Skip to main content
SpringerLink
Log in

Tree Drawings Revisited

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

Abstract

We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that (1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area \(n2^{O(\sqrt{\log \log n\log \log \log n})}\), improving the longstanding \(O(n\log n)\) bound; (2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area \(n\sqrt{\log n}(\log \log n)^{O(1)}\), improving the longstanding \(O(n\log n)\) bound; (3) every binary tree of size n has a straight-line orthogonal drawing with area \(n2^{O(\log ^*n)}\), improving the previous \(O(n\log \log n)\) bound; (4) every binary tree of size n has a straight-line order-preserving drawing with area \(n2^{O(\log ^*n)}\), improving the previous \(O(n\log \log n)\) bound; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area \(n2^{O(\sqrt{\log n})}\), improving the previous \(O(n^{3/2})\) bound.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. It is not clear to this author if their analysis assumed a much stronger property, that every subtree of size m has degree at most \(O(m^{1/2-\varepsilon })\).

  2. Constants c in different proofs may be different.

  3. Alternatively, one can see the solution directly without induction: The contribution of the \((n'/A)s\log ^3 s\) and \((n''/s)\log A\) terms cleary sums to \(O((n/A)s\log ^3 s + (n/s)\log A)\). The contribution of the first \(\log A\) term sums to at most \((2n/A-1)\log A\), because the number of nodes in the recursion tree is at most \(2n/A-1\). This is because we can charge at least A units to each leaf and each out-degree-1 node of the recursion tree in such a way that the total number of charges is at most n (since a leaf has \(n\ge A\), and a out-degree-1 node has \(m=1\) and \(n-n_1\ge A\)). This implies that the number of leaves and out-degree-1 nodes is at most n / A. The number of nodes of out-degree at least 2 is at most the number of leaves minus 1.

References

  1. Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51(4), 606–635 (2004). https://doi.org/10.1145/1008731.1008736

    Article  MATH  MathSciNet  Google Scholar 

  2. Bachmaier, C., Brandenburg, F.J., Brunner, W., Hofmeier, A., Matzeder, M., Unfried, T.: Tree drawings on the hexagonal grid. In: Proceedings of the 16th International Symposium on Graph Drawing (GD). Lecture Notes in Computer Science, vol. 5417, pp. 372–383. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-00219-9_36

  3. Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38(1), 91–109 (2001). https://doi.org/10.1006/jagm.2000.1127

    Article  MATH  MathSciNet  Google Scholar 

  4. Biedl, T.: Ideal drawings of rooted trees with approximately optimal width. J. Graph Algorithms Appl. 21(4), 631–648 (2017). https://doi.org/10.7155/jgaa.00432

    Article  MATH  MathSciNet  Google Scholar 

  5. Biedl, T.: Upward order-preserving 8-grid-drawings of binary trees. In: Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG), pp. 232–237 (2017). http://2017.cccg.ca/proceedings/Session6B-paper4.pdf

  6. Chan, T.M.: A near-linear area bound for drawing binary trees. Algorithmica 34(1), 1–13 (2002). https://doi.org/10.1007/s00453-002-0937-x

    Article  MATH  MathSciNet  Google Scholar 

  7. Chan, T.M., Goodrich, M.T., Kosaraju, S.R., Tamassia, R.: Optimizing area and aspect ratio in straight-line orthogonal tree drawings. Comput. Geom. 23(2), 153–162 (2002). https://doi.org/10.1016/S0925-7721(01)00066-9

    Article  MATH  MathSciNet  Google Scholar 

  8. Crescenzi, P., Di Battista, G., Piperno, A.: A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. 2(4), 187–200 (1992). https://doi.org/10.1016/0925-7721(92)90021-J

    Article  MATH  MathSciNet  Google Scholar 

  9. Crescenzi, P., Penna, P.: Strictly-upward drawings of ordered search trees. Theor. Comput. Sci. 203(1), 51–67 (1998). https://doi.org/10.1016/S0304-3975(97)00287-9

    Article  MATH  MathSciNet  Google Scholar 

  10. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990). https://doi.org/10.1007/BF02122694

    Article  MATH  MathSciNet  Google Scholar 

  11. Di Battista, G., Frati, F.: Drawing trees, outerplanar graphs, series-parallel graphs, and planar graphs in small area. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 121–165. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-0110-0_9

  12. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)

    MATH  Google Scholar 

  13. Frati, F.: Straight-line orthogonal drawings of binary and ternary trees. In: Proceedings of the 15th International Symposium on Graph Drawing (GD). Lecture Notes in Computer Science, vol. 4875, pp. 76–87. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-77537-9_11

