Author: Andi Qu

The official solution runs in ~\mathcal{O}(BN^2 \log N)~ time, but we can use bitsets to solve this problem in ~\mathcal{O}(BN^3 / 64)~ time.

Firstly, instead of changing the number of candies in a package, we can say that Kristian first discards a package and then adds a new one.

Observation 1

After discarding a package, if Kristian can fulfill ~K~ distinct orders, he can add a package so that he can fulfill ~2K~ distinct orders.

Why is this true?

Think about what happens when we use a bitset to do knapsack DP. Imagine the current bitset storing which orders Kristian can fulfill is B and we're at a package with a[i] candies.

To transition, we simply do B |= B << a[i].

Thus, if the added a[i] is sufficiently large, Kristian can double the number of orders he can fulfill.

This means that the package discarded must be the one that yields the most orders Kristian can fulfill when it's discarded.

We can do knapsack DP ~N~ times (considering discarding each package) to find this package in ~\mathcal{O}(BN^3 / 64)~ time.

Observation 2

If there are 2 subsets of candies ~A~ and ~B~, the new package can't have ~\sum_{i \in A} i - \sum_{i \in B} i~ candies, since the number of packages Kristian can fulfill won't double in that case. Note that ~B~ can be the empty set.

The number of candies in the new package must therefore be the smallest positive integer that can't be expressed as ~\sum_{i \in A} i - \sum_{i \in B} i~ for two subsets of candies ~A~ and ~B~.

To find this number, we can do knapsack DP again, but this time using both a[i] and -a[i] instead of just a[i].

This knapsack DP runs in ~\mathcal{O}(BN^3 / 64)~ as well, which is fast enough to pass.