1622E · Math Test

Problem: Petya is a math teacher. $n$ of his students has written a test consisting of $m$ questions. For each student, it is known which questions he has answered correctly and which he has not.

If the student answers the $j$-th question correctly, he gets $p_j$ points (otherwise, he gets $0$ points). Moreover, the points for the questions are distributed in such a way that the array $p$ is a permutation of numbers from $1$ to $m$.

For the $i$-th student, Petya knows that he expects to get $x_i$ points for the test. Petya wonders how unexpected the results could be. Petya believes that the surprise value of the results for students is equal to $\sum\limits_{i=1}^{n} |x_i - r_i|$, where $r_i$ is the number of points that the $i$-th student has got for the test.

Your task is to help Petya find such a permutation $p$ for which the surprise value of the results is maximum possible. If there are multiple answers, print any of them.

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

The first line of each test case contains two integers $n$ and $m$ ($1 \le n \le 10$; $1 \le m \le 10^4$) — the number of students and the number of questions, respectively.

The second line contains $n$ integers $x_1, x_2, \dots, x_n$ ($0 \le x_i \le \frac{m(m+1)}{2}$), where $x_i$ is the number of points that the $i$-th student expects to get.

This is followed by $n$ lines, the $i$-th line contains the string $s_i$ ($|s_i| = m; s_{i, j} \in \{0, 1\}$), where $s_{i, j}$ is $1$ if the $i$-th student has answered the $j$-th question correctly, and $0$ otherwise.

The sum of $m$ for all test cases does not exceed $10^4$.

Output Format: For each test case, print $m$ integers — a permutation $p$ for which the surprise value of the results is maximum possible. If there are multiple answers, print any of them.