networks of arbitrary structure. To illustrate this, we analyze consensus, set-agreement, and approximate agreement in networks, and derive lower bounds for these problems under classical computational settings, such as the LOCAL model and dynamic networks.

Keywords

Distributed computing Distributed graph algorithms Combinatorial topology 

Notes

Acknowledgments

Pierre Fraigniaud and Corentin Travers are supported by ANR projects DESCARTES and FREDA; Pierre Fraigniaud receives additional support from INRIA project GANG; Ami Paz is supported by the Fondation Sciences Mathématiques de Paris (FSMP); Sergio Rajsbaum is supported by project unam-papiit IN109917.

References

  1. 1.
    Alistarh, D., Aspnes, J., Ellen, F., Gelashvili, R., Zhu, L.: Why extension-based proofs fail. CoRR abs/1811.01421 http://arxiv.org/abs/1811.01421 (2018). To appear in STOC 2019
  2. 2.
    Attiya, H., Castañeda, A., Herlihy, M., Paz, A.: Bounds on the step and namespace complexity of renaming. SIAM J. Comput. 48(1), 1–32 (2019).  https://doi.org/10.1137/16M1081439MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balliu, A., Brandt, S., Hirvonen, J., Olivetti, D., Rabie, M., Suomela, J.: Lower bounds for maximal matchings and maximal independent sets. CoRR abs/1901.02441 http://arxiv.org/abs/1901.02441 (2019)
  4. 4.
    Barenboim, L., Elkin, M., Goldenberg, U.: Locally-iterative distributed (\(\delta + 1\))-coloring below szegedy-vishwanathan barrier, and applications to self-stabilization and to restricted-bandwidth models. In: Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, (PODC), pp. 437–446 (2018). https://dl.acm.org/citation.cfm?id=3212769
  5. 5.
    Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetry breaking. In: 53rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 321–330 (2012).  https://doi.org/10.1109/FOCS.2012.60
  6. 6.
    Bhadra, S., Ferreira, A.: Computing multicast trees in dynamic networks and the complexity of connected components in evolving graphs. J. Internet Services Appl. 3(3), 269–275 (2012).  https://doi.org/10.1007/s13174-012-0073-zCrossRefGoogle Scholar
  7. 7.
    Biely, M., Robinson, P., Schmid, U., Schwarz, M., Winkler, K.: Gracefully degrading consensus and k-set agreement in directed dynamic networks. Theor. Comput. Sci. 726, 41–77 (2018).  https://doi.org/10.1016/j.tcs.2018.02.019MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brandt, S., et al.: A lower bound for the distributed Lovász local lemma. In: 48th ACM Symposium on Theory of Computing (STOC), pp. 479–488 (2016).  https://doi.org/10.1145/2897518.2897570
  9. 9.
    Castañeda, A., Rajsbaum, S.: New combinatorial topology bounds for renaming: the lower bound. Distrib. Comput. 22(5–6), 287–301 (2010).  https://doi.org/10.1007/s00446-010-0108-2CrossRefzbMATHGoogle Scholar
  10. 10.
    Castañeda, A., Rajsbaum, S.: New combinatorial topology bounds for renaming: the upper bound. J. ACM 59(1), 3:1–3:49 (2012).  https://doi.org/10.1145/2108242.2108245MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casteigts, A., Flocchini, P., Godard, E., Santoro, N., Yamashita, M.: On the expressivity of time-varying graphs. Theor. Comput. Sci. 590, 27–37 (2015).  https://doi.org/10.1016/j.tcs.2015.04.004MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. Parallel Emergent Distrib. Syst. 27(5), 387–408 (2012).  https://doi.org/10.1080/17445760.2012.668546CrossRefGoogle Scholar
  13. 13.
    Chang, Y., Li, W., Pettie, S.: An optimal distributed\(({\varDelta }+1)\)-coloring algorithm? In: 50th ACM Symposium on Theory of Computing (STOC), pp. 445–456 (2018).  https://doi.org/10.1145/3188745.3188964
  14. 14.
    Charron-Bost, B., Függer, M., Nowak, T.: Approximate consensus in highly dynamic networks: the role of averaging algorithms. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 528–539. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47666-6_42CrossRefGoogle Scholar
  15. 15.
    Charron-Bost, B., Schiper, A.: The heard-of model: computing in distributed systems with benign faults. Distrib. Comput. 22(1), 49–71 (2009).  https://doi.org/10.1007/s00446-009-0084-6CrossRefzbMATHGoogle Scholar
  16. 16.
    Chaudhuri, S., Herlihy, M., Lynch, N.A., Tuttle, M.R.: Tight bounds for k-set agreement. J. ACM 47(5), 912–943 (2000).  https://doi.org/10.1145/355483.355489MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Coulouma, E., Godard, E., Peters, J.G.: A characterization of oblivious message adversaries for which consensus is solvable. Theor. Comput. Sci. 584, 80–90 (2015).  https://doi.org/10.1016/j.tcs.2015.01.024MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fischer, M., Ghaffari, M., Kuhn, F.: Deterministic distributed edge-coloring via hypergraph maximal matching. In: 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pp. 180–191 (2017).  https://doi.org/10.1109/FOCS.2017.25
  19. 19.
    Fischer, M.J., Lynch, N.A., Paterson, M.: Impossibility of distributed consensus with one faulty process. J. ACM 32(2), 374–382 (1985).  https://doi.org/10.1145/3149.214121MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ghaffari, M.: An improved distributed algorithm for maximal independent set. In: 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 270–277 (2016).  https://doi.org/10.1137/1.9781611974331.ch20
