1450D · Rating Compression

Problem: 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 CodeCook users will be happy after a $k$-compression of this array or not.

Input Format: 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 $3\cdot 10^5$.

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

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

Note: 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.