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Tree Drawings Revisited

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; (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; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area \(n2^{O(\sqrt{\log n})}\), improving the previous \(O(n^{3/2})\) bound.

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Notes

  1. 1.

    It is not clear to this author if their analysis assumed a much stronger property, that every subtree of size m has degree at most \(O(m^{1/2-\varepsilon })\).

  2. 2.

    Constants c in different proofs may be different.

  3. 3.

    Alternatively, one can see the solution directly without induction: The contribution of the \((n'/A)s\log ^3 s\) and \((n''/s)\log A\) terms cleary sums to \(O((n/A)s\log ^3 s + (n/s)\log A)\). The contribution of the first \(\log A\) term sums to at most \((2n/A-1)\log A\), because the number of nodes in the recursion tree is at most \(2n/A-1\). This is because we can charge at least A units to each leaf and each out-degree-1 node of the recursion tree in such a way that the total number of charges is at most n (since a leaf has \(n\ge A\), and a out-degree-1 node has \(m=1\) and \(n-n_1\ge A\)). This implies that the number of leaves and out-degree-1 nodes is at most n / A. The number of nodes of out-degree at least 2 is at most the number of leaves minus 1.

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Acknowledgements

I thank the reviewers for their careful reading and comments. Funding was provided by National Science Foundation [Grant No. CCF-1814026].

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Correspondence to Timothy M. Chan.

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A preliminary version of this paper appeared in Proc. 34th Symposium on Computational Geometry (SoCG), 23:1–23:15, 2018.

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Chan, T.M. Tree Drawings Revisited. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00106-w

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Keywords

  • Graph drawing
  • Trees
  • Recursion

Mathematics Subject Classification

  • 05C05
  • 68R10
  • 68U05
  • 68Q25