Large k-Gons in a 1.5D Terrain

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 object of exactly or at most k vertices inside $\mathcal {T}$. For a constant $k\ge 3$, we design a near linear time FPTAS that approximates the largest convex polygons with at most k vertices, within a factor $(1-\epsilon )$. For the case where $k=2$, we discuss an O(n) time exact algorithm for computing the longest line segment in $\mathcal {T}$, and for $k=3$, we design an $O(n^2)$ time exact algorithm for computing the largest-perimeter triangle that lies within $\mathcal {T}$.