# Count number of pairs with the given Comparator

• Difficulty Level : Hard
• Last Updated : 25 May, 2021

Given an array arr[], the task is to count the number of pairs (arr[i], arr[j]) on the right of every element with any custom comparator.

Comparator can be of any type, some of them are given below –

```arr[i] > arr[j],    where i < j
arr[i] < arr[j],    where i  2 * arr[j], where i < j```

Examples:

Input: arr[] = {5, 4, 3, 2, 1}, comp = arr[i] > arr[j]
Output: 10
Explanation:
There are 10 such pairs, in which right element is smaller than the left element –
{(5, 4), (5, 3), (5, 2), (5, 1), (4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1)}

Input: arr[] = {3, 4, 3}, comp = arr[i] == arr[j]
Output:
Explanation:
There is only one such pair such that elements are equal. That is (3, 3)

Naive Solution: Iterate over every pairs of the elements, such that i < j and check for each pair that the custom comparator is true or not. If yes, then increment the count.
Time Complexity: O(N2)

Efficient Approach: The idea is to customize merge sort, to compute such pairs at the time of merging two sub-arrays. There will be two types of count for every array that is –

• Inter-Array Pairs: Pairs those are present in the left subarray itself.
• Intra-Array Pairs: Pairs those are present in the right subarray.

For Left subarrays, the count can be calculated recursively from bottom to top whereas the main task will be to count the intra-array pairs.

Therefore, Total such pairs can be defined as –

```Total Pairs = Inter-Array pairs in Left Sub-array +
Inter-Array pairs in Right Sub-array +
Intra-Array pairs from left to right sub-array ```

Below is the illustration of the intra-array pairs of the array from left sub-array to right sub-array –

• Base Case: The base case for this problem will be when there is only one element in the two sub-arrays and we wanted to check the intra-array pairs. Then, check that those two elements form one such pair then increment the count and also place the smaller element at its position.
```if start1 == end1 and start2 == end2:
if compare(arr, start1, start2):
c += 1```
• Recursive Case: This problem can be divided into three types on the basis of the comparator function –
1. When comparison operator between pairs is of greater than or equal to.
2. When comparison operator between pairs is of less than or equal to.
3. When comparison operator between pairs is equal to.

Therefore, These all the three cases can be calculated individually for such pairs.

• Case 1: In case greater than or equal to, if we find any such pair then all the elements to right of that subarray will also form pair with the current element. Due to which the count of such pairs is incremented by the number of elements left in the left sub-array.
```if compare(arr, start1, start2):
count += end1 - start1 + 1```
• Case 2: In case less than or equal to, if we find any such pair then all the elements to right of that subarray will also form pair with the current element. Due to which the count of such pairs is incremented by the number of elements left in the right sub-array.
```if compare(arr, start1, start2):
count += end2 - start2 + 1```
• Case 3: In case equal to, If we find any such pair, then we try to find all such pairs possible in the left subarray with the help of a while loop. In each such possible pair increment the count by 1.
```if compare(arr, start1, start2):
while compare(arr, start1, start2):
count += 1
start1 += 1```
• Finally, Merge the two subarrays as it is done in the merge sort.

Below is the implementation of the above approach:

