Distribute Candies to people

Problem Overview

The challenge, "Distribute Candies to People," requires designing a strategy to distribute a given number of candies to a line of people. The distribution must follow a progressively increasing pattern, where people line up in a circular fashion, and the first person gets the smallest number of candies, gradually increasing as the line progresses.

Understanding the Problem

Given:

• n: The total number of candies.
• k: The number of people in a group, standing in a line.

The task is to distribute candies such that:

1. The first person receives 1 candy, the second gets 2 candies, and so forth.
2. Once the count reaches the k-th person, the distribution resumes, in a circular fashion, until all candies are distributed.

The expected output is an array representing the candies received by each person.

Key DSA Concepts and Theory

Binary Search

Binary search is not directly applied here since the problem can be addressed more effectively with a mathematical approach. However, knowing how to efficiently partition a search space can aid in understanding distribution-related problems.

Mathematical Summation

A key concept used here is the understanding of arithmetic series:

• The formula for the sum of the first m natural numbers is ( S_m = \frac{m(m + 1)}{2} ).

Our goal is to distribute candies in such a fashion where the increment and total calculation mimic a summative sequence split across a finite group.

Solution Approach

To solve this problem, follow these steps:

1. Initialize an Array: Create an array result of size k initialized to zeros to store the candies distributed to each person.

2. Simulate the Distribution: Use a loop to simulate the distribution, starting to give out candies from 1 to until no candies are left.

   • Each loop iteration is divided into rounds, considering the incrementing candy requirement starting from 1.
   • Use a loop index to track which person in the line currently receives candies.

3. Iterate and Distribute:

   • Maintain a variable to track the current number of candies to distribute.
   • Use modulo operation to reset current_index back to zero when it reaches k after one full round. This simulates the circular distribution.
   • Subtract the current number of candies from n (remaining candies), distribute it, and increment the current number.

Below is an implementation in C++ and Java.

C++ Code

#include <vector>

std::vector<int> distributeCandies(int candies, int num_people) {
    std::vector<int> result(num_people, 0);
    int current_candy = 1;
    int current_index = 0;
    
    while (candies > 0) {
        result[current_index] += std::min(current_candy, candies);
        candies -= current_candy;
        current_candy++;
        current_index = (current_index + 1) % num_people;
    }
    
    return result;
}

Java Code

class Solution {
    public int[] distributeCandies(int candies, int num_people) {
        int[] result = new int[num_people];
        int currentCandy = 1;
        int currentIndex = 0;
        
        while (candies > 0) {
            result[currentIndex] += Math.min(currentCandy, candies);
            candies -= currentCandy;
            currentCandy++;
            currentIndex = (currentIndex + 1) % num_people;
        }
        
        return result;
    }
}

Time and Space Complexity Analysis

• Time Complexity: (O(\sqrt{n})), where n is the total number of candies. This is derived by solving the inequality for when the number of candies distributed becomes less than zero.
• Space Complexity: (O(k)) where k is the number of people, as we need an array to store the distribution results.

Common Mistakes to Avoid

1. Incorrect Loop Bounds: Ensure the while-loop is tied to the correct condition with the remaining number of candies.
2. Modulo Operation Misuse: Ensure the circular iteration uses modulo correctly to avoid out-of-bounds errors.

Similar Problems on LeetCode

1. LeetCode 410 - Split Array Largest Sum
2. LeetCode 287 - Find the Duplicate Number
3. LeetCode 378 - Kth Smallest Element in a Sorted Matrix

Additional Resources and References

• Geeks for Geeks on Arithmetic Series: Understanding Arithmetic Progression
• Lecture Notes on Binary Search: Helpful in understanding partition search spaces for optimal sub-solution identification.