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Minimum pair sum operations to make array each element divisible by 4
  • Difficulty Level : Medium
  • Last Updated : 25 Jul, 2018
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Given an array of positive integers of length n. Our task is to find minimum number of operations to convert an array so that arr[i] % 4 is zero for each i. In each operation, we can take any two elements from the array, remove both of them and put back their sum in the array.

Examples:

Input : arr = {2 , 2 , 2 , 3 , 3}
Output : 3
Explanation: In 1 operation we pick 2 and 2 and put their sum back to the array , In 2 operation we pick 3 and 3 and do same for that ,now in 3 operation we pick 6 and 2 so overall 3 operation are required.

Input: arr = {4, 2, 2, 6, 6}
Output: 2
Explanation: In operation 1, we can take 2 and 2 and put back their sum i.e. 4. In operation 2, we can take 6 and 6 and put back their sum i.e. 12. And array becomes {4, 4, 12}.



Approach : Assume the count of elements leaving remainder 1, 2, 3 when divided by 4 are brr[1], brr[2] and brr[3].
If (brr[1] + 2 * brr[2] + 3 * brr[3]) is not a multiple of 4, solution does not exist.

Now greedily pair elements of brr[2] with brr[2] and elements of brr[1] with brr[3]. This helps us to achieve fixing a maximum of 2 elements at a time. Now, we can either we left with only 1 brr[2] element or none. If we are left with 1 brr[2] element, then we can pair with 2 remaining brr[1] or brr[3] elements. This will incur a total of 2 operations.

At last, we would be only left with brr[1] or brr[3] elements (if possible). This can only we fixed in one way. That is taking 4 of them and fixing them all together in 3 operations. Thus, we are able to fix all the elements of the array.

Below is the implementation:

C++




// CPP program to find Minimum number
// of operations to convert an array 
// so that arr[i] % 4 is zero.
#include <bits/stdc++.h>
using namespace std;
  
// Function to find minimum operations.
int minimumOperations(int arr[], int n)
{   
    // Counting of all the elements
    // leaving remainder 1, 2, 3 when
    // divided by 4 in the array brr.
    // at positions 1, 2 and 3 respectively.
    int brr[] = { 0, 0, 0, 0 };
    for (int i = 0; i < n; i++)
        brr[arr[i] % 4]++;
  
    // If it is possible to convert the 
    // array so that arr[i] % 4 is zero.
    if ((brr[1] + 2 * brr[2] + 3 * brr[3]) % 4 == 0) 
    {
        // Pairing the elements of brr3 and brr1.
        int min_opr = min(brr[3], brr[1]);
        brr[3] -= min_opr;
        brr[1] -= min_opr;
  
        // Pairing the brr2 elements.
        min_opr += brr[2] / 2;
  
        // Assigning the remaining brr2 elements.
        brr[2] %= 2;
  
        // If we are left with one brr2 element.
        if (brr[2]) {
  
            // Here we need only two operations
            // to convert the remaining one
            // brr2 element to convert it.
            min_opr += 2;
  
            // Now there is no brr2 element.
            brr[2] = 0;
  
            // Remaining brr3 elements.
            if (brr[3])             
                brr[3] -= 2;            
  
            // Remaining brr1 elements.
            if (brr[1]) 
                brr[1] -= 2;            
        }
  
        // If we are left with brr1 and brr2
        // elements then, we have to take four
        // of them and fixing them all together
        // in 3 operations.
        if (brr[1])        
            min_opr += (brr[1] / 4) * 3;        
        if (brr[3])        
            min_opr += (brr[3] / 4) * 3;        
  
        // Returns the minimum operations.
        return min_opr;
    }
  
    // If it is not possible to convert the array.
    return -1;    
}
  
// Driver function
int main()
{
    int arr[] = { 1, 2, 3, 1, 2, 3, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minimumOperations(arr, n);
}

Java




// Java program to find Minimum number
// of operations to convert an array 
// so that arr[i] % 4 is zero.
  
class GFG {
   
// Function to find minimum operations.
static int minimumOperations(int arr[], int n)
{   
  
    // Counting of all the elements
    // leaving remainder 1, 2, 3 when
    // divided by 4 in the array brr.
    // at positions 1, 2 and 3 respectively.
    int brr[] = { 0, 0, 0, 0 };
    for (int i = 0; i < n; i++)
        brr[arr[i] % 4]++;
   
