Count Set Bits Brian Kernighan Algorithm in DSA C++ - Time & Space Complexity

Time Complexity: Count Set Bits Brian Kernighan Algorithm

O(k)

Understanding Time Complexity

We want to understand how fast the Brian Kernighan algorithm counts the number of 1s in a number's binary form.

How does the time taken change as the number gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

This code counts how many bits are set to 1 in the binary form of the number n by repeatedly turning off the rightmost set bit.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop runs once for each set bit in n.
• How many times: Equal to the number of 1s in the binary representation of n.

How Execution Grows With Input

The number of operations depends on how many bits are set to 1, not the total number of bits.

Input Size (n)	Approx. Operations (set bits)
10 (binary 1010)	2
100 (binary 1100100)	3
1000 (binary 1111101000)	6

Pattern observation: The loop runs as many times as there are 1s, so it grows with the number of set bits, not the total bits.

Final Time Complexity

Time Complexity: O(k) where k is the number of set bits in n.

This means the time depends on how many 1s are in the number, not its size in bits.

Common Mistake

[X] Wrong: "The loop runs for all bits of the number, so it is O(log n)."

[OK] Correct: The loop only runs once per set bit, so if the number has few 1s, it runs faster than checking every bit.

Interview Connect

Understanding this algorithm shows you can optimize bit operations by focusing on set bits, a useful skill in many coding problems.

Self-Check

"What if we changed the input to always have all bits set (like 2^m - 1)? How would the time complexity change?"

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