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

The optimal solution has a time complexity of O(n).

What data structures are used to solve Number of Steps to Reduce a Number in Binary Representation to One?

This problem uses String, Bit Manipulation, Simulation. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Number of Steps to Reduce a Number in Binary Representation to One?

Insight 1: We don't need to convert the binary string to an integer. We can simulate the operations directly on the binary string by observing the last bit. Insight 2: Dividing by 2 in binary is equivalent to right-shifting (removing the last bit).

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Number of Steps to Reduce a Number in Binary Representation to One - LeetCode 1404 Solution

Q: Which companies ask LeetCode 1404. Number of Steps to Reduce a Number in Binary Representation to One in interviews?

This problem is frequently asked at Geico, Grab.

Number of Steps to Reduce a Number in Binary Representation to One - Complete Solution Guide

Number of Steps to Reduce a Number in Binary Representation to One is LeetCode problem 1404, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given the binary representation of an integer as a string s , return the number of steps to reduce it to 1 under the following rules : If the current number is even, you have to divide it by 2 . If the current number is odd, you have to add 1 to it. It is guaranteed that you can always reach one for all test cases. Example 1: Input: s = "1101" Output: 6 Explanation: "1101" corressponds to number 13 in their decimal representation. Step 1) 13 is odd, add 1 and obtain 14. Step 2) 14 is even, divid

Detailed Explanation

The problem requires us to find the number of steps to reduce a number represented in binary format (given as a string) to 1. The steps involve two operations: if the number is even, divide it by 2; if the number is odd, add 1 to it. We need to return the total number of steps taken to reach 1.

Solution Approach

The provided solution simulates the process of reducing the binary number to 1 step by step. It iterates through the binary string from right to left, keeping track of a `carry`. If the current bit plus carry is odd (equal to 1), it means adding 1 to the current number results in two steps: one for the addition and one implicit division by 2 (right shift). If the current bit plus carry is even (equal to 0 or 2), it means dividing the current number by 2, which needs only one step. The `carry` variable maintains the carry-over from previous additions. Finally, if there is a carry after processing all bits except the leftmost bit, one additional division step is required.

Step-by-Step Algorithm

Step 1: Initialize `steps` and `carry` to 0.
Step 2: Iterate through the binary string `s` from the second rightmost bit (index `len(s) - 1`) to the second bit (index 1).
Step 3: For each bit `s[i]`, check if the sum of the bit value (as an integer) and the `carry` is equal to 1. `if int(s[i]) + carry == 1:`
Step 4: If the sum is 1, increment `steps` by 2 (one for adding 1, the other for dividing by 2) and set `carry` to 1.
Step 5: If the sum is not 1 (meaning it's 0 or 2), increment `steps` by 1 (for dividing by 2). If the sum is 2, the `carry` is set to 1, else it remains 0.
Step 6: After the loop, add the final `carry` to `steps`. This accounts for the final division step if the carry is 1.
Step 7: Return the final `steps` count.

Key Insights

Insight 1: We don't need to convert the binary string to an integer. We can simulate the operations directly on the binary string by observing the last bit.
Insight 2: Dividing by 2 in binary is equivalent to right-shifting (removing the last bit).
Insight 3: Adding 1 to a binary number can result in carry-over operations, similar to addition in decimal.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: String, Bit Manipulation, Simulation.

Companies

Asked at: Geico, Grab.

Number of Steps to Reduce a Number in Binary Representation to One | LeetCode 1404

Medium

String Bit Manipulation Simulation

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int numSteps(String s) {
        int steps = 0;
        int carry = 0;
        
        // We iterate from the rightmost bit to the second bit (index 1).
        // The leftmost bit (s[0]) is handled after the loop.
        for (int i = s.length() - 1; i > 0; i--) {
            // If the current bit plus carry is odd, it's like an odd number.
            if (Character.getNumericValue(s.charAt(i)) + carry == 1) {
                // Odd number: add 1 (becomes even), then divide by 2.
                // This takes two steps.
                // Adding 1 to a '1' bit results in a '0' and a carry.
                steps += 2;
                carry = 1;
            } else {
                // Even number: just divide by 2.
                // This takes one step.
                // '0'+carry 0 -> '0', carry 0.
                // '1'+carry 1 -> '0', carry 1.
                // The carry propagates.
                steps += 1;
            }
        }
        
        // After the loop, we are left with the most significant bit s[0]
        // which is '1', and the final carry.
        // If carry is 1, the remaining number is "1" + 1 = "10" (2).
        // This requires one more step (division by 2) to become 1.
        return steps + carry;
    }
}

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Problem Info

Description

Problem 1404: Number of Steps to Reduce a Number in Binary Representation to One - Detailed explanation and solution approaches.

Topics

String Bit Manipulation Simulation

Companies

GeicoGrab

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:1404

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

7 steps

1
Step 1: Initialize `steps` and `carry` to 0.
2
Step 2: Iterate through the binary string `s` from the second rightmost bit (index `len(s) - 1`) to the second bit (index 1).
3
Step 3: For each bit `s[i]`, check if the sum of the bit value (as an integer) and the `carry` is equal to 1. `if int(s[i]) + carry == 1:`
4
Step 4: If the sum is 1, increment `steps` by 2 (one for adding 1, the other for dividing by 2) and set `carry` to 1.
5
Step 5: If the sum is not 1 (meaning it's 0 or 2), increment `steps` by 1 (for dividing by 2). If the sum is 2, the `carry` is set to 1, else it remains 0.
6
Step 6: After the loop, add the final `carry` to `steps`. This accounts for the final division step if the carry is 1.
7
Step 7: Return the final `steps` count.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The loop iterates `n-1` times, where `n` is the length of the string. Therefore, the time complexity is directly proportional to the length of the input string. The space used does not depend on the size of input so it's constant O(1).

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Convert to Integer and Simulate

Convert the binary string to an integer and then repeatedly perform the divide by 2 (if even) or add 1 (if odd) operations until the number becomes 1.

Time: O(n), where n is the value of the number. In worst case the value can be close to 2^length(s)Space: O(1)

Trade-offs: This approach might be intuitive, but converting the string to an integer might cause integer overflow issues for very large binary numbers (strings longer than 63 for Java, 31 for C++ for example). The main approach avoids overflow issues by directly operating on the string.

✨

Key Insights

1Insight 1: We don't need to convert the binary string to an integer. We can simulate the operations directly on the binary string by observing the last bit.
2Insight 2: Dividing by 2 in binary is equivalent to right-shifting (removing the last bit).
3Insight 3: Adding 1 to a binary number can result in carry-over operations, similar to addition in decimal.

📚

Key Terms Explained

For Students

String

A sequence of characters. Like a word or sentence in programming!

Edge Cases

Edge case 1: s = "1". The algorithm returns 0 as expected, since the number is already 1.
Edge case 2: s = "10". The algorithm handles this case correctly. It calculates 1 step. The number is even so we divide by 2, resulting in 1.
Edge case 3: Long binary strings. The solution handles arbitrarily long binary strings up to the constraint limit of 500 because the computations are bitwise, avoiding integer overflow.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer overflow when converting the binary string to an integer. This is especially problematic for long binary strings.

✗Common mistake 2: Forgetting to handle the carry-over correctly when adding 1. This will lead to an incorrect number of steps.

✗Common mistake 3: Not adding the final carry after the loop. The carry potentially requires one more step to divide by 2.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: This problem is relevant in cryptography where binary arithmetic and bit manipulations are fundamental operations.

🏦Real-world scenario 2: Computer architecture. Simulating the steps to reduce a binary number to 1 showcases the logic behind division and addition operations performed at the hardware level.

🎮Real-world scenario 3: Game development. Implementing game logic where resource management and algorithmic efficiency are crucial. The bitwise operations and carry handling demonstrate optimization techniques.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

String problems often involve sliding windows or two pointers. Consider if the character set is limited (lowercase letters = only 26 possibilities).

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Mutating strings directly in languages where strings are immutable. Use StringBuilder or arrays.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 23, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

❓

Frequently Asked Questions

How do I handle string manipulation efficiently?

In languages with immutable strings (Java, Python, C#), use StringBuilder/list to avoid O(n) concatenation cost. Consider using character arrays for in-place modifications. For pattern matching, consider KMP or Rabin-Karp for O(n+m) complexity.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Number of Steps to Reduce a Number in Binary Representation to One"?

For this medium-level String problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether bit manipulation techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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