Fair Distribution of Cookies

Identifying Problem Isomorphism

“Fair Distribution of Cookies” can be mapped to “Assign Cookies”.

Both problems revolve around a similar concept where we aim to distribute a set of items (cookies) to another set (children or people) based on certain constraints. The goal is to maximize the distribution in a fair or optimal manner.

In “Assign Cookies”, each child i has a greed factor g[i], which is the minimum size of a cookie that the child will be content with; and each cookie j has a size s[j]. If s[j] >= g[i], we can assign the cookie j to the child i, and the child i will be content. The goal is to maximize the number of content children and output the maximum number.

To map “Fair Distribution of Cookies” to “Assign Cookies”, we need to interpret the fair distribution constraint as the greed factor in the “Assign Cookies” problem. In both problems, we have to sort and iterate over the sets to find the optimal distribution.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

class Solution:
    def distributeCookies(self, cookies: List[int], k: int) -> int:
        min_unfairness = float('inf')
        distribution = [0] * k

        def backtrack(i):
            nonlocal min_unfairness, distribution

            if i == len(cookies):
                min_unfairness = min(min_unfairness, max(distribution))
                return

            # Bounding condition to stop a branch if unfairness already exceeds current optimal solution
            if min_unfairness <= max(distribution):
                return

            for j in range(k):
                distribution[j] += cookies[i]
                backtrack(i + 1)
                distribution[j] -= cookies[i]

        backtrack(0)
        return min_unfairness

All sections needs to be revised based on the working code. Read the editorial and ignore the rest of this document.

Problem Classification

This problem is a Combinatorial Optimization problem.

The problem involves distributing a set of items (cookie bags) among a certain number of recipients (children) in a way that minimizes the maximum total assigned to any single recipient (the unfairness).

The ‘What’ components:

1. The input data: An integer array “cookies”, where each element represents the number of cookies in a bag, and an integer “k” that represents the number of children to distribute the cookies to.

2. The task: The task is to distribute all the cookie bags among the children in a way that minimizes the maximum total cookies a single child gets.

3. The output: The output is the minimum possible unfairness, defined as the maximum total number of cookies a single child gets in the optimal distribution.

The problem can be classified as a Combinatorial Optimization problem because it involves finding an optimal object from a finite set of objects. In this case, the optimal object is the distribution of cookie bags. The solution needs to explore different ways of distributing the cookie bags (combinations) and find the one that minimizes the unfairness (optimization).

This problem can be seen as a Partitioning Problem, a special case of combinatorial optimization, where we partition a set into subsets. In this case, we are partitioning the set of cookie bags into subsets representing each child’s share.

It also has aspects of a Minimax Problem since we’re trying to minimize the maximum number of cookies any single child gets.

A good starting point for solving this kind of problem could be sorting the array and applying a greedy algorithm, or considering dynamic programming for optimal substructure and overlapping sub-problems. However, due to the constraints, exploring all possible combinations might also be feasible. It depends on the specifics of the problem and the properties of the solution space.

Problem Restatement

You’re given a list of bags, each containing a certain number of cookies, and a specific number of kids. Your task is to distribute the bags among the kids. The catch is that you can’t split a bag - it must go entirely to a single kid.

The “unfairness” of a distribution is determined by the child who ends up with the highest total number of cookies. Your goal is to arrange the distribution in such a way that this maximum value (the “unfairness”) is as small as possible.

You need to come up with a function or method that takes in the list of cookie bags and the number of children, and returns the minimum possible “unfairness” value.

Important to note is that the length of the cookies array is between 2 and 8 (inclusive), each element in the array can be between 1 and 105 (inclusive), and the number of children is between 2 and the length of the cookies array (inclusive). These constraints can help guide the solution strategy, as they limit the size of the problem space.

Abstract Representation of the Problem

You have an array of integers where each integer represents a quantity grouped together. You’re also given another integer which represents the number of distinct groups you have to distribute these quantities into. Each quantity must belong to a single group entirely and cannot be divided.

The disparity of a distribution is defined as the maximum total quantity of a single group in the distribution.

Your task is to find the minimum disparity possible in all possible distributions.

Constraints:

• The array has a length between 2 and 8 (inclusive)
• Each element in the array is between 1 and 105 (inclusive)
• The number of groups is between 2 and the length of the array (inclusive)

Terminology

In the context of this problem, the following terms and concepts are crucial:

1. Distribution: This refers to the process of dividing or allocating the cookies among the children. Each distribution represents a unique way of giving out the cookies.

2. Unfairness: This is a metric defined in this problem as the maximum total cookies obtained by a single child in the distribution. Essentially, the unfairness measures the disparity or inequality in the number of cookies received by the children.

3. Optimal Distribution: An optimal distribution in this context is the one that results in the minimum unfairness. The objective is to find such a distribution.

4. Constraints: These are the conditions that your solution must satisfy. In this problem, constraints include the number of cookies in each bag, the number of bags, and the number of children to distribute to. Understanding these constraints is crucial to developing a correct and efficient solution.

Understanding these terms and concepts is essential for formulating an effective strategy to solve the problem. They define the problem space and provide the criteria that a valid solution must meet.

Problem Simplification and Explanation

This problem is all about sharing cookies from different cookie bags among a number of children. The key thing to remember is that cookies from the same bag must be given to the same child - you can’t divide a single bag between two or more children.

Now, we need to distribute these cookie bags in such a way that the child who gets the most cookies (the “unfairness”) is as small a number as possible. We call this the minimum unfairness, and that’s what we want to find.

To put it in terms of an analogy, imagine you’re the host of a children’s party, and you’ve got a bunch of goody bags to hand out at the end. Each goody bag has a different number of candies in it. You want to distribute the goody bags among the children in a way that the child who ends up with the most candies has as few as possible. That way, no child feels overly favored or left out.

The key concepts here are the distribution of bags (how we give out the bags), and the unfairness (the maximum number of candies any child receives). We need to manipulate the distribution to minimize the unfairness.

Now, there are some restrictions or rules (the constraints) to keep in mind:

1. Each bag of cookies can only be given to one child - we can’t split a bag between two or more children.
2. We know exactly how many cookies are in each bag and how many children there are to distribute to.

In essence, this problem is a balance of fairness and logistics, similar to planning a children’s party with goody bags. It’s about finding the best way to distribute resources under certain rules and restrictions.

Constraints

Let’s go over the constraints:

1. 2 <= cookies.length <= 8: The number of cookie bags is fairly small, ranging from 2 to 8. This means that we can potentially use strategies that might not be as efficient with larger input sizes, such as trying all possible distributions of bags.

2. 1 <= cookies[i] <= 105: Each bag contains between 1 and 105 cookies. The maximum number of cookies is quite large relative to the number of bags, which means we could have significant disparity in the number of cookies per bag. This can affect how we choose to distribute the bags.

3. 2 <= k <= cookies.length: We have to distribute the cookie bags to at least two children and at most to as many children as there are bags. This means every child will get at least one bag, and no child will get more bags than there are available.

Given these constraints, we can identify a few potential approaches:

• Since the number of cookie bags is quite small, we could potentially use a brute-force approach to try all possible distributions and then pick the one with the smallest unfairness. We can do this efficiently because the number of possible distributions is manageable due to the small number of bags.

• The large range of possible cookies per bag means that distributing the bags with the most cookies first might not always lead to the smallest unfairness. It could be beneficial to explore strategies that consider the number of cookies in each bag, such as sorting the bags by the number of cookies.

These constraints and their implications provide a starting point for developing an algorithm to solve this problem.

Identification of Applicable Theoretical Concepts

There are several concepts that may come into play in solving this problem:

1. Combinatorics/Permutations: We are dealing with distributing items (cookie bags) into different groups (children). This inherently involves concepts from combinatorics. Specifically, we need to figure out all the possible ways to distribute the cookie bags among the children, which is a permutation problem.

2. Greedy Algorithms: We might think about distributing the bags in a certain order (for example, from most to least cookies) to minimize unfairness. This is a strategy often used in greedy algorithms, where we make the locally optimal choice at each step in the hopes that these local optimums will lead to a global optimum.

3. Dynamic Programming: Given the small input size, we might be able to use dynamic programming to remember and re-use answers to sub-problems (i.e., the minimum unfairness when distributing a subset of the bags) to build up the answer to the overall problem.

4. Sort: Sorting the array can help us organize the cookie bags in a certain way, such as in ascending or descending order, which could simplify the distribution process.

5. Divide and Conquer: We might be able to use a divide-and-conquer strategy, where we split the problem into smaller, more manageable sub-problems (for example, distributing a subset of the bags), solve each sub-problem, and then combine these solutions to solve the original problem.

