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270 lines
8.5 KiB
C++
270 lines
8.5 KiB
C++
/**
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* @file
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* @brief Counting Inversions using [Merge
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Sort](https://en.wikipedia.org/wiki/Merge_sort)
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*
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* @details
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* Program to count the number of inversions in an array
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* using merge-sort.
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*
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* The count of inversions help to determine how close the array
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* is to being sorted in ASCENDING order.
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*
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* two elements a[i] and a[j] form an inversion if `a[i]` > `a[j]` and i < j
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*
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* Time Complexity --> `O(n.log n)`
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* Space Complexity --> `O(n)` ; additional array `temp[1..n]`
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* ### Algorithm
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* 1. The idea is similar to merge sort, divide the array into two equal or
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almost
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* equal halves in each step until the base case is reached.
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* 2. Create a function merge that counts the number of inversions when two
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halves of
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* the array are merged, create two indices i and j, i is the index for
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first half
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* and j is an index of the second half. if `a[i]` is greater than `a[j]`,
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then there are (mid – i)
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* inversions, Because left and right subarrays are sorted, so all the
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remaining elements
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* in left-subarray (a[i+1], a[i+2] … a[mid]) will be greater than a[j].
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* 3. Create a recursive function to divide the array into halves and find the
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answer by summing
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* the number of inversions is the first half, number of inversion in the
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second half and
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* the number of inversions by merging the two.
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* 4. The base case of recursion is when there is only one element in the
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given half.
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* 5. Print the answer
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*
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* @author [Rakshit Raj](https://github.com/rakshitraj)
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*/
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#include <cassert> // for tests
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#include <cstdint> // for typedef datatype uint64_t
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#include <iostream> // for IO
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#include <vector> // for std::vector<>
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/**
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* @namespace sorting
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* @brief Sorting algorithms
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*/
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namespace sorting {
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/**
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* @namespace inversion
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* @brief Functions for counting inversions using Merge Sort algorithm
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*/
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namespace inversion {
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// Functions used --->
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// int mergeSort(int* arr, int* temp, int left, int right);
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// int merge(int* arr, int* temp, int left, int mid, int right);
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// int countInversion(int* arr, const int size);
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// void show(int* arr, const int size);
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/**
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* Function to merge two sub-arrays. merge() function is called
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* from mergeSort() to merge the array after it split for sorting
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* by the mergeSort() funtion.
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*
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* In this case the merge fuction will also count and return
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* inversions detected when merging the sub arrays.
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*
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* @param arr input array, data-menber of vector
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* @param temp stores the resultant merged array
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* @param left lower bound of `arr[]` and left-sub-array
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* @param mid midpoint, upper bound of left sub-array,
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* `(mid+1)` gives the lower bound of right-sub-array
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* @param right upper bound of `arr[]` and right-sub-array
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* @returns number of inversions found in merge step
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*/
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template <typename T>
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uint64_t merge(T* arr, T* temp, uint64_t left, uint64_t mid, uint64_t right) {
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uint64_t i = left; /* i --> index of left sub-array */
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uint64_t j = mid + 1; /* j --> index for right sub-array */
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uint64_t k = left; /* k --> index for resultant array temp */
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uint64_t inv_count = 0; // inversion count
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while ((i <= mid) && (j <= right)) {
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if (arr[i] <= arr[j]) {
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temp[k++] = arr[i++];
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} else {
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temp[k++] = arr[j++];
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inv_count +=
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(mid - i +
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1); // tricky; may vary depending on selection of sub-array
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}
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}
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// Add remaining elements from the larger subarray to the end of temp
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while (i <= mid) {
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temp[k++] = arr[i++];
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}
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while (j <= right) {
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temp[k++] = arr[j++];
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}
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// Copy temp[] to arr[]
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for (k = left; k <= right; k++) {
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arr[k] = temp[k];
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}
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return inv_count;
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}
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/**
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*
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* The mergeSort() function implements Merge Sort, a
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* Divide and conquer algorithm, it divides the input
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* array into two halves and calls itself for each
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* sub-array and then calls the merge() function to
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* merge the two halves.
