What is the time complexity of Number of 1 Bits?

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

What data structures are used to solve Number of 1 Bits?

This problem uses Divide and Conquer, Bit Manipulation. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Number of 1 Bits?

Insight 1: Understanding binary representation is crucial. The problem requires you to manipulate the bits directly. Insight 2: The clever bit manipulation trick `n &= (n - 1)` efficiently clears the least significant set bit. This is the core of the optimized solution.

Which companies ask LeetCode 191. Number of 1 Bits in interviews?

This problem is frequently asked at AMD, Box, Cisco, Qualcomm, Verkada.

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Number of 1 Bits - LeetCode 191 Solution

Number of 1 Bits - Complete Solution Guide

Number of 1 Bits is LeetCode problem 191, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given a positive integer n , write a function that returns the number of set bits in its binary representation (also known as the Hamming weight ). Example 1: Input: n = 11 Output: 3 Explanation: The input binary string 1011 has a total of three set bits. Example 2: Input: n = 128 Output: 1 Explanation: The input binary string 10000000 has a total of one set bit. Example 3: Input: n = 2147483645 Output: 30 Explanation: The input binary string 1111111111111111111111111111101 has a total of thirty

Detailed Explanation

The problem asks you to count the number of bits that are set to 1 in the binary representation of a given positive integer. The input is a positive integer `n`, and the output is the number of 1s in its binary representation. For example, the binary representation of 11 is 1011, which has three 1s. The constraints specify that the input integer `n` will always be between 1 and 2<sup>31</sup> - 1 (inclusive), meaning it fits within a 32-bit signed integer.

Solution Approach

The provided code uses a bit manipulation technique to efficiently count the set bits. It iteratively clears the least significant set bit in the number `n` and increments a counter. This process repeats until all set bits are cleared, and the counter then holds the total number of set bits.

Step-by-Step Algorithm

Step 1: Initialize a counter `count` to 0.
Step 2: Enter a `while` loop that continues as long as `n` is not 0.
Step 3: Inside the loop, perform the bit manipulation `n &= (n - 1)`. This operation clears the least significant set bit in `n`.
Step 4: Increment the `count` by 1.
Step 5: Repeat steps 3 and 4 until `n` becomes 0.
Step 6: Return the final value of `count`.

Key Insights

Insight 1: Understanding binary representation is crucial. The problem requires you to manipulate the bits directly.
Insight 2: The clever bit manipulation trick `n &= (n - 1)` efficiently clears the least significant set bit. This is the core of the optimized solution.
Insight 3: The loop continues until `n` becomes 0, meaning all set bits have been cleared. The count of iterations represents the total number of set bits.

Complexity Analysis

Time Complexity: O(log n)

Space Complexity: O(1)

Topics

This problem involves: Divide and Conquer, Bit Manipulation.

Companies

Asked at: AMD, Box, Cisco, Qualcomm, Verkada.

Number of 1 Bits | LeetCode 191

Easy

Divide and Conquer Bit Manipulation

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

Solution Code

Available in:

class Solution {
    public int hammingWeight(int n) {
        int count = 0;
        while (n != 0) {
            n &= (n - 1);
            count++;
        }
        return count;
    }
}

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Problem Info

Description

Problem 191: Number of 1 Bits - Detailed explanation and solution approaches.

Topics

Divide and Conquer Bit Manipulation

Companies

AMDBoxCiscoQualcommVerkada

Stats

Difficulty:Easy

Languages:4

Companies:5

Problem #:191

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

6 steps

1
Step 1: Initialize a counter `count` to 0.
2
Step 2: Enter a `while` loop that continues as long as `n` is not 0.
3
Step 3: Inside the loop, perform the bit manipulation `n &= (n - 1)`. This operation clears the least significant set bit in `n`.
4
Step 4: Increment the `count` by 1.
5
Step 5: Repeat steps 3 and 4 until `n` becomes 0.
6
Step 6: Return the final value of `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(log n)

🚀 Logarithmic - Very efficient! Halves the problem each step.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is O(log n) because the number of iterations in the `while` loop is proportional to the number of set bits in `n`. In the worst case (when all bits are set), the number of set bits is proportional to log₂(n). The space complexity is O(1) because only a constant amount of extra space is used to store the counter `count` and intermediate variables. The algorithm operates directly on the input integer; no additional data structures are needed.

📖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

Built-in function (if available)

Some languages have built-in functions (e.g., `bin()` in Python, `Integer.bitCount()` in Java) to count set bits. These functions are often highly optimized.

Time: O(log n) or better (depending on implementation)Space: O(1)

Trade-offs: Using built-in functions can be more concise and potentially faster than the manual bit manipulation approach. However, it relies on the availability of such functions in your programming language.

Iterative bit checking

Iterate through the bits one by one using bitwise AND and right bit shift operators to check if each bit is set.

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

Trade-offs: This approach is less concise and potentially less efficient than the `n &= (n-1)` method, but it's more straightforward to understand for beginners.

✨

Key Insights

1Insight 1: Understanding binary representation is crucial. The problem requires you to manipulate the bits directly.
2Insight 2: The clever bit manipulation trick `n &= (n - 1)` efficiently clears the least significant set bit. This is the core of the optimized solution.
3Insight 3: The loop continues until `n` becomes 0, meaning all set bits have been cleared. The count of iterations represents the total number of set bits.

📚

Key Terms Explained

For Students

Edge Cases

Edge case 1: n = 1. The algorithm correctly identifies the single set bit and returns 1.
Edge case 2: n = 2<sup>31</sup> - 1 (all bits are set). The algorithm correctly counts all 31 bits and returns 31.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly handling the loop termination condition. The loop must continue until `n` is 0, not 1.

✗Common mistake 2: Misunderstanding the bit manipulation operation. `n &= (n - 1)` is crucial; using other bitwise operations might not clear the least significant bit correctly.

✅

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: In networking, counting set bits in an IP address can be useful for network analysis or security protocols.

🏦Real-world scenario 2: In data compression, algorithms may use bit manipulation to represent data efficiently and counting set bits can aid in optimizing compression 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

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Don't rush. Explain your thought process clearly. Easy problems are often about communication.

⚠️ Watch Out

Always test your solution with edge cases: empty input, single element, and boundary values.

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: June 12, 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

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 6

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJun 29

❓

Frequently Asked Questions

What should I learn from easy problems?

Easy problems introduce core patterns that appear in harder problems. Master basic operations (iteration, conditionals, simple data structures), recognize common patterns (counting, searching, basic transformations), and practice explaining your thought process clearly.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Number of 1 Bits"?

For this easy-level Divide and Conquer 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.

🤖 Need help with this problem?