Count pairs (i, j) from an array such that |arr[i]| and |arr[j]| both lies between |arr[i] – arr[j]| and |arr[i] + arr[j]|

Given an array arr[] of size N, the task is to count the number of pairs (arr[i], arr[j]) such that |arr[i]| and |arr[j]| lies between |arr[i] – arr[j]| and |arr[i] + arr[j]|.

Examples:

Input: arr[] = {1, 3, 5, 7} 
Output: 2 
Explanation:
Pair (arr[1], arr[2]) (= (3, 5)) lies between |3 – 5| (= 2) and |3 + 5| (= 8).
Pair (arr[2], arr[3]) (= (5, 7)) lies between |5 – 7| (= 2) and |5 + 7| (= 12).    

Input: arr[] = {-4, 1, 9, 7, -1, 2, 8} 
Output: 9

Approach: The given problem can be solved by analyzing the following cases:

• If X is positive and Y is positive:
  • |X – Y| remains |X – Y|.
  • |X + Y| remains |X + Y|.
• If X is negative and Y is positive:
  • |X – Y| becomes |-(X + Y)|, i.e. |X + Y|.
  • |X + Y| becomes |-(X – Y)|, i.e. |X – Y|.
• If X is positive and Y is negative:
  • |X – Y| becomes |X + Y|.
  • |X + Y| becomes |X – Y|.
• If X is negative and Y is negative:
  • |X – Y| remains |X – Y|.
  • |X + Y| remains |X + Y|.

It is clear from the above cases, that |X – Y| and |X + Y| are at most swapping values, which does not change the solution.
Therefore, if a pair is valid for (X, Y), then it will also be valid for any of the above cases like (-X, Y). Therefore, the task reduces to just take absolute values of X and Y while finding the solution, i.e. to find (X, Y), where |X – Y| ≤ X, Y ≤ X + Y. 

Follow the steps below to solve the problem:

• Take absolute values of all elements present in the array arr[].
• Sort the array arr[].
• Initialize a variable, say left, as 0. 
• Initialize a variable, say ans, to store the count of valid pairs.
• Traverse the array arr[] using a variable, say right, and perform the following steps:
  • Increment left until 2 *arr[left] is less than arr[right].
  • Add the value (i – left) to ans to include the number of valid pairs.
• After completing the above steps, print the value of ans 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 find pairs (i, j) such that
// |arr[i]| and |arr[j]| lies in between
// |arr[i] - arr[j]| and |arr[i] + arr[j]|
void findPairs(int arr[], int N)
{
    // Calculate absolute value
    // of all array elements
    for (int i = 0; i < N; i++)
        arr[i] = abs(arr[i]);

    // Sort the array
    sort(arr, arr + N);

    int left = 0;

    // Stores the count of pairs
    int ans = 0;

    // Traverse the array
    for (int right = 0; right < N; right++) {

        while (2 * arr[left] < arr[right])

            // Increment left
            left++;

        // Add to the current
        // count of pairs
        ans += (right - left);
    }

    // Print the answer
    cout << ans;
}

// Driver Code
int main()
{
    int arr[] = { 1, 3, 5, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);

    findPairs(arr, N);

    return 0;
}

Java

// Java program for the above approach
import java.util.Arrays;

class GFG{

// Function to find pairs (i, j) such that
// |arr[i]| and |arr[j]| lies in between
// |arr[i] - arr[j]| and |arr[i] + arr[j]|
static void findPairs(int arr[], int N)
{

    // Calculate absolute value
    // of all array elements
    for(int i = 0; i < N; i++)
        arr[i] = Math.abs(arr[i]);

    // Sort the array
    Arrays.sort(arr);

    int left = 0;

    // Stores the count of pairs
    int ans = 0;

    // Traverse the array
    for(int right = 0; right < N; right++)
    {
        while (2 * arr[left] < arr[right])

            // Increment left
            left++;

        // Add to the current
        // count of pairs
        ans += (right - left);
    }

    // Print the answer
    System.out.print(ans);
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 3, 5, 7 };
    int N = arr.length;

    findPairs(arr, N);
}
}

// This code is contributed by AnkThon

Python3

# Python3 program for the above approach

# Function to find pairs (i, j) such that
# |arr[i]| and |arr[j]| lies in between
# |arr[i] - arr[j]| and |arr[i] + arr[j]|
def findPairs(arr,  N):

    # Calculate absolute value
    # of all array elements
    for i in range(N):
        arr[i] = abs(arr[i])

    # Sort the array
    arr.sort()
    left = 0

    # Stores the count of pairs
    ans = 0

    # Traverse the array
    for right in range(N):
        while (2 * arr[left] < arr[right]):

            # Increment left
            left += 1

        # Add to the current
        # count of pairs
        ans += (right - left)

    # Print the answer
    print(ans)

# Driver Code
if __name__ == "__main__":
    arr = [1, 3, 5, 7]
    N = len(arr)

    findPairs(arr, N)

    # This code is contributed by ukasp.

C#

// C# program for the above approach
using System;

class GFG{

// Function to find pairs (i, j) such that
// |arr[i]| and |arr[j]| lies in between
// |arr[i] - arr[j]| and |arr[i] + arr[j]|
static void findPairs(int []arr, int N)
{

    // Calculate absolute value
    // of all array elements
    for(int i = 0; i < N; i++)
        arr[i] = Math.Abs(arr[i]);

    // Sort the array
    Array.Sort(arr);

    int left = 0;

    // Stores the count of pairs
    int ans = 0;

    // Traverse the array
    for(int right = 0; right < N; right++)
    {
        while (2 * arr[left] < arr[right])

            // Increment left
            left++;

        // Add to the current
        // count of pairs
        ans += (right - left);
    }

    // Print the answer
    Console.Write(ans);
}

// Driver Code
public static void Main(string[] args)
{
    int []arr = { 1, 3, 5, 7 };
    int N = arr.Length;

    findPairs(arr, N);
}
}

// This code is contributed by AnkThon

Javascript

<script>
// JavaScript program for the above approach

// Function to find pairs (i, j) such that
// |arr[i]| and |arr[j]| lies in between
// |arr[i] - arr[j]| and |arr[i] + arr[j]|
function findPairs(arr, N)
{

    // Calculate absolute value
    // of all array elements
    for (let i = 0; i < N; i++)
        arr[i] = Math.abs(arr[i]);

    // Sort the array
    arr.sort((a, b) => a - b);

    let left = 0;

    // Stores the count of pairs
    let ans = 0;

    // Traverse the array
    for (let right = 0; right < N; right++) {

        while (2 * arr[left] < arr[right])

            // Increment left
            left++;

        // Add to the current
        // count of pairs
        ans += (right - left);
    }

    // Print the answer
    document.write(ans);
}

// Driver Code

    let arr = [ 1, 3, 5, 7 ];
    let N = arr.length;

    findPairs(arr, N);


// This code is contributed by Manoj.
</script>

Output:

2

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