CF1622E Math Test

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.

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 $ .

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.