Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

On the structure of DFS-forests on directed graphs and the dynamic maintenance of DFS on DAG's

  • Conference paper
  • First Online:
Algorithms — ESA '94 (ESA 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 855))

Included in the following conference series:

Abstract

In this paper we provide a characterization of a DFS-forest on directed graphs in terms of a relaxed planar embedding of its structure.

We propose an incremental algorithm, based on that characterization, to maintain a DFS-forest in a directed acyclic graph with n nodes and m edges, achieving O(nm) total time in the worst case for any sequence of arc insertions, that is O(n) amortized time per arc insertion in a sequence of Θ(m) such operations. This favorably compares with the time required to recompute DFS from scratch by using Tarjan's Θ(n+m) algorithm [19].

The graph is represented by means of adjacency lists and the dfs tree is maintained by using for each node a pointer to its parent, and a rank according to a suitable ordering: the whole data structure requires O(n+ m) space.

This is the first algorithm for dynamic DFS for nontrivial classes of graphs.

Partially supported by the ESPRIT Basic Research Action no.7141 (Alcom II) and by MURST national project “Algoritmi e Strutture di Calcolo”.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The design and analysis of computer algorithms. Addison-Wesley, Reading, MA, 1974.

    Google Scholar 

  2. G. Ausiello, G. F. Italiano, A. Marchetti-Spaccamela, and U. Nanni, Incremental Algorithms for Minimal Length Paths, Journal of Algorithms, 12, 4 (1991), 615–638.

    Article  MathSciNet  Google Scholar 

  3. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to algorithms. MIT Press, Cambridge, MA & McGraw-Hill, New York, NY, 1990.

    Google Scholar 

  4. S. Even. Graph algorithms. Pittman, London, UK, 1976.

    Google Scholar 

  5. S. Even, and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research 49 (1985), 371–387.

    MathSciNet  Google Scholar 

  6. D. Eppstein, Z. Galil, G. F. Italiano, A. Nissenzweig. Sparsification-A technique for speeding up dynamic graph algorithms. Proceedings 33th IEEE Symposium on Foundations of Computer Science (1992), 60–69.

    Google Scholar 

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

    Google Scholar 

  8. J. Hopcroft, R. E. Tarjan. Efficient algorithms for graph manipulation. Communications of the ACM, 16 (1973), 372–378.

    Article  Google Scholar 

  9. G. F. Italiano. Amortized efficiency of a path retrieval data structure. Theoretical Computer Science, 48 (1986), 273–281.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. F. Italiano. Finding paths and deleting edges in directed acyclic graphs. Information Processing Letters, 28 (1988), 5–11.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. N. Klein, S. Rao, M. Rauch and S. Subramanian. Faster shortest-path algorithms for planar graphs. Proceedings 26th ACM Symposium on Theory of Computing, 1994, to appear.

    Google Scholar 

  12. J. A. La Poutré. Alpha-algorithms for incremental planarity testing. Proceedings 26th ACM Symposium on Theory of Computing, 1994, to appear.

    Google Scholar 

  13. J. A. La Poutré and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. Proceedings International Workshop on Graph-Theoretic Concepts in Computer Science (WG 88), Lecture Notes in Computer Science, 314, Springer-Verlag (1988), 106–120.

    Google Scholar 

  14. A. Marchetti-Spaccamela, U. Nanni, H. Rohnert. On-line graph algorithms for incremental compilation. Proceedings International Workshop on Graph-Theoretic Concepts in Computer Science (WG 93), Utrecht (NL), June 16–18, 1993, Lecture Notes in Computer Science, Springer-Verlag.

    Google Scholar 

  15. J. H. Reif. Depth-First Search is Inherently Sequential. Information Processing Letters, 20, 1985.

    Google Scholar 

  16. J. H. Reif. A Topological Approach to Dynamic Graph Connectivity. Information Processing Letters, 25, 1987.

    Google Scholar 

  17. H. Rohnert, A dynamization of the all-pairs least cost path problem, Proceedings 2nd Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, 182, Springer-Verlag (1990), 279–286.

    Google Scholar 

  18. S. Subramanian. A fully dynamic data structure for reachability in planar graphs. Proceedings 1st Annual European Symposium on Algorithms (1993).

    Google Scholar 

  19. R. E. Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1 (1972), 146–160.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, via Salaria 113, I-00198, Roma, Italy

    Paolo G. Franciosa

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, I-00133, Roma, Italy

    Giorgio Gambosi

  3. Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, via Vetoio, I-67010, Coppito (AQ), Italy

    Umberto Nanni

Authors
  1. Paolo G. Franciosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Gambosi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Umberto Nanni
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jan van Leeuwen

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Franciosa, P.G., Gambosi, G., Nanni, U. (1994). On the structure of DFS-forests on directed graphs and the dynamic maintenance of DFS on DAG's. In: van Leeuwen, J. (eds) Algorithms — ESA '94. ESA 1994. Lecture Notes in Computer Science, vol 855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0049421

Download citation

  • DOI: https://doi.org/10.1007/BFb0049421

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58434-6

  • Online ISBN: 978-3-540-48794-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation