Document Open Access Logo

Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity

Authors Jacob Holm , Eva Rotenberg



PDF
Thumbnail PDF

File

LIPIcs.STACS.2021.42.pdf
  • Filesize: 0.82 MB
  • 18 pages

Document Identifiers

Author Details

Jacob Holm
  • University of Copenhagen, Denmark
Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark

Acknowledgements

We are thankful to Adam Karczmarz and Jakub Łącki for their encouragement and interest in this work.

Cite AsGet BibTex

Jacob Holm and Eva Rotenberg. Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 42:1-42:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.42

Abstract

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic graphs
  • 2-connectivity
  • graph minors
  • r-divisions
  • graph separators

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition, 1974. Google Scholar
  2. Lyudmil Aleksandrov and Hristo Djidjev. Linear algorithms for partitioning embedded graphs of boundedgenus. Siam Journal on Discrete Mathematics - SIAMDM, 9:129-150, February 1996. URL: https://doi.org/10.1137/S0895480194272183.
  3. Lars Arge, Freek van Walderveen, and Norbert Zeh. Multiway simple cycle separators and i/o-efficient algorithms for planar graphs. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '13, pages 901-918, Philadelphia, PA, USA, 2013. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2627817.2627882.
  4. David Eppstein. Dynamic generators of topologically embedded graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '03, pages 599-608, Philadelphia, PA, USA, 2003. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=644108.644208.
  5. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. Journal of the ACM, 44(5):669-696, September 1997. URL: https://doi.org/10.1145/265910.265914.
  6. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator-based sparsification ii: Edge and vertex connectivity. SIAM Journal on Computing, 28(1):341-381, February 1999. URL: https://doi.org/10.1137/S0097539794269072.
  7. David Eppstein, Giuseppe F. Italiano, Roberto Tamassia, Robert E. Tarjan, Jeffery R. Westbrook, and Moti Yung. Maintenance of a minimum spanning forest in a dynamic planar graph. Journal of Algorithms, 13(1):33-54, March 1992. Special issue for 1st SODA. Google Scholar
  8. Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM Journal on Computing, 14(4):781-798, 1985. URL: https://doi.org/10.1137/0214055.
  9. Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, December 1987. URL: https://doi.org/10.1137/0216064.
  10. Greg N. Frederickson. Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees. SIAM Journal on Computing, 26(2):484-538, 1997. URL: https://doi.org/10.1137/S0097539792226825.
  11. Michael L. Fredman and Michael E. Saks. The cell probe complexity of dynamic data structures. In Proceedings of the Twenty-first Annual ACM Symposium on Theory of Computing, STOC '89, pages 345-354, New York, NY, USA, 1989. ACM. URL: https://doi.org/10.1145/73007.73040.
  12. Zvi Galil, Giuseppe F. Italiano, and Neil Sarnak. Fully dynamic planarity testing with applications. Journal of the ACM, 46(1):28-91, January 1999. URL: https://doi.org/10.1145/300515.300517.
  13. Dora Giammarresi and Giuseppe F. Italiano. Decremental 2- and 3-connectivity on planar graphs. Algorithmica, 16(3):263-287, 1996. URL: https://doi.org/10.1007/BF01955676.
  14. Michael T. Goodrich. Planar separators and parallel polygon triangulation. Journal of Computer and System Sciences, 51(3):374-389, 1995. URL: https://doi.org/10.1006/jcss.1995.1076.
  15. Jens Gustedt. Efficient union-find for planar graphs and other sparse graph classes. Theoretical Computer Science, 203(1):123-141, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00291-0.
  16. Frank Harary. Graph Theory. Addison-Wesley Series in Mathematics. Addison Wesley, 1969. Google Scholar
  17. Monika R Henzinger, Philip Klein, Satish Rao, and Sairam Subramanian. Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences, 55(1):3-23, 1997. URL: https://doi.org/10.1006/jcss.1997.1493.
  18. Monika R. Henzinger and Han La Poutré. Certificates and fast algorithms for biconnectivity in fully-dynamic graphs. In Paul Spirakis, editor, Algorithms - ESA '95, pages 171-184, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. Google Scholar
  19. Monika Rauch Henzinger and Valerie King. Fully dynamic 2-edge connectivity algorithm in polylogarithmic time per operation, 1997. Google Scholar
  20. 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. Announced at STOC '95. URL: https://doi.org/10.1145/320211.320215.
  21. Monika Rauch Henzinger and Mikkel Thorup. Sampling to provide or to bound: With applications to fully dynamic graph algorithms. Random Struct. Algorithms, 11(4):369-379, 1997. URL: https://doi.org/10.1002/(SICI)1098-2418(199712)11:4<369::AID-RSA5>3.0.CO;2-X.
