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Improved Bounds for Drawing Trees on Fixed Points with L-Shaped Edges

  • Therese Biedl
  • Timothy M. Chan
  • Martin Derka
  • Kshitij Jain
  • Anna Lubiw
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)

Abstract

Let T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an “L-shaped” edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P for which such drawings are possible to: \(O(n^{1.55})\) for maximum degree 4 trees; \(O(n^{1.22})\) for maximum degree 3 (binary) trees; and \(O(n^{1.142})\) for perfect binary trees.

Drawing ordered trees with L-shaped edges is harder—we give an example that cannot be done and a bound of \(O(n \log n)\) points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.

Notes

Acknowledgments

We thank Jeffrey Shallit for investigating the alternating sequences discussed in Sect. 2. This work was done as part of a Problem Session in the Algorithms and Complexity group at the University of Waterloo. We thank the other participants for helpful discussions.

References

  1. 1.
    Aichholzer, O., Hackl, T., Scheucher, M.: Planar L-shaped point set embeddings of trees. In: European Workshop on Computational Geometry (EuroCG) (2016). http://www.eurocg2016.usi.ch/
  2. 2.
    Bárány, I., Buchin, K., Hoffmann, M., Liebenau, A.: An improved bound for orthogeodesic point set embeddings of trees. In: European Workshop on Computational Geometry (EuroCG) (2016). http://www.eurocg2016.usi.ch/
  3. 3.
    Bose, P.: On embedding an outer-planar graph in a point set. Comput. Geom. 23(3), 303–312 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. J. Graph Algorithms Appl. 1(2), 1–15 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorithms Appl. 10(2), 353–363 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. ACM SIGACT News 20(4), 83–86 (1989)CrossRefGoogle Scholar
  7. 7.
    Di Giacomo, E., Frati, F., Fulek, R., Grilli, L., Krug, M.: Orthogeodesic point-set embedding of trees. Comput. Geom. 46(8), 929–944 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fraysseix, H.D., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Monthly 98, 165–166 (1991)CrossRefGoogle Scholar
  10. 10.
    Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-geodesic embedding of planar graphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 207–218. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11805-0_21 CrossRefGoogle Scholar
  11. 11.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Scheucher, M.: Orthogeodesic point set embeddings of outerplanar graphs. Master’s thesis, Graz University of Technology (2015)Google Scholar
  13. 13.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 138–148 (1990)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Therese Biedl
    • 1
  • Timothy M. Chan
    • 2
  • Martin Derka
    • 1
  • Kshitij Jain
    • 1
  • Anna Lubiw
    • 1
  1. 1.University of WaterlooWaterlooCanada
  2. 2.University of Illinois at Urbana-ChampaignUrbanaUSA

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