Fair Distribution of Cookies

class Solution {
public:
    int recursiveDistribute(int idx, vector<int>& kids, vector<int>& sweets, int numKids, int zeroKids) {
        if (sweets.size() - idx < zeroKids) {
            return INT_MAX;
        }

        if (idx == sweets.size()) {
            return *max_element(kids.begin(), kids.end());
        }

        int minUnfairness = INT_MAX;
        for (int j = 0; j < numKids; ++j) {
            zeroKids -= kids[j] == 0 ? 1 : 0;
            kids[j] += sweets[idx];
            
            minUnfairness = min(minUnfairness, recursiveDistribute(idx + 1, kids, sweets, numKids, zeroKids));
            
            kids[j] -= sweets[idx];
            zeroKids += kids[j] == 0 ? 1 : 0;
        }
        
        return minUnfairness;
    }
    
    int distributeSweets(vector<int>& sweets, int numKids) {
        vector<int> kids(numKids, 0);

        return recursiveDistribute(0, kids, sweets, numKids, numKids);
    }
};

The provided C++ solution revolves around solving the problem of fair distribution of cookies to a certain number of kids. The key objective is to minimize the maximum number of cookies any kid receives, ensuring as fair a distribution as possible. The solution employs a recursive depth-first search (DFS) approach to distribute the cookies.

1. The distributeSweets function serves as the main entry point, initializing each kid's cookie count to zero and calling recursiveDistribute.

2. The recursiveDistribute function uses recursion to explore every possible way of distributing the cookies.

   • It terminates returning an infinitely large value if there are not enough cookies left to ensure each remaining kid gets at least one.
   • It returns the maximum cookies held by any kid when all cookies are successfully distributed, marking the base case of the recursion.
   • For each cookie, the function attempts to distribute it to every kid and recursively call itself to handle the next cookie, updating the minimum observed unfair distribution at each step.

3. An important aspect is ensuring that the distribution continues to track scenarios where any kid has yet to receive a cookie (managed by zeroKids counter), helping to optimize the recursive distribution by avoiding meaningless recursion paths where not every kid can get a cookie.

The solution heavily relies on efficiently managing recursive state transitions as cookies are distributed among kids and backtracking where necessary, ensuring it explores optimal and feasible distributions given the constraints. The complexity of the approach can grow substantially with increasing numbers of kids and cookies due to the combinatorial nature of the problem.

class Distribution {
    private int partition(int index, int[] childCookies, int[] cookies, int children, int countZeros) {
        if (cookies.length - index < countZeros) {
            return Integer.MAX_VALUE;
        }

        if (index == cookies.length) {
            int worstCase = Integer.MIN_VALUE;
            for (int amt : childCookies) {
                worstCase = Math.max(worstCase, amt);
            }
            return worstCase;
        }

        int minimumUnfairness = Integer.MAX_VALUE;
        for (int j = 0; j < children; ++j) {
            countZeros -= childCookies[j] == 0 ? 1 : 0;
            childCookies[j] += cookies[index];

            minimumUnfairness = Math.min(minimumUnfairness, partition(index + 1, childCookies, cookies, children, countZeros));

            childCookies[j] -= cookies[index];
            countZeros += childCookies[j] == 0 ? 1 : 0;
        }

        return minimumUnfairness;
    }

    public int distributeSnacks(int[] cookies, int children) {
        int[] childCookies = new int[children];

        return partition(0, childCookies, cookies, children, children);
    }
}

The provided Java code aims to evenly distribute cookies among children to minimize the maximum amount of cookies any single child receives, thus ensuring fairness. The Distribution class includes methods that work collaboratively to achieve this distribution through a recursive partitioning approach.

• The distributeSnacks function prepares for the recursive partitioning by initializing an array childCookies to store the distribution of cookies for each child and then calls the partition function.
• The partition function is a recursive function designed to distribute the cookies. It attempts various distributions by giving each child one cookie from the remaining cookies and recursively solving the smaller problem.
• The base case for the recursion occurs when all cookies have been considered (index == cookies.length). At this point, the function calculates and returns the maximum number of cookies any child has received, representing the "worst case" unfairness scenario.
• As the function iterates through different cookie distributions, it tracks the minimum value of the maximum cookies any child receives using minimumUnfairness, updating it each time a more fair distribution is found.
• The function also carefully manages scenarios with initially zero cookies using the countZeros parameter, ensuring that the logic handles cases where all children start with no cookies.

Ultimately, distributeSnacks returns the least unfair distribution of cookies among the children, ensuring that the difference between the largest and smallest number of cookies any child receives is as small as possible. This solution approach effectively uses recursion and backtracking to explore different distributions and find the optimal way to ensure fairness in cookie distribution.

class Distributor:
    def allocateCookies(self, sweets: List[int], children: int) -> int:
        child_sweets = [0] * children
        total_sweets = len(sweets)

        def recurse(index, zero_counter):
            if total_sweets - index < zero_counter:
                return float('inf')
            if index == total_sweets:
                return max(child_sweets)
            
            min_disparity = float('inf')
            for child in range(children):
                zero_counter -= int(child_sweets[child] == 0)
                child_sweets[child] += sweets[index]
                
                min_disparity = min(min_disparity, recurse(index + 1, zero_counter))
                
                child_sweets[child] -= sweets[index]
                zero_counter += int(child_sweets[child] == 0)
            
            return min_disparity
            
        return recurse(0, children)

The Fair Distribution of Cookies problem seeks to allocate a list of cookies among a specific number of children in a manner that minimizes the maximum number of cookies any child receives, aiming for an even distribution.

The Python implementation involves defining a class Distributor with the method allocateCookies, which takes two parameters: sweets, a list of integers representing the cookies, and children, the number of children. The solution leverages a recursive approach to explore all possible distributions and determines the one with the minimum disparity in cookie allocation among the children.

• Initialization: An array child_sweets initialized to zero is created to track the number of sweets each child receives. A variable total_sweets stores the number of different sweets available.

• Recursive Function: The inner function recurse is defined with parameters index and zero_counter, which keep track of the current sweet being distributed and the number of children who have received no sweets so far, respectively.

  • The base case checks if it's possible to distribute the remaining sweets; if not, it returns infinity to indicate an invalid distribution. If all sweets are accounted for, it returns the maximum number of sweets any child has.

  • The recursive step involves iterating through each child, attempting to add the current sweet to each child's total one at a time, then recursively calling itself with the next index. It then backtracks by removing the sweet just added—a depth-first search approach to explore all possible distributions.

  • During this process, a min_disparity variable captures the smallest maximum number of sweets any child has across all complete explorations of sweet distribution paths.

• Final Calculation: The allocateCookies method initiates the recursion starting from the first sweet and all children having received no sweets, and returns the minimum possible maximum number of sweets among children, given a fair distribution.

This approach ensures that every potential way of distributing cookies is considered, though it may become computationally intensive with a large number of cookies or children due to its exhaustive nature.