    // If it is possible to convert the 
    // array so that arr[i] % 4 is zero.
    if ((brr[1] + 2 * brr[2] + 3 * brr[3]) % 4 == 0
    {
        // Pairing the elements of brr3 and brr1.
        int min_opr = Math.min(brr[3], brr[1]);
        brr[3] -= min_opr;
        brr[1] -= min_opr;
   
        // Pairing the brr2 elements.
        min_opr += brr[2] / 2;
   
        // Assigning the remaining brr2 elements.
        brr[2] %= 2;
   
        // If we are left with one brr2 element.
        if (brr[2] == 1) {
   
            // Here we need only two operations
            // to convert the remaining one
            // brr2 element to convert it.
            min_opr += 2;
   
            // Now there is no brr2 element.
            brr[2] = 0;
   
            // Remaining brr3 elements.
            if (brr[3] == 1)             
                brr[3] -= 2;            
   
            // Remaining brr1 elements.
            if (brr[1]== 1
                brr[1] -= 2;            
        }
   
        // If we are left with brr1 and brr2
        // elements then, we have to take four
        // of them and fixing them all together
        // in 3 operations.
        if (brr[1] == 1)        
            min_opr += (brr[1] / 4) * 3;        
        if (brr[3] == 1)        
            min_opr += (brr[3] / 4) * 3;        
   
        // Returns the minimum operations.
        return min_opr;
    }
   
    // If it is not possible to convert the array.
    return -1;    
}
   
// Driver function
public static void main(String[] args)
{
    int arr[] = { 1, 2, 3, 1, 2, 3, 8 };
    int n = arr.length;
    System.out.println(minimumOperations(arr, n));
}
}
  
// This code is contributed by Prerna Saini.

Python3




# Python program to
# find Minimum number
# of operations to
# convert an array 
# so that arr[i] % 4 is zero.
  
# Function to find
# minimum operations.
def minimumOperations(arr,n):
  
    # Counting of all the elements
    # leaving remainder 1, 2, 3 when
    # divided by 4 in the array brr.
    # at positions 1, 2 and 3 respectively.
    brr = [ 0, 0, 0, 0 ]
    for i in range(n):
        brr[arr[i] % 4]+=1;
   
    # If it is possible to convert the 
    # array so that arr[i] % 4 is zero.
    if ((brr[1] + 2 * brr[2] + 3 * brr[3]) % 4 == 0): 
      
        # Pairing the elements
        # of brr3 and brr1.
        min_opr = min(brr[3], brr[1])
        brr[3] -= min_opr
        brr[1] -= min_opr
   
        # Pairing the brr2 elements.
        min_opr += brr[2] // 2
   
        # Assigning the remaining
        # brr2 elements.
        brr[2] %= 2
   
        # If we are left with
        # one brr2 element.
        if (brr[2]):
   
            # Here we need only two operations
            # to convert the remaining one
            # brr2 element to convert it.
            min_opr += 2
   
            # Now there is no brr2 element.
            brr[2] = 0
   
            # Remaining brr3 elements.
            if (brr[3]):             
                brr[3] -= 2            
   
            # Remaining brr1 elements.
            if (brr[1]): 
                brr[1] -= 2            
          
   
        # If we are left with brr1 and brr2
        # elements then, we have to take four
        # of them and fixing them all together
        # in 3 operations.
        if (brr[1]):        
            min_opr += (brr[1] // 4) * 3        
        if (brr[3]):        
            min_opr += (brr[3] // 4) * 3        
   
        # Returns the minimum operations.
        return min_opr
  
    # If it is not possible to convert the array.
    return -1    
  
# Driver function
  
arr = [ 1, 2, 3, 1, 2, 3, 8 ]
n =len(arr)
  
print(minimumOperations(arr, n))
  
# This code is contributed
# by Anant Agarwal.

C#




// C# program to find Minimum number
// of operations to convert an array 
// so that arr[i] % 4 is zero.
using System;
  
class GFG {
  
    // Function to find minimum operations.
    static int minimumOperations(int []arr, int n)
    
      
        // Counting of all the elements
        // leaving remainder 1, 2, 3 when
        // divided by 4 in the array brr.
        // at positions 1, 2 and 3 respectively.
        int []brr = { 0, 0, 0, 0 };
        for (int i = 0; i < n; i++)
            brr[arr[i] % 4]++;
      
