description: Petya has an array of integers a_1, a_2, …, 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 ≤ i, j, k ≤ n) and apply i → j → k → 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.
  
  Input
  
  Each test contains multiple test cases. The first line contains the number of test cases t (1 ≤ t ≤ 5 ⋅ 10^5). Description of the test cases follows.
  
  The first line of each test case contains a single integer n (1 ≤ n ≤ 5 ⋅ 10^5) — the length of the array a.
  
  The second line of each test case contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ n).
  
  It is guaranteed that the sum of n over all test cases does not exceed 5 ⋅ 10^5.
  
  Output
  
  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).
  
  Example
  
  Input
  
  
  7
  1
  1
  2
  2 2
  2
  2 1
  3
  1 2 3
  3
  2 1 3
  3
  3 1 2
  4
  2 1 4 3
  
  
  Output
  
  
  YES
  YES
  NO
  YES
  NO
  YES
  YES
  
  Note
  
  In the 6-th test case Petya can use the 3-cycle 1 → 3 → 2 → 1 to sort the array.
  
  In the 7-th test case Petya can apply 1 → 3 → 2 → 1 and make a = [1, 4, 2, 3]. Then he can apply 2 → 4 → 3 → 2 and finally sort the array.
solution: #include <bits/stdc++.h>
  using namespace std;
  const int MOD = 998244353;
  int32_t main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    int tt;
    cin >> tt;
    while (tt--) {
      int n;
      cin >> n;
      vector<int> a(n), cnt(n + 1);
      for (int i = 0; i < n; i++) {
        cin >> a[i];
        cnt[a[i]]++;
      }
      if (n == 1) {
        cout << "YES\n";
        continue;
      } else if (n == 2) {
        if (a[0] <= a[1])
          cout << "YES\n";
        else
          cout << "NO\n";
        continue;
      }
      bool done = false;
      for (int i = 1; i <= n; i++) {
        if (cnt[i] >= 2) {
          cout << "YES\n";
          done = true;
          break;
        }
      }
      if (done) continue;
      vector<int> ind(n + 1);
      for (int i = 0; i < n; i++) {
        ind[a[i]] = i;
      }
      for (int curr_ind = 0; curr_ind < n - 2; curr_ind++) {
        int idx1 = curr_ind;
        int idx2 = ind[curr_ind + 1];
        int idx3 = ind[curr_ind + 1] + 1;
        if (idx2 == n - 1) {
          idx3 = idx2;
          idx2--;
        }
        while (a[curr_ind] != curr_ind + 1) {
          swap(a[idx3], a[idx2]);
          swap(a[idx2], a[idx1]);
          ind[a[idx1]] = idx1;
          ind[a[idx2]] = idx2;
          ind[a[idx3]] = idx3;
        }
      }
      done = true;
      for (int i = 0; i < n - 1; i++) {
        if (a[i + 1] < a[i]) done = false;
      }
      if (done)
        cout << "YES\n";
      else
        cout << "NO\n";
    }
  }
  