Finding a Largest-Area Triangle in a Terrain in Near-Linear Time

TL;DR

This paper introduces an efficient algorithm to find the largest-area triangle within a terrain in near-linear time, significantly improving upon the previous quadratic-time solutions.

Contribution

The paper presents a novel $O(n \,\log\, n)$ algorithm for finding the largest-area triangle in an $x$-monotone terrain, advancing computational geometry methods.

Findings

Algorithm runs in $O(n \log n)$ time

Improves upon previous $O(n^2)$ algorithms

Efficiently finds the maximum-area triangle in terrains

Abstract

A terrain is an $x$-monotone polygon whose lower boundary is a single line segment. We present an algorithm to find in a terrain a triangle of largest area in $O(n \log n)$ time, where $n$ is the number of vertices defining the terrain. The best previous algorithm for this problem has a running time of $O(n^2)$.