1404. Number of Steps to Reduce a Number in Binary Representation to One - Detailed Explanation

Problem Statement

Given a binary string s representing a positive integer, the goal is to reduce the integer to 1 by performing the following operations:

• If the current number is even, divide it by 2.
• If the current number is odd, add 1.

Return the number of steps required to reduce the number to 1.

Example Inputs, Outputs, and Explanations

Example 1

• Input: s = "1101"
• Output: 6
• Explanation:
  1. "1101" (13 in decimal) is odd, so add 1 → "1110"
  2. "1110" (14) is even, divide by 2 → "111"
  3. "111" (7) is odd, add 1 → "1000"
  4. "1000" (8) is even, divide by 2 → "100"
  5. "100" (4) is even, divide by 2 → "10"
  6. "10" (2) is even, divide by 2 → "1"

Example 2

• Input: s = "10"
• Output: 1
• Explanation:
  1. "10" (2) is even, divide by 2 → "1"

Example 3

• Input: s = "1"
• Output: 0
• Explanation:
  The number is already 1, so no steps are required.

Constraints

• 1 <= s.length <= 500
• s consists only of the characters '0' and '1'.
• s[0] == '1' (i.e., there are no leading zeros).

Hints

1. Even Number Shortcut: Notice that when the binary number is even (ends with '0'), dividing by 2 is equivalent to removing the trailing zero.
2. Odd Number Handling: When the binary number is odd (ends with '1'), adding 1 may involve a cascade of changes due to carry propagation. Consider how adding 1 affects a sequence of trailing ones.

Approach 1: Brute Force Simulation

Explanation

• Conversion: Convert the binary string to an integer.
• Simulation:
  • While the number is greater than 1, check if it is even or odd.
  • If it is even, divide it by 2.
  • If it is odd, add 1.
  • Count each operation as a step.
• Drawbacks:
  • Although Python can handle large integers, continuously updating a potentially huge number might be less efficient when the binary string is long and the operations are many.
  • The worst-case time complexity may be higher due to the repeated arithmetic operations.

Step-by-Step Walkthrough (Brute Force)

Consider the input "1101":

1. Convert "1101" to integer → 13.
2. 13 is odd → 13 + 1 = 14. (Step count = 1)
3. 14 is even → 14 / 2 = 7. (Step count = 2)
4. 7 is odd → 7 + 1 = 8. (Step count = 3)
5. 8 is even → 8 / 2 = 4. (Step count = 4)
6. 4 is even → 4 / 2 = 2. (Step count = 5)
7. 2 is even → 2 / 2 = 1. (Step count = 6)

Approach 2: Optimal Simulation Using the Binary String Directly

Explanation

• Observation:
  • Dividing by 2 for an even binary number means simply removing the last digit (which is '0').
  • For an odd number, adding 1 can change a sequence of trailing '1's to '0's and propagate a carry to the left.
• Method:
  • Process the binary string from right to left while maintaining a carry flag.
  • For each bit (ignoring the most significant bit initially):
    • If the current bit plus any carry is even, it corresponds to a division by 2 (a single step).
    • If it is odd, perform an "add 1" operation (which in our simulation counts as 2 steps: one for addition and one for the subsequent division), and set the carry for future bits.
  • Finally, if a carry remains at the most significant bit, account for an extra step.
• Benefits:
  • This approach works in O(n) time (where n is the length of the binary string) and avoids costly arithmetic on potentially huge numbers.

Step-by-Step Walkthrough (Optimal Approach)

Consider the input "1101":

1. Start with a carry = 0 and a step count = 0.
2. Process from the least significant digit to the second digit:
   • Digit: '1' (plus carry 0) → odd, so add 2 steps (for add and division) and set carry = 1.
   • Next Digit: '0' (plus carry 1 gives 1) → odd, add 2 steps, carry remains 1.
   • Next Digit: '1' (plus carry 1 gives 2) → even, add 1 step.
3. Finally, process the most significant digit with the carry: add 1 extra step if a carry is present.
4. Total steps equal the accumulated count.

Code in Python