#1404

Medium

auto_awesomeString plus Bit Manipulation

LeetCode Problem Workspace

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

Determine the number of steps to reduce a binary representation of a number to 1 by following specific rules.

Problem Statement

Given a binary string representation of an integer, return the number of steps needed to reduce it to 1. In each step, check if the current number is even or odd. If it's odd, add 1; if it's even, divide it by 2. Repeat this process until the number equals 1.

The binary string is guaranteed to represent a valid integer, and the process will always eventually result in 1. For example, if the input is '1101' (representing the number 13), it takes 6 steps to reduce it to 1.

Examples

Example 1

Input: s = "1101"

Output: 6

"1101" corressponds to number 13 in their decimal representation. Step 1) 13 is odd, add 1 and obtain 14. Step 2) 14 is even, divide by 2 and obtain 7. Step 3) 7 is odd, add 1 and obtain 8. Step 4) 8 is even, divide by 2 and obtain 4. Step 5) 4 is even, divide by 2 and obtain 2. Step 6) 2 is even, divide by 2 and obtain 1.

Example 2

Input: s = "10"

Output: 1

"10" corresponds to number 2 in their decimal representation. Step 1) 2 is even, divide by 2 and obtain 1.

Example 3

Input: s = "1"

Output: 0

Example details omitted.

Constraints

1 <= s.length <= 500
s consists of characters '0' or '1'
s[0] == '1'

Solution Approach

Understanding the Problem

This problem involves using bit manipulation and string representation of numbers. You need to simulate the process of reducing a number represented as a binary string to 1. The key observation is that even numbers can be halved, while odd numbers require incrementing before halving.

Step-by-Step Simulation

To solve this, read the string from right to left. For each character, check if it corresponds to an even or odd number. Apply the appropriate operation—dividing by 2 for even and adding 1 for odd. Count each operation as a step until the number becomes 1.

Efficiency Considerations

This solution operates in O(N) time complexity, where N is the length of the binary string. Each operation (whether division or addition) is constant time, ensuring efficient handling even for the maximum input size.

Complexity Analysis

Metric	Value
Time	O(N)
Space	O(1)

The time complexity is O(N) due to iterating through the binary string once. Space complexity is O(1), as no additional data structures are required other than a counter for the steps.

What Interviewers Usually Probe

The candidate should demonstrate familiarity with string manipulation and bitwise operations.
Look for clarity in how they explain the iterative process and how they handle even/odd cases.
Check if they optimize the solution with minimal extra space usage.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the even/odd rule and applying the wrong operation (e.g., dividing by 2 when the number is odd).
Overcomplicating the solution by introducing unnecessary data structures or operations.
Not efficiently handling large binary strings, leading to incorrect time complexity analysis.

Follow-up variants

Modify the problem to handle decimal numbers directly instead of binary strings.
Allow multiple ways of representing the number (e.g., hexadecimal or octal).
Introduce variations like counting steps for different bases, or changing the reduction rules.

FAQ

How do I solve the 'Number of Steps to Reduce a Number in Binary Representation to One' problem?

Start by simulating the process: for odd numbers, increment by 1; for even numbers, divide by 2. Continue until you reach 1.

What is the time complexity of the 'Number of Steps to Reduce a Number in Binary Representation to One' problem?

The time complexity is O(N), where N is the length of the binary string, as each operation (add or divide) takes constant time.

What should I watch out for when solving this problem?

Be careful to correctly differentiate between even and odd numbers. Also, avoid unnecessary data structures to maintain optimal space usage.

Can I use bitwise operators for solving the 'Number of Steps to Reduce a Number in Binary Representation to One' problem?

Yes, bitwise operators can be used efficiently to handle even/odd checks and perform the required operations.

How can GhostInterview help me with this problem?

GhostInterview provides structured practice and feedback on bit manipulation and binary string handling, helping you solve this problem efficiently.

terminal

Solution

Solution 1: Simulation

We simulate operations $1$ and $2$, while maintaining a carry $\textit{carry}$ to indicate whether there is a current carry. Initially, $\textit{carry} = \text{false}$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

class Solution:
    def numSteps(self, s: str) -> int:
        carry = False
        ans = 0
        for c in s[:0:-1]:
            if carry:
                if c == '0':
                    c = '1'
                    carry = False
                else:
                    c = '0'
            if c == '1':
                ans += 1
                carry = True
            ans += 1
        if carry:
            ans += 1
        return ans

Continue Practicing

#1371 find the longest substring containing vowels in even counts #1461 check if a string contains all binary codes of size k #1324 print words vertically