Fair Distribution of Cookies

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RecursionDynamic Programming

You are given an integer array cookies, where cookies[i] denotes the number of cookies in the i^th bag. You are also given an integer k that denotes the number of children to distribute all the bags of cookies to. All the cookies in the same bag must go to the same child and cannot be split up.

The unfairness of a distribution is defined as the maximum total cookies obtained by a single child in the distribution.

Return the minimum unfairness of all distributions.

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 1^st child receives [8,15,8] which has a total of 8 + 15 + 8 = 31 cookies.
- The 2^nd 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 1^st child receives [6,1] which has a total of 6 + 1 = 7 cookies.
- The 2^nd child receives [3,2,2] which has a total of 3 + 2 + 2 = 7 cookies.
- The 3^rd 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

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the number of cookies and the number of children? Specifically, what are the minimum and maximum values for both?
Can the number of cookies be zero? Can any cookie have zero value?
If it's impossible to distribute the cookies fairly, should I return a specific value (e.g., -1, Integer.MAX_VALUE) or throw an exception?
What exactly constitutes a 'fair distribution'? Is the goal to minimize the maximum number of cookies any one child receives, or is there a different metric?
Are we allowed to re-distribute cookies once they are initially given out, or is it a one-time distribution?

Brute Force Solution

Approach

The brute force approach to fairly distributing cookies means we're going to try every single way to give the cookies to the children. We explore every possible distribution and find the one where the unhappiest child has the fewest cookies.

Here's how the algorithm would work step-by-step:

Consider the first cookie. Give it to the first child, then try giving it to the second child, then the third, and so on, until every child has been considered.
For each of those choices of where to give the first cookie, consider the second cookie. Again, give it to the first child, then the second, and so on for every child.
Continue this process for every single cookie, always trying every child as the recipient.
Once all cookies are distributed for a particular arrangement, find the child with the most cookies assigned to them.
Record the number of cookies that child has for that particular arrangement.
Repeat the process for every possible arrangement of cookies to children.
Finally, look at all the recorded 'most cookies' numbers. Choose the smallest one. That's the fairest distribution according to this brute force method.

Code Implementation

def distribute_cookies_brute_force(cookies, number_of_children):
    minimum_unfairness = float('inf')

    def find_minimum_unfairness(cookie_index, cookies_assigned):
        nonlocal minimum_unfairness

        # Base case: all cookies have been distributed
        if cookie_index == len(cookies):
            max_cookies_assigned = max(cookies_assigned)
            minimum_unfairness = min(minimum_unfairness, max_cookies_assigned)
            return

        # Try assigning the current cookie to each child
        for child_index in range(number_of_children):

            # Assign cookie to child and recurse
            cookies_assigned[child_index] += cookies[cookie_index]
            find_minimum_unfairness(cookie_index + 1, cookies_assigned)

            # Backtrack to explore other possibilities
            cookies_assigned[child_index] -= cookies[cookie_index]

    # Start the recursion with no cookies assigned
    initial_assignment = [0] * number_of_children
    find_minimum_unfairness(0, initial_assignment)

    # Find the minimum unfairness found
    return minimum_unfairness

Big(O) Analysis

Time Complexity

O(k^n) – The brute force approach explores every possible distribution of n cookies among k children. For each cookie, there are k possible children it can be assigned to. Since we repeat this assignment process for each of the n cookies, the total number of possible distributions is k * k * ... * k (n times), which equals k^n. Calculating the maximum number of cookies any child has for each distribution takes O(k) time. Therefore, the overall time complexity is O(k^n * k), which simplifies to O(k^n) as the k factor is dominated by k^n.

Space Complexity

O(K) – The brute force approach utilizes recursion to explore all possible cookie distributions. The depth of the recursion is equal to the number of cookies, let's call it N. At each level of the recursion, we have an array to store the number of cookies assigned to each child. The size of this array is equal to the number of children, K. Therefore, the space complexity is determined by the space needed to store the cookies that each child has, which is an array of size K that must be created for each level of recursion, but since we only need to store information about the current recursion, the space complexity will be O(K), where K is the number of children.

Optimal Solution

Approach

The best way to distribute cookies fairly is to think about trying all the ways to divide them among the children, but making smart choices along the way to avoid doing too much work. We explore different distributions but cut off branches that are clearly not leading to a good answer. This prevents unnecessary calculations.

