Skip to main content
SpringerLink
Log in

On Dynamic DFS Tree in Directed Graphs

  • Conference paper
  • First Online:
Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9235))

Abstract

Let \(G=(V,E)\) be a directed graph on n vertices and m edges. We address the problem of maintaining a depth first search (DFS) tree efficiently under insertion/deletion of edges in G.

  1. 1.

    We present an efficient randomized decremental algorithm for maintaining a DFS tree for a directed acyclic graph. For processing any arbitrary online sequence of edge deletions, this algorithm takes expected \(O(mn \log n)\) time.

  2. 2.

    We present the following lower bound results.

    1. (a)

      Any decremental (or incremental) algorithm for maintaining the ordered DFS tree explicitly requires \({\varOmega }(mn)\) total update time in the worst case.

    2. (b)

      Any decremental (or incremental) algorithm for maintaining the ordered DFS tree is at least as hard as computing all-pairs reachability in a directed graph.

Full version of this article is available at http://www.cse.iitk.ac.in/users/sbaswana/Papers-published/Dynamic-DFS-digraph.pdf.

S. Baswana—This research was partially supported by the India-Israel joint research project on dynamic graph algorithms, and the Indo-German Max Planck Center for Computer Science (IMPECS).

K. Choudhary—This research was partially supported by Google India under the Google India PhD Fellowship Award.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alstrup, S., Holm, J.: Improved algorithms for finding level ancestors in dynamic trees. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, p. 73. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  2. Baswana, S., Khan, S.: Incremental algorithm for maintaining DFS tree for undirected graphs. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 138–149. Springer, Heidelberg (2014)

    Google Scholar 

  3. Cole, R., Hariharan, R.: Dynamic LCA queries on trees. SIAM J. Comput. 34(4), 894–923 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Franciosa, P.G., Frigioni, D., Giaccio, R.: Semi-dynamic breadth-first search in digraphs. Theor. Comput. Sci. 250(1–2), 201–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Franciosa, P.G., Gambosi, G., Nanni, U.: The incremental maintenance of a depth-first-search tree in directed acyclic graphs. Inf. Process. Lett. 61(2), 113–120 (1997)

    Article  MathSciNet  Google Scholar 

  7. Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Miltersen, P.B., Subramanian, S., Vitter, J.S., Tamassia, R.: Complexity models for incremental computation. Theor. Comput. Sci. 130(1), 203–236 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Reif, J.H.: Depth-first search is inherently sequential. Inf. Process. Lett. 20(5), 229–234 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Reif, J.H.: A topological approach to dynamic graph connectivity. Inf. Process. Lett. 25(1), 65–70 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Roditty, L., Zwick, U.: On Dynamic Shortest Paths Problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 580–591. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. SIAM J. Comput. 37(5), 1455–1471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. SIAM J. Comput. 41(3), 670–683 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Robert Endre Tarjan: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Thorup, M.: Fully-dynamic min-cut. Combinatorica 27(1), 91–127 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: STOC, pp. 887–898 (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of CSE, IIT Kanpur, Kanpur, 208016, India

    Surender Baswana & Keerti Choudhary

Authors
  1. Surender Baswana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keerti Choudhary
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keerti Choudhary .

Editor information

Editors and Affiliations

  1. University of Rome "Tor Vergata", Rome, Italy

    Giuseppe F. Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Baswana, S., Choudhary, K. (2015). On Dynamic DFS Tree in Directed Graphs. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-48054-0_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48053-3

  • Online ISBN: 978-3-662-48054-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation