Morphing between two planar drawings using an extra dimension is relatively new area of research. Also, the morphs with polynomially-bounded resolution has gained a lot of attention in recent years. Recent results use linear number of morphing stepsto morph between two planar straight-line grid drawings of an n-vertex tree while maintaining the resolution.In this paper, we bring together both the paradigms to develop an algorithm that morphs(n) morphing steps and sometimes \Omega(n) steps is necessary even when G is a path~}[ (\log n) for an n-vertex tree }[ [  between two planar straight-line grid drawings of an n-vertex tree T in sublinear number of morphing steps such that each intermediate morphing step produces a 3D straight-line crossing free grid drawing of T while maintaining the resolution. To be precise, we first devise an algorithm that takes O(n) morphing steps but uses a grid of small volume. We then use this algorithm to devise a better algorithm that takes \mathcal{O}(\sqrt{n} \log n) morphing steps. We also ensure that each intermediate drawing lies in a 3D grid of polynomially-bounded volume. To the best of our knowledge, this is the first morphing algorithm that takes sublinear number of morphing steps while maintaining a polynomially-bounded grid size, and thus the resolution .
We study crossing-free grid morphs for planar tree drawings using 3D. A morph consists of morphing steps, where vertices move simultaneously along straight-line trajectories at constant speed. A crossing-free morph is known between two drawings of an n-vertex planar graph G with \mathcal{O(n) morphing steps and sometimes \Omega(n) steps is necessary even when G is a path~}[Alamdari et al.\ 2017 , while using 3D can reduce the number of steps to \mathcal{O(\log n) for an n-vertex tree }[Arseneva et al.\ 2019 . However, these morphs do not bound one practical parameter, the resolution. Recently[Barrera-Cruz et al.\ 2019 a 2D crossing-free morph was devised between two planar drawings of an n-vertex tree in O(n) steps with polynomial resolution, that is, all intermediate drawings are grid drawings on a grid of polynomial size. Can the number of steps in this setting be reduced substantially by using 3D? We answer this question in affirmative and present a 3D non-crossing morph between two planar grid drawings of an n-vertex tree in \mathcal{O}(\sqrt{n} \log n) morphing steps. Each intermediate drawing lies in a 3D grid of polynomial volume .
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Morphing between two planar drawings using an extra dimension is relatively new area of research. Also, the morphs with polynomially-bounded resolution has gained a lot of attention in recent years. Recent results use linear number of morphing stepsto morph between two planar
We study crossing-free grid morphs for planar tree drawings using 3D. A morph consists of morphing steps, where vertices move simultaneously along
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grid
trajectories at constant speed. A crossing-free morph is known between two
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tree while maintaining the resolution.In this paper, we bring together both the paradigms to develop an algorithm that morphs
planar graph G with \mathcal{O
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Alamdari et al.\ 2017
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, while using 3D can reduce the number of steps to \mathcal{O
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Arseneva et al.\ 2019
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. However, these morphs do not bound one practical parameter, the resolution. Recently
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Barrera-Cruz et al.\ 2019
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a 2D crossing-free morph was devised
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straight-line grid
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T in sublinear number of morphing steps such that each intermediate morphing step produces a 3D straight-line crossing free grid drawing of T while maintaining the resolution. To be precise, we first devise an algorithm that takes
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morphing steps but uses
steps with polynomial resolution, that is, all intermediate drawings are grid drawings on
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small volume. We then use this algorithm to devise a better algorithm that takes
polynomial size. Can the number of steps in this setting be reduced substantially by using 3D? We answer this question in affirmative and present a 3D non-crossing morph between two planar grid drawings of an n-vertex tree in
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We also ensure that each
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polynomially-bounded volume. To the best of our knowledge, this is the first morphing algorithm that takes sublinear number of morphing steps while maintaining a polynomially-bounded grid size, and thus the resolution
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