Acwing-----算法基础课之第五讲动态规划（一）

2. 01背包问题

• 链接：https://www.acwing.com/problem/content/2/

// 二维解法
#include <iostream>
using namespace std;

const int N = 1010;

int n, m, v[N], w[N], f[N][N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> v[i] >> w[i];
    f[0][0] = 0;
    
    for (int i = 1; i <= n; ++i) {
        for (int j = 0; j <= m; ++j) {
            f[i][j] = f[i - 1][j];
            if (j >= v[i]) f[i][j] = max(f[i][j], f[i - 1][j - v[i]] + w[i]);
        }
    }
    cout << f[n][m] << endl;
    return 0;
}

//优化解法
#include <iostream>
using namespace std;

const int N = 1010;

int n, m, v[N], w[N], f[N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> v[i] >> w[i];

    for (int i = 1; i <= n; ++i) {
        for (int j = m; j >= v[i]; --j) {
            f[j] = max(f[j], f[j - v[i]] + w[i]);
        }
    }
    cout << f[m] << endl;
    return 0;
}

3. 完全背包问题

• 链接：https://www.acwing.com/problem/content/3/

//二维解法
#include <iostream>
#include <algorithm>
using namespace std;
const int N = 1010;

int n, m, v[N], w[N], f[N][N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; ++i) {
        for (int j = 0; j <= m; ++j) {
            f[i][j] = f[i - 1][j];
            if (j >= v[i]) f[i][j] = max(f[i][j], f[i][j - v[i]] + w[i]);
        }
    }
    cout << f[n][m] << endl;
    return 0;
}

//优化解法
#include <iostream>
#include <algorithm>
using namespace std;
const int N = 1010;

int n, m, v[N], w[N], f[N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; ++i) {
        for (int j = v[i]; j <= m; ++j) {
            f[j] = max(f[j], f[j - v[i]] + w[i]);
        }
    }
    cout << f[m] << endl;
    return 0;
}

4. 多重背包问题 I

• 链接：https://www.acwing.com/problem/content/4/

#include <iostream>
#include <algorithm>

using namespace std;

const int N = 110;
int n, m;
int v[N], w[N], s[N];
int f[N][N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> v[i] >> w[i] >> s[i];
    
    for (int i = 1; i <= n; ++i) {
        for (int j = 0; j <= m; ++j) {
            for (int k = 0; k <= s[i] && k * v[i] <= j; ++k) {
                f[i][j] = max(f[i][j], f[i - 1][j - k * v[i]] + w[i] * k);
            }
        }
    }
    
    cout  << f[n][m] << endl;
    
    return 0;
}

5. 多重背包问题 II

• 链接：https://www.acwing.com/problem/content/5/

#include <iostream>
#include <algorithm>
using namespace std;

const int N = 25000, M = 2010;
int n, m, v[N], w[N], f[N];

int main() {
    cin >> n >> m;
    int cnt = 0;
    for (int i = 1; i <= n; ++i) {
        int a, b, s;
        cin >> a >> b >> s;
        int k = 1;
        while (k <= s) {
            cnt ++;
            v[cnt] = a * k;
            w[cnt] = b * k;
            s -= k;
            k *= 2;
        }
        if (s > 0) {
            cnt++;
            v[cnt] = a * s;
            w[cnt] = b * s;
        }
    }
    n = cnt;
    for (int i = 1; i <= n; ++i) {
        for (int j = m; j >= v[i]; --j) {
            f[j] = max(f[j], f[j - v[i]] + w[i]);
        }
    }
    cout << f[m] << endl;
    return 0;
}

9. 分组背包问题

• 链接：https://www.acwing.com/problem/content/9/

#include <iostream>
#include <algorithm>
using namespace std;

const int N = 110;
int n, m, v[N][N], w[N][N], f[N], s[N];

int main() {
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) {
        cin >> s[i];
        for (int j = 0; j < s[i]; ++j) cin >> v[i][j] >> w[i][j];
    }
    for (int i = 1; i <= n; ++i) {
        for (int j = m; j >= 0; --j) {
            for (int k = 0; k < s[i]; ++k) {
                if (v[i][k] <= j) f[j] = max(f[j], f[j- v[i][k]] + w[i][k]);
            }
        }
    }
    cout << f[m] << endl;
    return 0;
}