  14. Frati, F., Patrignani, M., Roselli, V.: LR-drawings of ordered rooted binary trees and near-linear area drawings of outerplanar graphs. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1980–1999. SIAM, Philadelphia (2017). https://doi.org/10.1137/1.9781611974782.129

  15. Garg, A., Goodrich, M.T., Tamassia, R.: Planar upward tree drawings with optimal area. Int. J. Comput. Geometry Appl. 6(3), 333–356 (1996). https://doi.org/10.1142/S0218195996000228. Preliminary version in SoCG’93

    Article  MATH  MathSciNet  Google Scholar 

  16. Garg, A., Rusu, A.: Area-efficient order-preserving planar straight-line drawings of ordered trees. Int. J. Comput. Geometry Appl. 13(6), 487–505 (2003). https://doi.org/10.1142/S021819590300130X

    Article  MATH  MathSciNet  Google Scholar 

  17. Garg, A., Rusu, A.: Straight-line drawings of general trees with linear area and arbitrary aspect ratio. In: Proceedings of the 3rd International Conference on Computational Science and Its Applications (ICCSA), Part III. Lecture Notes in Computer Science, vol. 2669, pp. 876–885. Springer, Berlin (2003). https://doi.org/10.1007/3-540-44842-X_89

  18. Garg, A., Rusu, A.: Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algorithms Appl. 8(2), 135–160 (2004). http://jgaa.info/accepted/2004/GargRusu2004.8.2.pdf

  19. Lee, S.: Upward octagonal drawings of ternary trees. Master’s thesis, University of Waterloo (2016). (Supervised by T. Biedl and T.M. Chan.) https://uwspace.uwaterloo.ca/handle/10012/10832

  20. Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: Proceedings of the 21st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 270–281. IEEE (1980). https://doi.org/10.1109/SFCS.1980.13

  21. Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993). https://doi.org/10.1007/BF02573972

    Article  MATH  MathSciNet  Google Scholar 

  22. Reingold, E.M., Tilford, J.S.: Tidier drawings of trees. IEEE Trans. Softw. Eng. 7(2), 223–228 (1981). https://doi.org/10.1109/TSE.1981.234519

    Article  Google Scholar 

  23. Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 138–148. SIAM, Philadelphia (1990). http://dl.acm.org/citation.cfm?id=320176.320191

  24. Shiloach, Y.: Linear and Planar Arrangement of Graphs. Ph.D. thesis, Weizmann Institute of Science (1976). https://lib-phds1.weizmann.ac.il/Dissertations/shiloach_yossi.pdf

  25. Shin, C., Kim, S.K., Chwa, K.-Y.: Area-efficient algorithms for straight-line tree drawings. Comput. Geom. 15(4), 175–202 (2000). https://doi.org/10.1016/S0925-7721(99)00053-X. Preliminary version in COCOON’96

    Article  MATH  MathSciNet  Google Scholar 

  26. Shin, C.-S., Kim, S.K., Kim, S.-H., Chwa, K.-Y.: Algorithms for drawing binary trees in the plane. Inf. Process. Lett. 66(3), 133–139 (1998). https://doi.org/10.1016/S0020-0190(98)00049-0

    Article  MATH  MathSciNet  Google Scholar 

  27. Trevisan, L.: A note on minimum-area upward drawing of complete and Fibonacci trees. Inf. Process. Lett. 57(5), 231–236 (1996). https://doi.org/10.1016/0020-0190(96)81422-0

    Article  MATH  MathSciNet  Google Scholar 

  28. Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Comput. 30(2), 135–140 (1981). https://doi.org/10.1109/TC.1981.6312176

    Article  MATH  MathSciNet  Google Scholar 

  29. Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001). https://www.cc.gatech.edu/fac/Vijay.Vazirani/book.pdf

Download references

Acknowledgements

I thank the reviewers for their careful reading and comments. Funding was provided by National Science Foundation [Grant No. CCF-1814026].

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Timothy M. Chan

Authors
  1. Timothy M. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Timothy M. Chan.

Additional information

Editor in Charge: Kenneth Clarkson

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper appeared in Proc. 34th Symposium on Computational Geometry (SoCG), 23:1–23:15, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chan, T.M. Tree Drawings Revisited. Discrete Comput Geom 63, 799–820 (2020). https://doi.org/10.1007/s00454-019-00106-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-019-00106-w

Keywords

Mathematics Subject Classification

Search

Navigation