Maximum-Area Rectangles in a Simple Polygon

TL;DR

This paper presents algorithms for finding maximum-area rectangles within simple polygons, including those with holes, with improved time complexity and the ability to report all such rectangles, extending previous work on convex and axis-aligned cases.

Contribution

It introduces an $O(n^3 ext{log} n)$ time algorithm for simple polygons with holes and a simpler $O(n^3)$ algorithm for convex polygons, advancing the computational methods for maximum-area rectangle problems.

Findings

Algorithm for simple polygons runs in $O(n^3 ext{log} n)$ time.

Algorithm for convex polygons runs in $O(n^3)$ time.

Can report all maximum-area rectangles efficiently.

Abstract

We study the problem of finding maximum-area rectangles contained in a polygon in the plane. There has been a fair amount of work for this problem when the rectangles have to be axis-aligned or when the polygon is convex. We consider this problem in a simple polygon with $n$ vertices, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle in $O(n^3\log n)$ time using $O(kn^2)$ space, where $k$ is the number of reflex vertices of $P$. Our algorithm can report all maximum-area rectangles in the same time using $O(n^3)$ space. We also present a simple algorithm that finds a maximum-area rectangle contained in a convex polygon with $n$ vertices in $O(n^3)$ time using $O(n)$ space.