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Maximize difference between the sum of absolute differences of each element with the remaining array

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Given an array arr[] consisting of N integers, the task is to maximize the difference between the sum of absolute difference of an element with the remaining elements in the array.

Examples:

Input: arr[] = {1, 2, 4, 7}
Output: 6
Explanation:
For i = 1, |1 – 2| + |1 – 4| + |1 – 7| = 1 + 3 + 6 =10.
For i = 2, |2 – 1| + |2 – 4| + |2 – 7| = 1 + 2 + 5 = 8.
For i = 3, |4 – 1| + |4 – 2| + |4 – 7| = 3 + 2 + 3 = 8.
For i = 4, |7 – 1| + |7 – 2| + |7 – 4| = 6 + 5 + 3 = 14.
Maximum=14, Minimum=8.
Therefore, the maximum difference = 14 – 8 = 6.

Input: arr[] = {2, 1, 5, 4, 3}
Output: 4

Naive Approach: The simplest idea is to traverse the array and for each array element, traverse the array using a nested loop and calculate and store the sum of its absolute difference with the remaining array. While calculating, keep track of the maximum and minimum sums obtained. Finally, print the difference between the maximum and minimum sums. 

Below is the implementation of the above approach:

C++




#include <bits/stdc++.h>
using namespace std;
 
void findMaxDifference(int arr[], int n)
{
    int Max = INT_MIN;
    int Min = INT_MAX;
  
    // Iterate through all elements of the array
    for (int i = 0; i < n; i++) {
        int sum = 0;
  
        // Find the sum of absolute differences
        // of arr[i] with all other elements
        for (int j = 0; j < n; j++) {
            sum += abs(arr[i] - arr[j]);
        }
  
        // Update the maximum and minimum
        Max = max(Max, sum);
        Min = min(Min, sum);
    }
  
    // Print the result
    cout << Max - Min << endl;
}
 
int main()
{
    int arr[] = {1, 2, 4, 7};
    int n = sizeof(arr) / sizeof(arr[0]);
  
    findMaxDifference(arr, n);
  
    return 0;
}


Java




import java.util.*;
 
public class GFG {
 
    // Function to find the maximum difference of absolute
    // differences
    static void findMaxDifference(int[] arr, int n)
    {
        int max = Integer.MIN_VALUE;
        int min = Integer.MAX_VALUE;
 
        // Iterate through all elements of the array
        for (int i = 0; i < n; i++) {
            int sum = 0;
 
            // Find the sum of absolute differences of
            // arr[i] with all other elements
            for (int j = 0; j < n; j++) {
                sum += Math.abs(arr[i] - arr[j]);
            }
 
            // Update the maximum and minimum
            max = Math.max(max, sum);
            min = Math.min(min, sum);
        }
 
        // Print the result
        System.out.println(max - min);
    }
 
    public static void main(String[] args)
    {
        int[] arr = { 1, 2, 4, 7 };
        int n = arr.length;
 
        findMaxDifference(arr, n);
    }
}


Python




def find_max_difference(arr):
    # Initialize variables to store maximum and minimum values
    max_val = float('-inf')
    min_val = float('inf')
 
    # Iterate through all elements of the array
    for i in range(len(arr)):
        total_sum = 0
 
        # Find the sum of absolute differences
        # of arr[i] with all other elements
        for j in range(len(arr)):
            total_sum += abs(arr[i] - arr[j])
 
        # Update the maximum and minimum values
        max_val = max(max_val, total_sum)
        min_val = min(min_val, total_sum)
 
    # Print the result (difference between maximum and minimum values)
    print(max_val - min_val)
 
 
# Driver code
if __name__ == "__main__":
    arr = [1, 2, 4, 7]
    find_max_difference(arr)
# sinudp5vi


C#




using System;
 
class GFG {
    static void FindMaxDifference(int[] arr, int n)
    {
        int max = int.MinValue;
        int min = int.MaxValue;
 
        // Iterate through all elements of the array
        for (int i = 0; i < n; i++) {
            int sum = 0;
 
            // Find the sum of absolute differences
            // of arr[i] with all other elements
            for (int j = 0; j < n; j++) {
                sum += Math.Abs(arr[i] - arr[j]);
            }
 
            // Update the maximum and minimum
            max = Math.Max(max, sum);
            min = Math.Min(min, sum);
        }
 
        // Print the result
        Console.WriteLine(max - min);
    }
 
    static void Main()
    {
        int[] arr = { 1, 2, 4, 7 };
        int n = arr.Length;
 
        FindMaxDifference(arr, n);
    }
}


Javascript




function findMaxDifference(arr, n)
{
    let Max = Number.MIN_SAFE_INTEGER;
    let Min = Number.MAX_SAFE_INTEGER;
  
    // Iterate through all elements of the array
    for (let i = 0; i < n; i++) {
        let sum = 0;
  
