Number of Steps to Reduce a Number in Binary Representation to One

The key to solving this problem efficiently is understanding how binary arithmetic works, especially the binary addition.

Observations:

1. If the least significant bit (rightmost bit) is 0, the number is even, and we divide by 2. This operation corresponds to simply removing the rightmost bit.
2. If the least significant bit is 1, the number is odd. Adding 1 to an odd binary number flips the rightmost bit and propagates the carry to the left until it encounters another 0 or runs out of bits.

Algorithm:

1. Initialize a counter to track the number of steps taken.
2. Start from the least significant bit (rightmost) of the string s.
3. Check the bit: a. If it’s 0, this is an even number. Increase the counter by 1 (divide by 2). b. If it’s 1: i. Increase the counter by 1 (for the addition). ii. Propagate the carry leftward, increasing the counter each time until we find another 0 or reach the left end. iii. If we reach the left end and still have a carry, it means we need to add another 1 to the leftmost position.
4. Repeat step 3 until we process all bits in s.

Implementation:

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class Solution:
    def numSteps(self, s: str) -> int:
        s = list(s)
        steps = 0
        while len(s) > 1:
            if s[-1] == '0':
                # Even number, just remove the last bit (divide by 2)
                s.pop()
                steps += 1
            else:
                # Odd number, add 1
                idx = len(s) - 1
                while idx >= 0 and s[idx] == '1':
                    s[idx] = '0'
                    idx -= 1
                if idx < 0:
                    s = ['1'] + s
                else:
                    s[idx] = '1'
                steps += 1
        return steps

Key Takeaways:

• Understanding the binary arithmetic operations and how they affect the bits can lead to efficient solutions.
• In this problem, binary addition and understanding even/odd numbers in binary representation helped in coming up with the optimal solution.