Morphing tree drawings in a small 3D grid
Elena Arseneva, Rahul Gangopadhyay, Aleksandra Istomina

TL;DR
This paper presents a new method for morphing planar tree drawings in 3D that significantly reduces the number of steps while maintaining bounded resolution, improving upon previous approaches.
Contribution
It introduces a 3D crossing-free morph between two planar grid drawings of a tree with fewer steps and bounded resolution, using a novel approach.
Findings
Achieves $ ext{O}(\sqrt{n} ext{log} n)$ morphing steps
Maintains polynomial volume in intermediate drawings
Improves step complexity over previous $ ext{O}( ext{log} n)$ bounds
Abstract
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 speeds. A crossing-free morph is known between two drawings of an -vertex planar graph with morphing steps and using the third dimension it can be reduced to for an -vertex tree [Arseneva et al.\ 2019]. However, these morphs do not bound one practical parameter, the resolution. Can the number of steps be reduced substantially by using the third dimension while keeping the resolution bounded throughout the morph? We answer this question in an affirmative and present a 3D non-crossing morph between two planar grid drawings of an -vertex tree in morphing steps. Each intermediate drawing lies in a grid of polynomial…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Cellular Automata and Applications · Advanced Materials and Mechanics
