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## B - RGB Matching Editorial by evima

If R, G, andB all appear even numbers of times, the answer is $$0$$. Below, we assume R and G appear odd numbers of times and B appears an even number of times. (We can handle the other cases similarly.)

Positive dissatisfactions only result from making pairs of RG, GB, BR.

We can see that there is an optimal solution with at most one RG pair, at most one GB pair, and at most one BR pair. This is because if a solution has two RG pairs, for example, we can recompose two of them into an RR pair and a GG pair, which will not increase the total dissatisfaction.

We can also see that there is an optimal solution with at most two pairs that are RG, GB, or BR. This is because if a solution has an RG pair, a GB pair, and a BR pair, we can recompose them into an RR pair, a GG pair, and a BB pair, which will not increase the total dissatisfaction.

From these two facts and our assumption, we can see that one of the following is true for the optimal solution:

• It has one RG pair, and all other pairs are RR, GG, or BB.
• It has one GB pair and one BR pair, and all other pairs are RR, GG, or BB.

To find the smallest dissatisfactions in these cases, we can simply find the smallest difference in cuteness between R and G, between G and B, and between B and R, because if the GB pair and the BR pair shares the same dog, it will not be better than making an RG pair. This solution will run fast enough in $$O(N \log N)$$ time, since the bottleneck is sorting.

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