Here's how the algorithm would work step-by-step:

Think of the problem as assigning each cookie to a child.
Start with the first cookie and consider giving it to each child, one at a time.
Keep track of the total number of cookies each child receives.
If, at any point, one of the children receives too many cookies (exceeding the unfairness limit), stop exploring that distribution path.
Continue assigning cookies, always checking for excessive unfairness, and stopping early if it's too high.
Once all cookies are assigned, calculate the 'unfairness' for this distribution (the maximum number of cookies any child received).
Keep track of the *smallest* unfairness you've found so far.
Repeat the process, trying different cookie assignments, cutting off paths that lead to high unfairness, and updating the smallest unfairness you've found.
The smallest unfairness you found is the fairest distribution possible.

Code Implementation

def distribute_cookies(cookies, number_of_children):
    minimum_unfairness = float('inf')

    def backtrack(cookie_index, distribution_so_far):
        nonlocal minimum_unfairness

        if cookie_index == len(cookies):
            # All cookies assigned, calculate unfairness and update min
            current_unfairness = max(distribution_so_far)
            minimum_unfairness = min(minimum_unfairness, current_unfairness)
            return

        for child_index in range(number_of_children):
            # Try assigning the current cookie to each child.
            distribution_so_far[child_index] += cookies[cookie_index]

            # Prune the search space. If current unfairness is already worse,
            # than the best unfairness found, no need to continue.
            if max(distribution_so_far) < minimum_unfairness:
                backtrack(cookie_index + 1, distribution_so_far)

            # Backtrack: remove the cookie from the current child's total.
            distribution_so_far[child_index] -= cookies[cookie_index]

    # Initialize cookie distribution array for each child.
    initial_distribution = [0] * number_of_children

    # Kick off recursive process of assigning each cookie.
    backtrack(0, initial_distribution)

    return minimum_unfairness

Big(O) Analysis

Time Complexity

O(k^n) – The algorithm explores all possible assignments of n cookies to k children. Each cookie can be assigned to any of the k children, leading to k choices for each of the n cookies. This results in a branching factor of k for each of the n levels in the decision tree of cookie assignments. Therefore, in the worst case, the algorithm explores k * k * ... * k (n times) possibilities, which is k^n. The pruning strategy helps to reduce the actual runtime, but the worst-case time complexity remains exponential with respect to the number of cookies.

Space Complexity

O(k^n) – The dominant space complexity comes from the recursion stack. In the worst-case scenario, each cookie (n cookies total) could potentially be assigned to each of the k children. This creates a branching factor of k at each level of recursion, and the depth of the recursion can go up to n (the number of cookies). Therefore, the maximum number of active call stacks is k * k * ... * k (n times) = k^n. Additionally, at each level of the recursion, we need to store the cookies assigned to each child (an array of size k at each level of recursion) adding up to k*k^n however, big O notation disregards constant factors so the overall space complexity is O(k^n).

Edge Cases

Case	How to Handle
Empty cookies array or k = 0	If cookies is empty or k is zero, return 0 since no distribution is possible and unfairness is zero.
k (number of children) is greater than the number of cookies.	Each child receives at most one cookie, so return the largest cookie value.
Cookies array with only one element.	If there's only one cookie, give it to the first child and return its value.
All cookies have the same value.	Regardless of distribution, each child with cookies will have the same unfairness so return that single cookie value
Cookies array contains large integer values.	Use long or appropriate data type to avoid integer overflow during sum calculation and other operations
k = 1 (only one child)	Give all cookies to the single child, so the unfairness is the sum of all cookie values.
Large cookies array and a large number of children, causing recursion depth issues.	Use iterative approach or implement memoization/dynamic programming to avoid stack overflow caused by deep recursion.
No valid distribution exists that minimizes unfairness (extremely skewed cookie values)	The algorithm should always find a valid (if not optimal) distribution and return a corresponding unfairness value based on that distribution.

Empty cookies array or k = 0

How to Handle:

If cookies is empty or k is zero, return 0 since no distribution is possible and unfairness is zero.

k (number of children) is greater than the number of cookies.

How to Handle:

Each child receives at most one cookie, so return the largest cookie value.

Cookies array with only one element.

How to Handle:

If there's only one cookie, give it to the first child and return its value.

All cookies have the same value.

How to Handle:

Regardless of distribution, each child with cookies will have the same unfairness so return that single cookie value

Cookies array contains large integer values.

How to Handle:

Use long or appropriate data type to avoid integer overflow during sum calculation and other operations

k = 1 (only one child)

How to Handle:

Give all cookies to the single child, so the unfairness is the sum of all cookie values.

Large cookies array and a large number of children, causing recursion depth issues.

How to Handle:

Use iterative approach or implement memoization/dynamic programming to avoid stack overflow caused by deep recursion.

No valid distribution exists that minimizes unfairness (extremely skewed cookie values)

How to Handle:

The algorithm should always find a valid (if not optimal) distribution and return a corresponding unfairness value based on that distribution.