UVA 108: Maximum Sum (solution)

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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.

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 x 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 / Output

The input consists of an NxN 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].

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

Sample Input

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

Sample Output



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