Making Mexes

You are given an array $a$ of $N$ non-negative integers $a_1,a_2,\dots ,a_N$ ($1\le N\le 2\cdot 10^5,0\le a_i\le N$). In one operation, you can change any element of $a$ to any non-negative integer.

The mex of an array is the minimum non-negative integer that it does not contain. For each $i$ in the range $0$ to $N$ inclusive, compute the minimum number of operations you need in order to make the mex of $a$ equal $i$.

• To make the mex of $a$ equal to $0$, we can change $a_4$ to $3$ (or any positive integer). In the resulting array, $[2,2,2,3]$, $0$ is the smallest non-negative integer that the array does not contain, so $0$ is the mex of the array.
• To make the mex of $a$ equal to $1$, we don't need to make any changes since $1$ is already the smallest non-negative integer that $a=[2,2,2,0]$ does not contain.
• To make the mex of $a$ equal to $2$, we need to change the first three elements of $a$. For example, we can change $a$ to be $[3,1,1,0]$.

Problem credits: Benjamin Qi

Source: USA Computing Olympiad.

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