Skip to main content

The State of the Art in Dynamic Graph Algorithms

  • Conference paper
  • First Online:
Book cover SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

Abstract

A dynamic graph algorithm is a data structure that supports operations on dynamically changing graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    There are still some openresearch question regarding the amortized versus the worst-case time per operation, but we will not discuss them here.

  2. 2.

    Note, however, that this does not exclude an algorithm that takes time \(O(m^{1/2})\) for both updates and queries.

References

  1. Abboud, A., Dahlgaard, S.: Popular conjectures as a barrier for dynamic planar graph algorithms. In: FOCS (2016)

    Google Scholar 

  2. Abboud, A., Williams, V.V.: Popular conjectures imply strong lower bounds for dynamic problems. In: FOCS (2014)

    Google Scholar 

  3. Abraham, I., Fiat, A., Goldberg, A.V., Werneck, R.F.: Highway dimension, shortest paths, and provably efficient algorithms. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 782–793. Society for Industrial and Applied Mathematics (2010)

    Google Scholar 

  4. Agarwal, P.K., Eppstein, D., Guibas, L.J., Henzinger, M.R.: Parametric and kinetic minimum spanning trees. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 596–605. IEEE (1998)

    Google Scholar 

  5. Anand, A., Baswana, S., Gupta, M., Sen, S.: Maintaining approximate maximum weighted matching in fully dynamic graphs. In: D’Souza, D., Kavitha, T., Radhakrishnan, J. (eds.) FSTTCS. LIPIcs, vol. 18, pp. 257–266. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)

    Google Scholar 

  6. Basch, J., Guibas, L.J., Hershberger, J.: Data structures for mobile data. J. Algorithms 31(1), 1–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baswana, S., Gupta, M., Sen, S.: Fully dynamic maximal matching in \(\cal{O}(\log n)\) update time. In: FOCS (2011). http://dx.doi.org/10.1137/130914140

  8. Bernstein, A., Karger, D.: A nearly optimal oracle for avoiding failed vertices and edges. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 101–110. ACM (2009)

    Google Scholar 

  9. Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 167–179. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47672-7_14

    Chapter  Google Scholar 

  10. Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 692–711. Society for Industrial and Applied Mathematics (2016)

    Google Scholar 

  11. Bhattacharya, S., Chakrabarty, D., Henzinger, M.: Deterministic fully dynamic approximate vertex cover and fractional matching in O(1) amortized update time. In: Eisenbrand, F., Koenemann, J. (eds.) IPCO 2017. LNCS, vol. 10328, pp. 86–98. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59250-3_8

    Chapter  Google Scholar 

  12. Bhattacharya, S., Henzinger, M., Italiano, G.F.: Deterministic fully dynamic data structures for vertex cover and matching. In: SODA (2015)

    Google Scholar 

  13. Bhattacharya, S., Henzinger, M., Nanongkai, D.: New deterministic approximation algorithms for fully dynamic matching. In: STOC 2016

    Google Scholar 

  14. Bhattacharya, S., Henzinger, M., Nanongkai, D.: Fully dynamic approximate maximum matching and minimum vertex cover in o(log3 n) worst case update time. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 470–489. SIAM (2017)

    Google Scholar 

  15. Chan, T.M.: Dynamic subgraph connectivity with geometric applications. SIAM J. Comput. 36(3), 681–694 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chechik, S., Langberg, M., Peleg, D., Roditty, L.: F-sensitivity distance oracles and routing schemes. Algorithmica 63(4), 861–882 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dahlgaard, S.: On the hardness of partially dynamic graph problems and connections to diameter. In: ICALP, pp. 48:1–48:14 (2016)

    Google Scholar 

  18. Duan, R., Pettie, S.: Connectivity oracles for failure prone graphs. In: Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pp. 465–474. ACM (2010)

    Google Scholar 

  19. Eppstein, D., Galil, Z., Italiano, G.F., Spencer, T.H.: Separator based sparsification for dynamic planar graph algorithms. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 208–217. ACM (1993)

    Google Scholar 

  20. Frigioni, D., Italiano, G.F.: Dynamically switching vertices in planar graphs. Algorithmica 28(1), 76–103 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Frigioni, D., Marchetti-Spaccamela, A., Nanni, U.: Fully dynamic algorithms for maintaining shortest paths trees. J. Algorithms 34(2), 251–281 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gupta, M., Peng, R.: Fully dynamic \((1+\epsilon )\)-approximate matchings. In: FOCS (2013)

    Google Scholar 

  23. Henzinger, M., Krinninger, S., Nanongkai, D.: Decremental single-source shortest paths on undirected graphs in near-linear total update time. In: 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 146–155. IEEE (2014)

    Google Scholar 

  24. Henzinger, M., Krinninger, S., Nanongkai, D., Saranurak, T.: Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In: STOC (2015)

    Google Scholar 

  25. Henzinger, M., Lincoln, A., Neumann, S., Williams, V.V.: Conditional hardness for sensitivity problems. In: ITCS (2017)

    Google Scholar 

  26. Henzinger, M., Neumann, S.: Incremental and fully dynamic subgraph connectivity for emergency planning. In: ESA (2016)

    Google Scholar 

  27. Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM (JACM) 46(4), 502–516 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Henzinger, M.R., Fredman, M.L.: Lower bounds for fully dynamic connectivity problems in graphs. Algorithmica 22(3), 351–362 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Holm, J., De Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM (JACM) 48(4), 723–760 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Italiano, G.F., La Poutré, J.A., Rauch, M.H.: Fully dynamic planarity testing in planar embedded graphs. In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 212–223. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57273-2_57

    Google Scholar 

  31. Klein, P.N., Subramanian, S.: A fully dynamic approximation scheme for shortest paths in planar graphs. Algorithmica 22(3), 235–249 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kopelowitz, T., Pettie, S., Porat, E.: Higher lower bounds from the 3 sum conjecture. In: SODA, pp. 1272–1287 (2016)

    Google Scholar 

  33. Larsen, K.G., Weinstein, O., Yu, H.: Crossing the logarithmic barrier for dynamic boolean data structure lower bounds. arXiv preprint arXiv:1703.03575 (2017)

  34. Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. In: STOC (2013)

    Google Scholar 

  35. Patrascu, M., Demaine, E.D.: Logarithmic lower bounds in the cell-probe model. SIAM J. Comput. 35(4), 932–963 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Patrascu, M., Thorup, M.: Planning for fast connectivity updates. In: 48th Annual IEEE Symposium on Foundations of Computer Science, 2007, FOCS 2007, pp. 263–271. IEEE (2007)

    Google Scholar 

  37. Peleg, D., Solomon, S.: Dynamic (1+\(\epsilon \))-approximate matchings: a density-sensitive approach. In: SODA (2016)

    Google Scholar 

  38. Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: SODA (2007)

    Google Scholar 

  39. Solomon, S.: Fully dynamic maximal matching in constant update time. In: 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pp. 325–334. IEEE (2016)

    Google Scholar 

  40. Subramanian, S.: A fully dynamic data structure for reachability in planar digraphs. In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 372–383. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57273-2_72

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Monika Henzinger .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Henzinger, M. (2018). The State of the Art in Dynamic Graph Algorithms. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73117-9_3

  • Published:

  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics