UVA 108: Maximum Sum (solution)

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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 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



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

Sample Output



15 

SOLUTION: C++
##Solution: C++ Problem: http://onlinejudge-problems.blogspot.com.co/2016/04/uva-108-maximum-sum-solution.html
#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <string>
#include <cctype>
#include <stack>
#include <queue>
#include <list>
#include <vector>
#include <map>
#include <sstream>
#include <utility>
#include <set>
#include <math.h>
using namespace std;
#define Dif(i,j,k) (Table[i+k][j] - Table[i][j])
#define MAXN 110
int N, MAX;
int Table[MAXN][MAXN];
void ReadCase()
{
int i, j;
for(i = 1; i<=N; i++)
for(j = 0; j<N; j++)
scanf("%d",&Table[i][j]);
}
void Cal()
{
int i, j, k, t;
for(i = 1; i<=N; i++)
{
for(j = 0; j<N; j++)
Table[i][j] = Table[i][j] + Table[i-1][j];
}
MAX = Table[1][0];
for(k = 1; k<=N; k++)
{
for(i = 0; i<=N-k; i++)
{
for(t = 0, j = 0; j<N; j++)
{
if(t>=0)
t+= Dif(i,j,k);
else
t = Dif(i,j,k);
if(t>MAX)
MAX = t;
}
}
}
printf("%d\n",MAX);
}
void FREE()
{
int i, j;
for(i = 0; i<=N; i++)
for(j = 0; j<=N; j++)
Table[i][j] = 0;
}
int main()
{
int f = 0;
while(scanf("%d",&N) == 1)
{
if(f++)
FREE();
ReadCase();
Cal();
}
return 0;
}
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