Bitwise AND, OR, XOR in C - Time & Space Complexity

Time Complexity: Bitwise AND, OR, XOR

O(n)

Understanding Time Complexity

We want to understand how the time it takes to do bitwise AND, OR, and XOR changes as the size of the numbers grows.

How does the number of steps grow when we work with bigger numbers?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

unsigned int bitwise_operations(unsigned int a, unsigned int b) {
    unsigned int and_result = a & b;
    unsigned int or_result = a | b;
    unsigned int xor_result = a ^ b;
    return and_result + or_result + xor_result;
}
    

This code performs bitwise AND, OR, and XOR on two unsigned integers and returns the sum of the results.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Bitwise operations on each bit of the input numbers.
• How many times: Once per bit in the numbers (usually fixed, like 32 or 64 bits).

How Execution Grows With Input

Each bit in the input numbers is processed once. If the number size grows, the steps grow linearly with the number of bits.

Input Size (bits)	Approx. Operations
8	8 steps
32	32 steps
64	64 steps

Pattern observation: The number of steps grows directly with the number of bits in the input numbers.

Final Time Complexity

Time Complexity: O(n)

This means the time to do bitwise AND, OR, and XOR grows linearly with the number of bits in the input numbers.

Common Mistake

[X] Wrong: "Bitwise operations are instant and take the same time no matter the number size."

[OK] Correct: While bitwise operations are very fast, they still process each bit. Larger numbers with more bits take proportionally more steps.

Interview Connect

Understanding how bitwise operations scale helps you reason about low-level code efficiency and is a useful skill in many programming tasks.

Self-Check

"What if we used bitwise operations on arrays of numbers instead of single numbers? How would the time complexity change?"