Document Open Access Logo

Dynamic Detours

Authors Daniel Dadush , Michał Pilipczuk , Amadeus Reinald , Marek Sokołowski , Michał Włodarczyk

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.35.pdf
  • Filesize: 1.02 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Data structures design and analysis
Keywords
  • Dynamic algorithms
  • Fixed-parameter tractability

Metrics

Abstract

Fix a parameter k ∈ ℕ. We give dynamic data structures that for a fully dynamic undirected graph G, updated over time by edge insertions and edge deletions, can answer the following queries:  
- Long (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of length at least k? 
- Long (u,v)-detour: Given u,v ∈ V(G), is there a path from u to v of length at least dist_G(u,v)+k? 
- Even/odd (u,v)-path: Given u,v ∈ V(G), is there a path from u to v of even/odd length?  The amortized time of executing an update or answering a query is 2^𝒪(k³) log n + 𝒪(log² n log² log n) in the first two cases, and 𝒪(log² n log² log n) in the last, where n is the number of vertices of G.
The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most k. Specifically, there is no data structure supporting queries about (u,v)-paths of length at most two in time n^o(1) unless the Triangle Conjecture fails.
Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.

Cite As Get BibTex

Daniel Dadush, Michał Pilipczuk, Amadeus Reinald, Marek Sokołowski, and Michał Włodarczyk. Dynamic Detours. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.35

Author Details

Daniel Dadush
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Amadeus Reinald
  • Institute of Informatics, University of Warsaw, Poland
Marek Sokołowski
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Michał Włodarczyk
  • Institute of Informatics, University of Warsaw, Poland

Funding

The work of MiP on this manuscript is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057). AR was supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143. MW was supported by Polish National Science Centre SONATA-19 grant number 2023/51/D/ST6/00155.

Acknowledgements

This research has been initiated/conducted at the AlgUW workshop (Bedlewo 09.2025), supported by the Excellence Initiative - Research University (IDUB) funds of the University of Warsaw.

