Fair Distribution of Cookies

Problem Statement

In this problem, you are provided with an array cookies, each element of which represents the number of cookies in a specific bag. Additionally, a number k signifies the total children available for distribution of these bags. The critical rule here is that cookies from the same bag must go entirely to a single child and cannot be divided among multiple children.

The goal is to distribute these bags among the k children such that the unfairness of the distribution is minimized. Unfairness is defined as the highest total number of cookies received by any single child from their allocated bags. Thus, the solution requires finding a distribution where the maximum cookies any child receives is the smallest possible when compared to all other feasible distributions.

Examples

Example 1

Input:

cookies = [8,15,10,20,8], k = 2

Output:

31

Explanation:

One optimal distribution is [8,15,8] and [10,20]
- The 1st child receives [8,15,8] which has a total of 8 + 15 + 8 = 31 cookies.
- The 2nd child receives [10,20] which has a total of 10 + 20 = 30 cookies.
The unfairness of the distribution is max(31,30) = 31.
It can be shown that there is no distribution with an unfairness less than 31.

Example 2

Input:

cookies = [6,1,3,2,2,4,1,2], k = 3

Output:

7

Explanation:

One optimal distribution is [6,1], [3,2,2], and [4,1,2]
- The 1st child receives [6,1] which has a total of 6 + 1 = 7 cookies.
- The 2nd child receives [3,2,2] which has a total of 3 + 2 + 2 = 7 cookies.
- The 3rd child receives [4,1,2] which has a total of 4 + 1 + 2 = 7 cookies.
The unfairness of the distribution is max(7,7,7) = 7.
It can be shown that there is no distribution with an unfairness less than 7.

Constraints

• 2 <= cookies.length <= 8
• 1 <= cookies[i] <= 105
• 2 <= k <= cookies.length

Approach and Intuition

Based on the examples provided and the constraints, the concept behind solving the problem can be outlined as follows:

1. The core challenge is to distribute the bags in such a way that the child who gets the maximum cookies receives as few as possible, compared to other distributions.

2. Given the constraints:

   • The maximum length for cookies is 8, meaning the problem size is relatively small, which hints that computational heavy operations like permutations or combinations might be feasible.
   • The constraints on the values within cookies ([1, 105]) establish that straight calculations on these during permutations won’t cause computational overhead.

3. From the examples:

   • Example 1 gives cookies = [8,15,10,20,8] and k = 2. An optimal grouping here turned out to be [8,15,8] and [10,20], resulting in a maximum of 31 cookies to one child. This instructively shows a split where a balance is somewhat maintained between groups despite one group having more bags.
   • Example 2 illustrates a case with more bags and one additional child (k = 3). The optimal distribution evenly splits the total cookies, hence achieving an outright balance across all groups.

4. The problem likely requires examining all possible distributions of the cookie bags to calculate the maximum cookies a child receives in each scenario, and then selecting the distribution where this maximum is the lowest.

5. Systematically, this can be approached by:

   • Generating all possible ways to distribute the n bags amongst k kids where each kid must get at least one bag.
   • For each distribution, compute the sum of cookies each kid would receive.
   • Record the maximum cookies received by any kid for each distribution.
   • The desired minimum unfairness would thus be the smallest of these recorded maximums.

In conclusion, tackling the unfairness minimization involves exploring combinations of how bags can be distributed among kids, each time calculating and then minimizing the cookies received by the most favored child in terms of cookie count. A suitable algorithm might involve recursive function calls with memorization to efficiently navigate through possible distributions.

Solutions

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);
    }
};