Minimum Increment / decrement to make array elements equal

Given an array of integers where 1 \leq A[i] \leq 10^{18}   . In one operation you can either Increment/Decrement any element by 1. The task is to find the minimum operations needed to be performed on the array elements to make all array elements equal.

Examples: 

Input : A[] = { 1, 5, 7, 10 }
Output : 11
Increment 1 by 4, 5 by 0.
Decrement 7 by 2, 10 by 5.
New array A = { 5, 5, 5, 5 } with 
cost of operations = 4 + 0 + 2 + 5 = 11.

Input : A = { 10, 2, 20 }
Output : 18

Approach:  

1. Sort the array of Integers in increasing order.
2. Now, to make all elements equal with min cost. We will have to make the elements equal to the middle element of this sorted array. So, select the middle value, Let it be K. 
   Note: In case of even numbers of elements, we will have to check for the costs of both middle elements and take the minimum.
3. If A[i] < K, Increment the element by K - A[i].
4. If A[i] > K, Decrement the element by A[i] - K.
5. Update cost of each operation performed.

Below is the implementation of above approach:  

// C++ program to find minimum Increment or
// decrement to make array elements equal
#include <bits/stdc++.h>
using namespace std;

// Function to return minimum operations need
// to be make each element of array equal
int minCost(int A[], int n)
{
    // Initialize cost to 0
    int cost = 0;

    // Sort the array
    sort(A, A + n);

    // Middle element
    int K = A[n / 2];

    // Find Cost
    for (int i = 0; i < n; ++i)
        cost += abs(A[i] - K);

    // If n, is even. Take minimum of the
    // Cost obtained by considering both
    // middle elements
    if (n % 2 == 0) {
        int tempCost = 0;

        K = A[(n / 2) - 1];

        // Find cost again
        for (int i = 0; i < n; ++i)
            tempCost += abs(A[i] - K);

        // Take minimum of two cost
        cost = min(cost, tempCost);
    }

    // Return total cost
    return cost;
}

// Driver Code
int main()
{
    int A[] = { 1, 6, 7, 10 };

    int n = sizeof(A) / sizeof(A[0]);

    cout << minCost(A, n);

    return 0;
}

// Java program to find minimum Increment or 
// decrement to make array elements equal 
import java.util.*;
class GfG { 

// Function to return minimum operations need 
// to be make each element of array equal 
static int minCost(int A[], int n) 
{ 
    // Initialize cost to 0 
    int cost = 0; 

    // Sort the array 
    Arrays.sort(A); 

    // Middle element 
    int K = A[n / 2]; 

    // Find Cost 
    for (int i = 0; i < n; ++i) 
        cost += Math.abs(A[i] - K); 

    // If n, is even. Take minimum of the 
    // Cost obtained by considering both 
    // middle elements 
    if (n % 2 == 0) { 
        int tempCost = 0; 

        K = A[(n / 2) - 1]; 

        // Find cost again 
        for (int i = 0; i < n; ++i) 
            tempCost += Math.abs(A[i] - K); 

        // Take minimum of two cost 
        cost = Math.min(cost, tempCost); 
    } 

    // Return total cost 
    return cost; 
} 

// Driver Code 
public static void main(String[] args) 
{ 
    int A[] = { 1, 6, 7, 10 }; 

    int n = A.length; 

    System.out.println(minCost(A, n)); 
}
} 

# Python3 program to find minimum Increment or
# decrement to make array elements equal
    
# Function to return minimum operations need
# to be make each element of array equal
def minCost(A, n):
    
    # Initialize cost to 0
    cost = 0
    
    # Sort the array
    A.sort();
    
    # Middle element
    K = A[int(n / 2)]
    
    #Find Cost
    for i in range(0, n):
        cost = cost + abs(A[i] - K)
    
    # If n, is even. Take minimum of the
    # Cost obtained by considering both
    # middle elements
    if n % 2 == 0:
        tempCost = 0
        K = A[int(n / 2) - 1]
        
        # FInd cost again
        for i in range(0, n):
            tempCost = tempCost + abs(A[i] - K)
        
        # Take minimum of two cost
        cost = min(cost, tempCost)
        
    # Return total cost
    return cost
    
# Driver code
A = [1, 6, 7, 10]
n = len(A)

print(minCost(A, n))
        
# This code is contributed 
# by Shashank_Sharma

// C# program to find minimum Increment or
// decrement to make array elements equal
using System;

class GFG {
    
// Function to return minimum operations need
// to be make each element of array equal
static int minCost(int []A, int n)
{
    // Initialize cost to 0
    int cost = 0;

    // Sort the array
    Array.Sort(A);

    // Middle element
    int K = A[n / 2];

    // Find Cost
    for (int i = 0; i < n; ++i)
        cost += Math.Abs(A[i] - K);

    // If n, is even. Take minimum of the
    // Cost obtained by considering both
    // middle elements
    if (n % 2 == 0) {
        int tempCost = 0;

        K = A[(n / 2) - 1];

        // Find cost again
        for (int i = 0; i < n; ++i)
            tempCost += Math.Abs(A[i] - K);

        // Take minimum of two cost
        cost = Math.Min(cost, tempCost);
    }

    // Return total cost
    return cost;
}

// Driver Code
public static void Main(String[] args)
{
    int []A = new int []{ 1, 6, 7, 10 };

    int n = A.Length;

    Console.WriteLine(minCost(A, n));
}
}

<?php
// PHP program to find minimum Increment or 
// decrement to make array elements equal 

// Function to return minimum operations need 
// to be make each element of array equal 
function minCost($A, $n) 
{ 
    // Initialize cost to 0 
    $cost = 0; 

    // Sort the array 
    sort($A); 

    // Middle element 
    $K = $A[$n / 2]; 
    // Find Cost 
    for ($i = 0; $i < $n; ++$i) 
        $cost += abs($A[$i] - $K); 

    // If n, is even. Take minimum of the 
    // Cost obtained by considering both 
    // middle elements 
    if ($n % 2 == 0)
    { 
        $tempCost = 0; 

        $K = $A[($n / 2) - 1]; 

        // Find cost again 
        for ($i = 0; $i < $n; ++$i) 
            $tempCost += abs($A[$i] - $K); 

        // Take minimum of two cost 
        $cost = min($cost, $tempCost); 
    } 

    // Return total cost 
    return $cost; 
} 

// Driver Code 
$A = array( 1, 6, 7, 10 ); 
$n = sizeof($A); 
echo minCost($A, $n); 

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

<script>

// Javascript program to find minimum Increment or
// decrement to make array elements equal

// Function to return minimum operations need
// to be make each element of array equal
function minCost(A,n)
{
    // Initialize cost to 0
    var cost = 0;

    // Sort the array
    A.sort();

    // Middle element
    var K = A[parseInt(n/2)];
    var i;
    // Find Cost
    for (i = 0; i < n; ++i)
        cost += Math.abs(A[i] - K);
    

    // If n, is even. Take minimum of the
    // Cost obtained by considering both
    // middle elements
    if (n % 2 == 0) {
        var tempCost = 0;

        K = A[parseInt(n / 2) - 1];

        // Find cost again
        for (i = 0; i < n; ++i)
            tempCost += Math.abs(A[i] - K);

        // Take minimum of two cost
        cost = Math.min(cost, tempCost);
    }

    // Return total cost
    return cost;
}

// Driver Code
    var A = [1, 6, 7, 10];

    var n = A.length;
    document.write(minCost(A, n));

</script>

10

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

Further Optimization We can find median in linear time and reduce the time complexity to O(N)