Skip to main content
SpringerLink
Log in

There are Plane Spanners of Degree 4 and Moderate Stretch Factor

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Let \({\fancyscript{E}}\) be the complete Euclidean graph on a set of points embedded in the plane. Given a constant \(t \ge 1\), a spanning subgraph \(G\) of \({\fancyscript{E}}\) is said to be a \(t\)-spanner, or simply a spanner, if for any pair of nodes \(u,v\) in \({\fancyscript{E}}\) the distance between \(u\) and \(v\) in \(G\) is at most \(t\) times their distance in \({\fancyscript{E}}\). The constant \(t\) is referred to as the stretch factor. A spanner is plane if its edges do not cross. This paper considers the question: “What is the smallest maximum degree that can always be achieved for a plane spanner of \({\fancyscript{E}}\)?” Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper, we show that the complete Euclidean graph always contains a plane spanner of maximum degree 4 and make a big step toward closing the question. The stretch factor of the spanner is bounded by \(156.82\). Our construction leads to an efficient algorithm for obtaining the spanner from Chew’s \(L_1\)-Delaunay triangulation.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. The first and the last edges in a cone of \(u\) are defined only if there are two or more edges incident to \(u\) in the cone.

References

  1. Bonichon, N., Gavoille, C., Hanusse, N., Perković, L.: Plane spanners of maximum degree six. In: Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP). Lecture Notes in Computer Science, vol. 6198, pp. 19–30. Springer, Berlin (2010)

  2. Bonichon, N., Gavoille, C., Hanusse, N., Perković, L.: The stretch factor of \({L}_1\)- and \({L}_\infty \)-Delaunay triangulations. In: Proceedings of the 20th Annual European Symposium on Algorithms (ESA). Lecture Notes in Computer Science, vol. 7501, pp. 205–216. Springer, Berlin (2012)

  3. Bose, P., Smid, M.: On plane geometric spanners: a survey and open problems. Comput. Geom. 46(7), 818–830 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bose, P., Morin, P., Stojmenović, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw. 7(6), 609–616 (2001)

    Article  MATH  Google Scholar 

  5. Bose, P., Gudmundsson, J., Smid, M.: Constructing plane spanners of bounded degree and low weight. Algorithmica 42(3–4), 249–264 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bose, P., Smid, M., Xu, D.: Delaunay and diamond triangulations contain spanners of bounded degree. Int. J. Comput. Geom. Appl. 19(2), 119–140 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bose, P., Carmi, P., Chaitman-Yerushalmi, L.: On bounded degree plane strong geometric spanners. J. Discrete Algorithms 15, 16–31 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bose, P., Damian, M., Douïeb, K., O’Rourke, J., Seamone, B., Smid, M.H.M., Wuhrer, S.: \(\pi /2\)-Angle Yao graphs are spanners. Int. J. Comput. Geom. Appl. 22(1), 61–82 (2012)

    Article  MATH  Google Scholar 

  9. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Competitive routing on a bounded-degree plane spanner. In: Proceedings of the 24th Canadian Conference on Computational Geometry (CCCG), pp. 299–304 (2012)

  10. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: On plane constrained bounded-degree spanners. In: Proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN). Lecture Notes in Computer Science, vol. 7256, pp. 85–96 (2012)

  11. Chew, L.P.: There is a planar graph almost as good as the complete graph. In: Proceedings of the Second Annual Symposium on Computational Geometry (SoCG), pp. 169–177 (1986)

  12. Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci. 39(2), 205–219 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Das, G., Heffernan, P.: Constructing degree-3 spanners with other sparseness properties. Int. J. Found. Comput. Sci. 7(2), 121–136 (1996)

    Article  MATH  Google Scholar 

  14. Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science (FOCS), pp. 20–26. IEEE Computer Society, Silver Spring, MD (1987)

  15. Kanj, I., Perković, L.: On geometric spanners of Euclidean and unit disk graphs. In. In Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science (STACS), vol. hal-00231084, pp. 409–420. HAL (2008)

  16. Keil, J.M., Gutwin, C.A.: Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7(1), 13–28 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, X.Y., Wang, Y.: Efficient construction of low weight bounded degree planar spanner. Int. J. Comput. Geom. Appl. 14(1–2), 69–84 (2004)

    Article  MATH  Google Scholar 

  18. Salowe, J.: Euclidean spanner graphs with degree four. Discrete Appl. Math. 54(1), 55–66 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, Y., Li, X.Y.: Localized construction of bounded degree and planar spanner for wireless ad hoc networks. Mob. Netw. Appl. 11(2), 161–175 (2006)

    Article  Google Scholar 

  20. Xia, G.: The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput. 42(4), 1620–1659 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was partially supported by ANR grant JCJC EGOS ANR-12-JS02-002-01.

Author information

Authors and Affiliations

  1. University of Bordeaux, LaBRI, UMR 5800, 33400, Talence, France

    Nicolas Bonichon

  2. CNRS, LaBRI, UMR 5800, F-33400, Talence, France

    Nicolas Bonichon

  3. School of Computing, DePaul University, 243 South Wabash Avenue, Chicago, IL, 60604, USA

    Iyad Kanj & Ljubomir Perković

  4. Department of Computer Science, Lafayette College, 506 Acopian Engineering Center, Easton, PA, 18042, USA

    Ge Xia

Authors
  1. Nicolas Bonichon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iyad Kanj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ljubomir Perković
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ge Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Bonichon.

Additional information

Editors-in-Charge: Siu-Wing Cheng and Olivier Devillers

Preliminary version in Proceedings of the 30th Annual Symposium on Computational Geometry, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonichon, N., Kanj, I., Perković, L. et al. There are Plane Spanners of Degree 4 and Moderate Stretch Factor. Discrete Comput Geom 53, 514–546 (2015). https://doi.org/10.1007/s00454-015-9676-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-015-9676-z

Keywords

Search

Navigation