## C++

 `// C++ implementation to find the``// elements on the right with the given``// custom comparator` `#include ` `using` `namespace` `std;` `// comparator to check``// if two elements are equal``bool` `compare(``int` `arr[], ``int` `s1, ``int` `s2){``    ``if` `(arr[s1] > arr[s2]){``        ``return` `true``;``    ``}``    ``else``{``        ``return` `false``;``    ``}``}` `// Function to find the Intra-Array``// Count in the two subarrays``int` `findIntraArrayCount(``int` `arr[], ``int` `s1,``       ``int` `e1, ``int` `s2, ``int` `e2, ``int` `g){``           ` `    ``// Base Case``    ``if` `(s1 == e1 && s2 == e2){``        ``int` `c = 0;``        ``if` `(compare(arr, s1, s2)){``            ``c += 1;``        ``}``        ``if` `(arr[s1] > arr[s2]){``            ``int` `temp = arr[s1];``            ``arr[s1] = arr[s2];``            ``arr[s2] = temp;``        ``}``        ``return` `c;``    ``}``    ` `    ``// Variable for keeping``    ``// the count of the pair``    ``int` `c = 0;``    ``int` `s = s1, e = e2, s3 = s1;``    ``int` `e3 = e1, s4 = s2, e4 = e2;``    ` `    ``while` `(s1 <= e1 && s2 <= e2){``        ` `        ``// Condition when we have to use the``        ``// Greater than comparator``        ``if` `(g == 1){``            ``if` `(compare(arr, s1, s2)){``                ``c += e1 - s1 + 1;``                ``s2 += 1;``            ``}``            ``else``{``                ``s1 += 1;``            ``}``        ``}``        ` `        ``// Condition when we have to use the``        ``// Less than comparator``        ``else` `if` `(g == 0){``            ``if` `(compare(arr, s1, s2)){``                ``c += e2 - s2 + 1;``                ``s1 += 1;``            ``}``            ``else` `{``                ``s2 += 1;``            ``}``        ``}``        ` `        ``// Condition when we have to use the``        ``// Equal to Comparator``        ``else` `if` `(g == -1){``            ``if` `(compare(arr, s1, s2)){``                ``int` `c1 = 0;``                ``while` `(s1 <= e1 &&``                       ``compare(arr, s1, s2)){``                    ``c1 += 1;``                    ``s1 += 1;``                ``}``                ``s1 -= 1;``                ``int` `c2 = 0;``                ``while` `(s2 <= e2 &&``                       ``compare(arr, s1, s2)){``                    ``c2 += 1;``                    ``s2 += 1;``                ``}``                ``c += c1 * c2;``            ``}``            ``else` `{``                ``if` `(arr[s1] > arr[s2]){``                    ``s2 += 1;``                ``}``                ``else``{``                    ``s1 += 1;``                ``}``            ``}``        ``}``    ``}``    ``s1 = s3; e1 = e3;``    ``s2 = s4; e2 = e4;``    ` `    ``// Array to store``    ``// the sorted subarray``    ``vector<``int``> aux;``    ` `    ``// Merge the two subarrays``    ``while` `(s1 <= e1 && s2 <= e2){``        ``if` `(arr[s1] <= arr[s2]){``            ``aux.push_back(arr[s1]);``            ``s1 += 1;``        ``}``        ``else``{``            ``aux.push_back(arr[s2]);``            ``s2 += 1;``        ``}``    ``}``    ` `    ``// Copy subarray 1 elements``    ``while` `(s1 <= e1){``        ``aux.push_back(arr[s1]);``        ``s1 += 1;``    ``}``    ` `    ``// Copy subarray 2 elements``    ``while` `(s2 <= e2){``        ``aux.push_back(arr[s2]);``        ``s2 += 1;``    ``}``    ` `    ``// Update the original array``    ``for` `(``int` `i = s; i <= e; i++){``        ``arr[i] = aux[i-s];``    ``}``    ``return` `c;``}` `// Function to find such pairs with``// any custom comparator function``int` `findElementsOnRight(``int` `arr[], ``int` `s,``                           ``int` `e, ``int` `g){``    ``if` `(s >= e){``        ``return` `0;``    ``}``    ``int` `mid = (s+e)/2;``    ` `    ``// Recursive call for inter-array``    ``// count of pairs in left subarrays``    ``int` `c1 = findElementsOnRight(``                      ``arr, s, mid, g);``    ` `    ``// Recursive call for inter-array``    ``// count of pairs in right sub-arrays``    ``int` `c2 = findElementsOnRight(``                   ``arr, mid + 1, e, g);``    ` `    ``// Call for intra-array pairs``    ``int` `c3 = findIntraArrayCount(``              ``arr, s, mid, mid+1, e, g);``    ` `    ``return` `c1 + c2 + c3;``}`  `// Driver code``int` `main()``{``    ``int` `arr[] = {4, 3, 2, 1};``    ``int` `g = 1;``    ``cout << findElementsOnRight(arr, 0, 3, g);``    ``return` `0;``}`