References

  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, pages 434-443. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.53.
  2. Shyan Akmal, Virginia Vassilevska Williams, Ryan Williams, and Zixuan Xu. Faster Detours in Undirected Graphs. In 31st Annual European Symposium on Algorithms, ESA 2023, volume 274 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:17, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.7.
  3. Josh Alman, Matthias Mnich, and Virginia Vassilevska Williams. Dynamic parameterized problems and algorithms. ACM Transactions on Algorithms, 16(4):45:1-45:46, 2020. URL: https://doi.org/10.1145/3395037.
  4. Shinwoo An, Kyungjin Cho, Leo Jang, Byeonghyeon Jung, Yudam Lee, Eunjin Oh, Donghun Shin, Hyeonjun Shin, and Chanho Song. Dynamic parameterized problems on unit disk graphs. In 35th International Symposium on Algorithms and Computation, ISAAC 2024, LIPIcs, pages 6:1-6:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2024.6.
  5. Max Bannach, Zacharias Heinrich, Rüdiger Reischuk, and Till Tantau. Dynamic kernels for hitting sets and set packing. Algorithmica, 84(11):3459-3488, 2022. URL: https://doi.org/10.1007/S00453-022-00986-0.
  6. Eli Berger, Paul D. Seymour, and Sophie Spirkl. Finding an induced path that is not a shortest path. Discrete Mathematics, 344(7):112398, 2021. URL: https://doi.org/10.1016/J.DISC.2021.112398.
  7. Christian Bertram, Deborah Haun, Mads Vestergaard Jensen, and Tuukka Korhonen. Dynamic meta-kernelization. CoRR, abs/2511.03461, 2025. Accepted to STOC 2026. URL: https://doi.org/10.48550/arXiv.2511.03461.
  8. Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V Fomin. Finding detours is fixed-parameter tractable. SIAM Journal on Discrete Mathematics, 33(4):2326-2345, 2019. URL: https://doi.org/10.1137/17M1148566.
  9. Etienne Birmele. Tree-width and circumference of graphs. Journal of Graph Theory, 43(1):24-25, 2003. URL: https://doi.org/10.1002/jgt.10099.
  10. Hans L. Bodlaender and Ton Kloks. Better algorithms for the pathwidth and treewidth of graphs. In 18th International Colloquium on Automata, Languages and Programming, ICALP 1991, Lecture Notes in Computer Science, pages 544-555. Springer, 1991. URL: https://doi.org/10.1007/3-540-54233-7_162.
  11. Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 796-809. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.50.
  12. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  13. Daniel Dadush, Michał Pilipczuk, Amadeus Reinald, Marek Sokołowski, and Michał Włodarczyk. Dynamic detours, 2026. URL: https://doi.org/10.48550/arXiv.2605.03225.
  14. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. 1.5-approximation for treewidth of graphs excluding a graph with one crossing as a minor. In 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX 2002, Lecture Notes in Computer Science, pages 67-80. Springer, 2002. URL: https://doi.org/10.1007/3-540-45753-4_8.
  15. Zdenek Dvořák, Martin Kupec, and Vojtech Tůma. A dynamic data structure for MSO properties in graphs with bounded tree-depth. In 22nd Annual European Symposium on Algorithms, ESA 2014, volume 8737 of Lecture Notes in Computer Science, pages 334-345. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_28.
  16. Zdenek Dvořák and Vojtech Tůma. A dynamic data structure for counting subgraphs in sparse graphs. In 13th International Symposium on Algorithms and Data Structures, WADS 2013, volume 8037 of Lecture Notes in Computer Science, pages 304-315. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40104-6_27.
  17. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification. I. Planarity testing and minimum spanning trees. Journal of Computer and System Sciences, 52(1):3-27, 1996. URL: https://doi.org/10.1006/jcss.1996.0002.
  18. Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov, Saket Saurabh, and Kirill Simonov. Detours in directed graphs. Journal of Computer and System Sciences, 137:66-86, 2023. URL: https://doi.org/10.1016/j.jcss.2023.05.001.
  19. Alejandro Grez, Filip Mazowiecki, Michał Pilipczuk, Gabriele Puppis, and Cristian Riveros. Dynamic data structures for timed automata acceptance. Algorithmica, 84(11):3223-3245, 2022. URL: https://doi.org/10.1007/s00453-022-01025-8.
  20. Maximilian Probst Gutenberg, Virginia Vassilevska Williams, and Nicole Wein. New algorithms and hardness for incremental single-source shortest paths in directed graphs. In 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 153-166. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384236.
  21. Frank Harary. Graph theory. Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. Google Scholar
  22. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM, 48(4):723-760, 2001. URL: https://doi.org/10.1145/502090.502095.
  23. Jacob Holm, Wojciech Nadara, Eva Rotenberg, and Marek Sokołowski. Fully dynamic biconnectivity in Õ(log² n) time. In 57th Annual ACM Symposium on Theory of Computing, STOC 2025, pages 156-165. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718270.
  24. Jacob Holm, Wojciech Nadara, Eva Rotenberg, and Marek Sokołowski. Fully dynamic biconnectivity in Õ(log² n) time. CoRR, abs/2503.21733, 2025. Full version of [Jacob Holm et al., 2025]. URL: https://doi.org/10.48550/arXiv.2503.21733.
  25. Jacob Holm and Eva Rotenberg. Fully-dynamic planarity testing in polylogarithmic time. In 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 167-180, New York, NY, USA, 2020. ACM. URL: https://doi.org/10.1145/3357713.3384249.
  26. Yoichi Iwata and Keigo Oka. Fast dynamic graph algorithms for parameterized problems. In 14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014, volume 8503 of Lecture Notes in Computer Science, pages 241-252. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-08404-6_21.
  27. Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi. Long directed detours: Reduction to 2-disjoint paths. Information Processing Letters, 186:106491, 2024. URL: https://doi.org/10.1016/J.IPL.2024.106491.
  28. Manas Jyoti Kashyop, N. S. Narayanaswamy, Meghana Nasre, and Sai Mohith Potluri. Trade-offs in dynamic coloring for bipartite and general graphs. Algorithmica, 85(4):854-878, 2023. URL: https://doi.org/10.1007/S00453-022-01050-7.
  29. Tuukka Korhonen. Dynamic treewidth in logarithmic time. In 66th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2025, pages 787-796. IEEE, 2025. URL: https://doi.org/10.1109/FOCS63196.2025.00042.
  30. Tuukka Korhonen, Konrad Majewski, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Dynamic treewidth. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, pages 1734-1744, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00105.
  31. Tuukka Korhonen, Konrad Majewski, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Dynamic treewidth. CoRR, abs/2304.01744, 2023. Full version of [Korhonen et al., 2023]. URL: https://doi.org/10.48550/arXiv.2304.01744.
  32. Tuukka Korhonen, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Fully dynamic approximation schemes on planar and apex-minor-free graphs. In David P. Woodruff, editor, 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, pages 296-313. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.12.
  33. Tuukka Korhonen and Marek Sokołowski. Almost-linear time parameterized algorithm for rankwidth via dynamic rankwidth. In 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1538-1549. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649732.
  34. Andrea S. LaPaugh and Christos H. Papadimitriou. The even-path problem for graphs and digraphs. Networks, 14(4):507-513, 1984. URL: https://doi.org/10.1002/NET.3230140403.
  35. Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski. Maintaining CMSO₂ properties on dynamic structures with bounded feedback vertex number. In 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 of LIPIcs, pages 46:1-46:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.46.
  36. Konrad Majewski, Michał Pilipczuk, and Anna Zych-Pawlewicz. Parameterized dynamic data structure for split completion. In 32nd Annual European Symposium on Algorithms, ESA 2024, LIPIcs, pages 87:1-87:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ESA.2024.87.
  37. Jędrzęj Olkowski, Michał Pilipczuk, Mateusz Rychlicki, Karol Węgrzycki, and Anna Zych-Pawlewicz. Dynamic data structures for parameterized string problems. In 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 of LIPIcs, pages 50:1-50:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.50.
  38. Mihai Pătraşcu and Erik D. Demaine. Lower bounds for dynamic connectivity. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 546-553. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007435.
  39. Monika Rauch Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. Journal of the ACM, 46(4):502-516, 1999. URL: https://doi.org/10.1145/320211.320215.
  40. Neil Robertson and Paul D. Seymour. Graph Minors. III. Planar tree-width. J. Comb. Theory, Ser. B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
  41. Robert Endre Tarjan and Uzi Vishkin. An efficient parallel biconnectivity algorithm. SIAM Journal on Computing, 14(4):862-874, 1985. URL: https://doi.org/10.1137/0214061.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact