Introduction

In the realm of software engineering, bit manipulation stands out as a fundamental skill, especially when navigating through coding interviews. A quintessential example of such a challenge is found in LeetCode's "Counting Bits" problem (LeetCode 338).

This task requires us to generate an array ans of length n + 1, where each element ans[i] represents the number of 1's in the binary representation of i. For instance, given n = 2, the output should be [0,1,1], as the binary representations of 0, 1, and 2 are 0, 1, and 10, respectively.

Given an integer n, return an array ans of length n + 1 such that for each i (0 <= i <= n), ans[i] is the number of 1's*in the binary representation of* i.

Example 1:

Input: n = 2
Output: [0,1,1]
Explanation:
0 --> 0
1 --> 1
2 --> 10

Example 2:

Input: n = 5
Output: [0,1,1,2,1,2]
Explanation:
0 --> 0
1 --> 1
2 --> 10
3 --> 11
4 --> 100
5 --> 101

Solving the Problem

To tackle this problem, understanding the binary representation of numbers is crucial. A number's binary form can be seen as a series of 1s and 0s, where each digit represents a power of 2. The key to solving this problem lies in efficiently counting the 1s in each number's binary form up to n.

Big O Notation Analysis

• Brute Force Approach: The time complexity is O(n*log(n)), primarily due to iterating through each number up to n and the bit count operation, which is O(log(n)) for each number.

• Count and Track Approach: Improves to O(n) by leveraging the pattern that the number of bits in current numbers is related to previously computed values.

• Extend Approach: Although seemingly efficient, this method also results in O(n) complexity due to the doubling pattern in binary representations but might have slightly worse constants due to array extension operations.

Brute Solution in Python

def countBits(self, n: int) -> List[int]:
    # Use list comprehension to iterate over each number up to n
    # Convert each number to binary with bin(), count '1's with .count('1')
    return [bin(i).count('1') for i in range(n+1)]

Count and Track Approach in Python

def countBits(self, n: int) -> List[int]:
    nextOrder = 2  # Initialize the next power of 2
    tracker = 0  # Track the index to refer back for bit counts
    counter = [0]*(n+1)  # Initialize counter list

    for i in range(1, n+1):
        if i == nextOrder:
            nextOrder *= 2  # Update next power of 2
            tracker = 0  # Reset tracker
        counter[i] = counter[tracker] + 1  # Count bits based on previous values
        tracker += 1

    return counter

Extend Approach in Python

def countBits(self, n: int) -> List[int]:
    counter = [0]  # Initialize counter list with zero's bit count
    while len(counter) < n + 1:
        # Double the size of counter by adding 1 to each current element
        # This leverages the pattern in binary representations
        counter.extend([i+1 for i in counter])
    return counter[:n+1]

Conclusion

Understanding and applying the right strategy for bit manipulation problems like "Counting Bits" can significantly enhance your problem-solving skills in coding interviews.

The brute force method provides a straightforward solution, while the count and track, and extend approaches offer more efficient alternatives by recognizing and leveraging underlying patterns.

Mastering these techniques not only aids in solving similar challenges but also deepens your comprehension of binary operations and their practical applications in software engineering.