CF1828B Permutation Swap

You are given an unsorted permutation $ p_1, p_2, \ldots, p_n $ . To sort the permutation, you choose a constant $ k $ ( $ k \ge 1 $ ) and do some operations on the permutation. In one operation, you can choose two integers $ i $ , $ j $ ( $ 1 \le j < i \le n $ ) such that $ i - j = k $ , then swap $ p_i $ and $ p_j $ . What is the maximum value of $ k $ that you can choose to sort the given permutation? A permutation is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ 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 ( $ n = 3 $ but there is $ 4 $ in the array). An unsorted permutation $ p $ is a permutation such that there is at least one position $ i $ that satisfies $ p_i \ne i $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 2 \le n \le 10^{5} $ ) — the length of the permutation $ p $ . The second line of each test case contains $ n $ distinct integers $ p_1, p_2, \ldots, p_n $ ( $ 1 \le p_i \le n $ ) — the permutation $ p $ . It is guaranteed that the given numbers form a permutation of length $ n $ and the given permutation is unsorted. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^{5} $ .

For each test case, output the maximum value of $ k $ that you can choose to sort the given permutation. We can show that an answer always exists.

In the first test case, the maximum value of $ k $ you can choose is $ 1 $ . The operations used to sort the permutation are: - Swap $ p_2 $ and $ p_1 $ ( $ 2 - 1 = 1 $ ) $ \rightarrow $ $ p = [1, 3, 2] $ - Swap $ p_2 $ and $ p_3 $ ( $ 3 - 2 = 1 $ ) $ \rightarrow $ $ p = [1, 2, 3] $ In the second test case, the maximum value of $ k $ you can choose is $ 2 $ . The operations used to sort the permutation are: - Swap $ p_3 $ and $ p_1 $ ( $ 3 - 1 = 2 $ ) $ \rightarrow $ $ p = [1, 4, 3, 2] $ - Swap $ p_4 $ and $ p_2 $ ( $ 4 - 2 = 2 $ ) $ \rightarrow $ $ p = [1, 2, 3, 4] $