6. Minimization/Maximization: The goal is to minimize the maximum total cookies obtained by a single child. This is a common type of optimization problem in both mathematics and computer science, and various strategies can be used to solve it, such as binary search, linear programming, etc.

These are the broad concepts that seem applicable to this problem. The final solution might involve one or more of these concepts, or perhaps even others that are not listed here.

Stepwise Refinement

Let’s explore a potential stepwise refinement for this problem. I’ll outline a general approach. Here’s a simple high-level plan to begin with:

1. Sort the Cookie Bags: First, sort the array of cookie bags in ascending order. This will allow us to distribute the cookies more evenly among the children, since we can start by giving the smallest bags to the children with the fewest cookies so far.

2. Initial Distribution: Distribute one bag of cookies to each child. If the number of children is less than or equal to the number of cookie bags, each child gets at least one bag. If not, some bags would be left undistributed for now.

3. Remaining Distribution: Continue distributing the remaining bags to the children. Always give the next smallest bag of cookies to the child who currently has the fewest total cookies. This is where we leverage a greedy approach.

4. Calculate Unfairness: After all bags have been distributed, the unfairness is the maximum total number of cookies any child received. This is because we’re aiming to minimize this maximum value.

Now, let’s refine this high-level plan into more specific steps:

1. Sort the Cookie Bags: Use any efficient sorting algorithm (like quicksort, mergesort, etc.) to sort the array in non-decreasing order.

2. Initial Distribution: Initialize an array (or a suitable data structure like a min heap) of size k (the number of children) to keep track of the total cookies each child has received. Distribute the first k cookie bags (if available) to each child.

3. Remaining Distribution: For the remaining cookie bags, always give the next smallest bag to the child who currently has the fewest total cookies. This can be achieved by extracting the child with the minimum cookies from the data structure, adding the next smallest bag of cookies to their total, and then inserting them back into the data structure.

4. Calculate Unfairness: After all bags have been distributed, extract all elements from the data structure and the maximum among these will be our answer. This maximum value represents the minimum possible unfairness, i.e., the maximum number of cookies received by a child in the most evenly distributed scenario.

Remember, this is just one potential approach. There may be other, perhaps more efficient, ways to solve this problem. The right approach often depends on the specific requirements and constraints of your situation. For example, if the number of cookie bags or children is very large, it may be more efficient to use a different strategy or data structure.

Solution Approach and Analysis

Let’s break down the problem-solving process in detail. This problem is essentially a minimization problem in which we are trying to minimize the maximum total number of cookies any child can receive.

In an abstract sense, this is similar to balancing a scale with unequal weights. We want to distribute the weights (cookies in this case) as evenly as possible to maintain balance (fairness).

Let’s discuss the approach using the four-step plan I mentioned earlier:

1. Sort the Cookie Bags: We sort the bags in ascending order so that we can start distributing from the smallest bag to ensure fairness. This is similar to first placing the lightest weights on the scale. For example, if we have the cookie bags as [8,15,10,20,8] and k = 2, after sorting we get [8,8,10,15,20].

2. Initial Distribution: Initialize an array or a priority queue (min-heap) to keep track of total cookies each child has received. Now distribute one bag of cookies to each child. For instance, we give one bag each to the two children, so our distribution may look like [8,8] and the remaining bags are [10,15,20].

3. Remaining Distribution: Now, we want to distribute the remaining bags. The goal is to always give the next smallest bag to the child who currently has the fewest total cookies. So we add the bag with 10 cookies to the first child, and the distribution becomes [18,8]. We continue this process with the remaining bags until all bags are distributed.

4. Calculate Unfairness: The unfairness is the maximum total number of cookies any child received. After all bags are distributed, our distribution becomes [33,28]. Hence, the unfairness in this case is 33.

So, how would changes in the problem’s parameters affect the solution?

1. Increase in the number of cookie bags: An increase in the number of cookie bags would simply mean more iterations of our distribution step. The nature of the problem remains the same.

2. Increase in the number of children (k): An increase in k would mean the initial distribution step (Step 2) would distribute bags to more children. This would also potentially decrease the unfairness as the cookies are distributed among more children.

3. Variations in the number of cookies in the bags: If the bags have a wide range of cookie counts, it may lead to a higher unfairness score since some children might end up with bags that have significantly more cookies. