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

class Solution {
public:
    int stepsToReduce(string binary) {
        int length = binary.length();

        int stepCount = 0;
        int extra = 0;
        for (int j = length - 1; j > 0; j--) {
            if (((binary[j] - '0') + extra) % 2) {
                stepCount += 2;
                extra = 1;
            } else {
                stepCount++;
            }
        }

        return stepCount + extra;
    }
};

This solution focuses on the problem of counting the number of steps required to reduce a binary number to one, using a C++ function. The function, stepsToReduce, takes a binary string as input and calculates the number of steps involved.

• Start by determining the length of the binary string.
• Initialize variables, stepCount for keeping track of the total steps and extra to manage cases when an operation on the binary string results in a carry over.
• Iterate through the binary string in reverse (excluding the last bit since the goal is to reduce it to one bit), adjusting the steps needed based on whether bits are odd or even, considering the extra carry over if applicable.
  • If a bit plus the extra (i.e., binary[j] - '0' + extra) is odd, it indicates a '1' bit. Two steps are needed: one for increment (causing a carry bit interference) and another for the actual sum operation, incrementing the stepCount by two and setting extra to 1.
  • If not, just increase the stepCount by one (for the division or removal of the last 0 bit).
• Finally, add any remaining extra to the stepCount.
• The function returns the total number of steps (stepCount + extra), representing the count from the input binary to reduce to '1'.

Effectively, this function leverages bitwise operations and control flow to calculate the minimal steps required by simulating the process of binary reduction through basic arithmetic and binary operations.

public class Solution {

    public int countSteps(String binaryStr) {
        int length = binaryStr.length();

        int stepCount = 0;
        int overflow = 0;
        for (int i = length - 1; i > 0; i--) {
            int value = Character.getNumericValue(binaryStr.charAt(i)) + overflow;

            if (value % 2 == 1) {
                stepCount += 2;
                overflow = 1;
            } else {
                stepCount++;
            }
        }

        return stepCount + overflow;
    }
}

The provided Java program aims to calculate the number of steps required to reduce a number in binary representation to one. Here's how the solution works:

• Start by determining the length of the binary string.
• Initialize stepCount to count each transformation step and overflow to handle carrying during step calculations.
• Iterate through the string from the last character to the first (ignoring the first since you reduce the number to one), converting each character to its numeric value and adjusting by the current overflow.
• For each iteration:
  • If the sum of the digit and overflow is odd (i.e., the binary digit is 1 and an overflow is added), increment stepCount by two and set overflow to 1 to account for the additional steps needed to handle binary carry over during the subtraction step.
  • For even sums, simply increment stepCount, as no carrying over is required.

Return the final stepCount including any residual overflow, which indicates the total operations needed to transform the input binary string to '1'.

This algorithm efficiently combines loop iteration and conditional checks to minimize the operations and accurately reflect the steps needed in binary subtraction and conversion processes.

class Solution:
    def countSteps(self, binary_string: str) -> int:
        string_length = len(binary_string)

        step_count = 0
        overflow = 0
        for index in range(string_length - 1, 0, -1):
            current_digit = int(binary_string[index]) + overflow
            if current_digit % 2 == 1:
                step_count += 2
                overflow = 1
            else:
                step_count += 1

        return step_count + overflow

The Python function countSteps is designed to compute the number of steps required to reduce a number, given in its binary representation, to one. The solution employs a direct approach that iterates backward from the least significant bit (rightmost) to the most significant bit (leftmost) of the binary string. Here's a succinct breakdown of how the solution works:

• Initialize step_count to count each operation and overflow to handle cases where a binary addition carries over.

• Iterate over the binary string's characters in reverse order (omitting the most significant bit for now):

  • Convert the current bit and the overflow to an integer, current_digit.
  • Determine operations based on the value of current_digit:
    • If current_digit is odd, it indicates a '1' that needs two operations: increment to introduce a carry and a subsequent division by two. Thus, increment step_count by two and set overflow to 1 for the next bit.
    • If current_digit is even, only a division operation is needed; thus, increment step_count by one.

• After processing all bits, add any remaining overflow to step_count since it represents a final increment operation which is needed to nullify the overflow.

By the time the loop exits, step_count will store the total operations needed to reduce the binary number to one. This approach leverages bit manipulation tricks and is efficient in cases where binary representation is directly dealt with. It calculates the minimal steps effectively by simulating each essential operation on the binary number.