Better Decremental and Fully Dynamic Sensitivity Oracles for Subgraph Connectivity

Authors Yaowei Long , Yunfan Wang

Thumbnail PDF


  • Filesize: 0.86 MB
  • 20 pages

Document Identifiers

Author Details

Yaowei Long
  • University of Michigan, Ann Arbor, MI, USA
Yunfan Wang
  • Tsinghua University, Beijing, China


We thank Thatchaphol Saranurak for helpful discussions.

Cite AsGet BibTex

Yaowei Long and Yunfan Wang. Better Decremental and Fully Dynamic Sensitivity Oracles for Subgraph Connectivity. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 109:1-109:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study the sensitivity oracles problem for subgraph connectivity in the decremental and fully dynamic settings. In the fully dynamic setting, we preprocess an n-vertices m-edges undirected graph G with n_{off} deactivated vertices initially and the others are activated. Then we receive a single update D ⊆ V(G) of size |D| = d ≤ d_{⋆}, representing vertices whose states will be switched. Finally, we get a sequence of queries, each of which asks the connectivity of two given vertices u and v in the activated subgraph. The decremental setting is a special case when there is no deactivated vertex initially, and it is also known as the vertex-failure connectivity oracles problem. We present a better deterministic vertex-failure connectivity oracle with Ô(d_{⋆}m) preprocessing time, Õ(m) space, Õ(d²) update time and O(d) query time, which improves the update time of the previous almost-optimal oracle [Long and Saranurak, 2022] from Ô(d²) to Õ(d²). We also present a better deterministic fully dynamic sensitivity oracle for subgraph connectivity with Ô(min{m(n_{off} + d_{⋆}),n^{ω}}) preprocessing time, Õ(min{m(n_{off} + d_{⋆}),n²}) space, Õ(d²) update time and O(d) query time, which significantly improves the update time of the state of the art [Bingbing Hu et al., 2023] from Õ(d⁴) to Õ(d²). Furthermore, our solution is even almost-optimal assuming popular fine-grained complexity conjectures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • connectivity
  • sensitivity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443. IEEE Computer Society, 2014. URL:
  2. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental SSSP and approximate min-cost flow in almost-linear time. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1000-1008. IEEE, 2021. URL:
  3. Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and Thatchaphol Saranurak. A deterministic algorithm for balanced cut with applications to dynamic connectivity, flows, and beyond. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1158-1167. IEEE, 2020. URL:
  4. Ran Duan and Seth Pettie. Connectivity oracles for failure prone graphs. In Proceedings of the forty-second ACM Symposium on Theory of Computing, pages 465-474, 2010. Google Scholar
  5. Ran Duan and Seth Pettie. Connectivity oracles for graphs subject to vertex failures. SIAM Journal on Computing, 49(6):1363-1396, 2020. Google Scholar
  6. Martin Furer and Balaji Raghavachari. Approximating the minimum-degree steiner tree to within one of optimal. Journal of Algorithms, 17(3):409-423, 1994. Google Scholar
  7. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 21-30. ACM, 2015. URL:
  8. Monika Henzinger and Stefan Neumann. Incremental and fully dynamic subgraph connectivity for emergency planning. In 24th Annual European Symposium on Algorithms (ESA 2016). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  9. Bingbing Hu, Evangelos Kosinas, and Adam Polak. Connectivity oracles for predictable vertex failures. CoRR, abs/2312.08489, 2023. URL:
  10. Arkady Kanevsky, Roberto Tamassia, Giuseppe Di Battista, and Jianer Chen. On-line maintenance of the four-connected components of a graph. In [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science, pages 793-801. IEEE Computer Society, 1991. Google Scholar
  11. Rohit Khandekar, Subhash Khot, Lorenzo Orecchia, and Nisheeth K Vishnoi. On a cut-matching game for the sparsest cut problem. Univ. California, Berkeley, CA, USA, Tech. Rep. UCB/EECS-2007-177, 6(7):12, 2007. Google Scholar
  12. Rohit Khandekar, Satish Rao, and Umesh V. Vazirani. Graph partitioning using single commodity flows. J. ACM, 56(4):19:1-19:15, 2009. URL:
  13. Evangelos Kosinas. Connectivity queries under vertex failures: Not optimal, but practical. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 75:1-75:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL:
  14. Yaowei Long and Thatchaphol Saranurak. Near-optimal deterministic vertex-failure connectivity oracles. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1002-1010. IEEE, 2022. Google Scholar
  15. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL:
  16. Merav Parter, Asaf Petruschka, and Seth Pettie. Connectivity labeling and routing with multiple vertex failures. In Proceedings 56th ACM Symposium on Theory of Computing (STOC), 2024. Google Scholar
  17. Michal Pilipczuk, Nicole Schirrmacher, Sebastian Siebertz, Szymon Torunczyk, and Alexandre Vigny. Algorithms and data structures for first-order logic with connectivity under vertex failures. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 102:1-102:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  18. Jan van den Brand and Thatchaphol Saranurak. Sensitive distance and reachability oracles for large batch updates. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 424-435. IEEE Computer Society, 2019. URL: