Problems/Bit Manipulation/Hamming Weights of Integers
Bit Manipulation
easy

Hamming Weights of Integers

Calculate the 'digital root' of each number from 0 to n, where the digital root is the result of repeatedly summing the digits of a number until a single-digit number is reached. This problem tests bit manipulation and dynamic programming skills, crucial for optimizing performance in numerical computations.

dynamic programmingmathematicsbit manipulationnumerical computationoptimization

Problem Statement

Given a non-negative integer 'n', determine the digital root for every number from 0 up to and including 'n'. Return an array where each element at index 'i' represents the digital root of 'i'. The digital root of a number is calculated by iteratively summing its digits until a single-digit number is obtained. For instance, the digital root of 38 is 2 because 3 + 8 = 11, and 1 + 1 = 2.

Example 1
Input: n = 10
Output: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1]
The digital roots for numbers 0 through 10 are: 0 -> 0, 1 -> 1, 2 -> 2, 3 -> 3, 4 -> 4, 5 -> 5, 6 -> 6, 7 -> 7, 8 -> 8, 9 -> 9, 10 -> 1+0 = 1.
Example 2
Input: n = 20
Output: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2]
The digital roots for numbers 0 through 20 are calculated similarly, with numbers greater than 9 requiring iterative summing of digits. For example, 19 -> 1+9 = 10 -> 1+0 = 1, and 20 -> 2+0 = 2.
Constraints
  • -0 <= n <= 1000
  • -Input 'n' will always be a non-negative integer.
  • -The output array must contain n+1 elements.

Brute Force Approach

The brute-force approach involves calculating the digital root for each number from 0 to 'n' independently. Imagine you're a cashier and have to manually sum the digits of each customer's total until you get a single digit. For each number, you'd repeatedly add the digits until a single-digit number remains. This method is straightforward but inefficient as it recalculates the digital root for each number without leveraging any previous computations.

TimeO(n * log(n))
SpaceO(n)

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