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Minimum operations to make XOR of array zero

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  • Difficulty Level : Hard
  • Last Updated : 19 Jul, 2022
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We are given an array of n elements. The task is to make XOR of whole array 0. We can do the following to achieve this.  

  1. We can select any one of the elements.
  2. After selecting an element, we can either increment or decrement it by 1.

We need to find the minimum number of increment/decrement operations required for the selected element to make the XOR sum of the whole array zero.

Examples: 

Input : arr[] = {2, 4, 8}
Output : Element = 8, 
         Operation required = 2
Explanation : Select 8 as element and perform 2 
              time decrement on it. So that it
              became 6, Now our array is {2, 4, 6} 
              whose XOR sum is 0.

Input : arr[] = {1, 1, 1, 1}
Output : Element = 1, 
         Operation required = 0
Explanation : Select any of 1 and you have already
              your XOR sum = 0. So, no operation 
              required.

Naive Approach: Select an element and then find the XOR of the rest of the array. If that element became equals to XOR obtained then our XOR of the whole array should become zero. Now, our cost for that will be the absolute difference between the selected element and obtained XOR. This process of finding cost will be done for each element and thus resulting in Time Complexity of (n^2).

Efficient Approach: Find the XOR of the whole array. Now, suppose we have selected element arr[i], so cost required for that element will be absolute(arr[i]-(XORsum^arr[i])). Calculating the minimum of these absolute values for each element will be our minimum required operation also the element corresponding to the minimum required operation will be our selected element. 

Implementation:

C++




// CPP to find min cost to make
// XOR of whole array zero
#include <bits/stdc++.h>
using namespace std;
 
// function to find min cost
void minCost(int arr[], int n)
{
    int cost = INT_MAX;
    int element;
 
    // calculate XOR sum of array
    int XOR = 0;
    for (int i = 0; i < n; i++)
        XOR ^= arr[i];
 
    // find the min cost and element corresponding
    for (int i = 0; i < n; i++) {
        if (cost > abs((XOR ^ arr[i]) - arr[i])) {
            cost = abs((XOR ^ arr[i]) - arr[i]);
            element = arr[i];
        }
    }
 
    cout << "Element = " << element << endl;
    cout << "Operation required = " << abs(cost);
}
 
// driver program
int main()
{
    int arr[] = { 2, 8, 4, 16 };
    int n = sizeof(arr) / sizeof(arr[0]);
    minCost(arr, n);
    return 0;
}

Java




// JAVA program to find min cost to make
// XOR of whole array zero
import java.lang.*;
 
class GFG
{
    // function to find min cost
    static void minCost(int[] arr, int n)
    {
        int cost = Integer.MAX_VALUE;
        int element=0;
 
        // calculate XOR sum of array
        int XOR = 0;
        for (int i = 0; i < n; i++)
            XOR ^= arr[i];
 
        // find the min cost and element
        // corresponding
        for (int i = 0; i < n; i++) {
            if (cost > Math.abs((XOR ^ arr[i])
                                - arr[i])) {
                cost = Math.abs((XOR ^ arr[i]) -
                                       arr[i]);
                element = arr[i];
            }
        }
 
    System.out.println("Element = " + element);
    System.out.println("Operation required = "+
                             Math.abs(cost));
    }
 
    // driver program
    public static void main (String[] args)
    {
        int[] arr = { 2, 8, 4, 16 };
        int n = arr.length;
        minCost(arr, n);
    }
}
/* This code is contributed by Kriti Shukla */

Python3




# python to find min cost to make
# XOR of whole array zero
 
# function to find min cost
def minCost(arr,n):
     
    cost = 999999;
     
    # calculate XOR sum of array
    XOR = 0;
    for i in range(0, n):
        XOR ^= arr[i];
 
    # find the min cost and element
    # corresponding
    for i in range(0,n):
        if (cost > abs((XOR ^ arr[i]) - arr[i])):
            cost = abs((XOR ^ arr[i]) - arr[i])
            element = arr[i]
 
    print("Element = ", element)
    print("Operation required = ", abs(cost))
 
 
# driver program
arr = [ 2, 8, 4, 16 ]
n = len(arr)
minCost(arr, n)
 
# This code is contributed by Sam007

C#




// C# program to find min cost to
// make XOR of whole array zero
using System;
 
class GFG
{
    // function to find min cost
    static void minCost(int []arr, int n)
    {
        int cost = int.MaxValue;
        int element=0;
 
        // calculate XOR sum of array
        int XOR = 0;
        for (int i = 0; i < n; i++)
            XOR ^= arr[i];
 
        // find the min cost and
        // element corresponding
        for (int i = 0; i < n; i++)
        {
            if (cost > Math.Abs((XOR ^ arr[i]) - arr[i]))
            {
                cost = Math.Abs((XOR ^ arr[i]) - arr[i]);
                element = arr[i];
            }
        }
 
    Console.WriteLine("Element = " + element);
    Console.Write("Operation required = "+
                          Math.Abs(cost));
    }
 
    // Driver program
    public static void Main ()
    {
        int []arr = {2, 8, 4, 16};
        int n = arr.Length;
        minCost(arr, n);
    }
}
 
// This code is contributed by nitin mittal.

PHP




<?php
// PHP to find min cost to make
// XOR of whole array zero
 
// function to find min cost
function minCost($arr, $n)
{
    $cost = PHP_INT_MAX;
    $element;
 
    // calculate XOR sum of array
    $XOR = 0;
    for ($i = 0; $i < $n; $i++)
        $XOR ^= $arr[$i];
 
    // find the min cost and
    // element corresponding
    for ($i = 0; $i < $n; $i++)
    {
        if ($cost > abs(($XOR ^ $arr[$i]) -
                                 $arr[$i]))
        {
            $cost = abs(($XOR ^ $arr[$i]) -
                                 $arr[$i]);
            $element = $arr[$i];
        }
    }
 
    echo "Element = " , $element ,"\n";
    echo "Operation required = " , abs($cost);
}
 
// Driver Code
$arr = array(2, 8, 4, 16) ;
$n = count($arr);
minCost($arr, $n);
 
// This code is contributed by vt_m.
?>

Javascript




<script>
 
// javascript to find min cost to make
// XOR of whole array zero
 
// function to find min cost
function minCost(arr, n)
{
    var cost = 1000000000;
    var element;
 
    // calculate XOR sum of array
    var XOR = 0;
    for (var i = 0; i < n; i++)
        XOR ^= arr[i];
 
    // find the min cost and element corresponding
    for (var i = 0; i < n; i++) {
        var x= Math.abs((XOR ^ arr[i]) - arr[i])
        if (cost > x) {
            cost = x;
            element = arr[i];
        }
    }
 
    document.write( "Element = " + element + "<br>");
    document.write( "Operation required = " + Math.abs(cost));
}
 
// driver program
var arr = [ 2, 8, 4, 16 ];
var n = arr.length;
minCost(arr, n);
 
</script>

Output

Element = 16
Operation required = 2

Time Complexity : O(n)

This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. 


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