CF1761C Set Construction

You are given a binary matrix $ b $ (all elements of the matrix are $ 0 $ or $ 1 $ ) of $ n $ rows and $ n $ columns. You need to construct a $ n $ sets $ A_1, A_2, \ldots, A_n $ , for which the following conditions are satisfied: - Each set is nonempty and consists of distinct integers between $ 1 $ and $ n $ inclusive. - All sets are distinct. - For all pairs $ (i,j) $ satisfying $ 1\leq i, j\leq n $ , $ b_{i,j}=1 $ if and only if $ A_i\subsetneq A_j $ . In other words, $ b_{i, j} $ is $ 1 $ if $ A_i $ is a proper subset of $ A_j $ and $ 0 $ otherwise. Set $ X $ is a proper subset of set $ Y $ , if $ X $ is a nonempty subset of $ Y $ , and $ X \neq Y $ . It's guaranteed that for all test cases in this problem, such $ n $ sets exist. Note that it doesn't mean that such $ n $ sets exist for all possible inputs. If there are multiple solutions, you can output any of them.

Each test contains multiple test cases. The first line contains a single integer $ t $ ( $ 1\le t\le 1000 $ ) — the number of test cases. The description of test cases follows. The first line contains a single integer $ n $ ( $ 1\le n\le 100 $ ). The following $ n $ lines contain a binary matrix $ b $ , the $ j $ -th character of $ i $ -th line denotes $ b_{i,j} $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 1000 $ . It's guaranteed that for all test cases in this problem, such $ n $ sets exist.

For each test case, output $ n $ lines. For the $ i $ -th line, first output $ s_i $ $ (1 \le s_i \le n) $ — the size of the set $ A_i $ . Then, output $ s_i $ distinct integers from $ 1 $ to $ n $ — the elements of the set $ A_i $ . If there are multiple solutions, you can output any of them. It's guaranteed that for all test cases in this problem, such $ n $ sets exist.

In the first test case, we have $ A_1 = \{1, 2, 3\}, A_2 = \{1, 3\}, A_3 = \{2, 4\}, A_4 = \{1, 2, 3, 4\} $ . Sets $ A_1, A_2, A_3 $ are proper subsets of $ A_4 $ , and also set $ A_2 $ is a proper subset of $ A_1 $ . No other set is a proper subset of any other set. In the second test case, we have $ A_1 = \{1\}, A_2 = \{1, 2\}, A_3 = \{1, 2, 3\} $ . $ A_1 $ is a proper subset of $ A_2 $ and $ A_3 $ , and $ A_2 $ is a proper subset of $ A_3 $ .