Find Xor-Beauty of Array

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Bit Manipulation

You are given a 0-indexed integer array nums.

The effective value of three indices i, j, and k is defined as ((nums[i] | nums[j]) & nums[k]).

The xor-beauty of the array is the XORing of the effective values of all the possible triplets of indices (i, j, k) where 0 <= i, j, k < n.

Return the xor-beauty of nums.

Note that:

val1 | val2 is bitwise OR of val1 and val2.
val1 & val2 is bitwise AND of val1 and val2.

Example 1:

Input: nums = [1,4]
Output: 5
Explanation: 
The triplets and their corresponding effective values are listed below:
- (0,0,0) with effective value ((1 | 1) & 1) = 1
- (0,0,1) with effective value ((1 | 1) & 4) = 0
- (0,1,0) with effective value ((1 | 4) & 1) = 1
- (0,1,1) with effective value ((1 | 4) & 4) = 4
- (1,0,0) with effective value ((4 | 1) & 1) = 1
- (1,0,1) with effective value ((4 | 1) & 4) = 4
- (1,1,0) with effective value ((4 | 4) & 1) = 0
- (1,1,1) with effective value ((4 | 4) & 4) = 4 
Xor-beauty of array will be bitwise XOR of all beauties = 1 ^ 0 ^ 1 ^ 4 ^ 1 ^ 4 ^ 0 ^ 4 = 5.

Example 2:

Input: nums = [15,45,20,2,34,35,5,44,32,30]
Output: 34
Explanation: The xor-beauty of the given array is 34.

Constraints:

1 <= nums.length <= 10⁵
1 <= nums[i] <= 10⁹

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values within the array (min and max) and what data type are they (integers, floats, etc.)?
How large can the input array be?
Can the input array be empty or null?
Are there any duplicate numbers in the array, and if so, how should they be handled in the calculation of the XOR-beauty?
Could you provide a more formal mathematical definition of XOR-beauty? Specifically, can you elaborate on how the XOR operation is applied to subsets?

Brute Force Solution

Approach

To find the Xor-Beauty, we explore every possible combination of selecting elements from the given group of numbers. We then perform a special operation (XOR) on each of these combinations. Finally, we combine the results of these XOR operations to obtain our final 'beauty'.

Here's how the algorithm would work step-by-step:

Consider all the ways you can pick some numbers from the group. You could pick just one number, or two numbers, or even all of them, or none at all.
For each of these 'picks', perform a special operation (XOR) on the numbers you picked. This operation combines the numbers in a specific way to produce a new number.
Now, take all the results you got from the XOR operations in the previous step and perform the XOR operation on all of those results together.
The final result you get after this last XOR operation is the 'Xor-Beauty' of the original group of numbers.

Code Implementation

def find_xor_beauty(numbers):
    xor_sum = 0

    # Iterate through all possible subsets of the input numbers
    for i in range(1 << len(numbers)):
        subset = []
        for j in range(len(numbers)):
            if (i >> j) & 1:
                subset.append(numbers[j])

        # Calculate the XOR of the current subset
        subset_xor = 0
        for number in subset:
            subset_xor ^= number

        # Accumulate the XOR sum
        xor_sum ^= subset_xor

    return xor_sum

Big(O) Analysis

Time Complexity

O(n) – The problem involves considering all possible subsets of the array. Each element can either be included or excluded in a subset. The XOR beauty is the XOR of XOR sums of all subsets. A crucial observation simplifies the calculation: only the elements that appear an odd number of times across all subsets contribute to the final XOR beauty. Each element appears in 2^(n-1) subsets. If n > 1, 2^(n-1) is always even. If any element x exists within an even amount of subsets, it can be expressed as zero. Therefore, the result boils down to XORing all original numbers, because each number is present in half of the subsets. Since each number is iterated only once through the XOR operations, the time complexity scales linearly with the number of elements in the array leading to O(n).

Space Complexity

O(1) – The problem description outlines XOR operations performed on combinations of array elements. A naive implementation might involve generating all possible subsets, but a more efficient solution, based on the properties of XOR, directly computes the XOR sum of the original array. Therefore, the algorithm requires no auxiliary data structures that scale with the input size N. It uses only a constant amount of extra memory to store the XOR sum, regardless of the size of the input array, leading to a space complexity of O(1).

Optimal Solution

Approach

The 'xor-beauty' of an array has a direct relationship with the elements of the array themselves. We can skip all the complex combinations and calculations. Instead, we'll use a clever insight to find the answer instantly.

Here's how the algorithm would work step-by-step:

Realize that the 'xor-beauty' of an array is simply the XOR of all the numbers in the array.
Combine all numbers in the array using the XOR operation. That result is the final answer.

Code Implementation

def find_xor_beauty(array_of_numbers):
    xor_sum_of_array = 0

    # The xor beauty is just the xor of all elements.
    for number in array_of_numbers:

        xor_sum_of_array ^= number

    # Return the XOR sum, which is the xor-beauty.
    return xor_sum_of_array

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the array once, performing a constant-time XOR operation for each element. The number of XOR operations is directly proportional to the number of elements in the input array (n). Therefore, the time complexity is linear with respect to the input size.

Space Complexity

O(1) – The provided solution calculates the XOR-beauty by iterating through the input array and accumulating the XOR of all elements into a single variable. This involves using a single variable to store the running XOR total, irrespective of the size of the input array (N). Therefore, the auxiliary space used is constant and independent of the input size.

Edge Cases

Case	How to Handle
Empty or null array	Return 0 since XOR-beauty is defined as 0 for empty input.
Array with a single element	Return the single element itself because XOR-beauty equals the element for a single-element array.
Array with all elements being 0	The XOR-beauty will be 0, handled correctly by the standard algorithm.
Array with identical elements (other than 0)	The XOR-beauty will be equal to the duplicated value itself, which the algorithm should handle correctly by calculating all possible pairs.
Array with large numbers causing potential integer overflow in XOR calculations.	Use a data type capable of holding the largest possible XOR result (e.g., long in Java/C++ or appropriately sized integer in Python).
Array with a very large size	Ensure that the algorithm has optimal time complexity (ideally O(n) or O(n log n)) to avoid exceeding time limits, and also that the memory usage does not cause the program to crash.
Array containing a mix of positive and negative numbers	XOR operations work correctly with negative numbers represented in two's complement so no specific handling is required.
Array containing only one unique element	The XOR-Beauty is the unique element, and the standard algorithm handles this correctly.

Empty or null array

How to Handle:

Return 0 since XOR-beauty is defined as 0 for empty input.

Array with a single element

How to Handle:

Return the single element itself because XOR-beauty equals the element for a single-element array.

Array with all elements being 0

How to Handle:

The XOR-beauty will be 0, handled correctly by the standard algorithm.

Array with identical elements (other than 0)

How to Handle:

The XOR-beauty will be equal to the duplicated value itself, which the algorithm should handle correctly by calculating all possible pairs.

Array with large numbers causing potential integer overflow in XOR calculations.

How to Handle:

Use a data type capable of holding the largest possible XOR result (e.g., long in Java/C++ or appropriately sized integer in Python).

Array with a very large size

How to Handle:

Ensure that the algorithm has optimal time complexity (ideally O(n) or O(n log n)) to avoid exceeding time limits, and also that the memory usage does not cause the program to crash.

Array containing a mix of positive and negative numbers

How to Handle:

XOR operations work correctly with negative numbers represented in two's complement so no specific handling is required.

Array containing only one unique element

How to Handle:

The XOR-Beauty is the unique element, and the standard algorithm handles this correctly.