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

Recall & Review

beginner

What does the Brian Kernighan algorithm count in a number?

It counts the number of set bits (1s) in the binary representation of the number.

intermediate

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

It repeatedly clears the rightmost set bit, reducing the number of iterations to the number of set bits instead of total bits.

intermediate

Explain the operation n = n & (n - 1) in the Brian Kernighan algorithm.

This operation removes the rightmost set bit from n by turning it to 0, effectively reducing the count of set bits by one.

intermediate

What is the time complexity of the Brian Kernighan algorithm for counting set bits?

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

beginner

Provide a simple Python code snippet using Brian Kernighan algorithm to count set bits.

def count_set_bits(n): count = 0 while n: n &= n - 1 count += 1 return count

What does the expression n & (n - 1) do in the Brian Kernighan 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 n & (n - 1) clears the rightmost set bit in n.

What is the main advantage of Brian Kernighan algorithm over checking each bit individually?

A It counts unset bits instead

B It runs in constant time

C It uses no loops

D It only loops as many times as there are set bits

The algorithm loops only as many times as there are set bits, making it faster for numbers with few set bits.

If n = 12 (binary 1100), how many times will the loop run in Brian Kernighan algorithm?

A 2

B 4

C 3

D 1

12 in binary is 1100, which has 2 set bits, so the loop runs 2 times.

Which bitwise operator is essential in Brian Kernighan algorithm?

A OR (|)

B AND (&)

C NOT (~)

D XOR (^)

The AND operator (&) is used to clear the rightmost set bit.

What will be the output of count_set_bits(0) using Brian Kernighan algorithm?

A 0

B 1

C Undefined

D Error

0 has no set bits, so the count is 0.

Describe how Brian Kernighan algorithm counts set bits in a number.

Think about how the number changes each loop.

Explain why Brian Kernighan algorithm is more efficient than checking each bit one by one.

Focus on the number of loop iterations.

Practice

1. What does the Brian Kernighan algorithm do when counting set bits in a number?

easy

A. It shifts the number to the right until it becomes zero without counting bits.

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

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

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 clears the least significant set bit in each iteration by doing n = n & (n-1).

2. Step 2: Effect of clearing bits

   Each clearing reduces the number of set bits by one until the number becomes zero, counting how many bits were set.

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 rightmost set bit repeatedly to count bits [OK]

Common Mistakes:

• Thinking it counts all digits, not just set bits
• Confusing bit shifting with bit clearing
• Counting characters in string form of number

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

easy

A. n = n & (n - 1)

B. n = n | (n - 1)

C. n = n ^ (n - 1)

D. n = n + (n - 1)

Solution

1. Step 1: Identify the bit clearing operation

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

2. Step 2: Understand why other options are incorrect

   Using OR, XOR, or addition does not clear the least significant set bit correctly.

3. Final Answer:

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

4. Quick Check:

   Bit clear uses AND with n-1 [OK]

Hint: Use AND with n minus one to clear bits [OK]

Common Mistakes:

• Using OR instead of AND
• Using XOR which toggles bits
• Adding numbers instead of bitwise operations

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

def count_set_bits(n):
    count = 0
    while n:
        n = n & (n - 1)
        count += 1
    return count

print(count_set_bits(13))

medium

A. 3

B. 5

C. 4

D. 2

Solution

1. Step 1: Convert 13 to binary

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

2. Step 2: Count set bits using the algorithm

   Each iteration clears one set bit: 1101 -> 1100 -> 1000 -> 0000, total 3 times.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Binary 1101 has 3 set bits [OK]

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

Common Mistakes:

• Counting total bits instead of set bits
• Miscounting binary digits
• Confusing decimal with binary

4. Find the error in the following code that tries to count set bits using Brian Kernighan's algorithm:

def count_set_bits(n):
    count = 0
    while n > 0:
        n = n & (n + 1)
        count += 1
    return count

print(count_set_bits(7))

medium

A. Loop condition should be n != 0

B. Missing initialization of count variable

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

D. Using addition instead of bitwise AND

Solution

1. Step 1: Identify the bit clearing operation

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

2. Step 2: Correct operation for clearing bits

   The correct operation is n = n & (n - 1) to clear the rightmost set bit.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

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

5. Given a list of integers, which code snippet correctly uses Brian Kernighan's algorithm to return a list of counts of set bits for each number?

nums = [3, 5, 7]

A) def count_bits_list(nums):
       return [bin(n).count('1') for n in nums]

B) def count_bits_list(nums):
       def count_set_bits(n):
           count = 0
           while n:
               n = n & (n - 1)
               count += 1
           return count
       return [count_set_bits(n) for n in nums]

C) def count_bits_list(nums):
       return [sum(int(d) for d in str(n)) for n in nums]

D) def count_bits_list(nums):
       return [len(str(n)) for n in nums]

hard

A. Option A uses built-in count of '1's in binary string.

B. Option B uses Brian Kernighan's algorithm correctly.

C. Option C sums digits of decimal string, not bits.

D. Option D counts decimal digits, not bits.

Solution

1. Step 1: Analyze the code snippets

   The first snippet uses Python's bin(n).count('1'), which counts set bits but not Brian Kernighan's algorithm. The second snippet implements the algorithm correctly using a while loop that clears the least significant set bit with n = n & (n - 1). The third sums the decimal digits, which is incorrect for bits. The fourth counts the length of the decimal string, also incorrect.

2. Step 2: Confirm correct implementation

   The second snippet defines count_set_bits using n = n & (n - 1) in a loop and applies it to each number via list comprehension, which is the proper use of Brian Kernighan's algorithm.

3. Final Answer:

   Option B uses Brian Kernighan's algorithm correctly. -> Option B

4. Quick Check:

   Correctly uses n & (n-1) loop [OK]

Hint: Use n & (n - 1) loop inside list comprehension [OK]

Common Mistakes:

• Using string digit sums instead of bits
• Counting decimal digits instead of set bits
• Confusing built-in binary count with algorithm