[Paper Review] Optimal Monotone Drawings of Trees

Dayu He, Xin He|arXiv (Cornell University)|Apr 13, 2016

Computational Geometry and Mesh Generation16 references3 citations

TL;DR

This paper presents an O(n)-time algorithm to construct monotone drawings of trees on a 12n×12n grid, achieving asymptotically optimal space by using a novel Path Draw algorithm and careful assignment of primitive vectors to tree paths. The key contribution is proving that such drawings are optimal up to a constant factor, as any monotone drawing of certain trees requires Ω(n)×Ω(n) grid space.

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 abpath 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}) x 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 x 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 Omega(n) x Omega(n). So the size of our monotone drawing of trees is asymptotically optimal.

Motivation & Objective

• To reduce the grid size required for monotone drawings of trees, improving upon prior O(n^1.205) bounds.
• To develop a practical and efficient algorithm for constructing monotone drawings with minimal space usage.
• To establish a theoretical lower bound showing that Ω(n)×Ω(n) grid is necessary for some trees, proving asymptotic optimality of the proposed construction.
• To close the gap between upper and lower bounds on monotone drawing size for trees by refining vector assignment and valid-pair analysis.

Proposed method

• Proposes a new Path Draw algorithm that assigns primitive vectors to tree paths based on a refined analysis of the Stern-Brocot tree structure.
• Uses a valid-pair technique to bound the size of fractions in the Stern-Brocot tree, ensuring that vector assignments remain within a controlled range.
• Applies a coordinate assignment strategy that maintains monotonicity by ensuring edge directions are consistent with monotone paths in specific directions.
• Employs a post-order traversal of the tree to assign vectors to edges, preserving monotonicity across all vertex pairs.
• Leverages properties of Fibonacci numbers and Farey sequences to compactly represent and bound the slopes of primitive vectors.
• Uses geometric and angular reasoning to prove that certain edge configurations in a tree force grid expansion, establishing the lower bound.

Experimental results

Research questions

• RQ1 Can monotone drawings of trees be constructed on a grid smaller than O(n^1.205)×O(n^1.205)?
• RQ2 Is it possible to achieve a monotone drawing of any n-vertex tree within a 12n×12n grid using a simple and efficient algorithm?
• RQ3 What is the minimum grid size required for monotone drawings of trees, and is this bound asymptotically tight?
• RQ4 Can the constant factor in the grid size be improved by refining the analysis of primitive vector assignments?

Key findings

• The proposed algorithm constructs monotone drawings of any n-vertex tree on a 12n×12n grid in O(n) time.
• The grid size of 12n×12n is asymptotically optimal, as there exists a tree T₀ requiring Ω(n)×Ω(n) grid space in any monotone drawing.
• The lower bound is proven by analyzing angular constraints in a tree with 12 edge-disjoint paths, showing that at least one path must span Ω(n) units in both x and y directions.
• The method improves upon prior work by reducing the grid size from O(n^1.205) to a linear factor of 12n, with a tighter analysis of vector assignment.
• The proof of the lower bound relies on angle monotonicity and the pigeonhole principle to force grid expansion in a quadrant.
• The authors identify that the current bound on valid-pairs in the Stern-Brocot tree is not tight, suggesting potential for further constant-factor improvements.