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>
  const long long MAXN = 500005;
  long long bit[MAXN];
  void add(long long x, long long v) {
    x += 1;
    while (x < MAXN) {
      bit[x] += v;
      x += x & (-x);
    }
  }
  long long sum(long long x) {
    x += 1;
    long long ret = 0;
    while (x > 0) {
      ret += bit[x];
      x -= x & (-x);
    }
    return ret;
  }
  signed main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    long long t;
    std::cin >> t;
    while (t--) {
      long long n;
      std::cin >> n;
      for (long long i = 0; i <= n; i++) bit[i] = 0;
      std::pair<long long, long long> help[n];
      for (long long i = 0; i < n; i++) {
        std::cin >> help[i].first;
        help[i].second = i;
      }
      if (n < 3) {
        bool sorted = true;
        for (long long i = 0; i + 1 < n; i++) {
          if (help[i] > help[i + 1]) sorted = false;
        }
        std::cout << (sorted ? "YES" : "NO") << std::endl;
        continue;
      }
      std::set<long long> set;
      long long done = 0;
      for (long long i = 0; i < n; i++) {
        if (set.count(help[i].first)) done = 1;
        set.insert(help[i].first);
      }
      if (done) {
        std::cout << "YES" << std::endl;
        continue;
      }
      std::sort(help, help + n);
      long long cur = 0;
      for (long long i = n - 1; i >= 0; i--) {
        cur += sum(help[i].second);
        add(help[i].second, 1);
      }
      std::cout << (cur % 2 == 0 ? "YES" : "NO") << std::endl;
    }
    return 0;
  }
  