MEX and Increments

Description:

Input:

The first line of input data contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases in the input.

The descriptions of the test cases follow.

The first line of the description of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the length of the array $$$a$$$.

The second line of the description of each test case contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$0 \le a_i \le n$$$) — elements of the array $$$a$$$.

It is guaranteed that the sum of the values $$$n$$$ over all test cases in the test does not exceed $$$2\cdot10^5$$$.

Output:

For each test case, output $$$n + 1$$$ integer — $$$i$$$-th number is equal to the minimum number of operations for which you can make the array $$$\mathrm{MEX}$$$ equal to $$$i$$$ ($$$0 \le i \le n$$$), or -1 if this cannot be done.

Sample Input:

5
3
0 1 3
7
0 1 2 3 4 3 2
4
3 0 0 0
7
4 6 2 3 5 0 5
5
4 0 1 0 4

Sample Output:

1 1 0 -1 
1 1 2 2 1 0 2 6 
3 0 1 4 3 
1 0 -1 -1 -1 -1 -1 -1 
2 1 0 2 -1 -1 

Note:

In the first set of example inputs, $$$n=3$$$:

• to get $$$\mathrm{MEX}=0$$$, it is enough to perform one increment: $$$a_1$$$++;
• to get $$$\mathrm{MEX}=1$$$, it is enough to perform one increment: $$$a_2$$$++;
• $$$\mathrm{MEX}=2$$$ for a given array, so there is no need to perform increments;
• it is impossible to get $$$\mathrm{MEX}=3$$$ by performing increments.