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- 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.
- 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 tex2html_wrap_inline33 or greater located within the whole array. As an example, the maximal sub-rectangle of the array:
- displaymath35
- is in the lower-left-hand corner:
- displaymath37
- and has the sum of 15.
- Input and Output
- The input consists of an tex2html_wrap_inline39 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 tex2html_wrap_inline43 integers separated by white-space (newlines and spaces). These tex2html_wrap_inline43 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
- 4
- 0 -2 -7 0 9 2 -6 2
- -4 1 -4 1 -1
- 8 0 -2
- Sample Output
- 15
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