  21. 21.
    Ghaffari, M., Kuhn, F., Maus, Y.: On the complexity of local distributed graph problems. In: 49th ACM Symposium on Theory of Computing (STOC), pp. 784–797 (2017).  https://doi.org/10.1145/3055399.3055471
  22. 22.
    Godard, E., Perdereau, E.: k-set agreement in communication networks with omission faults. In: 20th International Conference on Principles of Distributed Systems (OPODIS), pp. 8:1–8:17 (2016).  https://doi.org/10.4230/LIPIcs.OPODIS.2016.8
  23. 23.
    Göös, M., Hirvonen, J., Suomela, J.: Linear-in-\(\varDelta \) lowerbounds in the LOCAL model. Distrib. Comput. 30(5), 325–338 (2017).  https://doi.org/10.1007/s00446-015-0245-8MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Harris, D.G., Schneider, J., Su, H.: Distributed \(({\Delta }+1)\)-coloring in sublogarithmic rounds. In: 48th ACM Symposium on Theory of Computing (STOC), pp. 465–478 (2016).  https://doi.org/10.1145/2897518.2897533
  25. 25.
    Herlihy, M., Kozlov, D., Rajsbaum, S.: Distributed Computing Through Combinatorial Topology. Morgan Kaufmann, San Francisco (2013)zbMATHGoogle Scholar
  26. 26.
    Herlihy, M., Rajsbaum, S.: Set consensus using arbitrary objects. In: Proceedings of the Thirteenth Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 324–333 (1994).  https://doi.org/10.1145/197917.198119
  27. 27.
    Herlihy, M., Rajsbaum, S.: Algebraic spans. Math. Struct. Comput. Sci. 10(4), 549–573 (2000). http://journals.cambridge.org/action/displayAbstract?aid=54601MathSciNetCrossRefGoogle Scholar
  28. 28.
    Herlihy, M., Rajsbaum, S., Tuttle, M.R.: An axiomatic approach to computing the connectivity of synchronous and asynchronous systems. Electr. Notes Theor. Comput. Sci. 230, 79–102 (2009).  https://doi.org/10.1016/j.entcs.2009.02.018CrossRefzbMATHGoogle Scholar
  29. 29.
    Herlihy, M., Shavit, N.: The asynchronous computability theorem for t-resilient tasks. In: 25th ACM Symposium on Theory of Computing (STOC), pp. 111–120 (1993).  https://doi.org/10.1145/167088.167125
  30. 30.
    Herlihy, M., Shavit, N.: The topological structure of asynchronous computability. J. ACM 46(6), 858–923 (1999).  https://doi.org/10.1145/331524.331529MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kuhn, F., Lynch, N.A., Oshman, R.: Distributed computation in dynamic networks. In: 42nd ACM Symposium on Theory of Computing (STOC), pp. 513–522 (2010).  https://doi.org/10.1145/1806689.1806760
  32. 32.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local computation: lower and upper bounds. J. ACM 63(2), 17:1–17:44 (2016).  https://doi.org/10.1145/2742012MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kuhn, F., Moses, Y., Oshman, R.: Coordinated consensus in dynamic networks. In: 30th ACM Symposium on Principles of Distributed Computing (PODC), pp. 1–10 (2011).  https://doi.org/10.1145/1993806.1993808
  34. 34.
    Kuhn, F., Oshman, R.: Dynamic networks: models and algorithms. SIGACT News 42(1), 82–96 (2011).  https://doi.org/10.1145/1959045.1959064CrossRefGoogle Scholar
  35. 35.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992).  https://doi.org/10.1137/0221015MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mendes, H., Tasson, C., Herlihy, M.: Distributed computability in Byzantine asynchronous systems. In: 46th Symposium on Theory of Computing (STOC), pp. 704–713 (2014).  https://doi.org/10.1145/2591796.2591853
  37. 37.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  38. 38.
    Rajsbaum, S., Castañeda, A., Flores-Peñaloza, D., Alcantara, M.: Fault-tolerant robot gathering problems on graphs with arbitrary appearing times. In: 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 493–502 (2017).  https://doi.org/10.1109/IPDPS.2017.70
  39. 39.
    Rajsbaum, S., Raynal, M., Travers, C.: The iterated restricted immediate snapshot model. In: 14th International Conference on Computing and Combinatorics (COCOON), pp. 487–497 (2008).  https://doi.org/10.1007/978-3-540-69733-6_48
  40. 40.
    Sakavalas, D., Tseng, L.: Network topology and fault-tolerant consensus. Synth. Lect. Distrib. Comput. Theory 9, 1–151 (2019)CrossRefGoogle Scholar
  41. 41.
    Saks, M.E., Zaharoglou, F.: Wait-free k-set agreement is impossible: the topology of public knowledge. In: 25th ACM Symposium on Theory of Computing (STOC), pp. 101–110 (1993).  https://doi.org/10.1145/167088.167122
  42. 42.
    Suomela, J.: Survey of local algorithms. ACM Comput. Surv. 45(2), 24:1–24:40 (2013).  https://doi.org/10.1145/2431211.2431223CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Armando Castañeda
    • 1
  • Pierre Fraigniaud
    • 2
    Email author
  • Ami Paz
    • 2
  • Sergio Rajsbaum
    • 1
  • Matthieu Roy
    • 3
  • Corentin Travers
    • 4
  1. 1.UNAMMexico CityMexico
  2. 2.CNRS and Université de ParisParisFrance
  3. 3.CNRSToulouseFrance
  4. 4.CNRS and University of BordeauxBordeauxFrance

Personalised recommendations