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Faster Dynamic 2-Edge Connectivity in Directed Graphs

Authors Loukas Georgiadis , Konstantinos Giannis , Giuseppe F. Italiano

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Connectivity
  • dynamic algorithms
  • directed graphs

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Abstract

Let G be a directed graph with n vertices and m edges. We present a deterministic algorithm that maintains the 2-edge-connected components of G under a sequence of m edge insertions, with a total running time of O(n² log n). This significantly improves upon the previous best bound of O(mn) for graphs that are not very sparse. After each insertion, our algorithm supports the following queries with asymptotically optimal efficiency:  
- Test in constant time whether two query vertices v and w are 2-edge-connected in G. 
- Report in O(n) time all the 2-edge-connected components of G. 
Our approach builds on the recent framework of Georgiadis, Italiano, and Kosinas [FOCS 2024] for computing the 3-edge-connected components of a directed graph in linear time, which leverages the minset-poset technique of Gabow [TALG 2016].
Additionally, we provide a deterministic decremental algorithm for maintaining 2-edge-connectivity in strongly connected directed graphs. Given a sequence of m edge deletions, our algorithm maintains the 2-edge-connected components in total time n^(2+o(1)), while supporting the same queries as the incremental algorithm. This result assumes that the edges of a fixed spanning tree of G and of its reverse graph G^R are not deleted. Previously, the best known bound for the decremental problem was O(mn log n), obtained by a randomized algorithm without restrictions on the deletions.
In contrast to prior dynamic algorithms for 2-edge-connectivity in directed graphs, our method avoids the incremental computation of dominator trees, thereby circumventing the known conditional lower bound of Ω(mn).

Cite As Get BibTex

Loukas Georgiadis, Konstantinos Giannis, and Giuseppe F. Italiano. Faster Dynamic 2-Edge Connectivity in Directed Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.26

Author Details

Loukas Georgiadis
  • University of Ioannina, Greece
Konstantinos Giannis
  • Gran Sasso Science Institute, L’Aquila, Italy
Giuseppe F. Italiano
  • LUISS University, Rome, Italy

Acknowledgements

We thank the anonymous reviewers for their helpful comments.

