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 maxn = 5e5 + 10;
  struct node {
    int x, id;
    bool operator<(const node& y) const { return x < y.x; }
  } a[maxn];
  bool cmp(const node& x, const node& y) { return x.id < y.id; }
  bool v[maxn];
  int c[maxn];
  int n;
  void add(int x) {
    while (x <= n) {
      c[x]++;
      x += ((x) & (-(x)));
    }
  }
  int sum(int x) {
    int ans = 0;
    while (x >= 1) {
      ans += c[x];
      x -= ((x) & (-(x)));
    }
    return ans;
  }
  int main() {
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t;
    cin >> t;
    while (t--) {
      cin >> n;
      for (int i = 1; i <= n; ++i) {
        cin >> a[i].x;
        a[i].id = i;
        v[i] = 0;
        c[i] = 0;
      }
      bool flag = 0;
      for (int i = 1; i <= n; ++i) {
        if (v[a[i].x]) {
          flag = 1;
          break;
        }
        v[a[i].x] = 1;
      }
      if (flag) {
        cout << "YES" << endl;
      } else {
        int ans = 0;
        for (int i = 1; i <= n; ++i) {
          ans += sum(n) - sum(a[i].x);
          add(a[i].x);
        }
        cout << ((ans & 1) ? "NO" : "YES") << endl;
      }
    }
    return 0;
  }
  