Abstract
Valiant [1] showed how to embed any binary tree into the plane in linear area without crossovers. The edges in this embedding have a maximum length of 0(\( \sqrt {n} \)) With Paterson, we [2] showed that a complete binary tree can be embedded in the plane with maximum edge length of 0(\( \sqrt {n} \)/log n) and we argued the importance of short edge length for VLSI design and layout. Here we show that every binary tree can be embedded in the plane with all three properties: linear area, no crossovers, and 0(\( \sqrt {n} \)/log n) maximum edge length. This improves a result of Bhatt and Leiserson [3] — a graph with an n1/2-ε separator theorem can be embedded (perhaps with crossovers) in linear area and with a maximum edge length of 0(\( \sqrt {n} \)) — for the case of binary trees. In the paper we also observe that Valiant’s result can be extended to the case of oriented trees [7].
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References
L. G. Valiant Universality Consideration for VLSI Circuits IEEE Transactions on Computers, 1981
M. S. Paterson, W. L. Ruzzo, and L. Snyder Bounds on Minimax Edge Length for Complete Binary Trees Proceedings of the Thirteenth Annual Symposium on the Theory of Computing, 1981
S. N. Bhatt, and C. E. Leiserson Minimizing the Longest Edge in a VLSI Layout Manuscript, MIT, 1981
Hong Jia-Wei, Kurt Mehlhorn, and A. L. Rosenberg Cost Tradeoffs in Graph Embeddings ICALP 81, 1981
W. L. Ruzzo Embedding Trees in Balanced Trees Manuscript, University of Washington, 1981
R. P. Brent, and H. T. Kung On the Area of Binary Tree Layouts CMU Technical Report, 1979
D. E. Knuth Art of Computer Programming Volume I, Addison Wesley, 1968
Carver Mead, and Lynn Conway Introduction to VLSI Systems Addison Wesley, 1980
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© 1981 Carnegie-Mellon University
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Ruzzo, W.L., Snyder, L. (1981). Minimum Edge Length Planar Embeddings of Trees. In: Kung, H.T., Sproull, B., Steele, G. (eds) VLSI Systems and Computations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68402-9_14
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DOI: https://doi.org/10.1007/978-3-642-68402-9_14
Publisher Name: Springer, Berlin, Heidelberg
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