This problem is a Mooshak version of a UVA Online Judge Problem.

[PC021] Maximum Sum

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 1 × 1 or greater located within the whole array. As an example, the maximal sub-rectangle of the array:

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

is in the lower-left-hand corner:

 9 2
−4 1
−1 8

and has the sum of 15.

Input

The input consists of an \(N \times N\) 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 \(N\) lines describing the matrix, each with \(N\) integers \(x_{ij}\) with the contents of each cell.

Output

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

Constraints

• \( 1 \leq n \leq 300 \)
• \( -100 \leq x_{ij} \leq 100 \)

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

15