Large k-gons in a 1.5D Terrain
Given is a 1.5D terrain π―, i.e., an x-monotone polygonal chain in β^2. For a given 2β€ kβ€ n, our objective is to approximate the largest area or perimeter convex polygon of exactly or at most k vertices inside π―. 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-Ο΅). For the case where k=2, we design an O(n) time exact algorithm for computing the longest line segment in π―, and for k=3, we design an O(n log n) time exact algorithm for computing the largest-perimeter triangle that lies within π―.
READ FULL TEXT