Maximum Sum

Background

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.

Description

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 tex2html_wrap_inline33 or greater located within the whole array. As an example, the maximal sub-rectangle of the array:

$$ \begin{matrix}0 & −2 & −7 & 0 \\ 9 & 2 & −6 & 2 \\ −4 & 1 & −4 & 1 \\ −1 & 8 & 0 & −2\end{matrix} $$

is in the lower-left-hand corner:

$$ \begin{matrix} 9 & 2 \\ -4 & 1 \\ -1 & 8 \end{matrix} $$

and has the sum of 15.

Input format

The input consists of an $N*N$ 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 $N^{2}$ integers separated by white-space (newlines and spaces). These $N^{2}$ 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 $[-127, 127]$.

Output format

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