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 bits?

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

intermediate

Explain the key operation n = n & (n - 1) in Brian Kernighan's Algorithm.

This operation removes the rightmost set bit from n. It works because subtracting 1 flips all bits after the rightmost set bit, and ANDing with n clears that bit.

intermediate

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

Because it only loops as many times as there are set bits, not the total number of bits, making it faster especially for numbers with few set bits.

beginner

What is the time complexity of Brian Kernighan's Algorithm for counting set bits?

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

beginner

Show the result of applying Brian Kernighan's Algorithm on the number 13 (binary 1101). How many set bits does it have?

13 in binary is 1101. The algorithm removes set bits one by one: 1101 -> 1100 -> 1000 -> 0000. It counts 3 set bits.

What does the expression n & (n - 1) do in Brian Kernighan's Algorithm?

A Flips all bits of n

B Adds 1 to n

C Removes the rightmost set bit from n

D Checks if n is even

The expression clears the rightmost set bit, which is the core step in Brian Kernighan's Algorithm.

What is the main advantage of Brian Kernighan's Algorithm over checking each bit one by one?

A It runs in time proportional to the number of set bits

B It uses less memory

C It works only for even numbers

D It counts unset bits instead

The algorithm loops only as many times as there are set bits, making it faster for sparse bit patterns.

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

A 5

B Number of total bits

C 1

D Depends on the number's size

The loop runs once for each set bit, so 5 set bits mean 5 iterations.

What is the output of Brian Kernighan's Algorithm for the number 0?

A 1

B 0

C Undefined

D Depends on implementation

Zero has no set bits, so the count is zero.

Which of these is NOT true about Brian Kernighan's Algorithm?

A It counts set bits efficiently

B It modifies the input number during counting

C It uses bitwise AND operation

D It counts unset bits

The algorithm counts set bits, not unset bits.

Describe how Brian Kernighan's Algorithm counts the number of set bits in a number.

Think about how the algorithm changes the number each step.

Explain why Brian Kernighan's Algorithm is more efficient than checking each bit individually.

Consider the number of times the loop runs compared to total bits.

Practice

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

easy

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

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

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

D. It converts the number to a string and counts '1' characters.

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

   Each clearing reduces the number of set bits by one until the number becomes zero.

3. Final Answer:

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

4. Quick Check:

   Clearing bits one by one counts set bits [OK]

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

Common Mistakes:

• Counting all bits instead of only set bits
• Using string conversion unnecessarily
• Shifting bits without checking if set

2. Which of the following is the correct syntax for the main loop in Brian Kernighan's algorithm in C++?

easy

A. while (n > 0) { count += n % 2; n /= 2; }

B. for (int i = 0; i < n; i++) { count += n & 1; n >>= 1; }

C. while (n != 0) { n = n & (n - 1); count++; }

D. for (int i = 0; i < 32; i++) { count += (n >> i) & 1; }

Solution

1. Step 1: Identify Brian Kernighan's loop

   The algorithm uses while (n != 0) and clears the least significant set bit by n = n & (n - 1).

2. Step 2: Confirm increment of count

   Each iteration increments count by one for each cleared set bit.

3. Final Answer:

   while (n != 0) { n = n & (n - 1); count++; } -> Option C

4. Quick Check:

   Loop clears bits and counts [OK]

Hint: Use while with n = n & (n-1) to clear bits [OK]

Common Mistakes:

• Using for loop instead of while
• Counting bits by shifting instead of clearing
• Using modulo instead of bitwise operations

3. What is the output of the following C++ code snippet using Brian Kernighan's algorithm?

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

int main() {
  int x = 13;
  std::cout << countSetBits(x);
  return 0;
}

medium

A. 5

B. 3

C. 2

D. 4

Solution

1. Step 1: Convert 13 to binary

   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 count increments 3 times before n becomes 0.

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Number of 1s in 1101 = 3 [OK]

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

Common Mistakes:

• Counting total bits instead of set bits
• Miscounting binary 1s
• Confusing bitwise AND with OR

4. Identify the error in this Brian Kernighan algorithm implementation:

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

medium

A. Missing return statement.

B. Not initializing count to zero.

C. Using while (n > 0) instead of while (n != 0).

D. Using n + 1 instead of n - 1 in bit clearing step.

Solution

1. Step 1: Check bit clearing operation

   The correct operation to clear the least significant set bit is n = n & (n - 1), not n + 1.

2. Step 2: Effect of wrong operation

   Using n + 1 does not clear bits properly and causes incorrect count or infinite loop.

3. Final Answer:

   Using n + 1 instead of n - 1 in bit clearing step. -> Option D

4. Quick Check:

   Correct bit clearing uses n & (n - 1) [OK]

Hint: Use n & (n - 1) to clear bits, not n + 1 [OK]

Common Mistakes:

• Replacing n - 1 with n + 1
• Confusing loop condition syntax
• Forgetting to increment count

5. Given an array of integers, how can you use Brian Kernighan's algorithm to find the total number of set bits in all elements? Choose the best approach.

hard

A. Iterate over each element, apply Brian Kernighan's algorithm to count set bits, and sum all counts.

B. Convert all integers to strings and count '1's in each string, then sum.

C. Use a single Brian Kernighan's algorithm call on the sum of all integers.

D. Count set bits only in the first element and multiply by array size.

Solution

1. Step 1: Understand counting set bits per element

   Brian Kernighan's algorithm counts set bits in one integer at a time.

2. Step 2: Apply algorithm to each array element

   Iterate through the array, count set bits for each element, and sum these counts for total.

3. Final Answer:

   Iterate over each element, apply Brian Kernighan's algorithm to count set bits, and sum all counts. -> Option A

4. Quick Check:

   Sum of individual counts gives total set bits [OK]

Hint: Count bits per element, then sum for total [OK]

Common Mistakes:

• Counting bits of sum instead of each element
• Using string conversion unnecessarily
• Assuming uniform bit count per element