A14470 | Permutation Minimization by Deque

In fact, the problems E1 and E2 do not have much in common. You should probably think of them as two separate problems.

A permutation $p$ of size $n$ is given. A permutation of size $n$ is an array of size $n$ in which each integer from $1$ to $n$ occurs exactly once. For example, $[1, 4, 3, 2]$ and $[4, 2, 1, 3]$ are correct permutations while $[1, 2, 4]$ and $[1, 2, 2]$ are not.

Let us consider an empty [deque](https://tinyurl.com/pfeucbux) (double-ended queue). A deque is a data structure that supports adding elements to both the beginning and the end. So, if there are elements $[1, 5, 2]$ currently in the deque, adding an element $4$ to the beginning will produce the sequence $[\color{red}{4}, 1, 5, 2]$ , and adding same element to the end will produce $[1, 5, 2, \color{red}{4}]$ .

The elements of the permutation are sequentially added to the initially empty deque, starting with $p_1$ and finishing with $p_n$ . Before adding each element to the deque, you may choose whether to add it to the beginning or the end.

For example, if we consider a permutation $p = [3, 1, 2, 4]$ , one of the possible sequences of actions looks like this:

$\quad$ 1.add $3$ to the end of the deque:deque has a sequence $[\color{red}{3}]$ in it; $\quad$ 2.add $1$ to the beginning of the deque:deque has a sequence $[\color{red}{1}, 3]$ in it; $\quad$ 3.add $2$ to the end of the deque:deque has a sequence $[1, 3, \color{red}{2}]$ in it; $\quad$ 4.add $4$ to the end of the deque:deque has a sequence $[1, 3, 2, \color{red}{4}]$ in it;Find the lexicographically smallest possible sequence of elements in the deque after the entire permutation has been processed.

A sequence $[x_1, x_2, \ldots, x_n]$ is lexicographically smaller than the sequence $[y_1, y_2, \ldots, y_n]$ if there exists such $i \leq n$ that $x_1 = y_1$ , $x_2 = y_2$ , $\ldots$ , $x_{i - 1} = y_{i - 1}$ and $x_i < y_i$ . In other words, if the sequences $x$ and $y$ have some (possibly empty) matching prefix, and the next element of the sequence $x$ is strictly smaller than the corresponding element of the sequence $y$ . For example, the sequence $[1, 3, 2, 4]$ is smaller than the sequence $[1, 3, 4, 2]$ because after the two matching elements $[1, 3]$ in the start the first sequence has an element $2$ which is smaller than the corresponding element $4$ in the second sequence.

The first line contains an integer $t$ ( $1 \leq t \leq 1000$ ) — the number of test cases.

The next $2t$ lines contain descriptions of the test cases.

The first line of each test case description contains an integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — permutation size. The second line of the description contains $n$ space-separated integers $p_i$ ( $1 \le p_i \le n$ ; all $p_i$ are all unique) — elements of the permutation.

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

Print $t$ lines, each line containing the answer to the corresponding test case. The answer to a test case should contain $n$ space-separated integer numbers — the elements of the lexicographically smallest permutation that is possible to find in the deque after executing the described algorithm.