What is the time complexity of Counting Bits?

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

What data structures are used to solve Counting Bits?

This problem uses Dynamic Programming, Bit Manipulation. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Counting Bits?

Insight 1: Recognizing the pattern that the number of 1s in `i` is related to the number of 1s in `i/2` (integer division). Insight 2: Utilizing dynamic programming to store and reuse previously computed results. This avoids redundant calculations.

Which companies ask LeetCode 338. Counting Bits in interviews?

This problem is frequently asked at Nvidia.

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Counting Bits - LeetCode 338 Solution

Q: How to solve Counting Bits?

The provided solutions employ dynamic programming. They create an array `ans` of size `n+1` initialized with zeros. The algorithm iterates from 1 to `n`, calculating the number of 1s in the binary representation of each number `i`. It cleverly leverages the observation that the number of 1s in `i` is either the same as in `i/2` (if `i` is even) or one more than in `i/2` (if `i` is odd). This is efficiently determined using bitwise operations.

Counting Bits - Complete Solution Guide

Counting Bits is LeetCode problem 338, 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 an integer n , return an array ans of length n + 1 such that for each i ( 0 <= i <= n ) , ans[i] is the number of 1 's in the binary representation of i . Example 1: Input: n = 2 Output: [0,1,1] Explanation: 0 --> 0 1 --> 1 2 --> 10 Example 2: Input: n = 5 Output: [0,1,1,2,1,2] Explanation: 0 --> 0 1 --> 1 2 --> 10 3 --> 11 4 --> 100 5 --> 101 Constraints: 0 <= n <= 10 5 Follow up: It is very easy to come up with a solution with a runtime of O(n log n) . Can you do it in linear time O(n) a

Detailed Explanation

The problem asks you to create an array where each index `i` (from 0 to `n`) contains the count of 1s in the binary representation of `i`. For example, if `n` is 5, the binary representations are: 0 (0), 1 (1), 10 (1), 11 (2), 100 (1), 101 (2). Therefore, the output array should be `[0, 1, 1, 2, 1, 2]`. The challenge emphasizes achieving a linear time complexity O(n) and avoiding built-in functions that directly count set bits.

Solution Approach

The provided solutions employ dynamic programming. They create an array `ans` of size `n+1` initialized with zeros. The algorithm iterates from 1 to `n`, calculating the number of 1s in the binary representation of each number `i`. It cleverly leverages the observation that the number of 1s in `i` is either the same as in `i/2` (if `i` is even) or one more than in `i/2` (if `i` is odd). This is efficiently determined using bitwise operations.

Step-by-Step Algorithm

Step 1: Initialize an array `ans` of size `n+1` with all elements set to 0. `ans[i]` will store the number of 1s in the binary representation of `i`.
Step 2: Iterate from `i = 1` to `n`. For each `i`, calculate `ans[i]` using the formula `ans[i] = ans[i >> 1] + (i & 1)`. `i >> 1` performs a right bit shift (integer division by 2), and `i & 1` checks if `i` is odd (LSB is 1).
Step 3: Return the `ans` array.

Key Insights

Insight 1: Recognizing the pattern that the number of 1s in `i` is related to the number of 1s in `i/2` (integer division).
Insight 2: Utilizing dynamic programming to store and reuse previously computed results. This avoids redundant calculations.
Insight 3: Understanding that the least significant bit (LSB) can be efficiently extracted using the bitwise AND operator (`& 1`). The right bit shift (`>> 1`) effectively divides by 2 while discarding the LSB.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Dynamic Programming, Bit Manipulation.

Companies

Asked at: Nvidia.

Counting Bits | LeetCode 338

Easy

Dynamic Programming Bit Manipulation

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

Solution Code

Available in:

class Solution {
    public int[] countBits(int n) {
        int[] ans = new int[n + 1];
        int powerOf2 = 1;
        int count = 0;
        for (int i = 1; i <= n; i++) {
            if (i == powerOf2) {
                powerOf2 *= 2;
                count = 0;
            }
            ans[i] = ans[count] + 1;
            count++;
        }
        return ans;
    }
}

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Problem Info

Description

Problem 338: Counting Bits - Detailed explanation and solution approaches.

Topics

Dynamic Programming Bit Manipulation

Companies

Nvidia

Stats

Difficulty:Easy

Languages:4

Companies:1

Problem #:338

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

3 steps

1
Step 1: Initialize an array `ans` of size `n+1` with all elements set to 0. `ans[i]` will store the number of 1s in the binary representation of `i`.
2
Step 2: Iterate from `i = 1` to `n`. For each `i`, calculate `ans[i]` using the formula `ans[i] = ans[i >> 1] + (i & 1)`. `i >> 1` performs a right bit shift (integer division by 2), and `i & 1` checks if `i` is odd (LSB is 1).
3
Step 3: Return the `ans` array.

💡

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(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The algorithm iterates through the numbers from 1 to n once, performing constant time operations for each number. Therefore, the time complexity is linear, O(n). The space complexity is also O(n) because we need to store the `ans` array of size `n+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

Iterative Bit Counting

Iterate through each number, convert it to binary using string manipulation or bitwise operations, and count the number of 1s directly.

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

Trade-offs: Simpler to understand but less efficient than the dynamic programming approach. The conversion to binary and counting takes longer than the bitwise operations in the optimal solution.

Using Built-in Functions (if allowed)

Some languages provide built-in functions (like `__builtin_popcount` in C++) to count the set bits in an integer. This would significantly simplify the code but may not be allowed by the problem constraints.

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

Trade-offs: Fastest solution, but sacrifices the learning experience of bit manipulation and dynamic programming.

✨

Key Insights

1Insight 1: Recognizing the pattern that the number of 1s in `i` is related to the number of 1s in `i/2` (integer division).
2Insight 2: Utilizing dynamic programming to store and reuse previously computed results. This avoids redundant calculations.
3Insight 3: Understanding that the least significant bit (LSB) can be efficiently extracted using the bitwise AND operator (`& 1`). The right bit shift (`>> 1`) effectively divides by 2 while discarding the LSB.

📚

Key Terms Explained

For Students

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: n = 0. The solution correctly handles this by returning `[0]`.
Edge case 2: n = 1. The solution correctly returns `[0, 1]`.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Inefficiently counting bits by repeatedly dividing by 2 and checking remainders. This leads to O(n log n) time complexity.

✗Common mistake 2: Incorrectly handling the base case (n=0) or not understanding the relationship between the number of set bits in `i` and `i/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: Counting set bits is useful in various bit manipulation algorithms and data structures, such as analyzing network traffic patterns or processing image data.

🏦Real-world scenario 2: In database systems, analyzing data might involve counting the number of set bits in flags or status fields to determine the number of active features or processes.

💡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

Start with recursion, then memoize. The hardest part is identifying the state. Ask yourself: "What information do I need to make the current decision?"

🎯 Interview Tip

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

⚠️ Watch Out

Forgetting base cases or initializing the DP array incorrectly. Start by solving the smallest subproblems.

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: November 6, 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

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

@CodeNinjaJul 13

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

@TechLeadJul 6

❓

Frequently Asked Questions

How do I identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

How should I approach easy-level algorithm problems?

Easy problems build foundational skills. Focus on: 1) Understanding the problem completely before coding, 2) Writing clean, readable code, 3) Handling all edge cases (empty input, single element, etc.), 4) Analyzing time and space complexity. Don't skip complexity analysis even for simple solutions.

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 "Counting Bits"?

For this easy-level Dynamic Programming 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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