Finding the largest empty rectangle on a grated surface

Abstract

Suppose we are given a two-dimensional rectangular surface upon which is placed a grating of size n by m square elements. The (n+1) × (m+1) intersection points of this grid are either empty or occupied. We describe an O(n × m) algorithm for finding the largest, in area, empty subrectangle of the original rectangle. The algorithm was inspired by the dynamic programming [1] and plane-sweep [5] paradigms.

Previous algorithms have been given for the similar problem in which the occupied points are given as points within a two-dimensional continuum, i. e., not restricted to grid crossings. One previous algorithm [4] for this problem has O(F ²) worse case and O(F log² F) expected time, while another [3] has O(F log³ F) time, where F is the number of occupied points. We compare this result to our own, with special consideration to intended application areas and to the situation where the number of occupied points is proportional to the area of the original rectangle.