A14863 | MEX and Increments

Dmitry has an array of $n$ non-negative integers $a_1, a_2, \dots, a_n$ .

In one operation, Dmitry can choose any index $j$ ( $1 \le j \le n$ ) and increase the value of the element $a_j$ by $1$ . He can choose the same index $j$ multiple times.

For each $i$ from $0$ to $n$ , determine whether Dmitry can make the $\mathrm{MEX}$ of the array equal to exactly $i$ . If it is possible, then determine the minimum number of operations to do it.

The $\mathrm{MEX}$ of the array is equal to the minimum non-negative integer that is not in the array. For example, the $\mathrm{MEX}$ of the array $[3, 1, 0]$ is equal to $2$ , and the array $[3, 3, 1, 4]$ is equal to $0$ .

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 $2\cdot10^5$ .

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.