Restoring the Permutation

题意翻译

我们规定长度为 $n$ 的序列 $a$ 为一个长度为 $n$ 的排列仅当 $1$ 到 $n$ 的所有整数都在序列 $a$ 中存在。 进一步的，对于长度均为 $n$ 的两个排列 $a,b$，我们认为 $a$ 的字典序比 $b$ 的字典序大仅当存在整数 $i(1\leq i\leq n)$ 使得任意整数 $j\in[1,i-1]$ 有 $a_j=b_j$ 且 $a_i>b_i$。 你有一个长度为 $n$ 的序列 $q$，我们认为排列 $p$ 是合法排列仅当： - $p$ 为长度为 $n$ 的排列。 - 任意整数 $i\in[1,n]$ 有 $q_i=\max(p_1,p_2,\cdots,p_i)$。 求字典序最小的和字典序最大的合法排列，$T$ 组数据。 $1\leq T\leq10^4;1\leq n,\sum n\leq2\times10^5;1\leq q_i\leq n$ 且 $q_i$ 单调不减。

题目描述

A permutation is a sequence of $ n $ integers from $ 1 $ to $ n $ , in which all numbers occur exactly once. For example, $ [1] $ , $ [3, 5, 2, 1, 4] $ , $ [1, 3, 2] $ are permutations, and $ [2, 3, 2] $ , $ [4, 3, 1] $ , $ [0] $ are not. Polycarp was presented with a permutation $ p $ of numbers from $ 1 $ to $ n $ . However, when Polycarp came home, he noticed that in his pocket, the permutation $ p $ had turned into an array $ q $ according to the following rule: - $ q_i = \max(p_1, p_2, \ldots, p_i) $ . Now Polycarp wondered what lexicographically minimal and lexicographically maximal permutations could be presented to him. An array $ a $ of length $ n $ is lexicographically smaller than an array $ b $ of length $ n $ if there is an index $ i $ ( $ 1 \le i \le n $ ) such that the first $ i-1 $ elements of arrays $ a $ and $ b $ are the same, and the $ i $ -th element of the array $ a $ is less than the $ i $ -th element of the array $ b $ . For example, the array $ a=[1, 3, 2, 3] $ is lexicographically smaller than the array $ b=[1, 3, 4, 2] $ . For example, if $ n=7 $ and $ p=[3, 2, 4, 1, 7, 5, 6] $ , then $ q=[3, 3, 4, 4, 7, 7, 7] $ and the following permutations could have been as $ p $ initially: - $ [3, 1, 4, 2, 7, 5, 6] $ (lexicographically minimal permutation); - $ [3, 1, 4, 2, 7, 6, 5] $ ; - $ [3, 2, 4, 1, 7, 5, 6] $ ; - $ [3, 2, 4, 1, 7, 6, 5] $ (lexicographically maximum permutation). For a given array $ q $ , find the lexicographically minimal and lexicographically maximal permutations that could have been originally presented to Polycarp.

输入输出格式

输入格式

The first line contains one integer $ t $ ( $ 1 \le t \le 10^4 $ ). Then $ t $ test cases follow. The first line of each test case contains one integer $ n $ ( $ 1 \le n \le 2 \cdot 10^5 $ ). The second line of each test case contains $ n $ integers $ q_1, q_2, \ldots, q_n $ ( $ 1 \le q_i \le n $ ). It is guaranteed that the array $ q $ was obtained by applying the rule from the statement to some permutation $ p $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, output two lines: - on the first line output $ n $ integers — lexicographically minimal permutation that could have been originally presented to Polycarp; - on the second line print $ n $ integers — lexicographically maximal permutation that could have been originally presented to Polycarp;

输入输出样例

输入样例 #1

4
7
3 3 4 4 7 7 7
4
1 2 3 4
7
3 4 5 5 5 7 7
1
1

输出样例 #1

3 1 4 2 7 5 6 
3 2 4 1 7 6 5 
1 2 3 4 
1 2 3 4 
3 4 5 1 2 7 6 
3 4 5 2 1 7 6 
1 
1

题意翻译

我们规定长度为 $n$ 的序列 $a$ 为一个长度为 $n$ 的排列仅当 $1$ 到 $n$ 的所有整数都在序列 $a$ 中存在。 进一步的，对于长度均为 $n$ 的两个排列 $a,b$，我们认为 $a$ 的字典序比 $b$ 的字典序大仅当存在整数 $i(1\leq i\leq n)$ 使得任意整数 $j\in[1,i-1]$ 有 $a_j=b_j$ 且 $a_i>b_i$。 你有一个长度为 $n$ 的序列 $q$，我们认为排列 $p$ 是合法排列仅当： - $p$ 为长度为 $n$ 的排列。 - 任意整数 $i\in[1,n]$ 有 $q_i=\max(p_1,p_2,\cdots,p_i)$。 求字典序最小的和字典序最大的合法排列，$T$ 组数据。 $1\leq T\leq10^4;1\leq n,\sum n\leq2\times10^5;1\leq q_i\leq n$ 且 $q_i$ 单调不减。