## Java

 `// Java implementation to find the``// elements on the right with the given``// custom comparator` `import` `java.io.*;``import` `java.lang.*;``import` `java.util.*;` `class` `GFG {``    ` `    ``// comparator to check``    ``// if two elements are equal``    ``public` `static` `boolean` `compare(``          ``int``[] arr, ``int` `s1, ``int` `s2){``        ``if` `(arr[s1] > arr[s2]){``            ``return` `true``;``        ``}``        ``else``{``            ``return` `false``;``        ``}``    ``}``    ` `    ``// Function to find the Intra-Array``    ``// Count in the two subarrays``    ``public` `static` `int` `findIntraArrayCount(``        ``int``[] arr, ``int` `s1, ``int` `e1, ``int` `s2,``                           ``int` `e2, ``int` `g){``                               ` `        ``// Base Case``        ``if` `(s1 == e1 && s2 == e2){``            ``int` `c = ``0``;``            ``if` `(compare(arr, s1, s2)){``                ``c += ``1``;``            ``}``            ``if` `(arr[s1] > arr[s2]){``                ``int` `temp = arr[s1];``                ``arr[s1] = arr[s2];``                ``arr[s2] = temp;``            ``}``            ``return` `c;``        ``}``        ` `        ``// Variable for keeping``        ``// the count of the pair``        ``int` `c = ``0``;``        ``int` `s = s1, e = e2, s3 = s1;``        ``int` `e3 = e1, s4 = s2, e4 = e2;``        ` `        ``while` `(s1 <= e1 && s2 <= e2){``            ` `            ``// Condition when we have to use the``            ``// Greater than comparator``            ``if` `(g == ``1``){``                ``if` `(compare(arr, s1, s2)){``                    ``c += e1 - s1 + ``1``;``                    ``s2 += ``1``;``                ``}``                ``else``{``                    ``s1 += ``1``;``                ``}``            ``}``            ` `            ``// Condition when we have to use the``            ``// Equal to Comparator``            ``else` `if` `(g == ``0``){``                ``if` `(compare(arr, s1, s2)){``                    ``c += e2 - s2 + ``1``;``                    ``s1 += ``1``;``                ``}``                ``else` `{``                    ``s2 += ``1``;``                ``}``            ``}``            ` `            ``// Condition when we have to use the``            ``// Equal to Comparator``            ``else` `if` `(g == -``1``){``                ``if` `(compare(arr, s1, s2)){``                    ``int` `c1 = ``0``;``                    ``while` `(s1 <= e1 &&``                         ``compare(arr, s1, s2)){``                        ``c1 += ``1``;``                        ``s1 += ``1``;``                    ``}``                    ``s1 -= ``1``;``                    ``int` `c2 = ``0``;``                    ``while` `(s2 <= e2 &&``                          ``compare(arr, s1, s2)){``                        ``c2 += ``1``;``                        ``s2 += ``1``;``                    ``}``                    ``c += c1 * c2;``                ``}``                ``else` `{``                    ``if` `(arr[s1] > arr[s2]){``                        ``s2 += ``1``;``                    ``}``                    ``else``{``                        ``s1 += ``1``;``                    ``}``                ``}``            ``}``        ``}``        ``s1 = s3; e1 = e3;``        ``s2 = s4; e2 = e4;``        ` `        ``// Array to store``        ``// the sorted subarray``        ``ArrayList aux =``                     ``new` `ArrayList<>();``                     ` `        ``// Merge the two subarrays``        ``while` `(s1 <= e1 && s2 <= e2){``            ``if` `(arr[s1] <= arr[s2]){``                ``aux.add(arr[s1]);``                ``s1 += ``1``;``            ``}``            ``else``{``                ``aux.add(arr[s2]);``                ``s2 += ``1``;``            ``}``        ``}``        ` `        ``// Copy subarray 1 elements``        ``while` `(s1 <= e1){``            ``aux.add(arr[s1]);``            ``s1 += ``1``;``        ``}``        ` `        ``// Copy subarray 2 elements``        ``while` `(s2 <= e2){``            ``aux.add(arr[s2]);``            ``s2 += ``1``;``        ``}``        ` `        ``// Update the original array``        ``for` `(``int` `i = s; i <= e; i++){``            ``arr[i] = aux.get(i-s);``        ``}``        ``return` `c;``    ``}``    ` `    ``// Function to find such pairs with``    ``// any custom comparator function``    ``public` `static` `int` `findElementsOnRight(``        ``int``[] arr, ``int` `s, ``int` `e, ``int` `g){``        ``if` `(s >= e){``            ``return` `0``;``        ``}``        ``int` `mid = (s+e)/``2``;``        ` `        ``// Recursive call for inter-array``        ``// count of pairs in left subarrays``        ``int` `c1 = findElementsOnRight(arr, s,``                                     ``mid, g);``        ` `        ``// Recursive call for inter-array``        ``// count of pairs in right sub-arrays``        ``int` `c2 = findElementsOnRight(arr, mid + ``1``,``                                         ``e, g);``        ` `        ``// Call for intra-array pairs``        ``int` `c3 = findIntraArrayCount(arr, s,``                            ``mid, mid+``1``, e, g);``        ` `        ``return` `c1 + c2 + c3;                                  ``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `main (String[] args) {``        ``int``[] arr = {``4``, ``3``, ``2``, ``1``};``        ``int` `g = ``1``;``        ``System.out.println(``            ``findElementsOnRight(arr, ``0``, ``3``, g));``    ``}``}`

