Constructing the Array

1000ms

262144K

Description:

You are given an array $$$a$$$ of length $$$n$$$ consisting of zeros. You perform $$$n$$$ actions with this array: during the $$$i$$$-th action, the following sequence of operations appears:

Choose the maximum by length subarray (continuous subsegment) consisting only of zeros, among all such segments choose the leftmost one;
Let this segment be $$$[l; r]$$$. If $$$r-l+1$$$ is odd (not divisible by $$$2$$$) then assign (set) $$$a[\frac{l+r}{2}] := i$$$ (where $$$i$$$ is the number of the current action), otherwise (if $$$r-l+1$$$ is even) assign (set) $$$a[\frac{l+r-1}{2}] := i$$$.

Consider the array $$$a$$$ of length $$$5$$$ (initially $$$a=[0, 0, 0, 0, 0]$$$). Then it changes as follows:

Firstly, we choose the segment $$$[1; 5]$$$ and assign $$$a[3] := 1$$$, so $$$a$$$ becomes $$$[0, 0, 1, 0, 0]$$$;
then we choose the segment $$$[1; 2]$$$ and assign $$$a[1] := 2$$$, so $$$a$$$ becomes $$$[2, 0, 1, 0, 0]$$$;
then we choose the segment $$$[4; 5]$$$ and assign $$$a[4] := 3$$$, so $$$a$$$ becomes $$$[2, 0, 1, 3, 0]$$$;
then we choose the segment $$$[2; 2]$$$ and assign $$$a[2] := 4$$$, so $$$a$$$ becomes $$$[2, 4, 1, 3, 0]$$$;
and at last we choose the segment $$$[5; 5]$$$ and assign $$$a[5] := 5$$$, so $$$a$$$ becomes $$$[2, 4, 1, 3, 5]$$$.

Your task is to find the array $$$a$$$ of length $$$n$$$ after performing all $$$n$$$ actions. Note that the answer exists and unique.

You have to answer $$$t$$$ independent test cases.

Input:

The first line of the input contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Then $$$t$$$ test cases follow.

The only line of the test case contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the length of $$$a$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$ ($$$\sum n \le 2 \cdot 10^5$$$).

Output:

For each test case, print the answer — the array $$$a$$$ of length $$$n$$$ after performing $$$n$$$ actions described in the problem statement. Note that the answer exists and unique.

Sample Input:

Sample Output:

1 
1 2 
2 1 3 
3 1 2 4 
2 4 1 3 5 
3 4 1 5 2 6

Informação

Provedor Codeforces

Código CF1353D