A problem that is simple to solve in one dimension is often much more difficult to solve in more than one dimension. Consider satisfying a boolean expression in conjunctive normal form in which each conjunct consists of exactly 3 disjuncts. This problem (3-SAT) is NP-complete. The problem 2-SAT is solved quite efficiently, however. In contrast, some problems belong to the same complexity class regardless of the dimensionality of the problem.

Given a 2-dimensional array of positive and negative integers, find the
sub-rectangle with the largest sum. The sum of a rectangle is the sum of all the
elements in that rectangle. In this problem the sub-rectangle with the largest
sum is referred to as the *maximal sub-rectangle*. A sub-rectangle is any
contiguous sub-array of size or greater located
within the whole array. As an example, the maximal sub-rectangle of the array:

is in the lower-left-hand corner:

and has the sum of 15.

The input consists of an array of integers. The
input begins with a single positive integer *N* on a line by itself
indicating the size of the square two dimensional array. This is followed by
integers separated by white-space
(newlines and spaces). These integers make up the
array in row-major order (i.e., all numbers on the first row, left-to-right,
then all numbers on the second row, left-to-right, etc.). *N* may be as
large as 100. The numbers in the array will be in the range [-127, 127].

The output is the sum of the maximal sub-rectangle.

4 0 -2 -7 0 9 2 -6 2 -4 1 -4 1 -1 8 0 -2

15