CF2084F Skyscape

You are given a permutation $ a $ of length $ n $ $ ^{\text{∗}} $ . We say that a permutation $ b $ of length $ n $ is good if the two permutations $ a $ and $ b $ can become the same after performing the following operation at most $ n $ times (possibly zero): - Choose two integers $ l, r $ such that $ 1 \le l < r \le n $ and $ a_r = \min(a_l, a_{l + 1}, \ldots, a_r) $ . - Cyclically shift the subsegment $ [a_l, a_{l + 1}, \ldots, a_r] $ one position to the right. In other words, replace $ a $ with $ [a_1, \ldots, a_{l - 1}, \; a_r, a_l, a_{l + 1}, \ldots, a_{r - 1}, \; a_{r + 1}, \ldots, a_n] $ . You are also given a permutation $ c $ of length $ n $ where some elements are missing and represented by $ 0 $ . You need to find a good permutation $ b_1, b_2, \ldots, b_n $ such that $ b $ can be formed by filling in the missing elements of $ c $ (i.e., for all $ 1 \le i \le n $ , if $ c_i \ne 0 $ , then $ b_i = c_i $ ). If it is impossible, output $ -1 $ . $ ^{\text{∗}} $ A permutation of length $ n $ 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).

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 5 \cdot 10^5 $ ). The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le n $ ). It is guaranteed that $ a $ is a permutation of length $ n $ . The third line of each test case contains $ n $ integers $ c_1, c_2, \ldots, c_n $ ( $ 0 \le c_i \le n $ ). It is guaranteed that the elements of $ c $ that are not $ 0 $ are distinct. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 5 \cdot 10^5 $ .

For each test case: - If it is impossible to find such a good permutation $ b $ , output a single integer $ -1 $ . - Otherwise, output $ n $ integers $ b_1, b_2, \ldots, b_n $ — the good permutation $ b $ you've found. You need to ensure that for all $ 1 \le i \le n $ , if $ c_i \ne 0 $ , then $ b_i = c_i $ . If there are multiple answers, output any of them.

In the first test case, $ b = [1, 2] $ is a valid answer since after performing the following operation, $ a $ and $ b $ will become the same: - Choose $ l = 1, r = 2 $ and cyclically shift the subsegment $ [a_1, a_2] $ one position to the right. Then $ a $ will become $ [1, 2] $ . In the second test case, $ b = [2, 3, 4, 1] $ is a valid answer since after performing the following operation, $ a $ and $ b $ will become the same: - Choose $ l = 1, r = 2 $ and cyclically shift the subsegment $ [a_1, a_2] $ one position to the right. Then $ a $ will become $ [2, 3, 4, 1] $ . In the third test case, $ b = [1, 3, 2, 4, 5] $ is a valid answer since after performing the following operation, $ a $ and $ b $ will become the same: - Choose $ l = 1, r = 3 $ and cyclically shift the subsegment $ [a_1, a_2, a_3] $ one position to the right. Then $ a $ will become $ [1, 3, 2, 5, 4] $ . - Choose $ l = 4, r = 5 $ and cyclically shift the subsegment $ [a_4, a_5] $ one position to the right. Then $ a $ will become $ [1, 3, 2, 4, 5] $ . In the fourth test case, $ b = [3, 2, 1, 5, 4] $ is a valid answer since $ a $ and $ b $ are already the same. In the fifth test case, it is impossible to find such a good permutation $ b $ , so you should output $ -1 $ . In the sixth test case, $ b = [3, 2, 1, 5, 4, 6] $ is a valid answer since after performing the following operation, $ a $ and $ b $ will become the same: - Choose $ l = 2, r = 4 $ and cyclically shift the subsegment $ [a_2, a_3, a_4] $ one position to the right. Then $ a $ will become $ [3, 2, 5, 6, 1, 4] $ . - Choose $ l = 3, r = 5 $ and cyclically shift the subsegment $ [a_3, a_4, a_5] $ one position to the right. Then $ a $ will become $ [3, 2, 1, 5, 6, 4] $ . - Choose $ l = 5, r = 6 $ and cyclically shift the subsegment $ [a_5, a_6] $ one position to the right. Then $ a $ will become $ [3, 2, 1, 5, 4, 6] $ .