## Python3

 `# Python3 implementation to find the``# elements on the right with the given``# custom comparator` `import` `random, math``from` `copy ``import` `deepcopy as dc` `# comparator to check``# if two elements are equal``def` `compare(arr, s1, s2):``    ``if` `arr[s1] > arr[s2]:``        ``return` `True``    ``else``:``        ``return` `False`  `# Function to find the Intra-Array``# Count in the two subarrays``def` `findIntraArrayCount(arr, s1, \``                ``e1, s2, e2, g):``    ` `    ``# Base Case``    ``if` `s1 ``=``=` `e1 ``and` `s2 ``=``=` `e2:``        ``c ``=` `0``        ``if` `compare(arr, s1, s2):``            ``c ``+``=` `1``        ``if` `arr[s1] > arr[s2]:``            ``arr[s1], arr[s2] ``=` `arr[s2], arr[s1]``        ``return` `c`  `    ``# Variable for keeping``    ``# the count of the pair``    ``c ``=` `0``    ` `    ``# Total subarray length``    ``s ``=` `dc(s1); e ``=` `dc(e2)``    ` `    ``# length of subarray 1``    ``s3 ``=` `dc(s1); e3 ``=` `dc(e1)``    ` `    ``# length of subarray 2``    ``s4 ``=` `dc(s2); e4 ``=` `dc(e2)``    ` `    ``while` `s1 <``=` `e1 ``and` `s2 <``=` `e2:`  `        ``# Condition when we have to use the``        ``# Greater than comparator``        ``if` `g ``=``=` `1``:``            ``if` `compare(arr, s1, s2):``                ``c ``+``=` `e1 ``-` `s1 ``+` `1``                ``s2 ``+``=` `1``            ``else``:``                ``s1 ``+``=` `1` `        ``# Condition when we have to use the``        ``# Less than comparator``        ``elif` `g ``=``=` `0``:``            ``if` `compare(arr, s1, s2):``                ``c ``+``=` `e2 ``-` `s2 ``+` `1``                ``s1 ``+``=` `1``            ``else``:``                ``s2 ``+``=` `1``                ` `        ``# Condition when we have to use the``        ``# Equal to Comparator``        ``elif` `g ``=``=` `-``1``:``            ``if` `compare(arr, s1, s2):``                ``c1 ``=` `0``                ``while` `s1 <``=` `e1 ``and``\``                   ``compare(arr, s1, s2):``                    ``c1 ``+``=` `1``                    ``s1 ``+``=` `1``                ``s1 ``-``=` `1``                ``c2 ``=` `0``                ``while` `s2 <``=` `e2 ``and``\``                    ``compare(arr, s1, s2):``                    ``c2 ``+``=` `1``                    ``s2 ``+``=` `1``                ``c ``+``=` `c1 ``*` `c2``            ``else``:``                ``if` `arr[s1] > arr[s2]:``                    ``s2 ``+``=` `1``                ``else``:``                    ``s1 ``+``=` `1` `    ``s1 ``=` `dc(s3); e1 ``=` `dc(e3)``    ` `    ``s2 ``=` `dc(s4); e2 ``=` `dc(e4)``    ` `    ``# Array to store the sorted subarray``    ``aux ``=` `[]``    ` `    ` `    ``# Merge the two subarrays``    ``while` `s1 <``=` `e1 ``and` `s2 <``=` `e2:``        ``if` `arr[s1] <``=` `arr[s2]:``            ``aux.append(arr[s1])``            ``s1 ``+``=` `1``        ``else``:``            ``aux.append(arr[s2])``            ``s2 ``+``=` `1``    ` `    ``# Copy subarray 1 elements``    ``while` `s1 <``=` `e1:``        ``aux.append(arr[s1])``        ``s1 ``+``=` `1``        ` `    ` `    ``# Copy subarray 2 elements``    ``while` `s2 <``=` `e2:``        ``aux.append(arr[s2])``        ``s2 ``+``=` `1``        ` `    ` `    ``# Update the original array``    ``for` `i ``in` `range``(s, e ``+` `1``):``        ``arr[i] ``=` `aux[i``-``s]``    ``return` `c`   `# Function to find such pairs with``# any custom comparator function``def` `findElementsOnRight(arr, s, e, g):``    ``if` `s >``=` `e:``        ``return` `0``    ``mid ``=` `(s ``+` `e)``/``/``2``    ` `    ``# Recursive call for inter-array``    ``# count of pairs in left subarrays``    ``c1 ``=` `findElementsOnRight(arr, s, \``                            ``mid, g)``    ` `    ``# Recursive call for inter-array``    ``# count of pairs in right sub-arrays``    ``c2 ``=` `findElementsOnRight(arr, mid ``+` `1``, \``                                ``e, g)``    ` `    ``# Call for intra-array pairs``    ``c3 ``=` `findIntraArrayCount(arr, s, mid, \``                            ``mid ``+` `1``, e, g)``    ``return` `c1 ``+` `c2 ``+` `c3`    `# Driver Code``if` `__name__ ``=``=` `"__main__"``:``    ``arr ``=` `[``4``, ``3``, ``2``, ``1``]``    ``g ``=` `1``    ``out ``=` `findElementsOnRight(arr, ``0``, \``                        ``len``(arr)``-``1``, g)``    ``print``(out)`

