Abstract
We prove that every n-node ternary tree has a planar straight-line orthogonal drawing in \(O(n^{1.576})\) area, improving upon the previously best known \(O(n^{1.631})\) bound. Further, we present an upper bound, the outcomes of an experimental evaluation, and a conjecture on the area requirements of planar straight-line orthogonal drawings of complete ternary trees.
Partially supported by MIUR Project “MODE” under PRIN 20157EFM5C and by H2020-MSCA-RISE project 734922 – “CONNECT”.
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Notes
- 1.
Drawing algorithms usually assume trees to be rooted, that is, to have a distinguished node, called root. Trees of maximum degree 3 and 4 are then called binary and ternary trees, respectively. We will use standard terminology on rooted trees, like child, subtree, and leaf; refer to [5, 6, 11]. A tree is complete if every non-leaf node has the same number of children and every root-to-leaf path has the same length.
- 2.
We used the software at www.wolframalpha.com in order to solve the inequality.
- 3.
We claim that Theorem 4 can be generalized to ternary trees that are not necessarily complete. However, since our main interest in 1-2 drawings comes from the study of the area requirements of complete ternary trees, we opted for keeping the exposition simple and present the theorem and its proof for complete ternary trees only.
References
Ali, A.: Straight line orthogonal drawings of complete ternery trees. MIT Summer Program in Undergraduate Research Final Paper, July 2015
Chan, T.M.: Tree drawings revisited. In: Speckmann, B., Tóth, C.D. (eds.) Symposium on Computational Geometry (SoCG 2018) (2018)
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)
Crescenzi, P., Di Battista, G., Piperno, A.: A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. 2, 187–200 (1992)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood (1999)
Di Battista, G., Frati, F.: Drawing trees, outerplanar graphs, series-parallel graphs, and planar graphs in a 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
Frati, F.: Straight-line orthogonal drawings of binary and ternary trees. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 76–87. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77537-9_11
Frati, F., Patrignani, M., Roselli, V.: LR-drawings of ordered rooted binary trees and near-linear area drawings of outerplanar graphs. In: Klein, P.N. (ed.) Symposium on Discrete Algorithms (SODA 2017), pp. 1980–1999. SIAM (2017)
Garg, A., Rusu, A.: Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 320–332. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36151-0_30
Nomura, K., Tayu, S., Ueno, S.: On the orthogonal drawing of outerplanar graphs. IEICE Trans. 88-A(6), 1583–1588 (2005)
Rusu, A.: Tree drawing algorithms. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, pp. 155–192. CRC Press (2016). Chap. 5
Rusu, A., Santiago, C.: Grid drawings of binary trees: an experimental study. J. Graph Algorithms Appl. 12(2), 131–195 (2008)
Shiloach, Y.: Linear and Planar Arrangement of Graphs. Ph.D. thesis, Weizmann Institute of Science, Rehovot (1976)
Shin, C., Kim, S.K., Chwa, K.: Area-efficient algorithms for straight-line tree drawings. Comput. Geom. 15(4), 175–202 (2000)
Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)
Wolfram Research Inc.: Mathematica 10 (2014). http://www.wolfram.com
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Covella, B., Frati, F., Patrignani, M. (2018). On the Area Requirements of Straight-Line Orthogonal Drawings of Ternary Trees. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_11
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