Ideal Tree-drawings of Approximately Optimal Width (And Small Height)

TL;DR

This paper presents approximation algorithms for creating ideal, planar, straight-line, upward, order-preserving tree drawings with minimal width, achieving near-optimal solutions and bounded height for certain tree degrees.

Contribution

It introduces the first known 2-approximation algorithm for the minimum width of ideal tree drawings and a $2 riangle$-approximation with linear height for $ riangle$-ary trees.

Findings

A 2-approximation algorithm for minimum width ideal tree drawings.

A $2 riangle$-approximation with $O(n)$ height for $ riangle$-ary trees.

Optimal width solutions for trees with maximum degree 3.

Abstract

For rooted trees, an ideal drawing is one that is planar, straight-line, strictly-upward, and order-preserving. This paper considers ideal drawings of rooted trees with the objective of keeping the width of such drawings small. It is not known whether finding the minimum-possible width is NP-hard or polynomial. This paper gives a 2-approximation for this problem, and a $2\Delta$-approximation (for $\Delta$-ary trees) where additionally the height is $O(n)$. For trees with $\Delta\leq 3$, the former algorithm finds ideal drawings with minimum-possible width.