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Improved Spanning Ratio for Low Degree Plane Spanners


We describe an algorithm that builds a plane spanner with a maximum degree of 8 and a spanning ratio of \({\approx }4.414\) with respect to the complete graph. This is the best currently known spanning ratio for a plane spanner with a maximum degree of less than 14.

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  1. 1.

    Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, New York (1966)

  2. 2.

    Bonichon, N., Gavoille, C., Hanusse, N., Perković, L.: Plane spanners of maximum degree six. In: Abramsky, S., Gavoille, C., Kirchner, C., auf der Heide, F.M., Spirakis, P. (eds.) Automata, Languages and Programming, Volume 6198 of Lecture Notes in Computer Science, pp. 19–30. Springer, Berlin (2010)

  3. 3.

    Bonichon, N., Kanj, I., Perković, L., Xia, G.: There are plane spanners of degree 4 and moderate stretch factor. Discrete Comput. Geom. 53(3), 514–546 (2015)

  4. 4.

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

  5. 5.

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

  6. 6.

    Bose, P., Keil, J.M.: On the stretch factor of the constrained Delaunay triangulation. In: 3rd International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2006, Banff, Alberta, Canada, July 2–5, 2006, pp. 25–31. IEEE Computer Society (2006)

  7. 7.

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

  8. 8.

    Chew, P: There is a planar graph almost as good as the complete graph. In: Proceedings of the Second Annual Symposium on Computational Geometry, SCG ’86, pp. 169–177. ACM, New York, NY, USA (1986)

  9. 9.

    Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5(1), 399–407 (1990)

  10. 10.

    Kanj, I., Perkovic, L., Türkoglu, D.: Degree four plane spanners: simpler and better. In: Fekete, S., Lubiw, A. (eds.) 32nd International Symposium on Computational Geometry (SoCG 2016), Volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 45:1–45:15, Dagstuhl, Germany. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2016)

  11. 11.

    Kanj, I.A., Perković, L., Xia, G.: On spanners and lightweight spanners of geometric graphs. SIAM J. Comput. 39(6), 2132–2161 (2010)

  12. 12.

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

  13. 13.

    Li, X.-Y., Wang, Y.: Efficient construction of low weight bounded degree planar spanner. In: Warnow, T., Zhu, B. (eds.) Computing and Combinatorics, Volume 2697 of Lecture Notes in Computer Science, pp. 374–384. Springer, Berlin (2003)

  14. 14.

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

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Author information

Correspondence to Darryl Hill.

Additional information

This work was partially supported by the Natural Sciences and Engineering Research Council of Cananda (NSERC) and by the Ontario Graduate Scholarship (OGS).

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Bose, P., Hill, D. & Smid, M. Improved Spanning Ratio for Low Degree Plane Spanners. Algorithmica 80, 935–976 (2018).

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  • Computational geometry
  • Graphs
  • Graph theory
  • Plane
  • Spanners
  • Spanning graph
  • Spanning ratio
  • Degree
  • Bounded degree