Optimal Monotone Drawings of Trees

Abstract

A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l uw . (Namely, the order of the orthogonal projections of the vertices of P uw on l uw is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205 ) × O(n 1.205 ). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n × 12n. The smaller drawing size is achieved by a new simple Path Draw algorithm, and a procedure that carefully assigns primitive vectors to the paths of the input tree T .We also show that there exists a tree T 0 such that any monotone drawing of T 0 must use a grid of size Ω(n) × Ω(n). So the size of our monotone drawing of trees is asymptotically optimal.

Simple Compact Monotone Tree Drawings

“…The following problems on monotone tree drawings are worth studying: 1. He and He [6] described a tree that requires for its monotone drawing a grid of size at least n 9 × n 9 . Can this bound be improved?…”

“…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 × n grid. Despite its simplicity, our algorithm improves the 12n × 12n result of He and He [6]. By relaxing the drawing restrictions we can achieve smaller drawing area.…”

“…Recently, He and He [5] firstly reduced the grid size for a monotone tree drawing to O(n log(n)) × O(n log(n)) and, in a sequel paper, to O(n) × O(n) [6]. Their monotone tree drawing uses a grid of size at most 12n × 12n which turns out to be asymptotically optimal as there exist trees which require at least n 9 × n 9 area [6].…”

Drawing Graphs as Spanners

“…Even more restricted are angle-monotone drawings [10,20,37] in which, for every pair of vertices u and v, there is a path from u to v such that the angles of any two edges of the path differ by at most 90 • . Finally, monotone drawings [4,6,29,30,32,35] and strongly-monotone drawings [4,25,35] require, for every pair of vertices u and v, that a path from u to v exists that is monotone with respect to some direction or with respect to the direction of the straight line through u and v, respectively. While greedy, monotone, and strongly-monotone drawings might have unbounded spanning ratio, self-approaching, increasing-chord, and angle-monotone drawings are known to have spanning ratio at most 5.34 [33], at most 2.1 [44], and at most 1.42 [10], respectively.…”

Drawing a Rooted Tree as a Rooted $y-$Monotone Minimum Spanning Tree

“…see [15,Chapter 3], [4], [3]. Furthermore, drawing a rooted tree in the plane such that each pair of points is connected by a path that is monotone in some direction, in a grid of small area, is thoroughly investigated [2,9,13].…”