# 背包问题(二维求法)

## 求方案数

- 体积至多$j$
  - 初始：$f[0][i]=1$,$0<=i<=m$,其余为$0$
  - $01$背包
  -
    ```c++
      for (int j = 0; j <= m; j++) {
        f[i][j] = f[i - 1][j];
        if (j >= v) f[i][j] += f[i - 1][j - v];
      }
    ```  
  - 完全背包
  - 
    ```c++
      for (int j = 0; j <= m; j++) {
      f[i][j] = f[i - 1][j];
          if (j >= v) f[i][j] += f[i][j - v];
      }
    ```
- 体积恰好$j$
  - 初始：$f[i][0]=1$,$0<=i<=n$,其余为$0$
  - $01$背包
  - 
    ```c++
    for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] += f[i - 1][j - v];
        }
    ```
  - 完全背包
  - 
    ```c++
    for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] += f[i][j - v];
        }
    ``` 
- 体积至少$j$
  - 初始：$f[0][0]=1$,其余为$0$
  - $01$背包
  - 
    ```c++
    for (int j = 0; j <= m; j++)
       f[i][j] = f[i - 1][j] + f[i - 1][max(0, j - v)];
    ```
  - 完全背包代码
  -     
    <font color='red' size=4><b>有无穷多组方案数,不这么问</b></font>
    

## 求最大/小值

- 体积至多$j$
  - 初始：$f[i,j]=0，0 <= i <= n, 0 <= j <= m$
  - $01$背包
  - 
    ```c++
    for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = max(f[i][j], f[i - 1][j - v] + w);
        }
    ```
  - 完全背包
  -   
    ```c++
    for (int j = 1; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = max(f[i][j], f[i][j - v] + w);
        }
    ```
- 体积恰好$j$
  - 求最小值
    -  初始化$f[0][0]=0$,其它是$INF$
    -  01 背包
    -  
      ```c++
      for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = min(f[i][j], f[i - 1][j - v] + w);
        }
      ```
    - 完全背包
    - 
      ```c++
      for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = min(f[i][j], f[i][j - v] + w);
        }
      ``` 
  - 求最大值
    -  初始化$f[0][0]=0$, 其余是$−INF$
    -  01背包　
    -  
      ```c++
      for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = max(f[i][j], f[i - 1][j - v] + w);
        }
      ```
    - 完全背包
    - 
      ```c++
       for (int j = 0; j <= m; j++) {
            f[i][j] = f[i - 1][j];
            if (j >= v) f[i][j] = max(f[i][j], f[i][j - v] + w);
        }
      ```
- 体积至少$j$
  - 求最小值
    - 初始化$f[0][0]=0$, 其余是$INF$
    - 01背包　
    - 
      ```c++
        for (int j = 0; j <= m; j++)
            f[i][j] = min(f[i - 1][j], f[i - 1][max(0, j - v)] + w);
      ```
    - 完全背包
    - 
      ```c++
       for (int j = 0; j <= m; j++)
            f[i][j] = min(f[i - 1][j], f[i][max(0, j - v)] + w); 
      ```
  - 求最大值
    - <font color='red' size=4><b>没有求最大值的</b></font>