130 lines
3.1 KiB
Markdown
130 lines
3.1 KiB
Markdown
# 背包问题(二维求法)
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## 求方案数
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- 体积至多$j$
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- 初始：$f[0][i]=1$,$0<=i<=m$,其余为$0$
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- $01$背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] += f[i - 1][j - v];
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}
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```
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- 完全背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] += f[i][j - v];
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}
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```
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- 体积恰好$j$
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- 初始：$f[i][0]=1$,$0<=i<=n$,其余为$0$
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- $01$背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] += f[i - 1][j - v];
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}
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```
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- 完全背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] += f[i][j - v];
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}
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```
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- 体积至少$j$
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- 初始：$f[0][0]=1$,其余为$0$
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- $01$背包
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-
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```c++
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for (int j = 0; j <= m; j++)
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f[i][j] = f[i - 1][j] + f[i - 1][max(0, j - v)];
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```
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- 完全背包代码
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-
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<font color='red' size=4><b>有无穷多组方案数,不这么问</b></font>
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## 求最大/小值
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- 体积至多$j$
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- 初始：$f[i,j]=0，0 <= i <= n, 0 <= j <= m$
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- $01$背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = max(f[i][j], f[i - 1][j - v] + w);
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}
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```
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- 完全背包
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-
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```c++
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for (int j = 1; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = max(f[i][j], f[i][j - v] + w);
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}
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```
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- 体积恰好$j$
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- 求最小值
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- 初始化$f[0][0]=0$,其它是$INF$
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- 01 背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = min(f[i][j], f[i - 1][j - v] + w);
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}
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```
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- 完全背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = min(f[i][j], f[i][j - v] + w);
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}
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```
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- 求最大值
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- 初始化$f[0][0]=0$, 其余是$−INF$
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- 01背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = max(f[i][j], f[i - 1][j - v] + w);
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}
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```
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- 完全背包
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-
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```c++
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for (int j = 0; j <= m; j++) {
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f[i][j] = f[i - 1][j];
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if (j >= v) f[i][j] = max(f[i][j], f[i][j - v] + w);
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}
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```
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- 体积至少$j$
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- 求最小值
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- 初始化$f[0][0]=0$, 其余是$INF$
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- 01背包
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-
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```c++
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for (int j = 0; j <= m; j++)
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f[i][j] = min(f[i - 1][j], f[i - 1][max(0, j - v)] + w);
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```
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- 完全背包
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-
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```c++
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for (int j = 0; j <= m; j++)
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f[i][j] = min(f[i - 1][j], f[i][max(0, j - v)] + w);
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```
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- 求最大值
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- <font color='red' size=4><b>没有求最大值的</b></font>
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