动态规划（一）

是听了y总 算法基础课 的笔记，纯纯记录

背包问题

1、01背包  每件物品只有一个

2、完全背包  每件物品有无限个

3、多重背包  每件物品有有限个（个数已知）

4、分组背包  有若干组物品，每组有若干个，每组最多选一个

​​​​​​01背包

 未优化代码

#include 
#include 
using namespace std;

const int N = 1005;
int n ,m;
int v[N], w[N];
int 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 (v[i] <= j) f[i][j] = max(f[i][j], f[i-1][j-v[i]]+w[i]);
      }
    cout << f[n][m] << endl;
    return 0;
}

优化代码

#include 
#include 
using namespace std;

const int N = 1005;
int n ,m;
int v[N], w[N];
int 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;
}

完全背包

 未优化代码

#include 
#include 
using namespace std;

const int N = 1005;
int n ,m;
int v[N], w[N];
int 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;
}

优化代码

#include 
#include 
using namespace std;

const int N = 1005;
int n ,m;
int v[N], w[N];
int 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;
}

多重背包

多重背包1

#include 
#include 
using namespace std;

const int N = 105;
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*v[i]<=j && k<=s[i]; k++)
          f[i][j] =max(f[i][j], f[i-1][j-k*v[i]]+k*w[i]);
    cout << f[n][m] << endl;
    return 0;
}

多重背包2

#include 
#include 
using namespace std;

const int N = 12005, M = 2005;
int n,m;
int v[N], w[N];
int 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 = s-k;
            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;
}

分组背包

 代码

#include 
#include 
using namespace std;

const int N = 105;
int n, m;
int s[N];
int v[N][N], w[N][N];
int f[N];

int main()
{
    cin >> n >> m;
    for(int i=1; i<=n; i++)
    {
        cin >> s[i];
        for(int j=0; j> 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