Minimize value of x that minimizes value of |a1−x|^c+|a2−x|^c+···+|an−x|^c for value of c as 1 and 2

Given an array arr[] of N elements, the task is to find the value of x that minimizes the value of expression for c = 1.

|a₁−x|^c+|a₂−x|^c+···+|a_n−x|^c  = |a₁−x|+|a₂−x|+···+|a_n−x|

Examples:

Input: arr[] = { 1, 2, 9, 2, 6 }  
Output: 2
Explanation: The best solution is to select x = 2 which produces the sum  |1−2| + |2−2| + |9−2| + |2−2| + |6−2| = 12 , which is the minimum possible sum, for all other values, the sum so obtained will be greater than 2

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

Approach: In the general case, the best choice for x is the median of the given numbers, The median is an optimal choice, because if x is smaller than the median, the sum becomes smaller by increasing x, and if x is larger than the median, the sum becomes smaller by decreasing x. Hence, the optimal solution is that x is the median. 

Below is the implementation of the above approach:

C++

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

// Function to print the possible
// values of x that minimizes the sum
void findX(int arr[], int n)
{
    // Sort the numbers
    sort(arr, arr + n);

    // Stores the median
    int x;

    // Only one median if n is odd
    if (n % 2 != 0) {
        x = arr[n / 2];
    }

    // Two medians if n is even
    // and every value between them
    // is optimal print any of them
    else {
        int a = arr[n / 2 - 1];
        int b = arr[n / 2];
        x = a;
    }

    int sum = 0;

    // Find minimum sum
    for (int i = 0; i < n; i++) {
        sum += abs(arr[i] - x);
    }

    cout << sum;
}

// Driver Code
int main()
{
    int arr1[] = { 1, 2, 9, 2, 6 };
    int n1 = sizeof(arr1) / sizeof(arr1[0]);

    findX(arr1, n1);
    return 0;
}

Java

// Java code for the above approach
import java.util.*;

class GFG
{

  // Function to print the possible
// values of x that minimizes the sum
static void findX(int arr[], int n)
{

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

    // Stores the median
    int x;

    // Only one median if n is odd
    if (n % 2 != 0) {
        x = arr[(int)Math.floor(n / 2)];
    }

    // Two medians if n is even
    // and every value between them
    // is optimal print any of them
    else {
        int a = arr[n / 2 - 1];
        int b = arr[n / 2];
        x = a;
    }

    int sum = 0;

    // Find minimum sum
    for (int i = 0; i < n; i++) {
        sum += Math.abs(arr[i] - x);
    }

   System.out.println( sum);
}

    public static void main (String[] args) {

    int arr1[] = { 1, 2, 9, 2, 6 };
    int n1 = arr1.length;

    findX(arr1, n1);
    }
}

// This code is contributed by Potta Lokesh

Python3

# Python program for the above approach

# Function to print the possible
# values of x that minimizes the sum
def findX(arr, n):

  # Sort the numbers
  arr.sort();

  # Stores the median
  x = None;

  # Only one median if n is odd
  if (n % 2 != 0):
    x = arr[n // 2];

  # Two medians if n is even
  # and every value between them
  # is optimal print any of them
  else:
    a = arr[(n // 2) - 1];
    b = arr[n // 2];
    x = a;
  sum = 0;

  # Find minimum sum
  for i in range(n):
    sum += abs(arr[i] - x);


  print(sum);

# Driver Code
arr1 = [1, 2, 9, 2, 6];
n1 = len(arr1)

findX(arr1, n1);

# This code is contributed by gfgking.

C#

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

class GFG {

    // Function to print the possible
    // values of x that minimizes the sum
    static void findX(int[] arr, int n)
    {

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

        // Stores the median
        int x;

        // Only one median if n is odd
        if (n % 2 != 0) {
            x = arr[(int)Math.Floor((float)(n / 2))];
        }

        // Two medians if n is even
        // and every value between them
        // is optimal print any of them
        else {
            int a = arr[n / 2 - 1];

            x = a;
        }

        int sum = 0;

        // Find minimum sum
        for (int i = 0; i < n; i++) {
            sum += Math.Abs(arr[i] - x);
        }

        Console.WriteLine(sum);
    }

    public static void Main(string[] args)
    {

        int[] arr1 = { 1, 2, 9, 2, 6 };
        int n1 = arr1.Length;

        findX(arr1, n1);
    }
}

// This code is contributed by ukasp.

