Abstract
We present an algorithm to support the dynamic embedding in the plane of a dynamic graph. An edge can be inserted across a face between two vertices on the face boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable in the current embedding, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip- linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We support all updates and queries in \(\mathcal {O}(\log ^{2} n)\) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph. Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant interaction between top trees over the two trees via their common Euler tour.
Similar content being viewed by others
Notes
Jordan curve Theorem.
References
Alstrup, S., Holm, J., De Lichtenberg, K., Thorup, M.: Maintaining information in fully dynamic trees with top trees. ACM Trans. Algor. 1(2), 243–264 (2005)
Di Battista, G., Tamassia, R.: Incremental planarity testing. In: 30th Annual Symposium on Foundations of Computer Science, pp. 436–441. IEEE (1989)
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25, 956–997 (1996)
Eppstein, D.: Dynamic generators of topologically embedded graphs Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’03, pp 599–608. Society for Industrial and Applied Mathematics, Philadelphia (2003)
Eppstein, D., Galil, Z., Italiano, G.F., Spencer, T. H.: Separator based sparsification: I. Planarity testing and minimum spanning trees. J. Comput. Syst. Sci. 52(1), 3–27 (1996)
Eppstein, D., Italiano, G. F, Tamassia, R., Tarjan, R. E, Westbrook, J., Yung, M.: Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algor. 13(1), 33–54 (1992)
Frederickson, G. N.: Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14(4), 781–798 (1985)
Galil, Z., Italiano, G. F., Sarnak, N.: Fully dynamic planarity testing with applications. J. ACM 46, 28–91 (1999)
Hopcroft, J., Tarjan, R. E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)
Italiano, G. F., La Poutré, J. A., Rauch, M. H.: Fully dynamic planarity testing in planar embedded graphs. In: Lengauer, T. (ed.) European Symposium on Algorithms—ESA ’93: First Annual European Symposium Bad Honnef, Germany. Proceedings, pp 212–223. Springer, Berlin Heidelberg (1993)
Karger, D.R.: Random sampling in cut, flow, and network design problems. Math. Oper. Res. 648–657 (1994)
Klein, P. N.: Multiple-source shortest paths in planar graphs, pp 146–155. Society for Industrial and Applied Mathematics, Philadelphia (2005)
La Poutré, J.A.: Alpha-algorithms for incremental planarity testing (preliminary version) Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, STOC ’94, pp 706–715. ACM, New York (1994)
Patraşcu, M., Demaine, E.D.: Logarithmic lower bounds in the cell-probe model. SIAM J. Comput. 35(4), 932–963 (2006). See also STOC’04, SODA’04
Patrascu, M, Thorup, M.: Planning for fast connectivity updates Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’07, pp 263–271. IEEE Computer Society, Washington (2007)
Sleator, D. D., Tarjan, R. E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
Tutte, W. T.: How to Draw a Graph. Proce. London Math. Soc. s3–13(1), 743–767 (1963)
Tutte, W. T.: Graph Theory. Encyclopedia of Mathematics and its Applications. Addison-Wesley Pub. Co. Advanced Book Program (1984)
Westbrook, J.: Fast incremental planarity testing. In: Kuich, W. (ed.) Automata, Languages and Programming (ICALP), Volume 623 of Lecture Notes in Computer Science, pp 342–353. Springer, Berlin Heidelberg (1992)
Acknowledgments
We would like to thank Christian Wulff-Nilsen and Mikkel Thorup for many helpful and interesting discussions and ideas.
Author information
Authors and Affiliations
Corresponding author
Additional information
Announced at STACS 2015
Rights and permissions
About this article
Cite this article
Holm, J., Rotenberg, E. Dynamic Planar Embeddings of Dynamic Graphs. Theory Comput Syst 61, 1054–1083 (2017). https://doi.org/10.1007/s00224-017-9768-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-017-9768-7