Find Xor-Beauty of Array

Find Xor-Beauty of Array

1. What is this problem about?

The Find Xor-Beauty of Array interview question defines the "Xor-beauty" as the XOR sum of all possible values (nums[i] AND nums[j]) AND nums[k] for all triples of indices $(i, j, k)$. Given an array of integers, you need to calculate this value. With $N^3$ possible triples, a brute-force approach is impossible for large $N$.

2. Why is this asked in interviews?

Amazon and other companies ask the Find Xor-Beauty coding problem to test a candidate's ability to simplify complex bitwise expressions. It evaluations whether you can use the algebraic properties of XOR, AND, and OR to reduce an $O(N^3)$ problem to $O(N)$. It is a test of Bit Manipulation interview patterns and logical deduction.

3. Algorithmic pattern used

This problem relies on Bitwise Algebraic Simplification.

• The Expression: $\bigoplus_{i,j,k} ((nums[i] ext{ AND } nums[j]) ext{ AND } nums[k])$.
• Commutativity: This is equivalent to $\bigoplus_{i,j,k} (nums[i] ext{ AND } (nums[j] ext{ AND } nums[k]))$.
• The Insight: Due to the symmetry of XOR, any terms where $(i, j)$ are not equal to $(j, i)$ will appear an even number of times and cancel out. The only terms that remain are where the indices provide a unique contribution.
• The Result: The entire expression simplifies to just the XOR sum of the original array: nums[0] ^ nums[1] ^ ... ^ nums[n-1].

4. Example explanation

Array: [1, 4]

• Triples: (1,1,1), (1,1,4), (1,4,1), (1,4,4), (4,1,1), (4,1,4), (4,4,1), (4,4,4).
• Look at Bit 0 (values are 1 and 0):
• (1 AND 1) AND 1 = 1.
• All other triples involving bit 0 will have at least one 0, resulting in 0.
• Only (1,1,1) contributes to bit 0.
• Result: $1 \oplus 4 = 5$.

5. Common mistakes candidates make

• Simulation: Trying to actually compute $N^3$ values.
• Over-counting bits: Attempting to count bit frequencies for triples, which is still $O(N^3)$ or $O(N^2)$ if not careful.
• Lack of Proof: Failing to explain why the expression simplifies, even if they guess the correct answer.

6. Interview preparation tip

XOR properties are your best friend. Remember that $A \oplus A = 0$. If an expression is symmetric and you are XORing all permutations, the "asymmetric" parts almost always cancel out. This is a recurring theme in Math interview patterns.