References

  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In FOCS, pages 434-443, 2014. URL: https://doi.org/10.1109/FOCS.2014.53.
  2. Stephen Alstrup, Dov Harel, Peter W. Lauridsen, and Mikkel Thorup. Dominators in linear time. SIAM Journal on Computing, 28(6):2117-32, 1999. URL: https://doi.org/10.1137/S0097539797317263.
  3. Michael A. Bender, Jeremy T. Fineman, Seth Gilbert, and Robert E. Tarjan. A new approach to incremental cycle detection and related problems. ACM Trans. Algorithms, 12(2), December 2015. URL: https://doi.org/10.1145/2756553.
  4. Aaron Bernstein, Maximilian Probst, and Christian Wulff-Nilsen. Decremental strongly-connected components and single-source reachability in near-linear time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 365-376, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316335.
  5. Adam L. Buchsbaum, Loukas Georgiadis, Haim Kaplan, Anne Rogers, Robert E. Tarjan, and Jeffery R. Westbrook. Linear-time algorithms for dominators and other path-evaluation problems. SIAM Journal on Computing, 38(4):1533-1573, 2008. URL: https://doi.org/10.1137/070693217.
  6. Li Chen, Rasmus Kyng, Yang P. Liu, Simon Meierhans, and Maximilian Probst Gutenberg. Almost-linear time algorithms for incremental graphs: Cycle detection, sccs, s-t shortest path, and minimum-cost flow. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1165-1173, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649745.
  7. Harold N. Gabow. Path-based depth-first search for strong and biconnected components. Information Processing Letters, 74:107-114, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00051-X.
  8. Harold N. Gabow. The minset-poset approach to representations of graph connectivity. ACM Transactions on Algorithms, 12(2):24:1-24:73, February 2016. URL: https://doi.org/10.1145/2764909.
  9. Harold N. Gabow and Robert E. Tarjan. A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences, 30(2):209-21, 1985. URL: https://doi.org/10.1016/0022-0000(85)90014-5.
  10. Zvi Galil and Giuseppe F. Italiano. Reducing edge connectivity to vertex connectivity. SIGACT News, 22(1):57-61, March 1991. URL: https://doi.org/10.1145/122413.122416.
  11. Loukas Georgiadis, Thomas Dueholm Hansen, Giuseppe F. Italiano, Sebastian Krinninger, and Nikos Parotsidis. Decremental data structures for connectivity and dominators in directed graphs. In ICALP, pages 42:1-42:15, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.42.
  12. Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-edge-connected components of a graph in linear time. In Proc. 29th European Symposium on Algorithms, pages 47:1-47:17, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.47.
  13. Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 3-edge-connected components of directed graphs in linear time. In 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, pages 62-85, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00015.
  14. Loukas Georgiadis, Giuseppe F. Italiano, Luigi Laura, and Nikos Parotsidis. 2-edge connectivity in directed graphs. ACM Transactions on Algorithms, 13(1):9:1-9:24, 2016. URL: https://doi.org/10.1145/2968448.
  15. Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. Incremental 2-edge-connectivity in directed graphs. In ICALP, pages 49:1-49:15, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.49.
  16. Loukas Georgiadis, Giuseppe F. Italiano, and Nikos Parotsidis. Strong connectivity in directed graphs under failures, with applications. SIAM J. Comput., 49(5):865-926, 2020. URL: https://doi.org/10.1137/19M1258530.
  17. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In STOC, pages 21-30, 2015. URL: https://doi.org/10.1145/2746539.2746609.
  18. Monika Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edge connectivity. SIAM Journal on Computing, 49(1):1-36, 2020. URL: https://doi.org/10.1137/18M1180335.
  19. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, July 2001. URL: https://doi.org/10.1145/502090.502095.
  20. John E. Hopcroft and Robert E. Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  21. Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, and Seth Pettie. Fully dynamic connectivity in O(logn(loglogn)²) amortized expected time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 510-520, USA, 2017. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.5555/3039686.3039718.
  22. Wenyu Jin and Xiaorui Sun. Fully dynamic s-t edge connectivity in subpolynomial time (extended abstract). In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 861-872, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00088.
  23. Ken-Ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time. Journal of the ACM, 66(1), December 2018. URL: https://doi.org/10.1145/3274663.
  24. Tuukka Korhonen. Linear-time algorithms for k-edge-connected components, k-lean tree decompositions, and more. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC '25, pages 111-119, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3717823.3718123.
  25. Evangelos Kosinas. Computing the 5-edge-connected components in linear time. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1887-2119, 2024. URL: https://doi.org/10.1137/1.9781611977912.76.
  26. Jakub Łącki. Improved deterministic algorithms for decremental reachability and strongly connected components. ACM Transactions on Algorithms, 9(3):27, 2013. URL: https://doi.org/10.1145/2483699.2483707.
  27. Karl Menger. Zur allgemeinen kurventheorie. Fundamenta Mathematicae, 10(1):96-115, 1927. Google Scholar
  28. Wojciech Nadara, Mateusz Radecki, Marcin Smulewicz, and Marek Sokołowski. Determining 4-edge-connected components in linear time. In Proc. 29th European Symposium on Algorithms, pages 71:1-71:15, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.71.
  29. Hiroshi Nagamochi and Toshihide Ibaraki. A linear time algorithm for computing 3-edge-connected components in a multigraph. Japan J. Indust. Appl. Math, 9(163), 1992. URL: https://doi.org/10.1007/BF03167564.
  30. Hiroshi Nagamochi and Toshihide Ibaraki. Algorithmic Aspects of Graph Connectivity. Cambridge University Press, 2008. 1st edition. URL: https://doi.org/10.1017/CBO9780511721649.
  31. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time . In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 950-961, Los Alamitos, CA, USA, October 2017. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2017.92.
  32. Daniel D. Sleator and Robert E. Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26:362-391, 1983. URL: https://doi.org/10.1016/0022-0000(83)90006-5.
  33. Robert E. Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
  34. Robert E. Tarjan. Finding dominators in directed graphs. SIAM Journal on Computing, 3(1):62-89, 1974. URL: https://doi.org/10.1137/0203006.
  35. Robert E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215-225, 1975. URL: https://doi.org/10.1145/321879.321884.
  36. Robert E. Tarjan and Jan van Leeuwen. Worst-case analysis of set union algorithms. Journal of the ACM, 31(2):245-81, 1984. URL: https://doi.org/10.1145/62.2160.
  37. Mikkel Thorup. Fully-dynamic min-cut. Combinatorica, 27(1):91-127, February 2007. URL: https://doi.org/10.1007/s00493-007-0045-2.
  38. Yung H. Tsin. Yet another optimal algorithm for 3-edge-connectivity. Journal of Discrete Algorithms, 7(1):130-146, 2009. Selected papers from the 1st International Workshop on Similarity Search and Applications (SISAP). URL: https://doi.org/10.1016/j.jda.2008.04.003.
  39. Jan van den Brand, Li Chen, Rasmus Kyng, Yang P. Liu, Simon Meierhans, Maximilian Probst Gutenberg, and Sushant Sachdeva. Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality . In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 2010-2032, Los Alamitos, CA, USA, October 2024. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS61266.2024.00120.
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