Number of indices pair such that element pair sum from first Array is greater than second Array

Given two integer arrays A[] and B[] of equal sizes, the task is to find the number of pairs of indices {i, j} in the arrays such that A[i] + A[j] > B[i] + B[j] and i < j.

Examples:

Input: A[] = {4, 8, 2, 6, 2}, B[] = {4, 5, 4, 1, 3}
Output: 7
Explanation:
There are a total of 7 pairs of indices {i, j} in the array following the condition. They are:
{0, 1}: A[0] + A[1] > B[0] + B[1]
{0, 3}: A[0] + A[3] > B[0] + B[3]
{1, 2}: A[1] + A[2] > B[1] + B[2]
{1, 3}: A[1] + A[3] > B[1] + B[3]
{1, 4}: A[1] + A[4] > B[1] + B[4]
{2, 3}: A[2] + A[3] > B[2] + B[3]
{3, 4}: A[3] + A[4] > B[3] + B[4]

Input: A[] = {1, 3, 2, 4}, B[] = {1, 3, 2, 4}
Output: 0
Explanation:
No such possible pairs of {i, j} can be found that satisfies the given condition

Naive Approach: The naive approach is to consider all the possible pairs of {i, j} in the given arrays and check if A[i] + A[j] > B[i] + B[j]. This can be done by using the concept of nested loops.



Time Complexity: O(N2)

Efficient Approach: The key observation from the problem is that the given condition can also be visualised as (ai-bi) + (aj-bj)> 0 so we can make another array to store the difference of both arrays. let this array be D . Therefore, the problem reduces to finding pairs with Di+Dj>0. Now we can sort D array and for each corresponding element Di we will find the no of good pairs that Di can make and add this no of pairs to a count variable.For each element Di to find the no of good pairs it can make we can use the upper_bound function of the standard template library to find the upper bound of -Di. since the array is sorted so all elements present after -Di will also make good pair with Di .thus,if upper bound of -Di is x and n be the total size of array then total pairs corresponding to Di will be n-x. This approach takes O(NlogN) time.

  • The given condition in the question can be rewritten as:
    A[i] + A[j] > B[i] + B[j]
    A[i] + A[j] - B[i] - B[j] > 0
    (A[i] - B[i]) + (A[j] - B[j]) > 0
    
  • Create another array, say D, to store the difference between elements at the corresponding index in both array, i.e.
    D[i] = A[i] - B[i]
  • Now to make sure that the constraint i < j is satisfied, sort the difference array D, so that each element i is smaller than elements to its right.
  • If at some index i, the value in the difference array D is negative, then we only need to find the nearest position ‘j’ at which the value is just greater than -D[i], so that on summation the value becomes > 0.

    Inorder to find such index ‘j’, upper_bound() function or Binary Search can be used, since the array is sorted.

Below is the implementation of the above approach:

C++

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// C++ program to find the number of indices pair
// such that pair sum from first Array
// is greater than second Array
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to get the number of pairs of indices
// {i, j} in the given two arrays A and B such that
// A[i] + A[j] > B[i] + B[j]
int getPairs(vector<int> A, vector<int> B, int n)
{
    // Intitializing the difference array D
    vector<int> D(n);
  
    // Computing the difference between the
    // elements at every index and storing
    // it in the array D
    for (int i = 0; i < n; i++) {
        D[i] = A[i] - B[i];
    }
  
    // Sort the array D
    sort(D.begin(), D.end());
  
    // Variable to store the total
    // number of pairs that satisfy
    // the given condition
    long long total = 0;
  
    // Loop to iterate through the difference
    // array D and find the total number
    // of pairs of indices that follow the
    // given condition
    for (int i = n - 1; i >= 0; i--) {
  
        // If the value at the index i is positive,
        // then it remains positive for any pairs
        // with j such that j > i.
        if (D[i] > 0) {
            total += n - i - 1;
        }
  
        // If the value at that index is negative
        // then we need to find the index of the
        // value just greater than -D[i]
        else {
            int k = upper_bound(D.begin(),
                                D.end(), -D[i])
                    - D.begin();
            total += n - k;
        }
    }
    return total;
}
  
// Driver code
int main()
{
    int n = 5;
    vector<int> A;
    vector<int> B;
  
    A.push_back(4);
    A.push_back(8);
    A.push_back(2);
    A.push_back(6);
    A.push_back(2);
  
    B.push_back(4);
    B.push_back(5);
    B.push_back(4);
    B.push_back(1);
    B.push_back(3);
  
    cout << getPairs(A, B, n);
}

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Java

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// Java program to find the number of indices pair
// such that pair sum from first Array
// is greater than second Array
import java.util.*;
  
class GFG{
  
// Function to get the number of pairs of indices
// {i, j} in the given two arrays A and B such that
// A[i] + A[j] > B[i] + B[j]
static long getPairs(Vector<Integer> A, Vector<Integer> B, int n)
{
    // Intitializing the difference array D
    int []D = new int[n];
  
    // Computing the difference between the
    // elements at every index and storing
    // it in the array D
    for (int i = 0; i < n; i++)
    {
        D[i] = A.get(i) - B.get(i);
    }
  
    // Sort the array D
    Arrays.sort(D);
  
