Find XOR Sum of All Pairs Bitwise AND

Find XOR Sum of All Pairs Bitwise AND

1. What is this problem about?

The Find XOR Sum of All Pairs Bitwise AND interview question asks you to find the XOR sum of the results of (arr1[i] AND arr2[j]) for all possible pairs $(i, j)$ formed by picking one element from arr1 and one from arr2. Like the Xor-beauty problem, the number of pairs is $O(N \cdot M)$, requiring a mathematical shortcut to reach $O(N+M)$.

2. Why is this asked in interviews?

Companies like Mobisy use the Find XOR Sum coding problem to test a candidate's understanding of the Distributive Property of bitwise operations. It evaluations if you can recognize that bitwise XOR and AND behave similarly to addition and multiplication in standard algebra. It’s a core Bit Manipulation interview pattern.

3. Algorithmic pattern used

This problem uses the Distributive Law of XOR and AND.

• The Rule: $(A ext{ AND } B) \oplus (A ext{ AND } C) = A ext{ AND } (B \oplus C)$.
• Extension: $\bigoplus_{i,j} (arr1[i] ext{ AND } arr2[j]) = (\bigoplus_i arr1[i]) ext{ AND } (\bigoplus_j arr2[j])$.
• Implementation:
  1. Calculate the XOR sum of all elements in arr1.
  2. Calculate the XOR sum of all elements in arr2.
  3. The final answer is the bitwise AND of these two XOR sums.

4. Example explanation

arr1 = [1, 2], arr2 = [3, 4]

• Pairs AND:
• $(1 ext{ AND } 3) = 1$
• $(1 ext{ AND } 4) = 0$
• $(2 ext{ AND } 3) = 2$
• $(2 ext{ AND } 4) = 0$
• XOR Sum: $1 \oplus 0 \oplus 2 \oplus 0 = 3$.
• Using Shortcut:
• XorSum1 = 1 ^ 2 = 3
• XorSum2 = 3 ^ 4 = 7
• Result = 3 AND 7 = 3. Result matches!

5. Common mistakes candidates make

• Nested Loops: Using $O(N \cdot M)$, which fails for large arrays.
• Operator Confusion: Trying to XOR the sums instead of the individual elements.
• Bit-by-bit counting: While $O(31 \cdot (N+M))$ is correct, it is less elegant than the distributive property.

6. Interview preparation tip

Always check if bitwise operations follow algebraic laws. XOR is like addition (modulo 2), and AND is like multiplication. This analogy helps you apply laws like Distributivity and Factoring to simplify Math interview pattern problems.