Simple Compact Monotone Tree Drawings

TL;DR

This paper presents simple algorithms for creating compact monotone drawings of various types of trees, improving grid size efficiency while maintaining key properties like ordering and rootedness.

Contribution

The paper introduces new, straightforward algorithms for monotone tree drawings that are more space-efficient and adaptable to different tree types and constraints.

Findings

Rooted trees can be drawn in an n x n grid.

Unrooted ordered trees fit in an (n+1) x (n/2+1) grid.

Unrooted non-ordered trees fit in a roughly (3/4)(n+2) by (3/4)(n+2) grid.

Abstract

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He~\cite{mt:4} showed how to produce a monotone drawing of an arbitrary $n$-vertex tree that is contained in a $12n \times 12n$ grid. All monotone tree drawing algorithms that have appeared in the literature consider rooted ordered trees and they draw them so that (i) the root of the tree is drawn at the origin of the drawing, (ii) the drawing is confined in the first quadrant, and (iii) the ordering/embedding of the tree is respected. In this paper, we provide a simple algorithm that has the exact same characteristics and, given an $n$-vertex rooted tree $T$, it outputs a monotone drawing of $T$ that fits on a $n \times n$ grid. For unrooted ordered trees, we present an algorithms that produces monotone drawings that respect the ordering and fit in an $(n+1) \times (\frac{n}{2} +1)$ grid, while, for unrooted non-ordered trees we produce monotone drawings of good aspect ratio which fit on a grid of size at most $\left\lfloor \frac{3}{4} \left(n+2\right)\right\rfloor \times \left\lfloor \frac{3}{4} \left(n+2\right)\right\rfloor$.