``````/**
* @file
* @brief Counting Inversions using [Merge
Sort](https://en.wikipedia.org/wiki/Merge_sort)
*
* @details
* Program to count the number of inversions in an array
* using merge-sort.
*
* The count of inversions help to determine how close the array
* is to be sorted in ASCENDING order.
*
* two elements a[i] and a[j] form an inversion if `a[i]` > `a[j]` and i < j
*
* Time Complexity --> `O(n.log n)`

* Space Complexity --> `O(n)` ; additional array `temp[1..n]`
* ### Algorithm

*   1. The idea is similar to merge sort, divide the array into two equal or
almost
*      equal halves in each step until the base case is reached.
*   2. Create a function merge that counts the number of inversions when two
halves of
*      the array are merged, create two indices i and j, i is the index for
first half
*      and j is an index of the second half. if `a[i]` is greater than `a[j]`,
then there are (mid – i)
*      inversions, Because left and right subarrays are sorted, so all the
remaining elements
*      in left-subarray (a[i+1], a[i+2] … a[mid]) will be greater than a[j].
*   3. Create a recursive function to divide the array into halves and find the
*      the number of inversions is the first half, number of inversion in the
second half and
*      the number of inversions by merging the two.
*   4. The base case of recursion is when there is only one element in the
given half.
*
* @author [Rakshit Raj](https://github.com/rakshitraj)
*/
#include <cassert>   /// for assert
#include <cstdint>   /// for typedef datatype uint64_t
#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector

/**
* @namespace sorting
* @brief Sorting algorithms
*/
namespace sorting {
/**
* @namespace inversion
* @brief Functions for counting inversions using Merge Sort algorithm
*/
namespace inversion {

// Functions used --->
// int mergeSort(int* arr, int* temp, int left, int right);
// int merge(int* arr, int* temp, int left, int mid, int right);
// int countInversion(int* arr, const int size);
// void show(int* arr, const int size);

/**
* @brief Function to merge two sub-arrays.
*
* @details
* merge() function is called from mergeSort()
* to merge the array after it split for sorting
* by the mergeSort() funtion.
*
* In this case the merge fuction will also count and return
* inversions detected when merging the sub arrays.
*
* @param arr    input array, data-menber of vector
* @param temp   stores the resultant merged array
* @param left   lower bound of `arr[]` and left-sub-array
* @param mid    midpoint, upper bound of left sub-array,
*               `(mid+1)` gives the lower bound of right-sub-array
* @param right  upper bound of `arr[]` and right-sub-array
* @returns number of inversions found in merge step
*/
template <typename T>
uint32_t merge(T* arr, T* temp, uint32_t left, uint32_t mid, uint32_t right) {
uint32_t i = left;       /* i --> index of left sub-array */
uint32_t j = mid + 1;    /* j --> index for right sub-array */
uint32_t k = left;       /* k --> index for resultant array temp */
uint32_t inv_count = 0;  // inversion count

while ((i <= mid) && (j <= right)) {
if (arr[i] <= arr[j]) {
temp[k++] = arr[i++];
} else {
temp[k++] = arr[j++];
inv_count +=
(mid - i +
1);  // tricky; may vary depending on selection of sub-array
}
}
// Add remaining elements from the larger subarray to the end of temp
while (i <= mid) {
temp[k++] = arr[i++];
}
while (j <= right) {
temp[k++] = arr[j++];
}
// Copy temp[] to arr[]
for (k = left; k <= right; k++) {
arr[k] = temp[k];
}
return inv_count;
}

/**
* @brief Implement merge Sort and count inverions while merging
*
* @details
* The mergeSort() function implements Merge Sort, a
* Divide and conquer algorithm, it divides the input
* array into two halves and calls itself for each
* sub-array and then calls the merge() function to
* merge the two halves.