Javascript

<script>
// Javascript program for the above approach

// Function to print the possible
// values of x that minimizes the sum
function findX(arr, n) {
  // Sort the numbers
  arr.sort((a, b) => a - b);

  // Stores the median
  let x;

  // Only one median if n is odd
  if (n % 2 != 0) {
    x = arr[(Math.floor(n / 2))];
  }

  // Two medians if n is even
  // and every value between them
  // is optimal print any of them
  else {
    let a = arr[(Math.floor(n / 2) - 1)];
    let b = arr[(Math.floor(n / 2))];
    x = a;
  }

  let sum = 0;

  // Find minimum sum
  for (let i = 0; i < n; i++) {
    sum += Math.abs(arr[i] - x);
  }

  document.write(sum);
}

// Driver Code

let arr1 = [1, 2, 9, 2, 6];
let n1 = arr1.length;

findX(arr1, n1);

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

Output

12

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

Given an array arr[] of N elements, the task is to find the value of x that minimizes the value of expression for c = 2.

|a₁−x|^c+|a₂−x|^c+···+|a_n−x|^c  = (a₁−x)²+(a₂−x)²+···+(a_n−x)².

Examples :

Input: arr[] = { 1, 2, 9, 2, 6 }  
Output: 4
Explanation:  The best solution is to select x = 4 which produces the sum  (1−4)^2 + (2−4)^2 + (9−4)^2 + (2−4)^2 + (6−4)^2 = 46, which is the minimum possible sum.

Input: arr[] = { 1, 2, 2, 4, 6 }  
Output: 3

Approach: In the general case, the best choice for x is the average of the numbers. This result can be derived by expanding the sum as follows:

nx²−2x(a₁+a₂+···+a_n) + (a₁²+a₂²+···+a_n²) 

The last part does not depend on x. The remaining parts form a function nx² − 2xs where s=a₁+a₂+···+a_n. Applying derivative to this equation w.r.t x and equating the result to zero gives us x = s / n, which is the value that minimizes the sum.

Below is the implementation of the above approach:

C++

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

// Function to find the value of x
// that minimizes the sum
void findX(int arr[], int n)
{
    // Store the sum
    double sum = 0;
    for (int i = 0; i < n; i++) {
        sum += arr[i];
    }

    // Store the average of numbers
    double x = sum / n;

    double minSum = 0;

    // Find minimum sum
    for (int i = 0; i < n; i++) {
        minSum += pow((arr[i] - x), 2);
    }

    cout << minSum;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 9, 2, 6 };
    int n = sizeof(arr) / sizeof(arr[0]);

    findX(arr, n);

    return 0;
}

Java

// Java implementation for the above approach
import java.util.*;
public class GFG
{
// Function to find the value of x
// that minimizes the sum
static void findX(int []arr, int n)
{
    // Store the sum
    int sum = 0;
    for (int i = 0; i < n; i++) {
        sum += arr[i];
    }

    // Store the average of numbers
    int x = sum / n;

    int minSum = 0;

    // Find minimum sum
    for (int i = 0; i < n; i++) {
        minSum += Math.pow((arr[i] - x), 2);
    }

    System.out.print(minSum);
}

// Driver Code
public static void main(String args[])
{
    int []arr = { 1, 2, 9, 2, 6 };
    int n = arr.length;

    findX(arr, n);
}
}
// This code is contributed by Samim Hossain Mondal.

Python3

# Python implementation for the above approach

# Function to find the value of x
# that minimizes the sum
def findX(arr, n):

    # Store the sum
    sum = 0;
    for i in range(n):
        sum += arr[i];

    # Store the average of numbers
    x = sum // n;

    minSum = 0;

    # Find minimum sum
    for i in range(n):
        minSum += pow((arr[i] - x), 2);
    print(minSum);

# Driver Code
if __name__ == '__main__':
    arr = [ 1, 2, 9, 2, 6 ];
    n = len(arr);

    findX(arr, n);

# This code is contributed by shikhasingrajput

C#

// C# implementation for the above approach
using System;
class GFG
{
// Function to find the value of x
// that minimizes the sum
static void findX(int []arr, int n)
{
    // Store the sum
    int sum = 0;
    for (int i = 0; i < n; i++) {
        sum += arr[i];
    }

    // Store the average of numbers
    int x = sum / n;

    int minSum = 0;

    // Find minimum sum
    for (int i = 0; i < n; i++) {
        minSum += (int)Math.Pow((arr[i] - x), 2);
    }

    Console.Write(minSum);
}

// Driver Code
public static void Main()
{
    int []arr = { 1, 2, 9, 2, 6 };
    int n = arr.Length;

    findX(arr, n);
}
}
// This code is contributed by Samim Hossain Mondal.

Javascript

<script>
// Javascript implementation for the above approach

// Function to find the value of x
// that minimizes the sum
function findX(arr, n)
{
    // Store the sum
    let sum = 0;
    for (let i = 0; i < n; i++) {
        sum += arr[i];
    }

    // Store the average of numbers
    let x = sum / n;

    let minSum = 0;

    // Find minimum sum
    for (let i = 0; i < n; i++) {
        minSum += Math.pow((arr[i] - x), 2);
    }

    document.write(minSum);
}

// Driver Code
let arr = [ 1, 2, 9, 2, 6 ];
let n = arr.length;

findX(arr, n);

// This code is contributed by Samim Hossain Mondal.
</script>

Output

46

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