Doremy's IQ
Description
Doremy is asked to test $n$ contests. Contest $i$ can only be tested on day $i$. The difficulty of contest $i$ is $a_i$. Initially, Doremy's IQ is $q$. On day $i$ Doremy will choose whether to test contest $i$ or not. She can only test a contest if her current IQ is strictly greater than $0$.
If Doremy chooses to test contest $i$ on day $i$, the following happens:
- if $a_i>q$, Doremy will feel she is not wise enough, so $q$ decreases by $1$;
- otherwise, nothing changes.
Doremy wants to test as many contests as possible. Please give Doremy a solution.
The input consists of multiple test cases. The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of the test cases follows.
The first line contains two integers $n$ and $q$ ($1 \le n \le 10^5$, $1 \le q \le 10^9$) — the number of contests and Doremy's IQ in the beginning.
The second line contains $n$ integers $a_1,a_2,\cdots,a_n$ ($1 \le a_i \le 10^9$) — the difficulty of each contest.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
For each test case, you need to output a binary string $s$, where $s_i=1$ if Doremy should choose to test contest $i$, and $s_i=0$ otherwise. The number of ones in the string should be maximum possible, and she should never test a contest when her IQ is zero or less.
If there are multiple solutions, you may output any.
Input
The input consists of multiple test cases. The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of the test cases follows.
The first line contains two integers $n$ and $q$ ($1 \le n \le 10^5$, $1 \le q \le 10^9$) — the number of contests and Doremy's IQ in the beginning.
The second line contains $n$ integers $a_1,a_2,\cdots,a_n$ ($1 \le a_i \le 10^9$) — the difficulty of each contest.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case, you need to output a binary string $s$, where $s_i=1$ if Doremy should choose to test contest $i$, and $s_i=0$ otherwise. The number of ones in the string should be maximum possible, and she should never test a contest when her IQ is zero or less.
If there are multiple solutions, you may output any.
Samples
5
1 1
1
2 1
1 2
3 1
1 2 1
4 2
1 4 3 1
5 2
5 1 2 4 3
1
11
110
1110
01111
Note
In the first test case, Doremy tests the only contest. Her IQ doesn't decrease.
In the second test case, Doremy tests both contests. Her IQ decreases by $1$ after testing contest $2$.
In the third test case, Doremy tests contest $1$ and $2$. Her IQ decreases to $0$ after testing contest $2$, so she can't test contest $3$.