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Improved Upper and Lower Bounds for LR Drawings of Binary Trees

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Graph Drawing and Network Visualization (GD 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12590))

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Abstract

In SODA’99, Chan introduced a simple type of planar straight-line upward order-preserving drawings of binary trees, known as LR drawings: such a drawing is obtained by picking a root-to-leaf path, drawing the path as a straight line, and recursively drawing the subtrees along the paths. Chan proved that any binary tree with n nodes admits an LR drawing with \(O(n^{0.48})\) width. In SODA’17, Frati, Patrignani, and Roselli proved that there exist families of n-node binary trees for which any LR drawing has \(\varOmega (n^{0.418})\) width. In this paper, we improve Chan’s upper bound to \(O(n^{0.437})\) and Frati et al.’s lower bound to \(\varOmega (n^{0.429})\).

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Notes

  1. 1.

    Technically, that survey asks about a different but related function: \(W^{**}(n) = \max _T \min _\pi \max _{\alpha ,\beta } (W^{**}(|\alpha |) + W^{**}(|\beta |) + 1)\), where the outer maximum is over all n-node binary trees T. This function may be larger than \(\max _{T: |T|=n} W^*(T)\).

  2. 2.

    All edges have slope from \(\{0,\pm 1,\pm \infty \}\).

  3. 3.

    All edges are horizontal or vertical.

  4. 4.

    Hölder’s inequality states that \(\sum _i |x_iy_i|\le \left( \sum _i |x_i|^s\right) ^{1/s} \left( \sum _i |y_i|^t\right) ^{1/t}\) for any \(s,t>1\) with \(\frac{1}{s}+\frac{1}{t}=1\). In our applications, it is more convenient to set \(s=\frac{1}{p}\), \(t=\frac{1}{1-p}\), \(x_i=X_i^p\), and \(y_i=c_i\), and rephrase the inequality as: \(\sum _i c_iX_i^p \le \left( \sum _i c_i^{1/(1-p)}\right) ^{1-p} \left( \sum _i X_i\right) ^p\) for any \(0<p<1\) and \(c_i,X_i\ge 0\).

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Champaign, USA

    Timothy M. Chan & Zhengcheng Huang

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  1. Timothy M. Chan
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  2. Zhengcheng Huang
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Correspondence to Timothy M. Chan .

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Editors and Affiliations

  1. LaBRI, University of Bordeaux, Talence, France

    David Auber

  2. Charles University, Prague, Czech Republic

    Pavel Valtr

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Chan, T.M., Huang, Z. (2020). Improved Upper and Lower Bounds for LR Drawings of Binary Trees. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-68766-3_6

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