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TangDou/AcWing/BeiBao/【总结】脑图形式(二维)总结.md

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+# 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>
+   
+  
+