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Count pairs from a given array whose sum lies from a given range

  • Difficulty Level : Medium
  • Last Updated : 18 Aug, 2021
Geek Week

Given an array arr[] consisting of N integers and two integers L and R, the task is to count the number of pairs whose sum lies in the range [L, R].

Examples:

Input: arr[] = {5, 1, 2}, L = 4, R = 7
Output: 2
Explanation:
The pairs satisfying the necessary conditions are as follows:

  1. (5, 1): Sum = 5 + 1 = 6, which lies in the range [4, 7].
  2. (5, 2): Sum = 5 + 2 = 7, which lies in the range [4, 7].

Therefore, the count of such pairs is 2.

Input: arr[] = {5, 1, 2, 4, 3}, L = 5, R = 8
Output: 7

Naive Approach: The simplest approach to solve the given problem is to generate all possible pairs and count those pairs whose sum lies over the range [L, R]. After checking for all the pairs, print the total count of pairs. 



Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized by sorting the array then performing a Binary Search to find the number of elements for each array element arr[i] such that the sum will not exceed the given range. Follow the steps to solve the problem:

  • Sort the array arr[] in increasing order.
  • Initialize variables say, right as N – 1 and count as 0 to store numbers of pairs whose sum lies over the range [L, R].
  • Iterate until the right is greater than 0 and perform the following steps:
    • Find the starting index of the element whose sum with arr[right] is greater than or equal to L, and store it in a variable, say start.
    • Find the ending index of the element before arr[right] whose sum with arr[right] is less than or equal to R, and then store it in a variable, say end.
    • If the end is greater than equal to the start, then add (end – start + 1) to the count representing the number of elements whose sum with the current element lies over the range [L, R] and then decrement right by 1.
  • After completing the above steps, print the value of count as the result.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count pairs whose
// sum lies over the range [L, R]
int countPairSum(int arr[], int L,
                 int R, int N)
{
    // Sort the given array
    sort(arr, arr + N);
 
    int right = N - 1, count = 0;
 
    // Iterate until right > 0
    while (right > 0) {
 
        // Starting index of element
        // whose sum with arr[right] >= L
        auto it1
            = lower_bound(arr, arr + N,
                          L - arr[right]);
 
        int start = it1 - arr;
 
        // Ending index of element
        // whose sum with arr[right] <= R
        auto it2
            = upper_bound(arr, arr + N,
                          R - arr[right]);
 
        --it2;
 
        int end = it2 - arr;
 
        // Update the value of end
        end = min(end, right - 1);
 
        // Add the count of elements
        // to the variable count
        if (end - start >= 0) {
            count += (end - start + 1);
        }
 
        right--;
    }
 
    // Return the value of count
    return count;
}
 
// Driver Code
int main()
{
    int arr[] = { 5, 1, 2, 4, 3 };
    int L = 5, R = 8;
    int N = sizeof(arr) / sizeof(arr[0]);
    cout << countPairSum(arr, L, R, N);
 
    return 0;
}

Python3




# Python3 program for the above approach
from bisect import bisect_left, bisect_right
 
# Function to count pairs whose
# sum lies over the range [L, R]
def countPairSum(arr, L, R, N):
     
    # Sort the given array
    arr.sort()
 
    right = N - 1
    count = 0
 
    # Iterate until right > 0
    while (right > 0):
 
        # Starting index of element
        # whose sum with arr[right] >= L
        it1 = bisect_left(arr, L - arr[right])
 
        start = it1
 
        # Ending index of element
        # whose sum with arr[right] <= R
        it2 = bisect_right(arr, R - arr[right])
 
        it2 -= 1
        end = it2
 
        # Update the value of end
        end = min(end, right - 1)
 
        # Add the count of elements
        # to the variable count
        if (end - start >= 0):
            count += (end - start + 1)
 
        right -= 1
 
    # Return the value of count
    return count
 
# Driver Code
if __name__ == '__main__':
     
    arr = [ 5, 1, 2, 4, 3 ]
    L = 5
    R = 8
    N = len(arr)
     
    print(countPairSum(arr, L, R, N))
     
# This code is contributed by SURENDRA_GANGWAR
Output: 
7

 

Time Complexity: O(N*log N)
Auxiliary Space: O(1)

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