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Tree Drawings Revisited

Author Timothy M. Chan

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  • graph drawing
  • trees
  • recursion

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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 by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);
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 by Garg and Rusu (2003);
5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2^{O(sqrt{log n})}, improving the O(n^{3/2}) previous bound by Frati (2007).

Cite As Get BibTex

Timothy M. Chan. Tree Drawings Revisited. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.23

Author Details

Timothy M. Chan

References

  1. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL: http://dx.doi.org/10.1145/1008731.1008736.
  2. Christian Bachmaier, Franz-Josef Brandenburg, Wolfgang Brunner, Andreas Hofmeier, Marco Matzeder, and Thomas Unfried. Tree drawings on the hexagonal grid. In Proc. 16th International Symposium on Graph Drawing (GD), pages 372-383, 2008. URL: http://dx.doi.org/10.1007/978-3-642-00219-9_36.
  3. Gill Barequet and Sariel Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms, 38(1):91-109, 2001. URL: http://dx.doi.org/10.1006/jagm.2000.1127.
  4. Therese Biedl. Ideal drawings of rooted trees with approximately optimal width. J. Graph Algorithms Appl., 21:631-648, 2017. URL: http://dx.doi.org/10.7155/jgaa.00432.
  5. Therese Biedl. Upward order-preserving 8-grid-drawings of binary trees. In Proc. 29th Canadian Conference on Computational Geometry (CCCG), pages 232-237, 2017. URL: http://2017.cccg.ca/proceedings/Session6B-paper4.pdf.
  6. Timothy M. Chan. A near-linear area bound for drawing binary trees. Algorithmica, 34(1):1-13, 2002. URL: http://dx.doi.org/10.1007/s00453-002-0937-x.
  7. Timothy M. Chan, Michael T. Goodrich, S. Rao Kosaraju, and Roberto Tamassia. Optimizing area and aspect ration in straight-line orthogonal tree drawings. Comput. Geom., 23(2):153-162, 2002. URL: http://dx.doi.org/10.1016/S0925-7721(01)00066-9.
  8. Pierluigi Crescenzi, Giuseppe Di Battista, and Adolfo Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom., 2:187-200, 1992. URL: http://dx.doi.org/10.1016/0925-7721(92)90021-J.
  9. Pierluigi Crescenzi and Paolo Penna. Strictly-upward drawings of ordered search trees. Theor. Comput. Sci., 203(1):51-67, 1998. URL: http://dx.doi.org/10.1016/S0304-3975(97)00287-9.
  10. Hubert de Fraysseix, János Pach, and Richard Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. URL: http://dx.doi.org/10.1007/BF02122694.
  11. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing. Prentice Hall, 1999. Google Scholar
  12. Giuseppe Di Battista and Fabrizio Frati. A survey on small-area planar graph drawing. CoRR, abs/1410.1006, 2014. URL: http://arxiv.org/abs/1410.1006.
  13. Fabrizio Frati. Straight-line orthogonal drawings of binary and ternary trees. In Proc. 15th International Symposium on Graph Drawing (GD), pages 76-87, 2007. URL: http://dx.doi.org/10.1007/978-3-540-77537-9_11.
  14. Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. LR-drawings of ordered rooted binary trees and near-linear area drawings of outerplanar graphs. In Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1980-1999, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.129.
  15. Ashim Garg, Michael T. Goodrich, and Roberto Tamassia. Planar upward tree drawings with optimal area. Int. J. Comput. Geometry Appl., 6(3):333-356, 1996. Preliminary version in SoCG'93. URL: http://dx.doi.org/10.1142/S0218195996000228.
  16. Ashim Garg and Adrian Rusu. Area-efficient order-preserving planar straight-line drawings of ordered trees. Int. J. Comput. Geometry Appl., 13(6):487-505, 2003. URL: http://dx.doi.org/10.1142/S021819590300130X.
  17. Ashim Garg and Adrian Rusu. Straight-line drawings of general trees with linear area and arbitrary aspect ratio. In Proc. 3rd International Conference on Computational Science and Its Applications (ICCSA), Part III, pages 876-885, 2003. URL: http://dx.doi.org/10.1007/3-540-44842-X_89.
  18. Ashim Garg and Adrian Rusu. Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algorithms Appl., 8(2):135-160, 2004. URL: http://jgaa.info/accepted/2004/GargRusu2004.8.2.pdf.
  19. Stephanie Lee. Upward octagonal drawings of ternary trees. Master’s thesis, University of Waterloo, 2016. (Supervised by T. Biedl and T. M. Chan.) URL: https://uwspace.uwaterloo.ca/handle/10012/10832.
  20. Charles E. Leiserson. Area-efficient graph layouts (for VLSI). In Proc. 21st IEEE Symposium on Foundations of Computer Science (FOCS), pages 270-281, 1980. URL: http://dx.doi.org/10.1109/SFCS.1980.13.
  21. Jiří Matoušek. Range searching with efficient hiearchical cutting. Discrete & Computational Geometry, 10:157-182, 1993. URL: http://dx.doi.org/10.1007/BF02573972.
  22. Edward M. Reingold and John S. Tilford. Tidier drawings of trees. IEEE Trans. Software Eng., 7(2):223-228, 1981. URL: http://dx.doi.org/10.1109/TSE.1981.234519.
  23. Walter Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 138-148, 1990. URL: http://dl.acm.org/citation.cfm?id=320176.320191.
  24. Yossi Shiloach. Linear and Planar Arrangement of Graphs. PhD thesis, Weizmann Institute of Science, 1976. URL: https://lib-phds1.weizmann.ac.il/Dissertations/shiloach_yossi.pdf.
  25. Chan-Su Shin, Sung Kwon Kim, and Kyung-Yong Chwa. Area-efficient algorithms for straight-line tree drawings. Comput. Geom., 15(4):175-202, 2000. Preliminary version in COCOON'96. URL: http://dx.doi.org/10.1016/S0925-7721(99)00053-X.
  26. Chan-Su Shin, Sung Kwon Kim, Sung-Ho Kim, and Kyung-Yong Chwa. Algorithms for drawing binary trees in the plane. Inf. Process. Lett., 66(3):133-139, 1998. URL: http://dx.doi.org/10.1016/S0020-0190(98)00049-0.
  27. Luca Trevisan. A note on minimum-area upward drawing of complete and fibonacci trees. Inf. Process. Lett., 57(5):231-236, 1996. URL: http://dx.doi.org/10.1016/0020-0190(96)81422-0.
  28. Leslie G. Valiant. Universality considerations in VLSI circuits. IEEE Trans. Computers, 30(2):135-140, 1981. URL: http://dx.doi.org/10.1109/TC.1981.6312176.
  29. Vijay V. Vazirani. Approximation Algorithms. Springer, 2001. URL: https://www.cc.gatech.edu/fac/Vijay.Vazirani/book.pdf.
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