Finding largest rectangle in 2D array

Question

I need an algorithm which can parse a 2D array and return the largest continuous rectangle. For reference, look at the image I made demonstrating my question.

Answer 1

Use dynamic programming approach. Consider a function S(x,y) such that S(x,y) holds the area of the largest rectangle where (x,y) are the lowest-right-most corner cell of the rectangle; x is the row co-ordinate and y is the column co-ordinate of the rectangle.

For example, in your figure, S(1,1) = 1, S(1,2)=2, S(2,1)=2, and S(2,2) = 4. But, S(3,1)=0, because this cell is filled. S(8,5)=40, which says that the largest rectangle for which the lowest-right cell is (8,5) has the area 40, which happens to be the optimum solution in this example.

You can easily write a dynamic programming equation of S(x,y) from the value of S(x-1,y), S(x,y-1) and S(x-1,y-1). Using that you can obtain the values of all S(x,y) in O(mn) time, where m and n are the row and column dimension of the given table. Once, S(x,y) are know for all 1<=x <= m, and for all 1 <= y <= n, we simply need to find the x, and y for which S(x,y) is the largest; this step also takes O(mn) time. By keeping addition data, you can also find the side-length of the largest rectangle.

The overall complexity is O(mn). To understand more on this, Read Chapter 15 or Cormen's algorithm book, specifically Section 15.4.