Buying Order (Hard)

**This is the hard version of the problem available for 100 points in all divisions. Note that here you need to find the minimum number of items possible. Further, you need to solve the problem for all prefixes of the input.** There's a shop with $N$ distinct items numbered $1$, $2$, .... $N$. There's also a boolean array $B$ of length $N$. You want to buy all $N$ items once each, in some order