Small Cuts and Connectivity Certificates: A Fault Tolerant Approach

Author Merav Parter

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Merav Parter
  • Weizmann Institute, Rehovot, Israel


I am thankful to DISC '19 reviewers for valuable input, and to Michal Dory for preliminary discussions on these problems. I am also grateful to Oded Goldriech for sparking my curiosity on the derandomization of the fault tolerant sampling approach. Finally, special thanks to Moni Naor and Karthik C. S. for useful discussions on universal sets.

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Merav Parter. Small Cuts and Connectivity Certificates: A Fault Tolerant Approach. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds). Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS '11], Ghaffari and Kuhn [DISC '13]. - Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG '11]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC '19] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design. - Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest. - Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds. Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. '97] and Dori [PODC '18] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC '18] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Distributed algorithms
  • Connectivity
  • Minimum Cut
  • Spanners


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  1. Noga Alon. Explicit construction of exponential sized families of k-independent sets. Discrete Mathematics, 58(2):191-193, 1986. Google Scholar
  2. Noga Alon, Shiri Chechik, and Sarel Cohen. Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece., pages 12:1-12:14, 2019. Google Scholar
  3. Keren Censor-Hillel and Michal Dory. Fast Distributed Approximation for TAP and 2-Edge-Connectivity. In 21st International Conference on Principles of Distributed Systems, OPODIS 2017, Lisbon, Portugal, December 18-20, 2017, pages 21:1-21:20, 2017. Google Scholar
  4. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. Fault tolerant spanners for general graphs. SIAM Journal on Computing, 39(7):3403-3423, 2010. Google Scholar
  5. Mohit Daga, Monika Henzinger, Danupon Nanongkai, and Saranurak. Distributed Edge Connectivity in Sublinear Time. In STOC, 2019. Google Scholar
  6. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, pages 169-178. ACM, 2011. Google Scholar
  7. Michal Dory. Distributed Approximation of Minimum k-edge-connected Spanning Subgraphs. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, Egham, United Kingdom, July 23-27, 2018, pages 149-158, 2018. Google Scholar
  8. J Friedman. Constructing O (n log n) size monotone formulae for the k-th elementary symmetric polynomial of n Boolean variables. In Foundations of Computer Science, 1984. 25th Annual Symposium on, pages 506-515. IEEE, 1984. Google Scholar
  9. Mohsen Ghaffari. Improved Distributed Algorithms for Fundamental Graph Problems. PhD thesis, MIT, USA, 2017. URL:
  10. Mohsen Ghaffari and Fabian Kuhn. Distributed minimum cut approximation. In International Symposium on Distributed Computing, pages 1-15. Springer, 2013. Google Scholar
  11. Mohsen Ghaffari and Fabian Kuhn. Distributed Minimum Cut Approximation. arXiv preprint, 2013. URL:
  12. Mohsen Ghaffari and Fabian Kuhn. Derandomizing Distributed Algorithms with Small Messages: Spanners and Dominating Set. In 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, pages 29:1-29:17, 2018. Google Scholar
  13. David R Karger. Random sampling in cut, flow, and network design problems. Mathematics of Operations Research, 24(2):383-413, 1999. Google Scholar
  14. Christos Levcopoulos, Giri Narasimhan, and Michiel Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 186-195. ACM, 1998. Google Scholar
  15. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparsek-connected spanning subgraph of ak-connected graph. Algorithmica, 7(1-6):583-596, 1992. Google Scholar
  16. Danupon Nanongkai and Hsin-Hao Su. Almost-tight distributed minimum cut algorithms. In International Symposium on Distributed Computing, pages 439-453. Springer, 2014. Google Scholar
  17. Merav Parter and Eylon Yogev. Low Congestion Cycle Covers and Their Applications. SODA, 2019. Google Scholar
  18. Merav Parter and Eylon Yogev. Optimal Short Cycle Decomposition in Almost Linear Time. ICALP, 2019. Google Scholar
  19. David Peleg. Distributed Computing: A Locality-sensitive Approach. SIAM, 2000. Google Scholar
  20. Seth Pettie. Distributed algorithms for ultrasparse spanners and linear size skeletons. Distributed Computing, 22(3):147-166, 2010. URL:
  21. David Pritchard and Ramakrishna Thurimella. Fast computation of small cuts via cycle space sampling. ACM Transactions on Algorithms (TALG), 7(4):46, 2011. Google Scholar
  22. Václav Rozhon and Mohsen Ghaffari. Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization. arXiv preprint, 2019. URL:
  23. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed verification and hardness of distributed approximation. SIAM Journal on Computing, 41(5):1235-1265, 2012. Google Scholar
  24. Ramakrishna Thurimella. Sub-linear distributed algorithms for sparse certificates and biconnected components. Journal of Algorithms, 23(1):160-179, 1997. Google Scholar
  25. Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, 2012. URL:
  26. Xiao Fan Wang and Guanrong Chen. Complex networks: small-world, scale-free and beyond. IEEE circuits and systems magazine, 3(1):6-20, 2003. Google Scholar
  27. Oren Weimann and Raphael Yuster. Replacement paths and distance sensitivity oracles via fast matrix multiplication. ACM Transactions on Algorithms (TALG), 9(2):14, 2013. Google Scholar
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