Constructing the Array (Codeforces 1353D)
Tags: Others, sortings

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 Inputs:
6
1
2
3
4
5
6

Sample Outputs:
1 
1 2 
2 1 3 
3 1 2 4 
2 4 1 3 5 
3 4 1 5 2 6

Solution (Python):
import heapq

def gogogo(n):
    out = [0] * n
    q = [(-n, 0, n - 1)]
    heapq.heapify(q)
    for i in range(n):
        (_count, left, right) = heapq.heappop(q)
        mid = (left + right) // 2
        out[mid] = i + 1
        if mid != left:
            heapq.heappush(q, (-mid + left, left, mid - 1))
        if mid != right:
            heapq.heappush(q, (-right + mid, mid + 1, right))
    return out
n = int(input())
for _ in range(n):
    cur = int(input())
    ans = gogogo(cur)
    print(' '.join([str(x) for x in ans]))

Solution (Python):
from sys import stdin
from itertools import *
from bisect import *
from collections import *
from heapq import *
from math import *
from random import *
read_int = lambda : int(input())
read_int_list = lambda : list(map(int, input().split()))
read_tuple = lambda : map(int, input().split())

def solve():
    n = read_int()
    h = [(-n, 1, n)]
    res = [0] * (n + 1)
    for i in range(1, n + 1):
        (_, l, r) = heappop(h)
        mid = l + r >> 1
        res[mid] = i
        if l != r:
            heappush(h, (-(mid - l), l, mid - 1))
            heappush(h, (-(r - mid), mid + 1, r))
    print(' '.join(map(str, res[1:])))
if __name__ == '__main__':
    t = int(input())
    for _ in range(t):
        solve()

Solution (Python):
nCasos = int(input())
for _ in range(nCasos):
    sub = int(input())
    result = [0] * (sub + 1)
    matriz = [[] for _ in range(sub + 1)]
    matriz[sub] = [1]
    count = 1
    for i in range(sub, 0, -1):
        if len(matriz[i]) > 0:
            subMatriz = sorted(matriz[i])
            for k in subMatriz:
                if (i - 1) // 2 > 0:
                    matriz[(i - 1) // 2].append(k)
                result[k + (i - 1) // 2] = count
                count += 1
                matriz[i // 2].append(k + (i + 1) // 2)
    print(' '.join(list(map(str, result[1:]))))