CF1585D.Yet Another Sorting Problem

Petya has an array of integers $a_1, a_2, \ldots, a_n$ . He only likes sorted arrays. Unfortunately, the given array could be arbitrary, so Petya wants to sort it.

Petya likes to challenge himself, so he wants to sort array using only $3$ -cycles. More formally, in one operation he can pick $3$ pairwise distinct indices $i$ , $j$ , and $k$ ( $1 \leq i, j, k \leq n$ ) and apply $i \to j \to k \to i$ cycle to the array $a$ . It simultaneously places $a_i$ on position $j$ , $a_j$ on position $k$ , and $a_k$ on position $i$ , without changing any other element.

For example, if $a$ is $[10, 50, 20, 30, 40, 60]$ and he chooses $i = 2$ , $j = 1$ , $k = 5$ , then the array becomes $[\underline{50}, \underline{40}, 20, 30, \underline{10}, 60]$ .

Petya can apply arbitrary number of $3$ -cycles (possibly, zero). You are to determine if Petya can sort his array $a$ , i. e. make it non-decreasing.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \leq t \leq 5 \cdot 10^5$ ). Description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 5 \cdot 10^5$ ) — the length of the array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq n$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $5 \cdot 10^5$ .

For each test case, print "YES" (without quotes) if Petya can sort the array $a$ using $3$ -cycles, and "NO" (without quotes) otherwise. You can print each letter in any case (upper or lower).

In the $6$ -th test case Petya can use the $3$ -cycle $1 \to 3 \to 2 \to 1$ to sort the array.

In the $7$ -th test case Petya can apply $1 \to 3 \to 2 \to 1$ and make $a = [1, 4, 2, 3]$ . Then he can apply $2 \to 4 \to 3 \to 2$ and finally sort the array.