Upward drawings of search trees
We prove that any logarithmic binary tree admits a linear-area straight-line strictly-upward planar grid drawing (in short, upward drawing), that is, a drawing in which (a) each edge is mapped into a single straight-line segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point with integer coordinates. Informally, a logarithmic tree has the property that the height of any (sufficiently high) subtree is logarithmic with respect to the number of nodes. As a consequence, we have that k-balanced trees, red-black trees, and BB[α]-trees admit linear-area upward drawings. We then generalize our results to logarithmic m-ary trees: as an application, we have that B-trees admit linear-area upward drawings.
Unable to display preview. Download preview PDF.
- 1.C.-S. Shin, S. K. Kim, and K.-Y. Chwa. Area-efficient algorithms for upward straight-line drawings. Proc. COCOON '96, to appear.Google Scholar
- 2.T.H. Cormen, C.E. Leierson, and R.L. Rivest. Introduction to Algorithms, Mc Graw Hill, 1990.Google Scholar
- 3.P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, 2:187–200, 1992.Google Scholar
- 4.P. Crescenzi, P. Penna, and A. Piperno. Linear area upward drawings of AVL trees. Computational Geometry: Theory and Applications, to appear.Google Scholar
- 5.P. Crescenzi and A. Piperno. Optimal-area upward drawings of AVL trees. Proc. Graph Drawing, 307–317, 1994.Google Scholar
- 6.G. Di Battista, P. Eades, R. Tamassia, and I. Tollis. Algorithms for drawing graphs: an annotated bibliography. Computational Geometry: Theory and Applications, 4:235–282, 1994.Google Scholar
- 7.P. Eades, T. Lin, and X. Lin. Minimum size h-v drawings. In Proc. Int. Workshop AVI '92, 386–394, 1992.Google Scholar
- 8.A. Garg, M.T. Goodrich, and R. Tamassia. Area-efficient upward tree drawing. In Proc. ACM Symp. on Computational Geometry, 359–368, 1993.Google Scholar
- 9.A. Garg and R. Tamassia. On the computational complexity of upward and rectilinear planarity testing. Proc. Graph Drawing, 286–297, 1994.Google Scholar
- 10.D.E. Knuth. The Art of Computer Programming: Sorting and Searching, Addison Wesley, 1975.Google Scholar
- 11.Y. Shiloach. Linear and planar arrangements of graphs. Ph.D. Thesis, Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel, 1976.Google Scholar
- 12.L. Valiant. Universality considerations in VLSI circuits. IEEE Trans. on Computers, C-30(2):135–140, 1981.Google Scholar