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