背包问题dp

代码如下：

for (int i = 1; i <= N; ++i) {
    for (int j = 0; j <= V; ++j) {
        if(j >= w[i]) {
            dp[i][j] = max(dp[i - 1][j - w[i]] + v[i], dp[i - 1][j]);
        }
        else {
            dp[i][j] = dp[i-1][j];
        }
    }
}

 

01背包问题 

模板

（二维数组）

#include <iostream>
#define V 500
using namespace std;
int weight[20 + 1];
int value[20 + 1];
int f[20 + 1][V + 1];
int main() {
    int n, m;
    cout << "请输入物品个数:";
    cin >> n;
    cout << "请分别输入" << n << "个物品的重量和价值:" << endl; 
    for (int i = 1; i <= n; i++) {
        cin >> weight[i] >> value[i];
    }
    cout << "请输入背包容量:";
    cin >> m;
    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= m; j++) {
            if (weight[i] > j) {
                f[i][j] = f[i - 1][j];
            }
            else {
                f[i][j] = f[i - 1][j] > f[i - 1][j - weight[i]] + value[i] ? f[i - 1][j] : f[i - 1][j - weight[i]] + value[i];
            }
        }
    }    
    cout << "背包能放的最大价值为:" << f[n][m] << endl;
}

一维数组

 

#include <iostream>
#define V 500
using namespace std;
int weight[20 + 1];
int value[20 + 1];
int f[V + 1];
int main() {
    int n, m;
    cout << "请输入物品个数:";
    cin >> n;
    cout << "请分别输入" << n << "个物品的重量和价值:" << endl; 
    for (int i = 1; i <= n; i++) {
        cin >> weight[i] >> value[i];
    }
    cout << "请输入背包容量:";
    cin >> m;
    for (int i = 1; i <= n; i++) {
        for (int j = m; j >= 1; j--) {
            if (weight[i] <= j) {
                f[j] = f[j] > f[j - weight[i]] + value[i] ? f[j] : f[j - weight[i]] + value[i];
            }
        }
    }
    cout << "背包能放的最大价值为:" << f[m] << endl;
}

循环可改为： 

for (int i = 1; i <= n; i++) {
    for (int j = m; j >= weight[i]; j--) {
         f[j] = f[j] > f[j - weight[i]] + value[i] ? f[j] : f[j - weight[i]] + value[i];
    }
}

 

练习题 https://mp.youkuaiyun.com/postedit/88965298

https://mp.youkuaiyun.com/postedit/88965596

2.完全背包问题

模板

#include <iostream>
#define V 500
using namespace std;
int weight[20 + 1];
int value[20 + 1];
int f[V + 1];
int max(int a, int b) {
    return a > b ? a : b;
}
int main() {
    int n, m;
    cout << "请输入物品个数:";
    cin >> n;
    cout << "请分别输入" << n << "个物品的重量和价值:" << endl; 
    for (int i = 1; i <= n; i++) {
        cin >> weight[i] >> value[i];
    }
    cout << "请输入背包容量:";
    cin >> m;
    for (int i = 1; i <= n; i++) {
        for (int j = weight[i]; j <= m; j++) {
            f[j] = max(f[j], f[j - weight[i]] + value[i]);
        }
    }
    cout << "背包能放的最大价值为:" << f[m] << endl;
}

练习题

https://mp.youkuaiyun.com/postedit/88966509

3.多重背包问题

模板

#include <iostream>
using namespace std;
#define V 1000
int weight[50 + 1];
int value[50 + 1];
int num[20 + 1];
int f[V + 1];
int max(int a, int b) {
    return a > b ? a : b;
}
int main() {
    int n, m;
    cout << "请输入物品个数:";
    cin >> n;
    cout << "请分别输入" << n << "个物品的重量、价值和数量:" << endl; 
    for (int i = 1; i <= n; i++) {
        cin >> weight[i] >> value[i] >> num[i];
    }
    int k = n + 1;
    for (int i = 1; i <= n; i++) {
        while (num[i] != 1) {
            weight[k] = weight[i];
            value[k] = value[i];
            k++;
            num[i]--;
        }
    }
    cout << "请输入背包容量:";
    cin >> m;
    for (int i = 1; i <= k; i++) {
        for (int j = m; j >= 1; j--) {
            if (weight[i] <= j) f[j] = max(f[j], f[j - weight[i]] + value[i]);
        }
    }
    cout << "背包能放的最大价值为:" << f[m] << endl;
}

练习题：

 https://mp.youkuaiyun.com/postedit/88971050