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>
  #pragma comment(linker, "/STACK:102400000,102400000")
  const double PI = 3.14159265359;
  using namespace std;
  const int N = 5e5 + 10;
  int a[N], tr[N];
  map<int, int> r;
  int n;
  int bitwise(int x) { return x & -x; }
  void add(int x, int c) {
    for (int i = x; i <= n; i += bitwise(i)) {
      tr[i] += c;
    }
  }
  int sum(int x) {
    int ans = 0;
    for (int i = x; i >= 1; i -= bitwise(i)) {
      ans += tr[i];
    }
    return ans;
  }
  int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    int t;
    cin >> t;
    while (t--) {
      r.clear();
      cin >> n;
      int maxn = 0;
      int flag = 0;
      int ans = 0;
      for (int i = 1; i <= n; i++) {
        cin >> a[i];
        maxn = max(a[i], maxn);
        r[a[i]]++;
        if (r[a[i]] > 1) flag = 1;
      }
      if (flag == 1) {
        cout << "YES"
             << "\n";
        continue;
      }
      for (int i = 1; i <= n; i++) {
        ans += sum(n) - sum(a[i] - 1);
        add(a[i], 1);
      }
      if (ans % 2 == 0) {
        cout << "YES"
             << "\n";
      } else {
        cout << "NO"
             << "\n";
      }
      for (int i = 0; i <= maxn; i++) {
        tr[i] = 0;
      }
    }
  }
  