Permutation Weight (Hard Version)

题意翻译

**这是本题的困难版本。** **简单版本和困难版本的区别在于对你所构造的排列的限制。** **在本题中你需要输出所有权值最小化的排列中字典序最小的那个。** ### 题目描述 给定一个 $1\sim n$ 的排列 $p$。 对于一个 $1\sim n$ 的排列 $q$，定义其权值为： $$|q_1-p_{q_2}|+|q_2-p_{q_3}|+|q_3-p_{q_4}|+\cdots+|q_{n-1}-p_{q_n}|+|q_n-p_{q_1}|$$ 找出所有满足权值最小化的 $1\sim n$ 的排列 $q$ 中 **字典序最小** 的那个。 我们称两个 $1\sim n$ 的排列 $a,b$ 有 $a$ 比 $b$ 字典序小仅当存在整数 $i(1\leq i\leq n)$ 满足 $a_i<b_i$ 且对于任意整数 $j(1\leq j<i)$ 都有 $a_j=b_j$。 每个测试点包含 $t$ 组数据。 ### 输入格式 第一行输入一个整数 $t(1\leq t\leq100)$ 表示数据组数，接下来对于每组数据： 第一行输入一个整数 $n(2\leq n\leq200;\sum n\leq400)$ 表示排列长度。 接下来一行输入 $n$ 个整数表示排列 $p(1\leq p_i\leq n)$。 ### 输出格式 对于每组数据： 输出 $n$ 个整数表示你求出的权值最小化且字典序最小的排列 $q$。

题目描述

This is a hard version of the problem. The difference between the easy and hard versions is that in this version, you have to output the lexicographically smallest permutation with the smallest weight. You are given a permutation $ p_1, p_2, \ldots, p_n $ of integers from $ 1 $ to $ n $ . Let's define the weight of the permutation $ q_1, q_2, \ldots, q_n $ of integers from $ 1 $ to $ n $ as $ $|q_1 - p_{q_{2}}| + |q_2 - p_{q_{3}}| + \ldots + |q_{n-1} - p_{q_{n}}| + |q_n - p_{q_{1}}| $ $ </p><p>You want your permutation to be as lightweight as possible. Among the permutations $ q $ with the smallest possible weight, find the lexicographically smallest.</p><p>Permutation $ a\_1, a\_2, \\ldots, a\_n $ is lexicographically smaller than permutation $ b\_1, b\_2, \\ldots, b\_n $ , if there exists some $ 1 \\le i \\le n $ such that $ a\_j = b\_j $ for all $ 1 \\le j &lt; i $ and $ a\_i&lt;b\_i$.

输入输出格式

输入格式

The first line of the input contains a single integer $ t $ ( $ 1 \le t \le 100 $ ) — the number of test cases. The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 2 \le n \le 200 $ ) — the size of the permutation. The second line of each test case contains $ n $ integers $ p_1, p_2, \ldots, p_n $ ( $ 1 \le p_i \le n $ , all $ p_i $ are distinct) — the elements of the permutation. The sum of $ n $ over all test cases doesn't exceed $ 400 $ .

输出格式

For each test case, output $ n $ integers $ q_1, q_2, \ldots, q_n $ ( $ 1 \le q_i \le n $ , all $ q_i $ are distinct) — the lexicographically smallest permutation with the smallest weight.

输入输出样例

输入样例 #1

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

输出样例 #1

1 2 
1 3 4 2 
1 3 4 2 5

说明

In the first test case, there are two permutations of length $ 2 $ : $ (1, 2) $ and $ (2, 1) $ . Permutation $ (1, 2) $ has weight $ |1 - p_2| + |2 - p_1| = 0 $ , and the permutation $ (2, 1) $ has the same weight: $ |2 - p_1| + |1 - p_2| = 0 $ . In this version, you have to output the lexicographically smaller of them — $ (1, 2) $ . In the second test case, the weight of the permutation $ (1, 3, 4, 2) $ is $ |1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_1| = |1 - 1| + |3 - 4| + |4 - 3| + |2 - 2| = 2 $ . There are no permutations with smaller weights. In the third test case, the weight of the permutation $ (1, 3, 4, 2, 5) $ is $ |1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_5| + |5 - p_1| = |1 - 3| + |3 - 2| + |4 - 4| + |2 - 1| + |5 - 5| = 4 $ . There are no permutations with smaller weights.