Rivalries
Contest: Codeforces Round 887 (Div. 1)
Difficulty rating: 3400
Tags: constructive algorithms, data structures, greedy

Ntarsis has an array $$$a$$$ of length $$$n$$$.

The power of a subarray $$$a_l \dots a_r$$$ ($$$1 \leq l \leq r \leq n$$$) is defined as:

- The largest value $$$x$$$ such that $$$a_l \dots a_r$$$ contains $$$x$$$ and neither $$$a_1 \dots a_{l-1}$$$ nor $$$a_{r+1} \dots a_n$$$ contains $$$x$$$.
- If no such $$$x$$$ exists, the power is $$$0$$$.

Call an array $$$b$$$ a rival to $$$a$$$ if the following holds:

- The length of both $$$a$$$ and $$$b$$$ are equal to some $$$n$$$.
- Over all $$$l, r$$$ where $$$1 \leq l \leq r \leq n$$$, the power of $$$a_l \dots a_r$$$ equals the power of $$$b_l \dots b_r$$$.
- The elements of $$$b$$$ are positive.

Ntarsis wants you to find a rival $$$b$$$ to $$$a$$$ such that the sum of $$$b_i$$$ over $$$1 \leq i \leq n$$$ is maximized. Help him with this task!

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^5$$$). The description of the test cases follows.

The first line of each test case has a single integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$).

The next line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$).

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

Output format:
For each test case, output $$$n$$$ integers $$$b_1, b_2, \ldots, b_n$$$ — a valid rival to $$$a$$$ such that $$$b_1 + b_2 + \cdots + b_n$$$ is maximal.

If there exist multiple rivals with the maximum sum, output any of them.

Constraints:
Time limit: 2.0 s
Memory limit: 512.0 MB

Note:
For the first test case, one rival with the maximal sum is $$$[2, 4, 2, 3, 3]$$$.

$$$[2, 4, 2, 3, 3]$$$ can be shown to be a rival to $$$[1, 4, 1, 3, 3]$$$.

All possible subarrays of $$$a$$$ and $$$b$$$ and their corresponding powers are listed below:

- The power of $$$a[1:1] = [1] = 0$$$, the power of $$$b[1:1] = [2] = 0$$$.
- The power of $$$a[1:2] = [1, 4] = 4$$$, the power of $$$b[1:2] = [2, 4] = 4$$$.
- The power of $$$a[1:3] = [1, 4, 1] = 4$$$, the power of $$$b[1:3] = [2, 4, 2] = 4$$$.
- The power of $$$a[1:4] = [1, 4, 1, 3] = 4$$$, the power of $$$b[1:4] = [2, 4, 2, 3] = 4$$$.
- The power of $$$a[1:5] = [1, 4, 1, 3, 3] = 4$$$, the power of $$$b[1:5] = [2, 4, 2, 3, 3] = 4$$$.
- The power of $$$a[2:2] = [4] = 4$$$, the power of $$$b[2:2] = [4] = 4$$$.
- The power of $$$a[2:3] = [4, 1] = 4$$$, the power of $$$b[2:3] = [4, 2] = 4$$$.
- The power of $$$a[2:4] = [4, 1, 3] = 4$$$, the power of $$$b[2:4] = [4, 2, 3] = 4$$$.
- The power of $$$a[2:5] = [4, 1, 3, 3] = 4$$$, the power of $$$b[2:5] = [4, 2, 3, 3] = 4$$$.
- The power of $$$a[3:3] = [1] = 0$$$, the power of $$$b[3:3] = [2] = 0$$$.
- The power of $$$a[3:4] = [1, 3] = 0$$$, the power of $$$b[3:4] = [2, 3] = 0$$$.
- The power of $$$a[3:5] = [1, 3, 3] = 3$$$, the power of $$$b[3:5] = [2, 3, 3] = 3$$$.
- The power of $$$a[4:4] = [3] = 0$$$, the power of $$$b[4:4] = [3] = 0$$$.
- The power of $$$a[4:5] = [3, 3] = 3$$$, the power of $$$b[4:5] = [3, 3] = 3$$$.
- The power of $$$a[5:5] = [3] = 0$$$, the power of $$$b[5:5] = [3] = 0$$$.

It can be shown there exists no rival with a greater sum than $$$2 + 4 + 2 + 3 + 3 = 14$$$.

Example input:
7
5
1 4 1 3 3
5
1 4 1 8 8
5
2 1 1 1 2
8
3 2 3 5 2 2 5 3
8
1 1 1 1 4 3 3 3
10
1 9 5 9 8 1 5 8 9 1
16
1 1 1 1 5 5 5 5 9 9 9 9 7 7 7 7

Example output:
2 4 2 3 3
3 4 3 8 8
2 1 2 1 2
4 2 4 5 5 2 5 4
1 2 2 1 4 3 2 3
7 9 5 9 8 9 5 8 9 7
1 8 8 1 5 8 8 5 9 9 9 9 7 8 8 7

Editorial: