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Optimal Decremental Connectivity in Planar Graphs

Authors Jakub Lacki, Piotr Sankowski

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Jakub Lacki
Piotr Sankowski

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Jakub Lacki and Piotr Sankowski. Optimal Decremental Connectivity in Planar Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 608-621, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form 'Are vertices u and v connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in O(n) time. This results improves over previously known O(n \log n) time algorithm.
  • decremental connectivity
  • planar graphs
  • dynamic connectivity
  • algorithms


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  1. Stephen Alstrup, Jens P. Secher, and Maz Spork. Optimal on-line decremental connectivity in trees. Inf. Process. Lett., 64(4):161-164, 1997. Google Scholar
  2. David Eppstein, Zvi Galil, and Giuseppe F Italiano. Improved sparsification. Information and Computer Science, University of California, Irvine, 1993. Google Scholar
  3. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. J. ACM, 44:669-696, 1997. Google Scholar
  4. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification: I. Planarity testing and minimum spanning trees. J. Comput. Syst. Sci., 52(1):3-27, 1996. Google Scholar
  5. David Eppstein, Giuseppe F. Italiano, Roberto Tamassia, Robert Endre Tarjan, Jeffery Westbrook, and Moti Yung. Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algorithms, 13(1):33-54, 1992. Google Scholar
  6. Greg N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput., 14(4):781-798, 1985. Google Scholar
  7. Jens Gustedt. Efficient union-find for planar graphs and other sparse graph classes. Theoretical Computer Science, 203(1):123 - 141, 1998. Google Scholar
  8. Monika R. Henzinger and Valerie King. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM, 46(4):502-516, July 1999. Google Scholar
  9. Monika Rauch Henzinger and Michael L. Fredman. Lower bounds for fully dynamic connectivity problems in graphs. Algorithmica, 22(3):351-362, 1998. Google Scholar
  10. 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, 2001. Google Scholar
  11. 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. SIAM, 2013. Google Scholar
  12. 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. SIAM, 2013. Google Scholar
  13. Philip N. Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decompositions for planar graphs in linear time. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 505-514. ACM, 2013. Google Scholar
  14. Mihai Pǎtraşcu and Erik D. Demaine. Logarithmic lower bounds in the cell-probe model. SIAM J. Comput., 35(4):932-963, 2006. Google Scholar
  15. Johannes A. La Poutré. Lower bounds for the union-find and the sp;it-find problem on pointer machines. J. Comput. Syst. Sci., 52(1):87-99, 1996. Google Scholar
  16. Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22(2):215-225, 1975. Google Scholar
  17. Robert Endre Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. J. Comput. Syst. Sci., 18(2):110-127, 1979. Google Scholar
  18. Mikkel Thorup. Decremental dynamic connectivity. J. Algorithms, 33(2):229-243, 1999. Google Scholar
  19. Mikkel Thorup. Near-optimal fully-dynamic graph connectivity. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 343-350. ACM, 2000. Google Scholar
  20. Freek van Walderveen, Norbert Zeh, and Lars Arge. Multiway simple cycle separators and I/O-efficient algorithms for planar graphs. In Khanna [12], pages 901-918. Google Scholar
  21. Christian Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In Khanna [12], pages 1757-1769. Google Scholar
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