# LeetCode 338. Counting Bits

## The Problem

Link to original problem on LeetCode.

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.

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 --> 10
3 --> 11
4 --> 100
5 --> 101

Constraints

0 <= n <= 105

• 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 $O(n)$ and possibly in a single pass?
• Can you do it without using any built-in function (i.e., like __builtin_popcount in C++)?

## My Solution

The problem is very simple if you've already got a solution to 191. Number of 1 Bits. All you need to do is loop over the numbers from 0 through n, computer the Hamming Weight, and push that to an array to be returned.

// To understand what this function is doing, see the// explanation linked in the preceding paragraph.const hammingWeight = (n) => {	let count = 0;	while (n) {		n = n & (n - 1);		count++;	}	return count;};const countBits = (n) => {	const result = [];	for (let i = 0; i <= n; i++) {		result.push(hammingWeight(i));	}	return result;};

Ok, so that's easy. Copy and paste! But can we do better? Yes, we can!

With the above example, we have to count the 1s from scratch for each value of i. But we can actually use previous computations in our loop over i to save us the trouble!

Think about the even numbers in binary. 10, 100, 110, 1000, etc. Notice that the final digit is always zero. If you shift the bits right by one, the count of 1s in the number does not change because you've only ever lopped off a 0.

The other thing to notice is that shifting each even number to the right by 1 bit is equivalent to dividing by 2. E.g., 1000 >> 1 === 100 is the same as 8 / 2 === 4. So if we want to know how many 1s are in 8, we can go back to our previous count of how many 1s are in 4. This means that we can look back in our own results array for the value at the index of half the current number and find the count of 1s not including the final digit. To then account for the final digit, we use AND comparison to see if it is a 1 or a 0. Putting it all together:

const countBits = (n) => {	const result = [0];	for (let i = 0; i <= n; i++) {		result[i] = result[i >> 1] + (i & 1);	}	return result;};