Problem 2. Making Mexes

USACO 2025 February Contest, Bronze

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 $i$.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N$.

The next line contains $a_1,a_2,\dots, a_N$.

OUTPUT FORMAT (print output to the terminal / stdout):

For each $i$ in the range $0$ to $N$, output the minimum number of operations for $i$ on a new line. Note that it is always possible to make the mex of $a$ equal to any $i$ in the range $0$ to $N$.

SAMPLE INPUT:

4
2 2 2 0

SAMPLE OUTPUT:

1
0
3
1
2

• 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 $[3, 1, 1, 0]$.

SCORING: Inputs 2-6: $N\le 10^3$Inputs 7-11: No additional constraints.

Problem credits: Benjamin Qi