Sparsifying Congested Cliques and Core-Periphery Networks

  • Alkida Balliu
  • Pierre Fraigniaud
  • Zvi Lotker
  • Dennis Olivetti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9988)


The core-periphery network architecture proposed by Avin et al. [ICALP 2014] was shown to support fast computation for many distributed algorithms, while being much sparser than the congested clique. For being efficient, the core-periphery architecture is however bounded to satisfy three axioms, among which is the capability of the core to emulate the clique, i.e., to implement the all-to-all communication pattern, in O(1) rounds in the CONGEST model. In this paper, we show that implementing all-to-all communication in k rounds can be done in n-node networks with roughly \(n^2/k\) edges, and this bound is tight. Hence, sparsifying the core beyond just saving a fraction of the edges requires to relax the constraint on the time to simulate the congested clique. We show that, for \(p\gg \sqrt{\log n/n}\), a random graph in \(\mathcal{G}_{n,p}\) can, w.h.p., perform the all-to-all communication pattern in \(O(\min \{\frac{1}{p^2},n p\})\) rounds. Finally, we show that if the core can emulate the congested clique in t rounds, then there exists a distributed MST construction algorithm performing in \(O(t\log n)\) rounds. Hence, for \(t=O(1)\), our (deterministic) algorithm improves the best known (randomized) algorithm for constructing MST in core-periphery networks by a factor \(\varTheta (\log n)\).


  1. 1.
    Avin, C., Borokhovich, M., Lotker, Z., Peleg, D.: Distributed computing on core-periphery networks: axiom-based design. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 399–410. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43951-7_34 Google Scholar
  2. 2.
    Broder, A.Z., Frieze, A.M., Upfal, E.: Existence and construction of edge-disjoint paths on expander graphs. SIAM J. Comput. 23(5), 976–989 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Censor-Hillel, K., Kaski, P., Korhonen, J.H., Lenzen, C., Paz, A., Suomela, J.: Algebraic methods in the congested clique. In: ACM Symposium on Principles of Distributed Computing (PODC), pp. 143–152 (2015)Google Scholar
  4. 4.
    Censor-Hillel, K., Toukan, T.: On fast and robust information spreading in the vertex-congest model. In: Scheideler, C. (ed.) Structural Information and Communication Complexity. LNCS, vol. 9439, pp. 270–284. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-25258-2_19 CrossRefGoogle Scholar
  5. 5.
    Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: ACM Symposium on Principles of Distributed Computing (PODC), pp. 367–376 (2014)Google Scholar
  6. 6.
    Elkin, M.: A faster distributed protocol for constructing a minimum spanning tree. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 359–368 (2004)Google Scholar
  7. 7.
    Feige, U., Peleg, D., Raghavan, P., Upfal, E.: Randomized broadcast in networks. Random Struct. Algorithms 1(4), 447–460 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frieze, A.M.: Disjoint paths in expander graphs via random walks: a short survey. In: Luby, M., Rolim, J.D.P., Serna, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 1–14. Springer, Heidelberg (1998). doi: 10.1007/3-540-49543-6_1 CrossRefGoogle Scholar
  9. 9.
    Frieze, A.M.: Edge-disjoint paths in expander graphs. In: 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 717–725 (2000)Google Scholar
  10. 10.
    Gallager, R.G., Humblet, P.A., Spira, P.M.: A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst. 5(1), 66–77 (1983)CrossRefzbMATHGoogle Scholar
  11. 11.
    Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM J. Comput. 27(1), 302–316 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ghaffari, M., Parter, M.: MST in log-star rounds of congested clique. In: 35th ACM Symposium on Principles of Distributed Computing (PODC) (2016)Google Scholar
  13. 13.
    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 ACM Symposium on Principles of Distributed Computing (PODC), pp. 91–100 (2015)Google Scholar
  14. 14.
    Hegeman, J.W., Pemmaraju, S.V.: Lessons from the congested clique applied to MapReduce. Theor. Comput. Sci. 608, 268–281 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hegeman, J.W., Pemmaraju, S.V., Sardeshmukh, V.B.: Near-constant-time distributed algorithms on a congested clique. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 514–530. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45174-8_35 Google Scholar
  16. 16.
    Kutten, S., Peleg, D.: Fast distributed construction of small k-dominating sets and applications. J. Algorithms 28(1), 40–66 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leighton, T., Rao, S., Srinivasan, A.: Multicommodity flow and circuit switching. In: 31st Hawaii International Conference on System Sciences, pp. 459–465 (1998)Google Scholar
  18. 18.
    Christoph Lenzen. Optimal deterministic routing and sorting on the congested clique. In ACM Symposium on Principles of Distributed Computing (PODC), pp. 42–50, (2013)Google Scholar
  19. 19.
    Lotker, Z., Patt-Shamir, B., Pavlov, E., Peleg, D.: Minimum-weight spanning tree construction in O(log log n) communication rounds. SIAM J. Comput. 35(1), 120–131 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for constant diameter graphs. In: 20th ACM Symposium on Principles of Distributed Computing (PODC), pp. 63–71 (2001)Google Scholar
  21. 21.
    Mitzenmacher, M., Upfal, E.: Probability and Computing - Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ookawa, H., Izumi, T.: Filling logarithmic gaps in distributed complexity for global problems. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015. LNCS, vol. 8939, pp. 377–388. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46078-8_31 Google Scholar
  23. 23.
    Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed minimum-weight spanning tree construction. SIAM J. Comput. 30(5), 1427–1442 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sarma, A.D., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G, Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. In: 43rd ACM Symposium on Theory of Computing (STOC), pp. 363–372 (2011)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Alkida Balliu
    • 1
    • 2
  • Pierre Fraigniaud
    • 1
  • Zvi Lotker
    • 3
  • Dennis Olivetti
    • 1
    • 2
  1. 1.CNRS, University Paris DiderotParisFrance
  2. 2.Gran Sasso Science InstituteL’AquilaItaly
  3. 3.Ben Gurion UniversityBeershebaIsrael

Personalised recommendations