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Optimal Decremental Connectivity in Non-Sparse Graphs

Authors Anders Aamand, Adam Karczmarz , Jakub Łącki , Nikos Parotsidis , Peter M. R. Rasmussen , Mikkel Thorup

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Author Details

Anders Aamand
  • MIT, Cambridge, MA, USA
Adam Karczmarz
  • University of Warsaw, Poland
  • IDEAS NCBR, Warsaw, Poland
Jakub Łącki
  • Google Research, New York, NY,USA
Nikos Parotsidis
  • Google Research, Zürich, Switzerland
Peter M. R. Rasmussen
  • BARC, University of Copenhagen, Denmark
Mikkel Thorup
  • BARC, University of Copenhagen, Denmark

Funding

  • Aamand, Anders: Supported by Thorup’s VILLUM Investigator Grant 16582 and by a DFF-International Postdoc Grant 0164-00022B from the Independent Research Fund Denmark.
  • Karczmarz, Adam: Partially supported by the ERC CoG grant TUgbOAT no 772346.
  • Rasmussen, Peter M. R.: Supported by Thorup’s VILLUM Investigator Grant 16582.
  • Thorup, Mikkel: Supported by Investigator Grant 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation.

Cite AsGet BibTex

Anders Aamand, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal Decremental Connectivity in Non-Sparse Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.6

Abstract

We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in O(m + n poly log n) total time. Interspersed with the deletions, it can answer queries whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental c-edge-connectivity to a variant of fully-dynamic c-edge-connectivity on a sparse graph. For non-sparse graphs with Ω(n poly log n) edges, our connectivity and 2-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an O(m log (n² / m) + n poly log n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n²) edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/log log n) worst-case time lower bound in the decremental setting [Alstrup, Husfeldt, and Rauhe FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Graph algorithms
Keywords
  • decremental connectivity
  • dynamic connectivity