*
* @param arr   - array to be sorted
* @param temp  - merged resultant array
* @param left  - lower bound of array
* @param right - upper bound of array
* @returns number of inversions in array
*/
template <typename T>
uint32_t mergeSort(T* arr, T* temp, uint32_t left, uint32_t right) {
uint32_t mid = 0, inv_count = 0;
if (right > left) {
// midpoint to split the array
mid = (right + left) / 2;
// Add inversions in left and right sub-arrays
inv_count += mergeSort(arr, temp, left, mid);  // left sub-array
inv_count += mergeSort(arr, temp, mid + 1, right);

// inversions in the merge step
inv_count += merge(arr, temp, left, mid, right);
}
return inv_count;
}

/**
* @brief Function countInversion() returns the number of inversion
* present in the input array. Inversions are an estimate of
* how close or far off the array is to being sorted.
*
* @details
* Number of inversions in a sorted array is 0.
* Number of inversion in an array[1...n] sorted in
* non-ascending order is n(n-1)/2, since each pair of elements
* contitute an inversion.
*
* @param arr   - array, data member of std::vector<int>, input for counting
* inversions
* @param array_size    - number of elementa in the array
* @returns number of inversions in input array, sorts the array
*/
template <class T>
uint32_t countInversion(T* arr, const uint32_t size) {
std::vector<T> temp;
temp.reserve(size);
temp.assign(size, 0);
return mergeSort(arr, temp.data(), 0, size - 1);
}

/**
* @brief UTILITY function to print array.
* @param arr[]   array to print
* @param array_size    size of input array arr[]
* @returns void
*
*/
template <typename T>
void show(T* arr, const uint32_t array_size) {
std::cout << "Printing array: \n";
for (uint32_t i = 0; i < array_size; i++) {
std::cout << " " << arr[i];
}
std::cout << "\n";
}

}  // namespace inversion
}  // namespace sorting

/**
* @brief Test implementations
* @returns void
*/
static void test() {
// Test 1
std::vector<uint64_t> arr1 = {
100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84,
83,  82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67,
66,  65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50,
49,  48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33,
32,  31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16,
15,  14, 13, 12, 11, 10, 9,  8,  7,  6,  5,  4,  3,  2,  1};
uint32_t size1 = arr1.size();
uint32_t inv_count1 = 4950;
uint32_t result1 = sorting::inversion::countInversion(arr1.data(), size1);
assert(inv_count1 == result1);
// Test 2
std::vector<int> arr2 = {22, 66, 75, 23, 11, 87, 2, 44, 98, 43};
uint32_t size2 = arr2.size();
uint32_t inv_count2 = 20;
uint32_t result2 = sorting::inversion::countInversion(arr2.data(), size2);
assert(inv_count2 == result2);
// Test 3
std::vector<double> arr3 = {33.1, 45.2, 65.4, 76.5, 1.0,
2.9,  5.4,  7.7,  88.9, 12.4};
uint32_t size3 = arr3.size();
uint32_t inv_count3 = 21;
uint32_t result3 = sorting::inversion::countInversion(arr3.data(), size3);
assert(inv_count3 == result3);
// Test 4
std::vector<char> arr4 = {'a', 'b', 'c', 'd', 'e'};
uint32_t size4 = arr4.size();
uint32_t inv_count4 = 0;
uint32_t result4 = sorting::inversion::countInversion(arr4.data(), size4);
assert(inv_count4 == result4);
}

// /**
//  * @brief Program Body contains all main funtionality
//  * @returns void
//  */
// template <typename T>
// static void body() {
//     // Input your own sequence
//     uint_t size;
//     T input;
//     std::cout << "Enter number of elements:";
//     std::cin >> size;
//
//     std::vector<T> arr;
//     arr.reserve(size);
//
//     std::cout << "Enter elements -->\n";
//     for (uint64_t i=1; i<=size; i++) {
//         std::cout << "Element "<< i <<" :";
//         std::cin >> input;
//         arr.push_back(input);
//     }
//
//     if (size != arr.size()) {
//         size = arr.size();
//     }
//
//     std::cout << "\n";
//     sorting::inversion::show(arr.data(), size);
//     std::cout << "\n";
//
//     // Counting inversions
//     std::cout << "\nThe number of inversions: "<<
//     sorting::inversion::countInversion(arr.data(), size) << "\n";
//
//     // Output sorted array
//     std::cout << "\nSorted array -->  \n";
//     sorting::inversion::show(arr.data(), size);
// }

/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
test();  // Run test implementations
// body(); // test your own array
return 0;
}
``````

#### Count Inversions

R  