Making Towers
Description
You have a sequence of $n$ colored blocks. The color of the $i$-th block is $c_i$, an integer between $1$ and $n$.
You will place the blocks down in sequence on an infinite coordinate grid in the following way.
- Initially, you place block $1$ at $(0, 0)$.
- For $2 \le i \le n$, if the $(i - 1)$-th block is placed at position $(x, y)$, then the $i$-th block can be placed at one of positions $(x + 1, y)$, $(x - 1, y)$, $(x, y + 1)$ (but not at position $(x, y - 1)$), as long no previous block was placed at that position.
A tower is formed by $s$ blocks such that they are placed at positions $(x, y), (x, y + 1), \ldots, (x, y + s - 1)$ for some position $(x, y)$ and integer $s$. The size of the tower is $s$, the number of blocks in it. A tower of color $r$ is a tower such that all blocks in it have the color $r$.
For each color $r$ from $1$ to $n$, solve the following problem independently:
- Find the maximum size of a tower of color $r$ that you can form by placing down the blocks according to the rules.
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$).
The second line of each test case contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le n$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, output $n$ integers. The $r$-th of them should be the maximum size of an tower of color $r$ you can form by following the given rules. If you cannot form any tower of color $r$, the $r$-th integer should be $0$.
Input
The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$).
The second line of each test case contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le n$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, output $n$ integers. The $r$-th of them should be the maximum size of an tower of color $r$ you can form by following the given rules. If you cannot form any tower of color $r$, the $r$-th integer should be $0$.
Samples
6
7
1 2 3 1 2 3 1
6
4 2 2 2 4 4
1
1
5
5 4 5 3 5
6
3 3 3 1 3 3
8
1 2 3 4 4 3 2 1
3 2 2 0 0 0 0
0 3 0 2 0 0
1
0 0 1 1 1
1 0 4 0 0 0
2 2 2 2 0 0 0 0
Note
In the first test case, one of the possible ways to form a tower of color $1$ and size $3$ is:
- place block $1$ at position $(0, 0)$;
- place block $2$ to the right of block $1$, at position $(1, 0)$;
- place block $3$ above block $2$, at position $(1, 1)$;
- place block $4$ to the left of block $3$, at position $(0, 1)$;
- place block $5$ to the left of block $4$, at position $(-1, 1)$;
- place block $6$ above block $5$, at position $(-1, 2)$;
- place block $7$ to the right of block $6$, at position $(0, 2)$.
The blocks at positions $(0, 0)$, $(0, 1)$, and $(0, 2)$ all have color $1$, forming an tower of size $3$.
In the second test case, note that the following placement is not valid, since you are not allowed to place block $6$ under block $5$:
It can be shown that it is impossible to form a tower of color $4$ and size $3$.