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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).

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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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