Maximum sum of absolute differences between distinct pairs of a triplet from an array

Given an array arr[] consisting of N integers, the task is to find the maximum sum of absolute differences between all distinct pairs of the triplet in the array.

Examples:

Input: arr[] = {1, 2, 3, 4}
Output: 6
Explanation:
The valid triplet is (1, 3, 4) as sum = |1 - 4| + |1 - 3| + |3 - 4| = 6, which is the maximum among all the triplets.

Input: arr[] = {2, 2, 2}
Output: 0

Approach: The idea to solve the given problem is to sort the array in ascending order and find the sum of absolute differences between the pairs of the first and the last two elements of the array. Follow the steps below to solve the problem:

• Initialize a variable, say sum, to store the maximum possible sum.
• Sort the given array arr[] in ascending order.
• Find the sum of differences between the pairs of the first and the last two elements of the array i.e., sum = (arr[N - 2] - arr[0]) + (arr[N - 1] - arr[0]) + (arr[N - 2] - arr[N - 1]).
• After completing the above steps, print the value of the sum as the result.

Below is the implementation of the above approach:

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

// Function to find the maximum sum of
// absolute differences between
// distinct pairs of triplet in array
void maximumSum(int arr[], int N)
{
    // Stores the maximum sum
    int sum;

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

    // Sum of differences between
    // pairs of the triplet
    sum = (arr[N - 1] - arr[0])
          + (arr[N - 2] - arr[0])
          + (arr[N - 1] - arr[N - 2]);

    // Print the sum
    cout << sum;
}

// Driver Code
int main()
{
    int arr[] = { 1, 3, 4, 2 };
    int N = sizeof(arr) / sizeof(arr[0]);
    maximumSum(arr, N);

    return 0;
}

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

  // Function to find the maximum sum of
  // absolute differences between
  // distinct pairs of triplet in array
  static void maximumSum(int[] arr, int N)
  {

    // Stores the maximum sum
    int sum;

    // Sort the array in
    // ascending order
    Arrays.sort(arr);

    // Sum of differences between
    // pairs of the triplet
    sum = (arr[N - 1] - arr[0]) + (arr[N - 2] - arr[0])
      + (arr[N - 1] - arr[N - 2]);

    // Print the sum
    System.out.println(sum);
  }

  // Driver Code
  public static void main(String[] args)
  {
    int[] arr = { 1, 3, 4, 2 };
    int N = arr.length;
    maximumSum(arr, N);
  }
}

// This code is contributed by susmitakundugoaldanga.

# Python program for the above approach

# Function to find the maximum sum of
# absolute differences between
# distinct pairs of triplet in array
def maximumSum(arr, N):
  
    # Stores the maximum sum
    sum = 0
    
    # Sort the array in
    # ascending order
    arr.sort()
    
    # Sum of differences between
    # pairs of the triplet
    sum = (arr[N - 1] - arr[0]) + (arr[N - 2] - arr[0]) + (arr[N - 1] - arr[N - 2]);

    # Print the sum
    print(sum)

# Driver Code
arr = [ 1, 3, 4, 2 ]
N = len(arr)
maximumSum(arr, N)

# This code is contributed by rohitsingh07052.

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG 
{

    // Function to find the maximum sum of
    // absolute differences between
    // distinct pairs of triplet in array
    static void maximumSum(int[] arr, int N)
    {
      
        // Stores the maximum sum
        int sum;

        // Sort the array in
        // ascending order
        Array.Sort(arr);

        // Sum of differences between
        // pairs of the triplet
        sum = (arr[N - 1] - arr[0]) + (arr[N - 2] - arr[0])
              + (arr[N - 1] - arr[N - 2]);

        // Print the sum
        Console.Write(sum);
    }

    // Driver Code
    public static void Main()
    {
        int[] arr = { 1, 3, 4, 2 };
        int N = arr.Length;
        maximumSum(arr, N);
    }
}

// This code is contributed by chitranayal.

<script>

// Javascript program for the above approach

// Function to find the maximum sum of
// absolute differences between
// distinct pairs of triplet in array
function maximumSum(arr, N)
{
    // Stores the maximum sum
    let sum;

    // Sort the array in
    // ascending order
    arr.sort();

    // Sum of differences between
    // pairs of the triplet
    sum = (arr[N - 1] - arr[0])
        + (arr[N - 2] - arr[0])
        + (arr[N - 1] - arr[N - 2]);

    // Print the sum
    document.write(sum);
}

// Driver Code

    let arr = [ 1, 3, 4, 2 ];
    let N = arr.length;
    maximumSum(arr, N);

// This code is contributed by Mayank Tyagi

</script>

6

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