Morphing tree drawings in a small 3D grid

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 $n$-vertex planar graph $G$ with $\mathcal{O}(n)$ morphing steps and using the third dimension it can be reduced 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. 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 $n$-vertex tree in $\mathcal{O}(\sqrt{n} \log n)$ morphing steps. Each intermediate drawing lies in a $3D$ grid of polynomial volume.