Average of pairwise difference of all pairs formed from given N integers

Given an array arr[] containing N integers, the task is to calculate the average of the difference between both elements in pairs formed from given N integers.

Examples:

Input: arr[] = {-1, 3, -5, 4}
Output: 5.166667
Explanation: There are 6 possible pair of points in the given array with the pairwise difference as: diff(-1, 3) = 4, diff(-1, -5) = 4, diff(-1, 4) = 5, diff(3, -5) = 8, diff(3, 4) = 1, diff(-5, 4) = 9. Therefore, average pairwise difference is (4 + 4 + 5 + 8 + 1 + 9)/6 = 31/6 = 5.166667.

Input: arr[] = { -1, 2, -3, 7, -6 }
Output: 6.2

Approach: This problem can be solved by using the Greedy Approach and prefix sum method. If the points in the array arr[] are in sorted order, then the sum of distances of i^th point to all the greater points can be calculated as: (arr[i+1] – arr[i]) + (arr[i+2] – arr[i]) … + (arr[N-1] – arr[i]) => (arr[i+1] + arr[i+2]… + arr[N-1]) – arr[i] * (N – 1 – i). Using this observation, the given problem can be solved using the following steps:

• Initially sort the array arr[] in non-decreasing order.
• Create a prefix sum array pre[] of the array arr[].
• Iterate through every index i and add (pre[N – 1] – pre[i]) – arr[i] * (N – 1 – i) into a variable ans.
• The required answer is ans / count of pairs => ans / (N*(N-1)/2).

Below is the implementation of the above approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find average distance
// between given points on a line
long double averageDistance(
  vector<int> arr, int N)
{
    // Sorting the array arr[]
    sort(arr.begin(), arr.end());

    // Stores the prefix sum
    // array of arr[]
    int pre[N] = { 0 };
    pre[0] = arr[0];

    // Loop to calculate prefix sum
    for (int i = 1; i < N; i++) {
        pre[i] = pre[i - 1] + arr[i];
    }

    // Initialising the answer variable
    long double ans = 0;

    // Loop to iterate through arr[]
    for (int i = 0; i < N - 1; i++) {

        // Adding summation of all
        // distances from ith point
        ans += (pre[N - 1] - pre[i])
          - arr[i] * (N - 1 - i);
    }

    // Return Average
    return ans / ((N * (N - 1)) / 2);
}

// Driver Code
int main()
{
    vector<int> arr = { -1, 3, -5, 4 };
    cout << averageDistance(arr, arr.size());

    return 0;
}

Java

// Java implementation for the above approach
import java.util.*;
class GFG {

    // Function to find average distance
    // between given points on a line
    static double averageDistance(int[] arr, int N)
    {
        // Sorting the array arr[]
        Arrays.sort(arr);

        // Stores the prefix sum
        // array of arr[]
        int[] pre = new int[N];
        pre[0] = arr[0];

        // Loop to calculate prefix sum
        for (int i = 1; i < N; i++) {
            pre[i] = pre[i - 1] + arr[i];
        }

        // Initialising the answer variable
        double ans = 0;

        // Loop to iterate through arr[]
        for (int i = 0; i < N - 1; i++) {

            // Adding summation of all
            // distances from ith point
            ans += (pre[N - 1] - pre[i])
                   - arr[i] * (N - 1 - i);
        }

        // Return Average
        ans = (ans / ((N * (N - 1)) / 2));
        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { -1, 3, -5, 4 };

        System.out.print(String.format(
            "%.5f", averageDistance(arr, arr.length)));
    }
}

// This code is contributed by ukasp.

Python3

# Python3 program for above approach

# Function to find average distance
# between given points on a line
def averageDistance(arr, N):

    # Sorting the array arr[]
    arr.sort()

    # Stores the prefix sum
    # array of arr[]
    pre = [0 for _ in range(N)]
    pre[0] = arr[0]

    # Loop to calculate prefix sum
    for i in range(1, N):
        pre[i] = pre[i - 1] + arr[i]

    # Initialising the answer variable
    ans = 0

    # Loop to iterate through arr[]
    for i in range(0, N - 1):

        # Adding summation of all
        # distances from ith point
        ans += ((pre[N - 1] - pre[i]) -
               (arr[i] * (N - 1 - i)))

    # Return Average
    return ans / ((N * (N - 1)) / 2)

# Driver Code
if __name__ == "__main__":

    arr = [ -1, 3, -5, 4 ]

    print(averageDistance(arr, len(arr)))

# This code is contributed by rakeshsahni

C#

// C# implementation for the above approach
using System;
class GFG
{

// Function to find average distance
// between given points on a line
static double averageDistance(
  int []arr, int N)
{
    // Sorting the array arr[]
    Array.Sort(arr);

    // Stores the prefix sum
    // array of arr[]
    int []pre = new int[N];
    pre[0] = arr[0];

    // Loop to calculate prefix sum
    for (int i = 1; i < N; i++) {
        pre[i] = pre[i - 1] + arr[i];
    }

    // Initialising the answer variable
    double ans = 0;

    // Loop to iterate through arr[]
    for (int i = 0; i < N - 1; i++) {

        // Adding summation of all
        // distances from ith point
        ans += (pre[N - 1] - pre[i])
          - arr[i] * (N - 1 - i);
    }

    // Return Average
    ans  = Math.Round((ans / ((N * (N - 1)) / 2)), 5);
    return ans;
}

// Driver Code
public static void Main()
{
    int []arr = { -1, 3, -5, 4 };

    Console.Write(averageDistance(arr, arr.Length));
}
}

// This code is contributed by Samim Hossain Mondal.

Javascript

<script>

      // JavaScript Program to implement
      // the above approach

      // Function to find average distance
      // between given points on a line
      function averageDistance(
          arr, N)
      {

          // Sorting the array arr[]
          arr.sort(function (a, b) { return a - b })

          // Stores the prefix sum
          // array of arr[]
          let pre = new Array(N).fill(0);
          pre[0] = arr[0];

          // Loop to calculate prefix sum
          for (let i = 1; i < N; i++) {
              pre[i] = pre[i - 1] + arr[i];
          }

          // Initialising the answer variable
          let ans = 0;

          // Loop to iterate through arr[]
          for (let i = 0; i < N - 1; i++) {

              // Adding summation of all
              // distances from ith point
              ans += (pre[N - 1] - pre[i])
                  - arr[i] * (N - 1 - i);
          }

          // Return Average
          return ans / ((N * (N - 1)) / 2);
      }

      // Driver Code
      let arr = [-1, 3, -5, 4];
      document.write(averageDistance(arr, arr.length).toPrecision(6));

  // This code is contributed by Potta Lokesh
  </script>

Output

5.16667

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