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Distribute Candies to People

We distribute some number of candies, to a row of n = num_people people in the following way:

We then give 1 candy to the first person, 2 candies to the second person, and so on until we give n candies to the last person.

Then, we go back to the start of the row, giving n + 1 candies to the first person, n + 2 candies to the second person, and so on until we give 2 * n candies to the last person.

This process repeats (with us giving one more candy each time, and moving to the start of the row after we reach the end) until we run out of candies.  The last person will receive all of our remaining candies (not necessarily one more than the previous gift).

Return an array (of length num_people and sum candies) that represents the final distribution of candies.

 

Example 1:

Input: candies = 7, num_people = 4
Output: [1,2,3,1]
Explanation:
On the first turn, ans[0] += 1, and the array is [1,0,0,0].
On the second turn, ans[1] += 2, and the array is [1,2,0,0].
On the third turn, ans[2] += 3, and the array is [1,2,3,0].
On the fourth turn, ans[3] += 1 (because there is only one candy left), and the final array is [1,2,3,1].

Example 2:

Input: candies = 10, num_people = 3
Output: [5,2,3]
Explanation: 
On the first turn, ans[0] += 1, and the array is [1,0,0].
On the second turn, ans[1] += 2, and the array is [1,2,0].
On the third turn, ans[2] += 3, and the array is [1,2,3].
On the fourth turn, ans[0] += 4, and the final array is [5,2,3].

 

Constraints:

  • 1 <= candies <= 10^9
  • 1 <= num_people <= 1000

Solution Explanation: Distribute Candies to People

This problem involves distributing a given number of candies to a certain number of people in a specific pattern. The solution employs a simulation approach, iteratively distributing candies until all candies are exhausted.

Approach:

  1. Initialization: Create an array ans of size num_people (number of people), initialized with zeros. This array will store the number of candies each person receives.

  2. Iteration: Iterate until all candies (candies) are distributed. In each iteration:

    • Determine the amount of candies to give to the current person (min(candies, i + 1)), where i is the iteration number. This ensures that the person receives either the remaining candies or the next increment in the sequence (1, 2, 3...).
    • Update the candies received by the current person (ans[i % num_people] += ...). The modulo operator (%) ensures that the candies are distributed in a circular manner to the people.
    • Update the remaining candies (candies -= ...).

  3. Return: Once all candies are distributed, return the ans array.

Time Complexity Analysis:

The time complexity depends on the number of iterations required to distribute all candies. In the worst case, the number of iterations is proportional to the square root of the number of candies. This is because the number of candies distributed in each round increases linearly, forming a roughly triangular pattern. The sum of an arithmetic series is proportional to the square of the number of terms. Therefore, the number of rounds needed is roughly proportional to the square root of candies. However, the number of people also plays a factor limiting the number of iterations. Thus, a more accurate representation would be O(max(√candies, num_people)).

Space Complexity Analysis:

The space complexity is O(num_people) because we use an array of size num_people to store the candies distribution. The space used by other variables is constant and doesn't depend on the input size.

Code Examples (with explanations):

The provided code examples in Python, Java, C++, Go, and TypeScript all follow the same basic algorithmic approach. Let's examine the Python code in more detail:

class Solution:
    def distributeCandies(self, candies: int, num_people: int) -> List[int]:
        ans = [0] * num_people  # Initialize array to store candies for each person
        i = 0  # Iteration counter
        while candies:  # Continue until no candies left
            give = min(candies, i + 1) # determine candies to give to current person
            ans[i % num_people] += give  # Add candies to current person's total
            candies -= give  # Update remaining candies
            i += 1  # Increment iteration counter
        return ans

Each line of the Python code directly corresponds to a step in the algorithmic approach described above. The other languages' code implementations mirror this structure with only syntax differences.