## C#

 `// C# implementation to find the ``// elements on the right with the``// given custom comparator ``using` `System;``using` `System.Collections.Generic;` `class` `GFG{``    ` `// comparator to check ``// if two elements are equal ``public` `static` `bool` `compare(``int``[] arr, ``int` `s1,``                                      ``int` `s2)``{``    ``if` `(arr[s1] > arr[s2])``    ``{``        ``return` `true``;``    ``}``    ``else``    ``{``        ``return` `false``;``    ``}``}``  ` `// Function to find the Intra-Array ``// Count in the two subarrays ``public` `static` `int` `findIntraArrayCount(``int``[] arr, ``int` `s1,``                                      ``int` `e1, ``int` `s2, ``                                      ``int` `e2, ``int` `g)``{``    ` `    ``// Base Case``    ``if` `(s1 == e1 && s2 == e2)``    ``{``        ``int` `cc = 0;``        ``if` `(compare(arr, s1, s2))``        ``{``            ``cc += 1;``        ``}``        ``if` `(arr[s1] > arr[s2])``        ``{``            ``int` `temp = arr[s1];``            ``arr[s1] = arr[s2];``            ``arr[s2] = temp;``        ``}``        ``return` `cc;``    ``}``      ` `    ``// Variable for keeping ``    ``// the count of the pair``    ``int` `c = 0;``    ``int` `s = s1, e = e2, s3 = s1;``    ``int` `e3 = e1, s4 = s2, e4 = e2;``      ` `    ``while` `(s1 <= e1 && s2 <= e2)``    ``{``        ` `        ``// Condition when we have to use the ``        ``// Greater than comparator``        ``if` `(g == 1)``        ``{``            ``if` `(compare(arr, s1, s2))``            ``{``                ``c += e1 - s1 + 1;``                ``s2 += 1;``            ``}``            ``else``            ``{``                ``s1 += 1;``            ``}``        ``}``          ` `        ``// Condition when we have to use the ``        ``// Equal to Comparator ``        ``else` `if` `(g == 0)``        ``{``            ``if` `(compare(arr, s1, s2))``            ``{``                ``c += e2 - s2 + 1;``                ``s1 += 1;``            ``}``            ``else``            ``{``                ``s2 += 1;``            ``}``        ``}``          ` `        ``// Condition when we have to use the ``        ``// Equal to Comparator ``        ``else` `if` `(g == -1)``        ``{``            ``if` `(compare(arr, s1, s2))``            ``{``                ``int` `c1 = 0;``                ``while` `(s1 <= e1 && ``                       ``compare(arr, s1, s2))``                ``{``                    ``c1 += 1;``                    ``s1 += 1;``                ``}``                ``s1 -= 1;``                ``int` `c2 = 0;``                ``while` `(s2 <= e2 && ``                       ``compare(arr, s1, s2))``                ``{``                    ``c2 += 1;``                    ``s2 += 1;``                ``}``                ``c += c1 * c2;``            ``}``            ``else``            ``{``                ``if` `(arr[s1] > arr[s2])``                ``{``                    ``s2 += 1;``                ``}``                ``else``                ``{``                    ``s1 += 1;``                ``}``            ``}``        ``}``    ``}``    ``s1 = s3; e1 = e3;``    ``s2 = s4; e2 = e4;``      ` `    ``// Array to store ``    ``// the sorted subarray``    ``List<``int``> aux = ``new` `List<``int``>();``                   ` `    ``// Merge the two subarrays``    ``while` `(s1 <= e1 && s2 <= e2)``    ``{``        ``if` `(arr[s1] <= arr[s2])``        ``{``            ``aux.Add(arr[s1]);``            ``s1 += 1;``        ``}``        ``else``        ``{``            ``aux.Add(arr[s2]);``            ``s2 += 1;``        ``}``    ``}``      ` `    ``// Copy subarray 1 elements ``    ``while` `(s1 <= e1)``    ``{``        ``aux.Add(arr[s1]);``        ``s1 += 1;``    ``}``      ` `    ``// Copy subarray 2 elements``    ``while` `(s2 <= e2)``    ``{``        ``aux.Add(arr[s2]);``        ``s2 += 1;``    ``}``      ` `    ``// Update the original array``    ``for``(``int` `i = s; i <= e; i++)``    ``{``        ``arr[i] = aux[i-s];``    ``}``    ``return` `c;``}``  ` `// Function to find such pairs with ``// any custom comparator function ``public` `static` `int` `findElementsOnRight(``int``[] arr, ``int` `s,``                                      ``int` `e, ``int` `g)``{``    ``if` `(s >= e)``    ``{``        ``return` `0;``    ``}``    ``int` `mid = (s + e) / 2;``      ` `    ``// Recursive call for inter-array ``    ``// count of pairs in left subarrays``    ``int` `c1 = findElementsOnRight(arr, s, ``                                 ``mid, g);``      ` `    ``// Recursive call for inter-array ``    ``// count of pairs in right sub-arrays ``    ``int` `c2 = findElementsOnRight(arr, mid + 1, ``                                   ``e, g);``      ` `    ``// Call for intra-array pairs ``    ``int` `c3 = findIntraArrayCount(arr, s, mid,``                                 ``mid + 1, e, g);``      ` `    ``return` `c1 + c2 + c3;                                   ``}` `// Driver code``static` `public` `void` `Main()``{``    ``int``[] arr = { 4, 3, 2, 1 };``    ``int` `g = 1;``    ` `    ``Console.WriteLine(findElementsOnRight(``        ``arr, 0, 3, g));``}``}` `// This code is contributed by offbeat`

## Javascript

 ``

Output:

`6`

Time Complexity: The above method takes O(N*logN) time.
Auxiliary Space: O(N)

My Personal Notes arrow_drop_up