Pole Dancing: 3D Morphs for Tree Drawings

TL;DR

This paper investigates the minimal number of linear steps needed to morph between two tree drawings in 3D without crossings, providing bounds for both 2D and 3D cases.

Contribution

It establishes that crossing-free 3D morphs between tree drawings can be achieved in logarithmic steps in 2D and linear steps in 3D, with tight bounds.

Findings

O(log n) steps suffice in 2D

Theta(n) steps are necessary in 3D

Crossing-free morphs are always possible within these bounds

Abstract

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O(\log n)$ steps, while for the latter $\Theta(n)$ steps are always sufficient and sometimes necessary.