ZOJ 1074 Problem Set || nyoj 104 最大和
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Problem
Given a two-dimensional array of positive and negative integers, a sub-rectangle is any contiguous sub-array of size 1 x 1 or greater located within the whole array. 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.
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 corner:
9 2
-4 1
-1 8
and has a sum of 15.
The input consists of an N x 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 whitespace (spaces and newlines). These are the N 2 integers of the array, presented in row-major order. That is, all numbers in the first row, left to right, then all numbers in 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
Output the sum of the maximal sub-rectangle.
Example
Input
4
0 -2 -7 0 9 2 -6 2
-4 1 -4 1 -1
8 0 -2
Output
15
求最大子矩阵和, 可以用求最大和连续子序列解,将矩阵二维转化成一维, 压缩行或列都可以,毕竟n*n的都一样。
ps:一维数组最大连续子序列和
int a[n],b=0,sum=-0x3f3f3f3f; for(int i=0;i<n;i++) { if(b>0) b+=a[i]; else b=a[i]; if(b>sum) sum=b; } printf("%d\n",sum);
循环DP,共三层循环。
最外层i循环1-N,表明子矩阵是从第i列开始累加的。
第二层j循环i-N,表明子矩阵是从第i列累加到第j列。
第三层k从1到M做一维DP
#include<cstdio>#include<iostream>#include<cstring>#include<map>#include<algorithm>using namespace std;int main(){ int n,a[110][110],max,sum,i,j,k; while(scanf("%d",&n)!=EOF) { for(i = 1 ; i <= n ; i++) { for( j = 1 ; j <= n ; j++) { int t; scanf("%d",&t); a[i][j]=a[i][j-1]+t; //表示第i行从第1列到第j列之和 } } max = 0 ; for(i = 1; i <= n ; i++)//i表示列 { for(j = i ; j <= n ; j++)//j表示列,中间变量 { // j > i sum = 0; for(k = 1 ; k <= n ; k++)//表示行,从1到第n行 { int t; t = a[k][j]-a[k][i-1];//表示从第k行i列到k行j列之和, //j-(i-1)表示区间i--j sum +=t; //状态压缩至一维求最大和 if(sum < 0 ) sum = 0; if(sum > max) max = sum ; }//k每循环完一次,sum表示第i到j列在第1到n行中的最大连续子序列和 } } printf("%d\n",max); } return 0;}
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