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;
  int main() {
    int T;
    cin >> T;
    while (T--) {
      int N;
      cin >> N;
      int A[N];
      map<int, set<int> > M;
      bool res = false;
      for (int i = 0; i < N; i++) {
        cin >> A[i];
        M[A[i]].insert(i);
        if (M[A[i]].size() > 1) {
          res = true;
        }
      }
      if (res) {
        cout << "YES\n";
        continue;
      }
      for (int i = N - 1; i >= 2; i--) {
        auto itr = M.rbegin();
        int ele = (*itr).first;
        if (ele == A[i]) {
          if ((*itr).second.size() == 1) {
            M.erase(ele);
          } else {
            (*itr).second.erase(i);
          }
          continue;
        }
        int i1, i2, i3;
        auto itr2 = (*itr).second.begin();
        if ((*itr2) == 0) {
          i1 = 0;
          i2 = 1;
          i3 = i;
          int x = A[i1], y = A[i2], z = A[i3];
          if (M[x].size() == 1) {
            M.erase(x);
          } else {
            M[x].erase(i1);
          }
          M[z].insert(i2);
          M[z].erase(i3);
          M[y].erase(i2);
          M[y].insert(i1);
          swap(A[i2], A[i3]);
          swap(A[i1], A[i3]);
        } else {
          i1 = 0;
          i2 = *((*itr).second.begin());
          i3 = i;
          int x = A[i1], y = A[i2], z = A[i3];
          M[z].erase(i3);
          M[z].insert(i1);
          M[x].insert(i2);
          M[x].erase(i1);
          if (M[y].size() == 1) {
            M.erase(y);
          } else {
            M[y].erase(i2);
          }
          swap(A[i2], A[i3]);
          swap(A[i1], A[i2]);
        }
      }
      if (A[0] <= A[1]) {
        cout << "YES\n";
      } else {
        cout << "NO\n";
      }
    }
    return 0;
  }
  