Abstract
We present the first data structure to maintain an embedded planar graph under arbitrary edge insertions, arbitrary edge deletions and queries that test whether the insertion of a new edge would violate the planarity of the embedding. Our data structure supports online updates and queries on an n-vertex embedded planar graph in O(log2 n) worst-case time, it can be built in linear time and requires linear storage.
The first author was on leave from Università di Roma. The research of the second author was supported by a NATO Science Fellowship awarded by NWO (the Netherlands Organization for Scientific Research), and partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science — NSF-STC88-09648). This work was done while the third author was at Princeton University, Princeton, NJ 08544.
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References
A. V. Aho, J. E. Hopcroft, J. D. Ullman, “The Design and Analysis of Computer Algorithms” Addison-Wesley, Reading, Massachusetts, 1974.
D. Eppstein, G. F. Italiano, R. Tamassia, R. E. Tarjan, J. Westbrook, M. Yung, “Maintenance of a Minimum Spanning Forest in a Dynamic Plane Graph” J. Algorithms, 13 (1992), 33–54.
D. Eppstein, Z. Galil, G. F. Italiano, A. Nissenzweig, “Sparsification — A Technique for Speeding Up Dynamic Graph Algorithms” Proc. 33rd FOCS, 1992, 60–69.
D. Eppstein, Z. Galil, G. F. Italiano, T. H. Spencer, “Separator Based Sparsification for Dynamic Planar Graph Algorithms” Proc. 25th STOC, 1993, 208–217.
S. Even, R. E. Tarjan, “Computing an st-numbering” Theoretical Computer Science, 2 (1976), 339–344.
G. N. Frederickson, “Data Structures for On-line Updating of Minimum Spanning Trees” SIAM J. Comput. 14 (1985), 781–798.
G. N. Frederickson, “Ambivalent Data Structures for Dynamic 2-edge-connectivity and k smallest spanning trees” Proc. 32nd FOCS, 1991, 632–641.
Z. Galil, G. F. Italiano, “Maintaining Biconnected Components of Dynamic Planar Graphs” Proc. 18th ICALP, LNCS, Springer-Verlag, 1991, 339–350.
F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.
D. Harel, R. E. Tarjan, “Fast Algorithms for Finding Nearest Common Ancestors” SIAM J. Comput. 13 (1984), 338–355.
J. Hershberger, M. Rauch, S. Suri. “Fully Dynamic 2-Edge-Connectivity in Planar Graphs” Proc. 3rd Scandinavian Workshop on Algorithm Theory, LNCS 621, Springer-Verlag, 1992, 233–244.
M. Rauch, “Fully Dynamic Biconnectivity in Graphs” Proc. 33rd FOCS, 1992, 50–59.
R. Tamassia, “A Dynamic Data Structure for Planar Graph Embedding”, Proc. 15th ICALP, LNCS 317, Springer-Verlag, 1988, 576–590.
R. Tamassia, F. P. Preparata, “Dynamic Maintenance of Planar Digraphs, with Applications”, Algorithmica, 5 (1990), 509–527.
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© 1993 Springer-Verlag Berlin Heidelberg
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Italiano, G.F., La Poutré, J.A., Rauch, M.H. (1993). Fully dynamic planarity testing in planar embedded graphs. In: Lengauer, T. (eds) Algorithms—ESA '93. ESA 1993. Lecture Notes in Computer Science, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57273-2_57
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DOI: https://doi.org/10.1007/3-540-57273-2_57
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