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python/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
    •   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
    •   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
    • 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
    • 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
    • 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
    • 有无穷多组方案数,不这么问

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
    • 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
    • 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
      • 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
      • 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　
      • 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
      •  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　
      •   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
      •  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
      • 没有求最大值的