What is the optimal time complexity of solving Counting Bits?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Counting Bits?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Counting Bits?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Counting Bits?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Counting Bits

Learn how to solve the 'Counting Bits' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

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.

Write a function countBits(n: int) -> List[int].

Constraints

•0 <= n <= 10^5

Examples

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 -> 1; 3 -> 2; 4 -> 1; 5 -> 2.

Need a Hint?

Analyze the input constraints. Try sorting first (O(n log n)) or using a hash map/set to track seen elements in O(n) time.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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