Finding the largest area axis-parallel rectangle in a polygon

This paper considers the geometric optimization problem of finding the Largest area axis-parallel Rectangle (LR) in an n-vertex general polygon. We characterize the LR for general polygons by considering different cases based on the types of contacts between the rectangle and the polygon. A general framework is presented for solving a key subproblem of the LR problem which dominates the running time for a variety of polygon types. This framework permits us to transform an algorithm for orthogonal polygons into an algorithm for non-orthogonal polygons. Using this framework, we show that the LR in a general polygon (allowing holes) can be found in O(n log² n) time. This matches the running time of the best known algorithm for orthogonal polygons. References are given for the application of the framework to other types of polygons. For each type, the running time of the resulting algorithm matches the running time of the best known algorithm for orthogonal polygons of that type. A lower bound of time in Ω(n log n) is established for finding the LR in both self-intersecting polygons and general polygons with holes. The latter result gives us both a lower bound of Ω(n log n) and an upper bound of O(n log² n) for general polygons.