Linear time Minimum Area All-flush Triangles Circumscribing a Convex Polygon

TL;DR

This paper presents an $O(n)$ time algorithm for computing the minimum area triangle that circumscribes a convex polygon, touching edges in an edge-to-edge manner, improving over previous $O(n extlog n)$ solutions.

Contribution

It introduces a linear-time algorithm based on the Rotate-and-Kill technique for finding the minimum area circumscribing triangle.

Findings

Algorithm runs in $O(n)$ time.

Computes all locally minimal area circumscribing triangles.

Improves previous algorithms with $O(n extlog n)$ complexity.

Abstract

We study the problem of computing the minimum area triangle that circumscribes a given $n$-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of $n$ half-planes defined by a given convex polygon. Building on the Rotate-and-Kill technique {Arxiv:1707.04071}, we propose an algorithm that solves the problem in $O(n)$ time, improving the best-known $O(n\log n)$ time algorithms given in [A. Aggarwal et. al. DCG94; B. Schieber. SODA95}. Our algorithm computes all the locally minimal area circumscribing triangles touching edge-to-edge.