Skip to main content
SpringerLink
Log in
Book cover

European Symposium on Algorithms

ESA 1993: Algorithms—ESA '93 pp 372–383Cite as

A fully dynamic data structure for reachability in planar digraphs

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 726))

Abstract

In this paper we investigate the problem of maintaining all-pairs reachability information in a planar digraph G as it undergoes changes. We give a fully dynamic O(n)-space data structure to support an arbitrary sequence of operations that consist of adding new edges (or nodes), deleting some existing edge, and querying to find out if a given node v is reachable in G by a directed path from another node u.

We show that using our data structure a reachability query between two nodes u and v can be performed in O(n2/3 log n) time, where n is the number of nodes in G. Additions and deletions of edges and nodes can also be handled within the same time bounds. The time for deletion is worst-case while the time for edge-addition is amortized. This is the first fully dynamic algorithm for the planar reachability problem that uses only sublinear time for both queries and updates.

Research supported in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF research grant CCR-9007851, by Army Research Office grant DAAL03-91-G-0035, and by the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-91-J-4052 and ARPA order 8225. Email: ss@cs.brown.edu

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Birkhoff, “Lattice Theory,” American Mathematical Society Colloquium Publications 25 (1979).

    Google Scholar 

  2. 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.

    Google Scholar 

  3. D. Eppstein, Z. Galil, G.F. Italiano, and T. Spencer, “Separator Based Sparsification for Dynamic Planar Graph Algorithms,” Proc. 25th Annual ACM Symposium on Theory of Computing (1993).

    Google Scholar 

  4. G.N. Frederickson, “Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications,” SIAM J. Computing 14 (1985), 781–798.

    Google Scholar 

  5. G.N. Frederickson, “Fast Algorithms for Shortest Paths in Planar Graphs, with Applications,” SIAM Journal on Computing 16 (1987), 1004–1022.

    Google Scholar 

  6. Z. Galil and G. F. Italiano, “Maintaining biconnected components of dynamic planar graphs,” Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991), 339–350.

    Google Scholar 

  7. Z. Galil, G.F. Italiano, and N. Sarnak, “Fully Dynamic Planarity Testing,” Proc. 24th Annual ACM Symposium on Theory of Computing (1992), 495–506.

    Google Scholar 

  8. G.F. Italiano, “Amortized Efficiency of a Path Retrieval Data Structure,” Theoretical Computer Science 48 (1986), 273–281.

    Google Scholar 

  9. G.F. Italiano, “Finding Paths and Deleting Edges in Directed Acyclic Graphs,” Information Processing Letters 28 (1988), 5–11.

    Google Scholar 

  10. G.F. Italiano, A. Marchetti-Spaccamela, and U. Nanni, “Dynamic Data Structures for Series-Parallel Graphs,” Proc. WADS' 89, LNCS 382 (1989), 352–372.

    Google Scholar 

  11. T. Kameda, “On the Vector Representation of the Reachability in Planar Directed Graphs,” Information Processing Letters 3 (1975), 75–77.

    Google Scholar 

  12. D. Kelly, “On the Dimension of Partially Ordered Sets,” Discrete Mathematics 35 (1981), 135–156.

    Google Scholar 

  13. P. N. Klein and S. Subramanian, “A Fully Dynamic Approximation Scheme for All-Pairs Shortest Paths in Planar Graphs,” Proc. (to appear) 1993 Workshop on Algorithms and Data Structures (1993).

    Google Scholar 

  14. G. Miller, “Finding Small Simple Cycle Separators for 2-Connected Planar Graphs,” Journal of Computer and System Sciences 32 (1986), 265–279.

    Google Scholar 

  15. 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.

    Google Scholar 

  16. R. Tamassia and F.P. Preparata, “Dynamic Maintenance of Planar Digraphs, with Applications,” Algorithmica 5 (1990), 509–527.

    Google Scholar 

  17. R. Tamassia and I.G. Tollis, “Dynamic Reachability in Planar Digraphs,” Theoretical Computer Science (1993), (to appear).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brown University, USA

    Sairam Subramanian

Authors
  1. Sairam Subramanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Thomas Lengauer

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Subramanian, S. (1993). A fully dynamic data structure for reachability in planar digraphs. 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_72

Download citation

  • DOI: https://doi.org/10.1007/3-540-57273-2_72

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57273-2

  • Online ISBN: 978-3-540-48032-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation