Minimum sum of differences with an element in an array

Given an array, we need to find the sum of elements of an array after changing the element as arr[i] will become abs(arr[i]-x) where x is an array element.

Examples:

Input  : {2, 5, 1, 7, 4}
Output : 9
We get minimum sum when we choose
x = 4. The minimum sum is 
abs(2-4) + abs(5-4) + abs(1-4) + (7-4)
abs(4-4) = 9
Input  : {5, 11, 14, 10, 17, 15}
Output : 20
We can either choose x = 14 or x = 11

Naive Approach:

The naive approach to solve the problem can be like: We pick each element of the array and then traverse entire array to store sum of absolute difference of the picked element and ith element. After traversal of the array, we keep updating a variable say res with minimum of the current sum that we get after traversal and the previous value that variable res will be having. Finally, after going through each element of the array the same way we will be having our answer in res variable. 

Algorithm:

1. Define a function “findSum” which takes an integer array “arr”, its size “n” and an integer “x” as input.
2. Initialize a variable “res” to a very large number.
3. For each element “i” in the array “arr”, do the following:

                 a. Initialize a variable “sum” to 0.

                 b. For each element “j” in the array “arr”, do the following:

                      i. Calculate the absolute difference between the elements “arr[i]” and “arr[j]”. This can be done by subtracting “arr[j]”                          from “arr[i]” and taking the absolute value of the result using the “abs()” function.
                      ii. Add this absolute difference to the variable “sum”.

                 c. If the value of “sum” is less than the current value of “res”, update the value of “res” to “sum”.
       4. Return the value of “res”.

Below is the implementation of the approach:

C++

// C++ code for the approach

#include <bits/stdc++.h>

using namespace std;

// Function to calculate sum of absolute differences
int calculateSum(int arr[], int n, int x) {
    int sum = 0;

      // traverse to find sum of absolute differences
    for (int i = 0; i < n; i++) {
        sum += abs(arr[i] - x);
    }

    return sum;
}

// Function to find minimum sum of absolute differences
int findMinSum(int arr[], int n) {
      // variable to store the minimum result value
    int res = INT_MAX;

      // traverse through each element to
      // find sum of absolute difference
      // of all elements with it
    for (int i = 0; i < n; i++) {
          // Function call
        int sum = calculateSum(arr, n, arr[i]);
          // update res with the minimum one
        res = min(res, sum);
    }

    return res;
}

// Driver's Code
int main() {
      // Input
    int arr[] = { 5, 11, 14, 10, 17, 15 };
    int n = sizeof(arr) / sizeof(arr[0]);

      // Function call
    int minSum = findMinSum(arr, n);
    cout << minSum << endl;

    return 0;
}

Java

public class Main {
    // Function to calculate the sum of absolute differences
    static int calculateSum(int[] arr, int n, int x) {
        int sum = 0;

        // Traverse to find the sum of absolute differences
        for (int i = 0; i < n; i++) {
            sum += Math.abs(arr[i] - x);
        }

        return sum;
    }

    // Function to find the minimum sum of absolute differences
    static int findMinSum(int[] arr, int n) {
        // Variable to store the minimum result value
        int res = Integer.MAX_VALUE;

        // Traverse through each element to
        // find the sum of absolute difference
        // of all elements with it
        for (int i = 0; i < n; i++) {
            // Function call
            int sum = calculateSum(arr, n, arr[i]);

            // Update res with the minimum one
            res = Math.min(res, sum);
        }

        return res;
    }

    public static void main(String[] args) {
        // Input
        int[] arr = { 5, 11, 14, 10, 17, 15 };
        int n = arr.length;

        // Function call
        int minSum = findMinSum(arr, n);
        System.out.println(minSum);
    }
}

Python3

# Function to calculate sum of absolute differences
def calculateSum(arr, n, x):
    sum = 0

    # traverse to find sum of absolute differences
    for i in range(n):
        sum += abs(arr[i] - x)

    return sum

# Function to find minimum sum of absolute differences


def findMinSum(arr, n):
    # variable to store the minimum result value
    res = float('inf')

    # traverse through each element to
    # find sum of absolute difference
    # of all elements with it
    for i in range(n):
        # Function call
        sum = calculateSum(arr, n, arr[i])
        # update res with the minimum one
        res = min(res, sum)

    return res


# Driver Code
if __name__ == '__main__':
    # Input
    arr = [5, 11, 14, 10, 17, 15]
    n = len(arr)

    # Function call
    minSum = findMinSum(arr, n)
    print(minSum)

C#

using System;

class Program {
    // Function to calculate the sum of absolute differences
    static int CalculateSum(int[] arr, int n, int x)
    {
        int sum = 0;

        // Traverse to find the sum of absolute differences
        for (int i = 0; i < n; i++) {
            sum += Math.Abs(arr[i] - x);
        }

        return sum;
    }

    // Function to find the minimum sum of absolute
    // differences
    static int FindMinSum(int[] arr, int n)
    {
        // Variable to store the minimum result value
        int res = int.MaxValue;

        // Traverse through each element to
        // find the sum of absolute difference
        // of all elements with it
        for (int i = 0; i < n; i++) {
            // Function call
            int sum = CalculateSum(arr, n, arr[i]);

            // Update res with the minimum one
            res = Math.Min(res, sum);
        }

        return res;
    }

    static void Main()
    {
        // Input
        int[] arr = { 5, 11, 14, 10, 17, 15 };
        int n = arr.Length;

        // Function call
        int minSum = FindMinSum(arr, n);
        Console.WriteLine(minSum);
    }
}

Javascript

// Function to calculate sum of absolute differences
function calculateSum(arr, n, x) {
    let sum = 0;

    // Traverse to find sum of absolute differences
    for (let i = 0; i < n; i++) {
        sum += Math.abs(arr[i] - x);
    }

    return sum;
}

// Function to find minimum sum of absolute differences
function findMinSum(arr, n) {
    // Variable to store the minimum result value
    let res = Number.MAX_SAFE_INTEGER;

    // Traverse through each element to
    // find sum of absolute difference
    // of all elements with it
    for (let i = 0; i < n; i++) {
        // Function call
        let sum = calculateSum(arr, n, arr[i]);
        // Update res with the minimum one
        res = Math.min(res, sum);
    }

    return res;
}

// Driver's Code
// Input
let arr = [5, 11, 14, 10, 17, 15];
let n = arr.length;

// Function call
let minSum = findMinSum(arr, n);
console.log(minSum);

Output

20

Time Complexity: O(n*n) as two nested for loops is basically executing in total. Here, n is size of the input array. 

Auxiliary Space: O(1) as no extra space has been used.

Efficient Approach:

The idea is based on fact that middle element would cause minimum sum of differences. When there are even number of elements, we can take any of the middle two elements. We can verify this fact by taking few examples. 

Implementation:

C++

// C++ program to find minimum sum of absolute
// differences with an array element.
#include<bits/stdc++.h>
using namespace std;
    // function to find min sum after operation
    int absSumDidd(int a[],int n)
    {
        // Sort the array
        sort(a,a+n);

        // Pick middle value
        int midValue = a[(int)(n / 2)];

        // Sum of absolute differences.
        int sum = 0;
        for (int i = 0; i < n; i++) {
            sum = sum + abs(a[i] - midValue);
        }

        return sum;
    }

    // Driver Code
    int main()
    {
        int arr[] = { 5, 11, 14, 10, 17, 15 };
        int n=sizeof(arr)/sizeof(arr[0]);
        cout<< absSumDidd(arr,n);
    }
    // Contributed by mits

Java

// Java program to find minimum sum of absolute
// differences with an array element.
import java.lang.*;
import java.util.*;

public class GFG {

    // function to find min sum after operation
    static int absSumDidd(int a[])
    {
        // Sort the array
        Arrays.sort(a);

        // Pick middle value
        int midValue = a[a.length / 2];

        // Sum of absolute differences.
        int sum = 0;
        for (int i = 0; i < a.length; i++) {
            sum = sum + Math.abs(a[i] - midValue);
        }

        return sum;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 5, 11, 14, 10, 17, 15 };
        System.out.print(absSumDidd(arr));
    }
    // Contributed by Saurav Jain
}

Python3

# Python3 program to find minimum sum of
# absolute differences with an array element.

# function to find min sum after operation
def absSumDidd(a, n):

    # Sort the array
    a.sort()

    # Pick middle value
    midValue = a[(int)(n // 2)]

    # Sum of absolute differences.
    sum = 0
    for i in range(n):
        sum = sum + abs(a[i] - midValue)
    return sum

# Driver Code
arr = [5, 11, 14, 10, 17, 15]
n = len(arr)
print(absSumDidd(arr, n))

# This code is contributed
# by sahishelangia

C#

// C# program to find minimum sum of absolute
// differences with an array element.
using System;


 class GFG {

    // function to find min sum after operation
    static int absSumDidd(int []a)
    {
        // Sort the array
        Array.Sort(a);

        // Pick middle value
        int midValue = a[a.Length / 2];

        // Sum of absolute differences.
        int sum = 0;
        for (int i = 0; i < a.Length; i++) {
            sum = sum + Math.Abs(a[i] - midValue);
        }

        return sum;
    }

    // Driver Code
    public static void Main()
    {
        int []arr = { 5, 11, 14, 10, 17, 15 };
        Console.Write(absSumDidd(arr));
    }
    // Contributed by Subhadeep
}

Javascript

<script>

// Javascript program to find minimum sum of absolute
// differences with an array element.

// Function to find min sum after operation
function absSumDidd(a)
{

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

    // Pick middle value
    var midValue = a[a.length / 2];

    // Sum of absolute differences.
    var sum = 0;
    for(var i = 0; i < a.length; i++)
    {
        sum = sum + Math.abs(a[i] - midValue);
    }
    return sum;
}

// Driver Code
var arr = [ 5, 11, 14, 10, 17, 15 ];
document.write(absSumDidd(arr));

// This code is contributed by shikhasingrajput

</script>

PHP

<?php
// PHP program to find minimum
// sum of absolute differences
// with an array element.

// function to find min sum
// after operation
function absSumDidd($a, $n)
{
    // Sort the array
    sort($a);

    // Pick middle value
    $midValue = $a[($n / 2)];

    // Sum of absolute differences.
    $sum = 0;
    for ( $i = 0; $i < $n; $i++)
    {
        $sum = $sum + abs($a[$i] -
               $midValue);
    }

    return $sum;
}

// Driver Code
$arr = array(5, 11, 14, 10, 17, 15 );
$n = count($arr);
echo absSumDidd($arr, $n);

// This code is contributed
// by anuj_67
?>

Output

20

Time Complexity: O(n Log n) 
Auxiliary Space: O(1), as we are only using constant space to store some variables.

We can further optimize the above solution to O(n) by using linear time algorithm for median finding.