        // Find the sum of absolute differences
        // of arr[i] with all other elements
        for (let j = 0; j < n; j++) {
            sum += Math.abs(arr[i] - arr[j]);
        }
  
        // Update the maximum and minimum
        Max = Math.max(Max, sum);
        Min = Math.min(Min, sum);
    }
  
    // Prlet the result
    console.log(Max - Min);
}
 
let arr = [1, 2, 4, 7];
let n = arr.length;
 
findMaxDifference(arr, n);


Output

6







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

Efficient Approach: To optimize the above approach, the idea is based on the observation that in a sorted array, for any index i, the elements on its left will be smaller and elements on its right will be greater. The sum of absolute difference for any element arr[i] in this sorted array can be calculated using the following formula:

(Number of elements to its left)*(arr[i]) – Sum of elements to its left + Sum of elements to its right – (Number of elements to its right)*(arr[i]))

Follow the steps below to solve the problem:

  • Initialize totalSum as 0 to store the sum of all the element of the array and leftSum as 0 to store the sum of elements on the left of any index.
  • Initialize two variables, Max as INT_MIN and Min as INT_MAX.
  • Sort the array arr[] in ascending order.
  • Traverse the array, arr[] using the variable i and do the following:
    • Store the sum of absolute difference of arr[i] with the rest of the elements using the formula in Sum = (i * arr[i]) – leftSum + totalSum – ((N – i – 1) * arr[i]).
    • Update Max to the maximum of Max and Sum.
    • Update Min to the minimum of Min and Sum.
  • After the above steps, print the value of Max and Min 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 maximize difference of
// the sum of absolute difference of
// an element with the rest of the
// elements in the array
void findMaxDifference(int arr[], int n)
{
    // Sort the array in ascending order
    sort(arr, arr + n);
 
    // Stores prefix sum at any instant
    int Leftsum = 0;
 
    // Store the total array sum
    int Totalsum = 0;
 
    // Initialize minimum and maximum
    // absolute difference
    int Min = INT_MAX, Max = INT_MIN;
 
    // Traverse the array to find
    // the total array sum
    for (int i = 0; i < n; i++)
        Totalsum += arr[i];
 
    // Traverse the array arr[]
    for (int i = 0; i < n; i++) {
 
        // Store the number of
        // elements to its left
        int leftNumbers = i;
 
        // Store the number of
        // elements to its right
        int rightNumbers = n - i - 1;
 
        // Update the sum of elements
        // on its left
        Totalsum = Totalsum - arr[i];
 
        // Store the absolute difference sum
        int sum = (leftNumbers * arr[i])
                  - Leftsum
                  + Totalsum
                  - (rightNumbers * arr[i]);
 
        // Update the Minimum
        Min = min(Min, sum);
 
        // Update the Maximum
        Max = max(Max, sum);
 
        // Update sum of elements
        // on its left
        Leftsum += arr[i];
    }
 
    // Print the result
    cout << Max - Min;
}
 
// Driven Code
int main()
{
    int arr[] = { 1, 2, 4, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);
    findMaxDifference(arr, N);
 
    return 0;
}


Java




// Java program for the above approach
import java.util.*;
class GFG
{
 
// Function to maximize difference of
// the sum of absolute difference of
// an element with the rest of the
// elements in the array
static void findMaxDifference(int arr[], int n)
{
   
    // Sort the array in ascending order
    Arrays.sort(arr);
 
    // Stores prefix sum at any instant
    int Leftsum = 0;
 
    // Store the total array sum
    int Totalsum = 0;
 
    // Initialize minimum and maximum
    // absolute difference
    int Min = Integer.MAX_VALUE, Max = Integer.MIN_VALUE;
 
    // Traverse the array to find
    // the total array sum
    for (int i = 0; i < n; i++)
        Totalsum += arr[i];
 
    // Traverse the array arr[]
    for (int i = 0; i < n; i++)
    {
 
        // Store the number of
        // elements to its left
        int leftNumbers = i;
 
        // Store the number of
        // elements to its right
        int rightNumbers = n - i - 1;
 
        // Update the sum of elements
        // on its left
        Totalsum = Totalsum - arr[i];
 
        // Store the absolute difference sum
        int sum = (leftNumbers * arr[i])
                  - Leftsum
                  + Totalsum
                  - (rightNumbers * arr[i]);
 
        // Update the Minimum
        Min = Math.min(Min, sum);
 
        // Update the Maximum
        Max = Math.max(Max, sum);
 
        // Update sum of elements
        // on its left
        Leftsum += arr[i];
    }
 
    // Print the result
    System.out.print(Max - Min);
}
 
// Driven Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 4, 7 };
    int N = arr.length;
    findMaxDifference(arr, N);
}
}
 
// This code is contributed by 29AjayKumar


Python3




# Python3 program for the above approach
 
# Function to maximize difference of
# the sum of absolute difference of
# an element with the rest of the
# elements in the array
def findMaxDifference(arr, n):
   
    # Sort the array in ascending order
    arr = sorted(arr)
 
    # Stores prefix sum at any instant
    Leftsum = 0
 
    # Store the total array sum
    Totalsum = 0
 
    # Initialize minimum and maximum
    # absolute difference
    Min, Max = 10**8, -10**8
 
    # Traverse the array to find
    # the total array sum
    for i in range(n):
        Totalsum += arr[i]
 
    # Traverse the array arr[]
    for i in range(n):
 
        # Store the number of
        # elements to its left
        leftNumbers = i
 
        # Store the number of
        # elements to its right
        rightNumbers = n - i - 1
 
        # Update the sum of elements
        # on its left
        Totalsum = Totalsum - arr[i]
 
        # Store the absolute difference sum
        sum = (leftNumbers * arr[i])- Leftsum + Totalsum - (rightNumbers * arr[i])
 
        # Update the Minimum
        Min = min(Min, sum)
 
        # Update the Maximum
        Max = max(Max, sum)
 
        # Update sum of elements
        # on its left
        Leftsum += arr[i]
 
    # Prthe result
    print (Max - Min)
 
# Driven Code
if __name__ == '__main__':
    arr = [1, 2, 4, 7]
    N = len(arr)
    findMaxDifference(arr, N)
 
# This code is contributed by mohit kumar 29.


C#




// C# Program to implement
// the above approach
using System;
class GFG
{
 
  // Function to maximize difference of
  // the sum of absolute difference of
  // an element with the rest of the
  // elements in the array
  static void findMaxDifference(int[] arr, int n)
  {
 
    // Sort the array in ascending order
    Array.Sort(arr);
 
    // Stores prefix sum at any instant
    int Leftsum = 0;
 
    // Store the total array sum
    int Totalsum = 0;
 
    // Initialize minimum and maximum
    // absolute difference
    int Minn = Int32.MaxValue, Maxx = Int32.MinValue;
 
    // Traverse the array to find
    // the total array sum
    for (int i = 0; i < n; i++)
      Totalsum += arr[i];
 
    // Traverse the array arr[]
    for (int i = 0; i < n; i++)
    {
 
      // Store the number of
      // elements to its left
      int leftNumbers = i;
 
      // Store the number of
      // elements to its right
      int rightNumbers = n - i - 1;
 
      // Update the sum of elements
      // on its left
      Totalsum = Totalsum - arr[i];
 
      // Store the absolute difference sum
      int sum = (leftNumbers * arr[i])
        - Leftsum
        + Totalsum
        - (rightNumbers * arr[i]);
 
      // Update the Minimum
      Minn = Math.Min(Minn, sum);
 
      // Update the Maximum
      Maxx = Math.Max(Maxx, sum);
 
      // Update sum of elements
      // on its left
      Leftsum += arr[i];
    }
 
    // Print the result
    Console.WriteLine(Maxx - Minn);
  }
 
  // Driver Code
  public static void Main(String[] args)
  {
    int[] arr = { 1, 2, 4, 7 };
    int N = arr.Length;
    findMaxDifference(arr, N);
  }
}
 
// This code is contributed by sanjoy_62.


Javascript




<script>
 
// javascript program of the above approach
 
// Function to maximize difference of
// the sum of absolute difference of
// an element with the rest of the
// elements in the array
function findMaxDifference(arr, n)
{
    
    // Sort the array in ascending order
    arr.sort();
  
    // Stores prefix sum at any instant
    let Leftsum = 0;
  
    // Store the total array sum
    let Totalsum = 0;
  
    // Initialize minimum and maximum
    // absolute difference
    let Min = Number.MAX_VALUE, Max = Number.MIN_VALUE;
  
    // Traverse the array to find
    // the total array sum
    for (let i = 0; i < n; i++)
        Totalsum += arr[i];
  
    // Traverse the array arr[]
    for (let i = 0; i < n; i++)
    {
  
        // Store the number of
        // elements to its left
        let leftNumbers = i;
  
        // Store the number of
        // elements to its right
        let rightNumbers = n - i - 1;
  
        // Update the sum of elements
        // on its left
        Totalsum = Totalsum - arr[i];
  
        // Store the absolute difference sum
        let sum = (leftNumbers * arr[i])
                  - Leftsum
                  + Totalsum
                  - (rightNumbers * arr[i]);
  
        // Update the Minimum
        Min = Math.min(Min, sum);
  
        // Update the Maximum
        Max = Math.max(Max, sum);
  
        // Update sum of elements
        // on its left
        Leftsum += arr[i];
    }
  
    // Prlet the result
    document.write(Max - Min);
}
 
    // Driver Code
     
     // Given array
        let arr = [ 1, 2, 4, 7 ];
    let N = arr.length;
    findMaxDifference(arr, N);
 
// This code is contributed by target_2.
</script>


Output

6







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



Last Updated : 21 Sep, 2023
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