Solution: Number of Steps to Reduce a Binary Number to One

Statement

You are given a string, str, as a binary representation of an integer. Your task is to return the number of steps needed to reduce it to $1$ by following these rules:

• If the number is even, divide it by $2$.

• If the number is odd, add $1$ to it.

You can always reach 1 for all provided test cases.

Constraints:

• $1 <=$str.length $<= 500$

• str consists of characters $'0'$ or $'1'$

• $str[0] ==$ $'1'$

Solution

The idea of the solution is to use a greedy approach to count the steps needed to reduce a binary number to $1$ because, at each stage of the decision-making process, it takes the most immediate and optimal step—if the digit is odd, increment and divide; if the digit is even, divide to minimize the number of operations needed to reduce it to $1$. We start from the least significant digit(the rightmost one), and we determine if each digit (after accounting for any previous carry) is even or odd. For an odd digit, we add two steps (one to make it even and one to divide by $2$), and for an even digit, we add one step (to divide by $2$). This process continues until we reach the most significant digit(the leftmost one). Any carry-over from adding 1 is counted in the final step count. This ensures the total steps needed to reduce the binary number.

Now, let’s walk-through the steps of the solution.

• Initialize the variable, steps, with $0$.

• Initialize the variable, c, to store carryover with $0$.

• Iterate through each digit from the end of the str to the second digit:

  • Add the current digit and c.

  • If the result is odd:

    • Increment steps by $2$.

    • Set c to $1$.

  • Otherwise, If the result is even:

    • Increment steps by $1$.

• Return the sum of steps and c as the total step count.

Let’s look at the following illustration to get a better understanding of the solution: