CF1450D.Rating Compression

On the competitive programming platform CodeCook, every person has a rating graph described by an array of integers $a$ of length $n$ . You are now updating the infrastructure, so you've created a program to compress these graphs.

The program works as follows. Given an integer parameter $k$ , the program takes the minimum of each contiguous subarray of length $k$ in $a$ .

More formally, for an array $a$ of length $n$ and an integer $k$ , define the $k$ -compression array of $a$ as an array $b$ of length $n-k+1$ , such that $$$$b_j =\min_{j\le i\le j+k-1}a_i $$

For example, the $3$ -compression array of \[1, 3, 4, 5, 2\] is \[\\min\\{1, 3, 4\\}, \\min\\{3, 4, 5\\}, \\min\\{4, 5, 2\\}\]=\[1, 3, 2\].

A permutation of length $m$ is an array consisting of $m$ distinct integers from $1$ to $m$ in arbitrary order. For example, \[2,3,1,5,4\] is a permutation, but \[1,2,2\] is not a permutation ( $2$ appears twice in the array) and \[1,3,4\] is also not a permutation ( $m=3$ but there is $4$ in the array).

A $k$ -compression array will make CodeCook users happy if it will be a permutation. Given an array $a$ , determine for all $1\\leq k\\leq n$ if CookCook users will be happy after a $k$$$-compression of this array or not.

The first line contains a single integer $t$ ( $1\leq t\leq 10^4$ ) — the number of test cases.

The first line of the description of each test case contains a single integer $n$ ( $1\leq n\leq 3\cdot 10^5$ ) — the length of the array.

The second line of the description of each test case contains $n$ integers $a_1,\ldots,a_n$ ( $1\leq a_i\leq n$ ) — the elements of the array.

It is guaranteed, that the sum of $n$ for all test cases does not exceed $3\cdot 10^5$ .

For each test case, print a binary string of length $n$ .

The $k$ -th character of the string should be $1$ if CookCook users will be happy after a $k$ -compression of the array $a$ , and $0$ otherwise.

In the first test case, $a=[1, 5, 3, 4, 2]$ .

• The $1$ -compression of $a$ is $[1, 5, 3, 4, 2]$ and it is a permutation.
• The $2$ -compression of $a$ is $[1, 3, 3, 2]$ and it is not a permutation, since $3$ appears twice.
• The $3$ -compression of $a$ is $[1, 3, 2]$ and it is a permutation.
• The $4$ -compression of $a$ is $[1, 2]$ and it is a permutation.
• The $5$ -compression of $a$ is $[1]$ and it is a permutation.