Skip to main content
SpringerLink
Log in

Finding a Minimum-depth Embedding of a Planar Graph in O(n 4) Time

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Consider an n-vertex planar graph G. The depth of an embedding Γ of G is the maximum distance of its internal faces from the external one. Several researchers pointed out that the quality of a planar embedding can be measured in terms of its depth. We present an O(n 4)-time algorithm for computing an embedding of G with minimum depth. This bound improves on the best previous bound by an O(nlog n) factor. As a side effect, our algorithm improves the bounds of several algorithms that require the computation of a minimum-depth embedding.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)

    Article  MATH  Google Scholar 

  2. Bienstock, D., Monma, C.L.: On the complexity of covering vertices by faces in a planar graph. SIAM J. Comput. 17, 53–76 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1), 93–109 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Computing radial drawings on the minimum number of circles. J. Graph Algorithms Appl. 9, 365–389 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Dolev, D., Leighton, F.T., Trickey, H.: Planar embedding of planar graphs. Adv. Comput. Res. 2, 147–161 (1984)

    Google Scholar 

  7. Kammer, F.: Determining the smallest k such that g is k-outerplanar. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA’07. LNCS, vol. 4698, pp. 359–370. Springer, Berlin (2007)

    Google Scholar 

  8. Pizzonia, M.: Minimum depth graph embeddings and quality of the drawings: An experimental analysis. In: Healy, P., Nikolov, N.S. (eds.) Graph Drawing ’05. LNCS, vol. 3843, pp. 397–408. Springer, Berlin (2005)

    Chapter  Google Scholar 

  9. Pizzonia, M., Tamassia, R.: Minimum depth graph embedding. In: Paterson, M. (ed.) ESA ’00. LNCS, vol. 1879, pp. 356–367. Springer, Berlin (2000)

    Chapter  Google Scholar 

  10. Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B 36(1), 49–64 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Thorup, M.: Undirected single-source shortest path with positive integer weights in linear time. J. ACM 46(3), 362–394 (1999)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università Roma Tre, Rome, Italy

    Patrizio Angelini, Giuseppe Di Battista & Maurizio Patrignani

Authors
  1. Patrizio Angelini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Di Battista
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maurizio Patrignani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Di Battista.

Additional information

A shorter version of this work appears in Proceedings of the 10th Workshop on Algorithms and Data Structures (WADS 2007), vol. 4619 of LNCS, pp. 287–299. Springer, 2007, under the title “Computing a Minimum-Depth Planar Graph Embedding in O(n 4) Time”. Work partially supported by EC–Fet Project DELIS–Contract no 001907 and by MUR under Project “MAINSTREAM: Algoritmi per strutture informative di grandi dimensioni e data streams”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Angelini, P., Di Battista, G. & Patrignani, M. Finding a Minimum-depth Embedding of a Planar Graph in O(n 4) Time. Algorithmica 60, 890–937 (2011). https://doi.org/10.1007/s00453-009-9380-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-009-9380-6

Search

Navigation