Abstract
We present the SPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph with respect to its triconnected components, and show its application to a variety of on-line graph algorithms dealing with triconnectivity, transitive closure, minimum spanning tree, and planarity testing. The results are further extended to general graphs by means of another data structure, the BC-tree.
Extended Abstract
This work was supported in party by the Office of Naval Research under contract N00014-83-K-0146 and ARPA Order No. 4786, and by the ESPRIT II Basic Research Actions Program of the European Communities under Contract No. 3075 (project ALCOM).
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References
G. Ausiello, G.F. Italiano, A. Marchetti-Spaccamela and U. Nanni, “Incremental Algorithms for Minimal Length Paths,” Proc. ACM-SIAM Symp. on Discrete Algorithms (1990), 12–21.
D. Bienstock and C.L. Monma, “On the Complexity of Covering Vertices by Faces in a Planar Graph,” SIAM J. Computing 17 (1988), 53–76.
A.L. Buchsbaum, P.C. Kanellakis and J.S. Vitter, “A Data Structure for Arc Insertion and Regular Path Finding,” Proc. ACM-SIAM Symp. on Discrete Algorithms (1990), 22–31.
G. Di Battista and R. Tamassia, “Algorithms for Plane Representations of Acyclic Digraphs,” Theoretical Computer Science 61 (1988), 175–198.
G. Di Battista and R. Tamassia, “Incremental Planarity Testing,” Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436–441.
G. Di Battista, R. Tamassia and I.G. Tollis, “Area Requirement and Symmetry Display in Drawing Graphs,” Proc. ACM Symp. on Computational Geometry (1989), 51–60.
D. Eppstein, G.F. Italiano, R. Tamassia, R.E. Tarjan, J. Westbrook and M. Yung, “Maintenance of a Minimum Spanning Forest in a Dynamic Planar Graph,” Proc. First ACM-SIAM Symp. on Discrete Algorithms (1990), 1–11.
S. Even and 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, with Applications,” SIAM J. Computing 14 (1985), 781–798.
D. Fussell, V. Ramachandran and R. Thurimella, “Finding Triconnected Components by Local Replacements,” Proc. 16th ICALP, LNCS, 372 (1989), 379–393.
H.N. Gabow and R.e. Tarjan, “A Linear Time Algorithm for a Special Case of Disjoint Set Union,” J. Computer Systems Sciences 30 (1985).
J. Hopcroft and R.E. Tarjan, “Dividing a Graph into Triconnected Components,” SIAM J. Computing 2 (1973), 135–158.
H. Imai and T. Asano, “Dynamic Orthogonal Segment Intersection Search,” J. Algorithms 8 (1987), 1–18.
G.F. Italiano, “Amortized Efficiency of a Path Retrieval Data Structure,” Theoretical Computer Science 48 (1986), 273–281.
G.F. Italiano, A. Marchetti-Spaccamela and U. Nanni, “Dynamic Data Structures for Series-Parallel Graphs,” Proc. WADS' 89, LNCS 382 (1989), 352–372.
A. Kanevsky and V. Ramachandran, “A Characterization of Separating Pairs and Triplets in a Graph,” Coordinated Science Laboratory, Univ. of Illinois, Technical Report ACT-79, 1987.
A. Lempel, S. Even and I. Cederbaum, “An Algorithm for Planarity Testing of Graphs,” Theory of Graphs, Int. Symposium (1966), 215–232.
J.A. La Poutré, Personal Communication, 1990.
J.A. La Poutré and J. van Leeuwen, “Maintenance of Transitive Closures and Transitive Reductions of Graphs,” Proc. WG '87, LNCS 314 (1988), 106–120.
F.P. Preparata and R. Tamassia, “Fully Dynamic Techniques for Point Location and Transitive Closure in Planar Structures,” Proc. 29th IEEE Symp. on Foundations of Computer Science (1988), 558–567.
F.P. Preparata and R. Tamassia, “Fully Dynamic Point Location in a Monotone Subdivision,” SIAM J. Computing 18 (1989), 811–830.
D.D. Sleator and R.E. Tarjan, “A Data Structure for Dynamic Trees,” J. Computer Systems Sciences 24 (1983), 362–381.
R. Tamassia, “A Dynamic Data Structure for Planar Graph Embedding,” Proc. 15th ICALP, LNCS 317 (1988), 576–590.
R.E. Tarjan and J. van Leeuwen, “Worst-Case Analysis of Set-Union Algorithms,” J. ACM 31 (1984), 245–281.
J. Westbrook, “Algorithms and Data Structures for Dynamic Graph Problems,” Dept. Computer Science, Princeton Univ., Ph.D. dissertation (Technical Report CSTR-229-89), 1989.
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Di Battista, G., Tamassia, R. (1990). On-line graph algorithms with SPQR-trees. In: Paterson, M.S. (eds) Automata, Languages and Programming. ICALP 1990. Lecture Notes in Computer Science, vol 443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0032061
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DOI: https://doi.org/10.1007/BFb0032061
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