Minimize XOR of Pairs to Make the Array Palindrome Using C++

In the field of computer science, solving optimization problems efficiently is crucial for developing optimal algorithms and systems. One such problem is minimizing the XOR (exclusive OR) of pairs in an array to make it a palindrome. This situation holds significance because it offers a chance to determine the optimal approach towards reordering items within an array, which can lead to lower XOR value and creation of palindromes. This essay examines two methods that utilize C++ programming language for resolving this predicament.

Syntax

To begin with, let's define the syntax of the function we will be using in the following code examples −

int minimizeXORToPalindrome(int arr[], int n);

Algorithm

The algorithm we will be using aims to minimize the XOR of pairs in the array to transform it into a palindrome. Here are the general steps −

• Sort the array in non-decreasing order.

• Initialize two pointers, left and right, pointing to the first and last elements of the array, respectively.

• Initialize a variable, xorSum, to store the XOR sum of pairs.

• While left is less than or equal to right, do the following −

• Calculate the XOR of the elements at left and right, and add it to xorSum.

• Move left one step forward and right one step backward.

• Return xorSum.

Approach 1: Reverse and XOR

The first approach involves reversing the array and XORing the corresponding elements to minimize the XOR value.

Example

#include <algorithm>
#include <iostream>

int minimizeXORToPalindrome(int arr[], int n) {
   std::sort(arr, arr + n);  // Step 1: Sort the array
   int xorSum = 0;
   int left = 0, right = n - 1;
    
   while (left <= right) {
      xorSum += arr[left] ^ arr[right];  // Step 4: XOR the elements
      left++;
      right--;
   }
    
   return xorSum;  // Step 5: Return the XOR sum
}

int main() {
   int arr[] = {5, 3, 8, 6, 2};
   int n = sizeof(arr) / sizeof(arr[0]);
    
   int result = minimizeXORToPalindrome(arr, n);
   std::cout << "Minimized XOR value to make the array palindrome: " << result << std::endl;
    
   return 0;
}

Output

Minimized XOR value to make the array palindrome: 15

Explanation

The minimizeXORToPalindrome function takes an array (arr) and its size (n) as input.

Inside the function, we first sort the array in non-decreasing order using std::sort(arr, arr + n). This is step 1 of the algorithm.

Initially, let us set variable xorSum equal to zero since its function would be storing XOR sum of pairs later on.

Additionally, two new variables named left and right are defined such that they respectively point towards the first and last element in our input array.

Finally, as long as an input condition states that left should be lesser than or equal to right; it keeps executing iteratively using a while statement within its body block.

Inside the loop, we calculate the XOR of the elements at left and right using the ^ operator and add it to xorSum. This is step 4 of the algorithm.

We increment left by one and decrement right by one to move closer towards the center of the array.

Once left becomes greater than right, the while loop exits.

Finally, we return the xorSum, which represents the minimized XOR value to make the array a palindrome.

In the main function, we provide an example usage of the minimizeXORToPalindrome function −

• We create an array arr with values {5, 3, 8, 6, 2}.

• We calculate the size of the array (n) using the sizeof operator.

• We call the minimizeXORToPalindrome function, passing the array and its size as arguments, and store the result in the result variable.

• Finally, we print the minimized XOR value to make the array a palindrome.

Approach 2: Efficient Pairing

The second approach focuses on pairing the elements in an efficient way to minimize the XOR value.

Example

#include <algorithm>
#include <iostream>  // Include the <iostream> header for std::cout and std::endl
#include <ostream>   // Include the <ostream> header for std::endl

int minimizeXORToPalindrome(int arr[], int n) {
   std::sort(arr, arr + n);  // Step 1: Sort the array
   int xorSum = 0;
    
   for (int i = 0; i < n / 2; i++) {
      xorSum += arr[i] ^ arr[n - 1 - i];  // Step 4: XOR the elements
   }
    
   return xorSum;  // Step 5: Return the XOR sum
}

int main() {
   int arr[] = {5, 3, 8, 6, 2};
   int n = sizeof(arr) / sizeof(arr[0]);
    
   int result = minimizeXORToPalindrome(arr, n);
   std::cout << "Minimized XOR value to make the array palindrome: " << result << std::endl;
    
   return 0;
}

Output

Minimized XOR value to make the array palindrome: 15

Explanation

The minimizeXORToPalindrome function takes an array (arr) and its size (n) as input.

Inside the function, we first sort the array in non-decreasing order using std::sort(arr, arr + n). This is step 1 of the algorithm.

We commence by setting up our xorSum variable and assigning a value of zero. It is worth noting that this variable will be crucial in storing all our XOR sum pairs.

Next up is utilizing a for loop that cycles through half of our total array size -from index 0 up until n/2-1.

Within this loop lies Step four where we proceed with calculating XOR values between arr[i] (ith element from start) and arr[n-1-i] (ith value starting from endpoint).

We add this XOR value to xorSum.

Once the loop finishes, we have efficiently paired all the elements to minimize the XOR value.

Finally, we return the xorSum, which represents the minimized XOR value to make the array a palindrome.

Our presented code exhibits effective utilization of minimizeXORToPalindrome functionality akin to prior examples displayed in previous iterations of this feature set.

Our process begins by instigating creation of an array in conjunction with size calculation encompassed within main method operations accentuated further by invocation via minimizseXORTOPalindrome.

We conclude with feedback conveying most efficient negotiated XOR level necessary for converting original set elements into palindromically structured configurations

Conclusion

In this article, we explored two approaches to minimize the XOR of pairs in an array and transform it into a palindrome. By employing these approaches in C++, programmers can efficiently reorder the elements in the array, reducing the XOR value and creating a palindrome. This optimization technique is valuable in various domains, such as data manipulation and algorithm design, allowing for more efficient solutions to related problems.