LR-Drawings of Ordered Rooted Binary Trees and Near-Linear Area Drawings of Outerplanar Graphs

TL;DR

This paper advances the understanding of LR-drawings for ordered rooted binary trees by providing optimal width algorithms, establishing new lower bounds, and connecting these results to outerplanar graph drawings with near-linear area.

Contribution

It introduces an algorithm for constructing minimum-width LR-drawings in sub-quadratic time and relates LR-drawings to outerplanar graph drawings, improving area bounds.

Findings

Optimal LR-drawings with minimum width can be constructed in O(n^1.48) time.

There exist trees requiring Omega(n^0.418) width in any LR-drawing.

Outerplanar graphs can be drawn in near-linear area using the structural decomposition of trees.

Abstract

In this paper we study a family of algorithms, introduced by Chan [SODA 1999] and called LR-algorithms, for drawing ordered rooted binary trees. In particular, we are interested in constructing LR-drawings (that are drawings obtained via LR-algorithms) with small width. Chan showed three different LR-algorithms that achieve, for an ordered rooted binary tree with $n$ nodes, width $O(n^{0.695})$, width $O(n^{0.5})$, and width $O(n^{0.48})$. We prove that, for every $n$-node ordered rooted binary tree, an LR-drawing with minimum width can be constructed in $O(n^{1.48})$ time. Further, we show an infinite family of $n$-node ordered rooted binary trees requiring $\Omega(n^{0.418})$ width in any LR-drawing; no lower bound better than $\Omega(\log n)$ was previously known. Finally, we present the results of an experimental evaluation that allowed us to determine the minimum width of all the ordered rooted binary trees with up to $451$ nodes. Our interest in LR-drawings is mainly motivated by a result of Di Battista and Frati [Algorithmica 2009], who proved that $n$-vertex outerplanar graphs have outerplanar straight-line drawings in $O(n^{1.48})$ area by means of a drawing algorithm which resembles an LR-algorithm. We deepen the connection between LR-drawings and outerplanar straight-line drawings by proving that, if $n$-node ordered rooted binary trees have LR-drawings with $f(n)$ width, for any function $f(n)$, then $n$-vertex outerplanar graphs have outerplanar straight-line drawings in $O(f(n))$ area. Finally, we exploit a structural decomposition for ordered rooted binary trees introduced by Chan in order to prove that every $n$-vertex outerplanar graph has an outerplanar straight-line drawing in $O(n\cdot 2^{\sqrt{2 \log_2 n}} \sqrt{\log n})$ area.