Finding the Maximum Area Axis-parallel Rectangle in a Polygon

We consider the geometric optimization problem of nding the maximum area axis-parallel rectangle (MAAPR) in an n-vertex general polygon. We characterize the MAAPR for general polygons by considering di erent cases based on the types of contacts between the rectangle and the polygon. We present a general framework for solving a key subcase of the MAAPR problem which dominates the running time for a variety of polygon types. Using this framework, we obtain the following MAAPR time results: (n) for xy-monotone polygons, O(n (n)) for orthogonally convex polygons, O(n (n) logn) for horizontally (vertically) convex polygons, (where (n) is the slowly growing inverse of Ackermann's function), O(n logn) for a special type of horizontally convex polygon, and O(n log2 n) for general polygons. For all these types of non-rectilinear polygons, we match the running time of the best known algorithms for their rectilinear counterparts. We prove a lower bound of time in (n logn) for nding the MAAPR in both selfintersecting polygons and general polygons with holes. The latter result gives us both a lower bound of (n logn) and an upper bound of O(n log2 n) for general polygons with holes.