Simple Compact Monotone Tree Drawings

TL;DR

The paper addresses monotone straight-line drawings of trees and introduces a hierarchy of algorithms that drastically tighten grid-area bounds. It starts with a simple one-quadrant method for rooted ordered trees on an $n \times n$ grid, then extends to a two-quadrant approach for unrooted ordered trees and a four-quadrant strategy for unrooted non-ordered trees, achieving grid sizes of $n \times \frac{n+1}{2}$ (odd $n$) or $(n+1) \times (\frac{n}{2}+1)$ (even $n$) and $\left\lfloor \frac{3}{4}(n+2) \right\rfloor \times \left\lfloor \frac{3}{4}(n+2) \right\rfloor$, respectively. The authors also present convex-monotone drawings on the $n \times n$ grid via a convexification procedure and discuss rooting choices (gravity roots) to control subtree sizes, ultimately delivering a compact, hierarchy-preserving suite of algorithms that improves prior $12n \times 12n$ results and opens questions about lower bounds and angular-resolution trade-offs. The work leverages the slope-disjoint framework and geometric grid-location lemmas to guarantee monotonicity and planarity while achieving strong area bounds.

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 $\left\lfloor \frac{3}{4} \left(n+2\right)\right\rfloor \times \left\lfloor \frac{3}{4} \left(n+2\right)\right\rfloor$.

Figures (10)

• Figure 1: Geometric representation of \ref{['lemma:smallAngle']}.
• Figure 1: A full binary tree (31 vertices) as drawn by \ref{['algo:1_quadrant']}. Grid size: $23 \times 22$.
• Figure 1: Non-convex monotone drawing of a small tree (12 vertices) by \ref{['algo:1_quadrant']}: the indicated angle at the central fork-highlighted via a red arc-is non-convex.
• Figure 1: A full binary tree (31 vertices) as drawn by \ref{['algo:2_quadrants']}. Grid size: $23 \times 12$.
• Figure 1: Example of how does \ref{['algo:4_quadrants']} places $T_1$ and $T_2$.

Theorems & Definitions (43)

• Theorem 1: Angelini et al.m:1
• Theorem 2: Angelini et al.m:1
• Definition 3
• Lemma 4
• Proof 1
• Theorem 5
• Proof 2
• Lemma 6
• Proof 3
• Lemma 7