Neighboring Bitwise XOR

Neighboring Bitwise XOR

What is this problem about?

The Neighboring Bitwise XOR problem gives you a binary array derived where derived[i] = original[i] XOR original[(i+1) % n]. Given this derived array, determine whether a valid original binary array could have produced it. This Neighboring Bitwise XOR coding problem tests circular XOR constraint reasoning and bit manipulation.

Why is this asked in interviews?

Amazon and Google ask this to test the ability to reason about circular XOR constraints mathematically, rather than by simulation. The key insight — that the XOR of all derived values must equal 0 for a valid original to exist — requires understanding XOR properties deeply. The array and bit manipulation interview pattern is directly demonstrated.

Algorithmic pattern used

XOR parity check. If we XOR all values in derived, each original[i] appears exactly twice in the expansion (once as derived[i-1]'s second term and once as derived[i]'s first term). Since x XOR x = 0, the XOR of all derived values equals 0 if and only if a valid original exists. Simply return XOR(derived) == 0.

Example explanation

derived = [1, 1, 0]. XOR all: 1^1^0 = 0 → true (valid original exists). Verification: if original[0]=0, then original[1] = 0^1=1, original[2]=1^1=0. Check: derived[2]=0^0=0 ✓.

derived = [1, 1]. XOR: 1^1 = 0 → true. derived = [1, 0]. XOR: 1^0 = 1 ≠ 0 → false.

Common mistakes candidates make

• Trying all 2 possible starting values (0 and 1) for original[0] and simulating (works but O(n) extra space and misses the elegant O(1) insight).
• Not recognizing the circular nature (the last derived value involves original[n-1] and original[0]).
• Incorrect XOR expansion — thinking each original appears once instead of twice.
• Returning false for the trivial case of a single-element array.

Interview preparation tip

Circular XOR problems often have elegant parity arguments. When a problem defines a derived sequence as XOR of consecutive pairs in a circular array, the XOR of the entire derived sequence collapses to 0 (if valid). Practice similar XOR parity problems — "does a binary array of operations have a consistent solution?" — to build intuition for when XOR constraints can be verified in O(1) using global parity rather than simulation.