Simple Distributed Spanners in Dense Congest Networks

  • Leonid BarenboimEmail author
  • Tzalik Maimon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)


The problem of computing a sparse spanning subgraph is a well-studied problem in the distributed setting, and a lot of research was done in the direction of computing spanners or solving the more relaxed problem of connectivity. Still, efficiently constructing a linear-size spanner deterministically remains a challenging open problem even in specific topologies.

In this paper we provide several simple spanner constructions of linear size, for various graph families. Our first result shows that the connectivity problem can be solved deterministically using a linear size spanner within constant running time on graphs with bounded neighborhood independence. This is a very wide family of graphs that includes unit-disk graphs, unit-ball graphs, line graphs, claw-free graphs and many others. Moreover, our algorithm works in the \(\mathcal {CONGEST}\) model. It also immediately leads to a constant time deterministic solution for the connectivity problem in the Congested-Clique.

Our second result provides a linear size spanner in the \(\mathcal {CONGEST}\) model for graphs with bounded diversity. This is a subtype of graphs with bounded neighborhood independence that captures various types of networks, such as wireless networks and social networks. Here too our result has constant running time and is deterministic. Moreover, the latter result has an additional desired property of a small stretch.


Spanners Distributed computing Diversity 



The authors are grateful to Michael Elkin for fruitful discussions and helpful remarks.


  1. 1.
    Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Near-linear cost sequential and distribured constructions of sparse neighborhood covers. In: FOCS 1993, pp. 638–647 (1993)Google Scholar
  2. 2.
    Barenboim, L., Elkin, M.: Distributed deterministic edge coloring using bounded neighborhood independence. Distrib. Comput. 26(5–6), 273–287 (2013)CrossRefGoogle Scholar
  3. 3.
    Barenboim, L., Elkin, M., Gavoille, C.: A fast network-decomposition algorithm and its applications to constant-time distributed computation. In: Scheideler, C. (ed.) Structural Information and Communication Complexity. LNCS, vol. 9439, pp. 209–223. Springer, Cham (2015). Scholar
  4. 4.
    Barenboim, L., Elkin, M., Maimon, T.: Deterministic distributed \((\varDelta + o(\varDelta ))\)-edge-coloring, and vertex-coloring of graphs with bounded diversity. In: PODC 2017, pp. 175–184 (2017)Google Scholar
  5. 5.
    Barenboim, L., Maimon, T.: Distributed symmetry breaking in graphs with bounded diversity. In: IPDPS 2018, 723–732 (2018)Google Scholar
  6. 6.
    Baswana, S., Sen, S.: A simple linear time algorithm for computing a \((2k - 1)\)-spanner of \({O}({n}^{1+1/{k}})\) size in weighted graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 384–396. Springer, Heidelberg (2003). Scholar
  7. 7.
    Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: PODC 2008, pp. 273–282 (2008)Google Scholar
  8. 8.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. In: FOCS, pp. 452–461 (1996)Google Scholar
  9. 9.
    Derbel, B., Mosbah, M., Zemmari, A.: Sublinear fully distributed partition with applications. Theory Comput. Syst. 47(2), 368–404 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Elkin, M.: Computing almost shortest paths. ACM Algorithms 1(2), 283–323 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Erdos, P.: Extremal problems in graph theory. In: Theory of Graphs and its Applications. Proceedings of Symposium Smolenice, pp. 29–36 (1963)Google Scholar
  12. 12.
    Elkin, M., Peleg, D.: \((1 + \epsilon ,\beta )\)-spanner constructions for general graphs. In: STOC 2001, pp. 173–182 (2001)Google Scholar
  13. 13.
    Grossman, O., Parter, M.: Improved deterministic distributed construction of spanners. In: DISC 2017, 24:1–24:16 (2017)Google Scholar
  14. 14.
    Hegeman, J.W., Pandurangan, G., Pemmaraju, S.V., Sardeshmukh, V.B., Scquizzato, M.: Toward optimal bounds in the congested clique: graph connectivity and MST. In: PODC 2015, pp. 91–100 (2015)Google Scholar
  15. 15.
    Jurdzinski, T., Nowicki, K.: MST in \(O(1)\) Rounds of the Congested Clique (2017).
  16. 16.
    Korhonen, J.H.: Deterministic MST Sparsification in the Congested Clique (2016).
  17. 17.
    Kuhn, F.: Faster Deterministic Distributed Coloring Through Recursive List Coloring (2019).
  18. 18.
    Lenzen, C.: Optimal deterministic routing and sorting on the congested clique. In: PODC 2013, pp. 42–50 (2013)Google Scholar
  19. 19.
    Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST construction in \(O(\log \log n)\) communication rounds. In: SPAA 2003, pp. 94–100 (2003)Google Scholar
  20. 20.
    Peleg, D.: Distributed computing: a locality-sensitive approach. In: SIAM 2000 (2000)Google Scholar
  21. 21.
    Peleg, A., Schaffer, A.: Graph spenners. J. Graph Theory 13(1), 99–116 (1989)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Peleg, D., Solomon, S.: Dynamic \((1+\epsilon )\)-approximate matchings: a density-sensitive approach. In: SODA 2016, pp. 712–729 (2016)Google Scholar
  23. 23.
    Peleg, D., Ullman, J.: An optimal synchronizer for the hypercube. In: PODC, pp. 77–85 (1987)Google Scholar
  24. 24.
    Schneider, J., Wattenhofer, R.: A log-star distributed Maximal Independent Set algorithm for Growth Bounded Graphs. In: Proceedings of the 27th ACM Symposium on Principles of Distributed Computing, pp. 35–44 (2008)Google Scholar
  25. 25.
    Woodruff, D.P.: Lower bounds for additive spanners, emulators, and more. In: FOCS 2006, pp. 389–398 (2006)Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.The Open University of IsraelRaananaIsrael
  2. 2.Ben-Gurion University of The NegevBeer-ShevaIsrael

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