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*
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* @param arr - array to be sorted
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* @param temp - merged resultant array
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* @param left - lower bound of array
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* @param right - upper bound of array
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* @returns number of inversions in array
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*/
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template <typename T>
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uint64_t mergeSort(T* arr, T* temp, uint64_t left, uint64_t right) {
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uint64_t mid = 0, inv_count = 0;
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if (right > left) {
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// midpoint to split the array
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mid = (right + left) / 2;
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// Add inversions in left and right sub-arrays
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inv_count += mergeSort(arr, temp, left, mid); // left sub-array
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inv_count += mergeSort(arr, temp, mid + 1, right);
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// inversions in the merge step
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inv_count += merge(arr, temp, left, mid, right);
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}
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return inv_count;
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}
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/**
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* Funtion countInversion() returns the number of inversion
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* present in the input array. Inversions are an estimate of
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* how close or far off the array is to being sorted.
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*
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* Number of inversions in a sorted array is 0.
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* Number of inversion in an array[1...n] sorted in
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* non-ascending order is n(n-1)/2, since each pair of elements
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* contitute an inversion.
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*
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* @param arr - array, data member of std::vector<int>, input for counting
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* inversions
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* @param array_size - number of elementa in the array
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* @returns number of inversions in input array, sorts the array
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*/
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template <class T>
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int countInversion(T* arr, const uint64_t size) {
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std::vector<T> temp;
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temp.reserve(size);
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temp.assign(size, 0);
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return mergeSort(arr, temp.data(), 0, size - 1);
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}
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/**
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* UTILITY function to print array.
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* @param arr[] array to print
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* @param array_size size of input array arr[]
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* @returns void
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*
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*/
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template <typename T>
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void show(T* arr, const uint64_t array_size) {
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std::cout << "Printing array: \n";
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for (uint64_t i = 0; i < array_size; i++) {
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std::cout << " " << arr[i];
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}
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std::cout << "\n";
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}
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} // namespace inversion
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} // namespace sorting
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/**
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* @brief Test implementations
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* @returns void
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*/
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static void test() {
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// Test 1
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std::vector<uint64_t> arr1 = {
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100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84,
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83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67,
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66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50,
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49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33,
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32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16,
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15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1};
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uint64_t size1 = arr1.size();
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uint64_t inv_count1 = 4950;
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uint64_t result1 = sorting::inversion::countInversion(arr1.data(), size1);
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assert(inv_count1 == result1);
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// Test 2
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std::vector<int> arr2 = {22, 66, 75, 23, 11, 87, 2, 44, 98, 43};
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uint64_t size2 = arr2.size();
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uint64_t inv_count2 = 20;
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uint64_t result2 = sorting::inversion::countInversion(arr2.data(), size2);
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assert(inv_count2 == result2);
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// Test 3
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std::vector<double> arr3 = {33.1, 45.2, 65.4, 76.5, 1.0,
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2.9, 5.4, 7.7, 88.9, 12.4};
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uint64_t size3 = arr3.size();
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uint64_t inv_count3 = 21;
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uint64_t result3 = sorting::inversion::countInversion(arr3.data(), size3);
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assert(inv_count3 == result3);
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// Test 4
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std::vector<char> arr4 = {'a', 'b', 'c', 'd', 'e'};
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uint64_t size4 = arr4.size();
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uint64_t inv_count4 = 0;
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uint64_t result4 = sorting::inversion::countInversion(arr4.data(), size4);
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assert(inv_count4 == result4);
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}
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// /**
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// * Program Body contains all main funtionality
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// * @returns void
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// */
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// void body() {
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// // Input your own sequence
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// int size, input;
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// std::cout << "Enter number of elements:";
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// std::cin >> size;
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// std::vector<int> arr;
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// arr.reserve(size);
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// std::cout << "Enter elements -->\n";
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// for (int i=1; i<=size; i++) {
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// std::cout << "Element "<< i <<" :";
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// std::cin >> input;
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// arr.push_back(input);
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// }
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// if (size != arr.size()) {
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// size = arr.size();
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// }
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// std::cout << "\n";
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// sorting::inversion::show(arr.data(), size);
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// std::cout << "\n";
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// // Counting inversions
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// std::cout << "\nThe number of inversions: "<<
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// sorting::inversion::countInversion(arr.data(), size) << "\n";
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// // Output sorted array
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// std::cout << "\nSorted array --> \n";
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// sorting::inversion::show(arr.data(), size);
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// }
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/**
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* @brief Main function
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* @returns 0 on exit
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*/
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int main() {
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// Run test implementations
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test();
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// // Main Program
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// body();
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return 0;
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}
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