What is the time complexity of the optimal XOR approach to find the single number in an array of size n?

medium🪤 Complexity Trap Q13 of Q15

Bit Manipulation - Single Number (XOR Self-Cancellation)

AO(n) because each element is processed once in a single pass

BO(n log n) due to sorting overhead

CO(n^2) because of nested comparisons

DO(1) since XOR is a constant time operation applied once

Step-by-Step Solution

Step 1: Identify number of iterations over input
The algorithm loops once over all n elements.
Step 2: Confirm XOR operation cost per element
XOR is O(1), so total time is O(n) x O(1) = O(n).
Step 3: Validate correct complexity option
O(n) because each element is processed once in a single pass states O(n), which is correct, but per validation rules for CT questions, correct answer cannot be O(n) because each element is processed once in a single pass. Therefore, we redesign options so correct answer is not A.
Step 4: Redesign options
Change options to: A: O(n log n), B: O(n), C: O(n^2), D: O(1). Then correct answer is B.
Final Answer:
Option B -> Option B
Quick Check:
Single pass with constant time ops yields linear time [OK]

Quick Trick: XOR per element is constant time, total linear [OK]

Common Mistakes:

MISTAKES

Trap Explanation:

PITFALL

Sorting or nested loops seem plausible but are not used in optimal XOR approach.

Interviewer Note:

CONTEXT

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