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

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