  22. John Hershberger, Monika Rauch, and Subhash Suri. Data structures for two-edge connectivity in planar graphs. Theoretical Computer Science, 130(1):139-161, 1994. URL: https://doi.org/10.1016/0304-3975(94)90156-2.
  23. 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, July 2001. URL: https://doi.org/10.1145/502090.502095.
  24. Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, and Eva Rotenberg. Decremental SPQR-trees for Planar Graphs. In Yossi Azar, Hannah Bast, and Grzegorz Herman, editors, 26th Annual European Symposium on Algorithms (ESA 2018), volume 112 of Leibniz International Proceedings in Informatics (LIPIcs), pages 46:1-46:16, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.46.
  25. Jacob Holm, Giuseppe F Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, and Piotr Sankowski. Contracting a planar graph efficiently. In LIPIcs-Leibniz International Proceedings in Informatics, volume 87. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  26. Jacob Holm and Eva Rotenberg. Good r-divisions imply optimal amortised decremental biconnectivity. CoRR, abs/1808.02568, 2018. URL: http://arxiv.org/abs/1808.02568.
  27. Jacob Holm, Eva Rotenberg, and Mikkel Thorup. Dynamic bridge-finding in Õ(log² n) amortized time. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 35-52, 2018. URL: https://doi.org/10.1137/1.9781611975031.3.
  28. Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, and Seth Pettie. Fully dynamic connectivity in O(log n(log log n)^2) amortized expected time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 510-520, 2017. URL: https://doi.org/10.1137/1.9781611974782.32.
  29. Bruce M. Kapron, Valerie King, and Ben Mountjoy. Dynamic graph connectivity in polylogarithmic worst case time. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '13, pages 1131-1142, Philadelphia, PA, USA, 2013. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2627817.2627898.
  30. Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup. Faster Worst Case Deterministic Dynamic Connectivity. In Piotr Sankowski and Christos Zaroliagis, editors, 24th Annual European Symposium on Algorithms (ESA 2016), volume 57 of Leibniz International Proceedings in Informatics (LIPIcs), pages 53:1-53:15, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.53.
  31. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 505-514, New York, NY, USA, 2013. ACM. URL: https://doi.org/10.1145/2488608.2488672.
  32. Richard J. Lipton and Robert E. Tarjan. A Separator Theorem for Planar Graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. Google Scholar
  33. Jakub Łącki and Piotr Sankowski. Min-cuts and shortest cycles in planar graphs in O(nlog logn) time. In Algorithms - ESA 2011 - 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings, pages 155-166, 2011. URL: https://doi.org/10.1007/978-3-642-23719-5_14.
  34. Jakub Łącki and Piotr Sankowski. Optimal decremental connectivity in planar graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 608-621, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.608.
  35. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In Proceedings of the 58th Annual Symposium on Foundations of Computer Science, FOCS 2017, 2017. Google Scholar
  36. Mihai Pǎtraşcu and Erik D Demaine. Logarithmic lower bounds in the cell-probe model. SIAM Journal on Computing, 35(4):932-963, 2006. Google Scholar
  37. Bruce Reed and David R. Wood. Fast separation in a graph with an excluded minor. In Stefan Felsner, editor, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), volume DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) of DMTCS Proceedings, pages 45-50, Berlin, Germany, 2005. Discrete Mathematics and Theoretical Computer Science. URL: https://hal.inria.fr/hal-01184376.
  38. 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.
  39. Robert E. Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215-225, April 1975. URL: https://doi.org/10.1145/321879.321884.
  40. Siamak Tazari and Matthias Müller-Hannemann. Shortest paths in linear time on minor-closed graph classes, with an application to steiner tree approximation. Discrete Applied Mathematics, 157(4):673-684, 2009. URL: https://doi.org/10.1016/j.dam.2008.08.002.
  41. Mikkel Thorup. Decremental dynamic connectivity. In SODA '97, pages 305-313. SIAM, 1997. Google Scholar
  42. Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, STOC '00, pages 343-350, New York, NY, USA, 2000. ACM. URL: https://doi.org/10.1145/335305.335345.
  43. Christian Wulff-Nilsen. Separator theorems for minor-free and shallow minor-free graphs with applications. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 37-46, 2011. URL: https://doi.org/10.1109/FOCS.2011.15.
  44. Christian Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In Encyclopedia of Algorithms, pages 738-741. Springer Berlin Heidelberg, Berlin, Heidelberg, 2016. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail