Large $k$-gons in a 1.5D Terrain

TL;DR

This paper develops efficient algorithms to approximate or exactly compute the largest convex polygons with a fixed number of vertices inside a 1.5D terrain, including special cases for line segments and triangles.

Contribution

It introduces an FPTAS for approximating largest convex polygons with fixed vertices and provides exact algorithms for longest segments and largest triangles within the terrain.

Findings

FPTAS achieves (1-ε) approximation for fixed k

Exact linear-time algorithm for longest segment (k=2)

Efficient n log n algorithm for largest triangle (k=3)

Abstract

Given is a 1.5D terrain $\mathcal{T}$, i.e., an $x$-monotone polygonal chain in $\mathbb{R}^2$. For a given $2\le k\le n$, our objective is to approximate the largest area or perimeter convex polygon of exactly or at most $k$ vertices inside $\mathcal{T}$. For a constant $k>3$, we design an FPTAS that efficiently approximates the largest convex polygons with at most $k$ vertices, within a factor $(1-\epsilon)$. For the case where $k=2$, we design an $O(n)$ time exact algorithm for computing the longest line segment in $\mathcal{T}$, and for $k=3$, we design an $O(n \log n)$ time exact algorithm for computing the largest-perimeter triangle that lies within $\mathcal{T}$.