So far, you have probably brainstormed some approaches and have an idea of how to solve this problem. Let’s explore some of these approaches and figure out which one to follow based on considerations such as time complexity and any implementation constraints.

The naive approach for this solution would be to iterate through the string character by character. During the traversal, we convert each number to its binary representation, count the number of $1$ bits in each binary representation, and store the result in an array.

This approach would have a time complexity of $O(n \log n)$ and a space complexity of $O(n)$.

The problem involves using a dynamic programming approach to calculate and store the count of $1$ bits and use it for future iterations. Let’s take an example to see how the previous count of $1$ bits helps to find the count of $1$ bits in future iterations.

For every digit, we know it can be an even number or an odd number. Let’s see how we can compute each by using binary manipulation:

1. For even numbers, the count can be calculated using the count for half of that number. For example, the count of $1$ bit for ...