Count Set Bits Brian Kernighan Algorithm in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the Brian Kernighan algorithm do in the context of counting set bits?

It counts the number of 1s (set bits) in the binary representation of a number by repeatedly turning off the rightmost set bit until the number becomes zero.

intermediate

How does the Brian Kernighan algorithm reduce the number of operations compared to checking each bit individually?

It only loops as many times as there are set bits, not the total number of bits, by removing the rightmost set bit in each iteration.

beginner

What is the key operation used in Brian Kernighan's algorithm to remove the rightmost set bit?

The operation is: n = n & (n - 1), which clears the rightmost set bit of n.

advanced

Why does the expression n & (n - 1) remove the rightmost set bit?

Because subtracting 1 flips all bits after the rightmost set bit, and ANDing with n clears that rightmost set bit while keeping others unchanged.

beginner

Write the time complexity of the Brian Kernighan algorithm for counting set bits.

The time complexity is O(k), where k is the number of set bits in the number.

What does the Brian Kernighan algorithm do to count set bits?

A Uses a lookup table for each byte

B Checks each bit from left to right

C Repeatedly clears the rightmost set bit until the number is zero

D Converts the number to a string and counts '1's

The algorithm repeatedly clears the rightmost set bit using n = n & (n - 1) until the number becomes zero.

Which bitwise operation is central to Brian Kernighan's algorithm?

A n | (n - 1)

B n & (n - 1)

C n ^ (n - 1)

D n << 1

The operation n & (n - 1) removes the rightmost set bit, which is the key step in the algorithm.

If a number has 5 set bits, how many times will Brian Kernighan's loop run?

A 5 times

B Once

C Equal to the total number of bits in the number

D Depends on the number's value

The loop runs exactly as many times as there are set bits, so 5 times in this case.

What is the time complexity of Brian Kernighan's algorithm?

A O(k) where k is number of set bits

B O(n) where n is total bits

C O(1)

D O(log n)

The algorithm runs in O(k) time, where k is the number of set bits, making it efficient for sparse bit sets.

Why is Brian Kernighan's algorithm more efficient than checking each bit?

A It converts number to decimal

B It uses recursion

C It uses multiplication

D It skips unset bits by removing set bits directly

It directly removes set bits, so it only loops as many times as there are set bits, skipping unset bits.

Explain how Brian Kernighan's algorithm counts the number of set bits in a number.

Think about how the rightmost 1 bit is cleared each time.

Describe why Brian Kernighan's algorithm is more efficient than checking each bit individually.

Focus on how the algorithm reduces the number of loop iterations.

Practice

1. What does the Brian Kernighan algorithm do when counting set bits in an integer?

easy

A. It converts the number to a string and counts characters.

B. It counts the total number of digits in the integer.

C. It shifts the bits to the right one by one and counts zeros.

D. It repeatedly clears the least significant set bit until the number becomes zero.

Solution

1. Step 1: Understand the algorithm's main operation

   The Brian Kernighan algorithm works by clearing the least significant set bit in each iteration.

2. Step 2: Effect of clearing bits on counting set bits

   Each time it clears a set bit, it increments the count until the number becomes zero.

3. Final Answer:

   It repeatedly clears the least significant set bit until the number becomes zero. -> Option D

4. Quick Check:

   Clearing bits counts set bits [OK]

Hint: Clear least set bit repeatedly to count ones [OK]

Common Mistakes:

• Counting zeros instead of ones
• Using string conversion to count bits
• Shifting bits without clearing

2. Which of the following is the correct syntax to clear the least significant set bit in C# using Brian Kernighan's method?

easy

A. n = n & (n - 1);

B. n = n | (n - 1);

C. n = n ^ (n + 1);

D. n = n << 1;

Solution

1. Step 1: Identify the bit clearing operation

   The expression n = n & (n - 1) clears the least significant set bit in n.

2. Step 2: Verify other options

   Other options perform different bitwise operations that do not clear the least significant set bit.

3. Final Answer:

   n = n & (n - 1); -> Option A

4. Quick Check:

   Use n & (n-1) to clear least set bit [OK]

Hint: Use n & (n-1) to clear least set bit [OK]

Common Mistakes:

• Using OR instead of AND
• Using XOR with n+1 incorrectly
• Using left shift which moves bits

3. What is the output of the following C# code snippet?

int countSetBits(int n) {
    int count = 0;
    while (n > 0) {
        n = n & (n - 1);
        count++;
    }
    return count;
}

Console.WriteLine(countSetBits(13));

medium

A. 3

B. 4

C. 2

D. 5

Solution

1. Step 1: Convert 13 to binary and count set bits

   13 in binary is 1101, which has three set bits (1s).

2. Step 2: Apply Brian Kernighan's algorithm

   Each iteration clears one set bit, so the loop runs 3 times, incrementing count to 3.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Binary 1101 has 3 ones [OK]

Hint: Count 1s in binary of number [OK]

Common Mistakes:

• Counting total bits instead of set bits
• Misunderstanding bitwise AND operation
• Off-by-one errors in loop

4. Identify the error in the following C# code that attempts to count set bits using Brian Kernighan's algorithm:

int countSetBits(int n) {
    int count = 0;
    while (n != 0) {
        n = n & (n + 1);
        count++;
    }
    return count;
}

medium

A. Missing return statement

B. Using n & (n + 1) instead of n & (n - 1)

C. Count variable not initialized

D. Incorrect loop condition; should be n > 0

Solution

1. Step 1: Check the bit clearing expression

   The code uses n & (n + 1), which does not clear the least significant set bit correctly.

2. Step 2: Correct expression for clearing bits

   The correct expression is n & (n - 1) to clear the least significant set bit.

3. Final Answer:

   Using n & (n + 1) instead of n & (n - 1) -> Option B

4. Quick Check:

   Use n & (n-1) to clear bits [OK]

Hint: Always use n & (n-1) to clear least set bit [OK]

Common Mistakes:

• Using n & (n + 1) which is incorrect
• Confusing loop conditions
• Forgetting to initialize count

5. You want to count set bits in a large list of integers efficiently using Brian Kernighan's algorithm in C#. Which approach is best to optimize performance?

hard

A. Convert each integer to binary string and count '1' characters sequentially.

B. Use a simple for loop shifting bits right and counting set bits sequentially.

C. Use Brian Kernighan's algorithm inside a parallel loop to count bits for each integer concurrently.

D. Store all integers in a list and count set bits only for the first element.

Solution

1. Step 1: Understand the need for efficiency

   Counting set bits for many integers can be slow if done sequentially.

2. Step 2: Use parallel processing with Brian Kernighan's algorithm

   Running the efficient bit counting algorithm concurrently on multiple integers speeds up processing.

3. Final Answer:

   Use Brian Kernighan's algorithm inside a parallel loop to count bits for each integer concurrently. -> Option C

4. Quick Check:

   Parallel + efficient algorithm = best performance [OK]

Hint: Combine parallelism with Brian Kernighan's method [OK]

Common Mistakes:

• Using string conversion which is slow
• Sequential bit shifting which is less efficient
• Ignoring concurrency for large data