        // If it is possible to convert the 
        // array so that arr[i] % 4 is zero.
        if ((brr[1] + 2 * brr[2] + 3 * brr[3]) % 4 == 0) 
        {
            // Pairing the elements of brr3 and brr1.
            int min_opr = Math.Min(brr[3], brr[1]);
            brr[3] -= min_opr;
            brr[1] -= min_opr;
      
            // Pairing the brr2 elements.
            min_opr += brr[2] / 2;
      
            // Assigning the remaining brr2 elements.
            brr[2] %= 2;
      
            // If we are left with one brr2 element.
            if (brr[2] == 1) {
      
                // Here we need only two operations
                // to convert the remaining one
                // brr2 element to convert it.
                min_opr += 2;
      
                // Now there is no brr2 element.
                brr[2] = 0;
      
                // Remaining brr3 elements.
                if (brr[3] == 1)             
                    brr[3] -= 2;         
      
                // Remaining brr1 elements.
                if (brr[1]== 1) 
                    brr[1] -= 2;         
            }
      
            // If we are left with brr1 and brr2
            // elements then, we have to take four
            // of them and fixing them all together
            // in 3 operations.
            if (brr[1] == 1)     
                min_opr += (brr[1] / 4) * 3;     
            if (brr[3] == 1)     
                min_opr += (brr[3] / 4) * 3;     
      
            // Returns the minimum operations.
            return min_opr;
        }
      
        // If it is not possible to convert the array.
        return -1; 
    }
      
    // Driver function
    public static void Main()
    {
        int []arr = { 1, 2, 3, 1, 2, 3, 8 };
        int n = arr.Length;
        Console.WriteLine(minimumOperations(arr, n));
    }
}
  
// This code is contributed by  vt_m

PHP




<?php
// PHP program to find 
// Minimum number of 
// operations to convert 
// an array so that 
// arr[i] % 4 is zero.
  
// Function to find 
// minimum operations.
function minimumOperations($arr, $n)
    // Counting of all the 
    // elements leaving remainder
    // 1, 2, 3 when divided by 4 
    // in the array brr at positions
    // 1, 2 and 3 respectively.
    $brr = array(0, 0, 0, 0);
    for ($i = 0; $i < $n; $i++)
        $brr[$arr[$i] % 4]++;
  
    // If it is possible to 
    // convert the array so 
    // that arr[i] % 4 is zero.
    if (($brr[1] + 2 * 
         $brr[2] + 3 * 
         $brr[3]) % 4 == 0) 
    {
        // Pairing the elements 
        // of brr3 and brr1.
        $min_opr = min($brr[3], 
                       $brr[1]);
        $brr[3] -= $min_opr;
        $brr[1] -= $min_opr;
  
        // Pairing the
        // brr2 elements.
        $min_opr += $brr[2] / 2;
  
        // Assigning the remaining
        // brr2 elements.
        $brr[2] %= 2;
  
        // If we are left with 
        // one brr2 element.
        if ($brr[2]) 
        {
  
            // Here we need only two 
            // operations to convert 
            // the remaining one brr2
            // element to convert it.
            $min_opr += 2;
  
            // Now there is no 
            // brr2 element.
            $brr[2] = 0;
  
            // Remaining brr3 elements.
            if ($brr[3])             
                $brr[3] -= 2;         
  
            // Remaining brr1 elements.
            if ($brr[1]) 
                $brr[1] -= 2;         
        }
  
        // If we are left with brr1 
        // and brr2 elements then, 
        // we have to take four of 
        // them and fixing them all 
        // together in 3 operations.
        if ($brr[1])     
            $min_opr += ($brr[1] / 4) * 3;     
        if ($brr[3])     
            $min_opr += ($brr[3] / 4) * 3;     
  
        // Returns the
        // minimum operations.
        return $min_opr;
    }
      
    // If it is not possible 
    // to convert the array.
    return -1; 
}
  
// Driver Code
$arr = array(1, 2, 3, 
             1, 2, 3, 8);
$n = count($arr);
echo (minimumOperations($arr, $n));
  
// This code is contributed by 
// Manish Shaw(manishshaw1)
?>


Output:
3

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