Count Set Bits Brian Kernighan Algorithm in DSA C - 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 clears the rightmost set bit of 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 patterns.

beginner

Write the basic C code snippet for counting set bits using Brian Kernighan's algorithm.

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

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

A Clears the rightmost set bit of n

B Sets the rightmost unset bit of n

C Inverts all bits of n

D Shifts n to the right by one bit

The expression 'n & (n - 1)' clears the rightmost set bit in n, which is the key step in Brian Kernighan's algorithm.

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

A It counts unset bits instead of set bits

B It uses less memory

C It works only for positive numbers

D It runs in time proportional to the number of set bits, not total bits

Brian Kernighan's algorithm runs in O(k) time where k is the number of set bits, making it faster than checking all bits.

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

A Number of total bits in the number

B 5

C Depends on the number's value

D Always 1

The loop runs exactly as many times as there are set bits, so 5 times for 5 set bits.

What initial value should the count variable have in the Brian Kernighan algorithm?

A Number of bits in the integer

B 1

C 0

D Undefined

Count starts at 0 and increments each time a set bit is cleared.

Which of these is a correct output of countSetBits(13) using Brian Kernighan's algorithm?

A 3

B 4

C 2

D 5

13 in binary is 1101, which has 3 set bits.

Describe how Brian Kernighan's algorithm counts the set bits in an integer.

Think about how the number changes each loop iteration.

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

Consider the number of loop iterations.

Practice

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

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 binary representation.

C. It shifts the number right and counts zeros.

D. It converts the number to decimal and counts digits.

Solution

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

   The algorithm uses n = n & (n-1) to clear the least significant set bit in each iteration.

2. Step 2: Effect of clearing bits

   Each operation 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 = It repeatedly clears the least significant set bit until the number becomes zero. [OK]

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

Common Mistakes:

• Counting zeros instead of ones
• Shifting without clearing bits
• Confusing decimal digits with bits

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 ^ (n - 1);

Solution

1. Step 1: Recall the bit clearing operation

   The Brian Kernighan algorithm clears the least significant set bit by n = n & (n - 1).

2. Step 2: Verify other options

   Options A, B, and D do not clear the least significant set bit correctly.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

• Using +1 instead of -1
• Using OR or XOR instead of AND
• Confusing bitwise operators

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

unsigned int n = 13; // binary 1101
int count = 0;
while (n) {
  n = n & (n - 1);
  count++;
}
printf("%d", count);

medium

A. 5

B. 2

C. 4

D. 3

Solution

1. Step 1: Analyze initial value and binary form

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

2. Step 2: Count iterations clearing bits

   Each loop clears one set bit: 1101 -> 1100 -> 1000 -> 0000, so loop runs 3 times.

3. Final Answer:

   3 -> Option D

4. Quick Check:

   Number of set bits in 13 = 3 [OK]

Hint: Count set bits by clearing one bit per loop [OK]

Common Mistakes:

• Counting total bits instead of set bits
• Off-by-one in loop count
• Misreading binary representation

4. Identify the error in this C code snippet that tries 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 initialization of count variable.

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

C. Incorrect return type of the function.

D. Loop condition should be while (n > 0).

Solution

1. Step 1: Check the bit clearing operation

   The code uses n = 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 least significant set bit.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

• Using +1 instead of -1
• Ignoring loop condition correctness
• Forgetting to initialize count

5. Given an array of unsigned integers, which approach using Brian Kernighan's algorithm is most efficient to find the total count of set bits in all elements?

hard

A. Shift each number bit by bit and count set bits using bitwise AND with 1.

B. Convert each number to binary string and count '1's using string functions.

C. For each element, repeatedly clear least significant set bit and count until zero, then sum counts.

D. Use a lookup table for each byte and sum counts for all bytes in each number.

Solution

1. Step 1: Identify Brian Kernighan's approach

   For each element, repeatedly clear least significant set bit and count until zero, then sum counts.: For each element, repeatedly clear the least significant set bit until zero and count (using n = n & (n-1)), then sum the counts across elements. This is the direct application of Brian Kernighan's algorithm.

2. Step 2: Why it is most efficient using Brian Kernighan

   B (strings) is slow; C (shifts) is O(word size) always; D (lookup) is fast but a separate method, not Brian Kernighan's. Brian Kernighan excels for sparse bits (O(set bits) time).

3. Final Answer:

   For each element, repeatedly clear least significant set bit and count until zero, then sum counts. -> Option C

4. Quick Check:

   Brian Kernighan on array: apply n & (n-1) loop per element [OK]

Hint: Brian Kernighan per element: clear LSB until zero, sum counts [OK]

Common Mistakes:

• Using string conversion which is slow
• Bit shifting each bit without optimization
• Confusing lookup table with Brian Kernighan method