    // Variable to store the total
    // number of pairs that satisfy
    // the given condition
    long total = 0;
  
    // Loop to iterate through the difference
    // array D and find the total number
    // of pairs of indices that follow the
    // given condition
    for (int i = n - 1; i >= 0; i--) {
  
        // If the value at the index i is positive,
        // then it remains positive for any pairs
        // with j such that j > i.
        if (D[i] > 0) {
            total += n - i - 1;
        }
  
        // If the value at that index is negative
        // then we need to find the index of the
        // value just greater than -D[i]
        else {
            int k = upper_bound(D,0, D.length, -D[i]);
            total += n - k;
        }
    }
    return total;
}
static int upper_bound(int[] a, int low, 
                        int high, int element)
{
    while(low < high){
        int middle = low + (high - low)/2;
        if(a[middle] > element)
            high = middle;
        else
            low = middle + 1;
    }
    return low;
  
// Driver code
public static void main(String[] args)
{
    int n = 5;
    Vector<Integer> A = new Vector<Integer>();
    Vector<Integer> B= new Vector<Integer>();
  
    A.add(4);
    A.add(8);
    A.add(2);
    A.add(6);
    A.add(2);
  
    B.add(4);
    B.add(5);
    B.add(4);
    B.add(1);
    B.add(3);
  
    System.out.print(getPairs(A, B, n));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python 3 program to find the number of indices pair
# such that pair sum from first Array
# is greater than second Array
import bisect 
  
# Function to get the number of pairs of indices
# {i, j} in the given two arrays A and B such that
# A[i] + A[j] > B[i] + B[j]
def getPairs(A,  B, n):
  
    # Intitializing the difference array D
    D = [0]*(n)
   
    # Computing the difference between the
    # elements at every index and storing
    # it in the array D
    for i in range(n):
        D[i] = A[i] - B[i]
   
    # Sort the array D
    D.sort()
   
    # Variable to store the total
    # number of pairs that satisfy
    # the given condition
    total = 0
   
    # Loop to iterate through the difference
    # array D and find the total number
    # of pairs of indices that follow the
    # given condition
    for i in range(n - 1, -1, -1):
   
        # If the value at the index i is positive,
        # then it remains positive for any pairs
        # with j such that j > i.
        if (D[i] > 0):
            total += n - i - 1
   
        # If the value at that index is negative
        # then we need to find the index of the
        # value just greater than -D[i]
        else:
            k = bisect.bisect_right(D, -D[i], 0, len(D))
            total += n - k
    return total
   
# Driver code
if __name__ == "__main__":
      
    n = 5
    A = []
    B = []
   
    A.append(4);
    A.append(8);
    A.append(2);
    A.append(6);
    A.append(2);
   
    B.append(4);
    B.append(5);
    B.append(4);
    B.append(1);
    B.append(3);
   
    print(getPairs(A, B, n))
  
# This code is contributed by chitranayal

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C#

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// C# program to find the number of indices pair
// such that pair sum from first Array
// is greater than second Array
using System;
using System.Collections.Generic;
  
class GFG{
   
// Function to get the number of pairs of indices
// {i, j} in the given two arrays A and B such that
// A[i] + A[j] > B[i] + B[j]
static long getPairs(List<int> A, List<int> B, int n)
{
    // Intitializing the difference array D
    int []D = new int[n];
   
    // Computing the difference between the
    // elements at every index and storing
    // it in the array D
    for (int i = 0; i < n; i++)
    {
        D[i] = A[i] - B[i];
    }
   
    // Sort the array D
    Array.Sort(D);
   
    // Variable to store the total
    // number of pairs that satisfy
    // the given condition
    long total = 0;
   
    // Loop to iterate through the difference
    // array D and find the total number
    // of pairs of indices that follow the
    // given condition
    for (int i = n - 1; i >= 0; i--) {
   
        // If the value at the index i is positive,
        // then it remains positive for any pairs
        // with j such that j > i.
        if (D[i] > 0) {
            total += n - i - 1;
        }
   
        // If the value at that index is negative
        // then we need to find the index of the
        // value just greater than -D[i]
        else {
            int k = upper_bound(D,0, D.Length, -D[i]);
            total += n - k;
        }
    }
    return total;
}
static int upper_bound(int[] a, int low, 
                        int high, int element)
{
    while(low < high){
        int middle = low + (high - low)/2;
        if(a[middle] > element)
            high = middle;
        else
            low = middle + 1;
    }
    return low;
   
// Driver code
public static void Main(String[] args)
{
    int n = 5;
    List<int> A = new List<int>();
    List<int> B= new List<int>();
   
    A.Add(4);
    A.Add(8);
    A.Add(2);
    A.Add(6);
    A.Add(2);
   
    B.Add(4);
    B.Add(5);
    B.Add(4);
    B.Add(1);
    B.Add(3);
   
    Console.Write(getPairs(A, B, n));
}
}
  
// This code is contributed by sapnasingh4991

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Output:

7

Time Complexity Analysis:

  • The sorting of the array takes O(N * log(N)) time.
  • The time taken to find the index which is just greater than a specific value is O(Log(N)). Since in the worst case, this can be executed for N elements in the array, the overall time complexity for this is O(N * log(N)).
  • Therefore, the overall time complexity is O(N * log(N)).

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