Improved Bounds for Drawing Trees on Fixed Points with L-shaped Edges

TL;DR

This paper improves bounds on the size of point sets needed to draw trees with L-shaped edges in the plane, providing new methods and bounds for various tree degrees and types, including ordered trees.

Contribution

The paper introduces new drawing methods that significantly improve bounds on point set sizes for L-shaped tree drawings, especially for trees with maximum degree 4 and ordered caterpillars.

Findings

Bound of O(n^{1.55}) points for degree 4 trees

Bound of O(n^{1.22}) points for degree 3 trees

Bound of O(n^{1.142}) points for perfect binary trees

Abstract

Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for maximum degree 4 trees; $O(n^{1.22})$ for maximum degree 3 (binary) trees; and $O(n^{1.142})$ for perfect binary trees. Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.