In search of the Giant Convex Quadrilateral hidden in the Mountains

TL;DR

This paper presents an $O(n^2)$ algorithm to find the largest-area convex quadrilateral within a 1.5D terrain and shows that the largest axis-parallel rectangle provides a 50% approximation for this problem.

Contribution

The paper introduces a quadratic time algorithm for the maximum-area convex quadrilateral in 1.5D terrains and establishes an approximation method using axis-parallel rectangles.

Findings

The algorithm runs in $O(n^2)$ time.

Largest axis-parallel rectangle approximates the maximum convex quadrilateral within 50%.

Provides a practical approach to a geometric optimization problem.

Abstract

A $1.5$D terrain is a simple polygon bounded by a line segment $\ell$ and a polygonal chain monotone with respect to the line segment $\ell$. Usually, $\ell$ is chosen aligned to the $x$-axis, and is called the base of the terrain. In this paper, we consider the problem of finding a convex quadrilateral of largest area inside a $1.5$D terrain in $\mathbb{R}^2$. We present an $O(n^2)$ time algorithm for this problem, where $n$ is the number of vertices of the terrain. Finally, we show that the largest area axis-parallel rectangle inside the terrain yields a $\frac{1}{2}$-approximation result to the largest convex quadrilateral problem.