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References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 459-467. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Michael Benedikt, Markus Krötzsch, and Maurizio Lenzerini, editors, Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5-14. ACM, 2012. URL: https://doi.org/10.1145/2213556.2213560.
  3. S. Alstrup, T. Husfeldt, and T. Rauhe. Marked ancestor problems. In Proceedings 39th Annual Symposium on Foundations of Computer Science (FOCS), pages 534-543, 1998. URL: https://doi.org/10.1109/SFCS.1998.743504.
  4. Stephen Alstrup, Jens P. Secher, and Mikkel Thorup. Word encoding tree connectivity works. In David B. Shmoys, editor, Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA, pages 498-499. ACM/SIAM, 2000. URL: http://dl.acm.org/citation.cfm?id=338219.338598.
  5. Stephen Alstrup, Jens Peter Secher, and Maz Spork. Optimal on-line decremental connectivity in trees. Information Processing Letters, 64(4):161-164, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00170-1.
  6. Joshua D. Batson, Daniel A. Spielman, Nikhil Srivastava, and Shang-Hua Teng. Spectral sparsification of graphs: theory and algorithms. Commun. ACM, 56(8):87-94, 2013. URL: https://doi.org/10.1145/2492007.2492029.
  7. Sayan Bhattacharya, Fabrizio Grandoni, Janardhan Kulkarni, Quanquan C. Liu, and Shay Solomon. Fully dynamic (Δ +1)-coloring in O(1) update time. ACM Trans. Algorithms, 18(2):10:1-10:25, 2022. URL: https://doi.org/10.1145/3494539.
  8. 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. CoRR, abs/1910.08025, 2019. URL: https://arxiv.org/abs/1910.08025.
  9. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - A technique for speeding up dynamic graph algorithms. J. ACM, 44(5):669-696, 1997. URL: https://doi.org/10.1145/265910.265914.
  10. Shimon Even and Yossi Shiloach. An on-line edge-deletion problem. J. ACM, 28(1):1-4, 1981. URL: https://doi.org/10.1145/322234.322235.
  11. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2-3):207-216, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.013.
  12. Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781-798, 1985. URL: https://doi.org/10.1137/0214055.
  13. M. Fredman and M. Saks. The cell probe complexity of dynamic data structures. In Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing (STOC), STOC '89, pages 345-354, New York, NY, USA, 1989. Association for Computing Machinery. URL: https://doi.org/10.1145/73007.73040.
  14. Michael L. Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34(3):596-615, 1987. URL: https://doi.org/10.1145/28869.28874.
  15. Harold N. Gabow, Haim Kaplan, and Robert Endre Tarjan. Unique maximum matching algorithms. J. Algorithms, 40(2):159-183, 2001. URL: https://doi.org/10.1006/jagm.2001.1167.
  16. Harold N. Gabow and Robert Endre Tarjan. A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci., 30(2):209-221, 1985. URL: https://doi.org/10.1016/0022-0000(85)90014-5.
  17. Zvi Galil and Giuseppe F. Italiano. Maintaining the 3-edge-connected components of a graph on-line. SIAM J. Comput., 22(1):11-28, 1993. URL: https://doi.org/10.1137/0222002.
  18. 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.
  19. Michael T Goodrich and Michael Mitzenmacher. Invertible bloom lookup tables. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 792-799. IEEE, 2011. Google Scholar
  20. Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn, and Shay Solomon. (1 + ε)-approximate incremental matching in constant deterministic amortized time. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1886-1898. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.114.
  21. Monika Henzinger and Pan Peng. Constant-time dynamic (Δ +1)-coloring. ACM Trans. Algorithms, 18(2):16:1-16:21, 2022. URL: https://doi.org/10.1145/3501403.
  22. Monika Rauch Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM, 46(4):502-516, 1999. URL: https://doi.org/10.1145/320211.320215.
  23. 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.
  24. Jacob Holm and Eva Rotenberg. Good r-divisions imply optimal amortised decremental biconnectivity. CoRR, abs/1808.02568, 2018. URL: https://arxiv.org/abs/1808.02568.
  25. Jacob Holm, Eva Rotenberg, and Mikkel Thorup. Dynamic bridge-finding in Õ(log^2 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.
  26. 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.
  27. Wenyu Jin and Xiaorui Sun. Fully dynamic s-t edge connectivity in subpolynomial time (extended abstract). In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 861-872. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00088.
  28. 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 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1131-1142, 2013. URL: https://doi.org/10.1137/1.9781611973105.81.
  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, USA, 2013. Society for Industrial and Applied Mathematics. Google Scholar
  30. David R. Karger. Random sampling in cut, flow, and network design problems. Math. Oper. Res., 24(2):383-413, 1999. URL: https://doi.org/10.1287/moor.24.2.383.
  31. Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup. Faster worst case deterministic dynamic connectivity. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 53:1-53:15, 2016. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.53.
  32. Jakub Lacki and Piotr Sankowski. Optimal decremental connectivity in planar graphs. Theory Comput. Syst., 61(4):1037-1053, 2017. URL: https://doi.org/10.1007/s00224-016-9709-x.
  33. 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: https://doi.org/10.1007/BF01758778.
  34. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 950-961, 2017. URL: https://doi.org/10.1109/FOCS.2017.92.
  35. Mihai Pǎtraşcu and Erik D. Demaine. Lower bounds for dynamic connectivity. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’04, pages 546-553, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1007352.1007435.
  36. Mihai Pǎtraşcu and Mikkel Thorup. Don't rush into a union: take time to find your roots. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 559-568, 2011. URL: https://doi.org/10.1145/1993636.1993711.
  37. Shay Solomon. Fully dynamic maximal matching in constant update time. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 325-334. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.43.
  38. Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22(2):215-225, April 1975. URL: https://doi.org/10.1145/321879.321884.
  39. Mikkel Thorup. Decremental dynamic connectivity. Journal of Algorithms, 33(2):229-243, 1999. URL: https://doi.org/10.1006/jagm.1999.1033.
  40. Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 343-350, 2000. URL: https://doi.org/10.1145/335305.335345.
  41. Mikkel Thorup. Fully-dynamic min-cut. Comb., 27(1):91-127, 2007. URL: https://doi.org/10.1007/s00493-007-0045-2.
  42. Zhengyu Wang. An improved randomized data structure for dynamic graph connectivity. CoRR, abs/1510.04590, 2015. URL: https://arxiv.org/abs/1510.04590.
  43. Christian Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1757-1769. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.126.
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