Count Triplets with Even XOR Set Bits II

Count Triplets with Even XOR Set Bits II

What is this problem about?

This is the "Medium" version of the previous problem. In the Count Triplets with Even XOR Set Bits II coding problem, the arrays can be much larger, making an $O(N)$ parity counting approach mandatory. The goal remains the same: count triplets across three arrays where the XOR sum has an even number of 1s in binary.

Why is this asked in interviews?

Amazon uses this version to ensure the candidate has found the optimal $O(N)$ solution rather than just a "cleaner" $O(N^3)$ solution. It evaluations the ability to apply combinatorics to bitwise properties. It tests if you can scale your logic to handle high-volume data streams where only the statistical properties (like parity distribution) matter.

Algorithmic pattern used

This uses Frequency Counting and Combinatorial Parity.

1. Pass 1: Iterate through arrays a, b, and c once.
2. For each number, calculate its parity (number of set bits modulo 2).
3. Increment the respective even_count or odd_count for each array.
4. Pass 2: Use the combinatorial formula for an even XOR result:
   • $EvenSum = E_a E_b E_c + E_a O_b O_c + O_a E_b O_c + O_a O_b E_c$.
   • This formula covers all ways to have an even number of "odd" parities (0 or 2).

Example explanation

a = [1, 2], b = [3, 4], c = [5]

• Array a: 1(O), 2(O) $o E=0, O=2$
• Array b: 3(E), 4(O) $o E=1, O=1$
• Array c: 5(E) $o E=1, O=0$ Formula:
• $(0 imes 1 imes 1) + (0 imes 1 imes 0) + (2 imes 1 imes 0) + (2 imes 1 imes 1) = 0 + 0 + 0 + 2 = 2$. Triplets are (1, 4, 5) and (2, 4, 5).

Common mistakes candidates make

• Scaling: Not realizing the problem is essentially the same as the Easy version but with stricter time limits, thus failing to apply the pre-counting optimization.
• Bit Counting Complexity: Using an inefficient bit-counting method inside the loop. In most languages, Integer.bitCount() or bin(x).count('1') is preferred.
• Handling large results: Forgetting to return the count as a long integer.

Interview preparation tip

Master the "Parity of sum" and "Parity of XOR" rules. Sum is even if: (Even+Even) or (Odd+Odd). XOR set bits are even if: (ParityX == ParityY). These simple boolean logic rules are the foundation for optimizing many complex bitwise problems.