Count Inversions
D
/**
* @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
answer by summing
* 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.
* 5. Print the answer
*
* @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;
}
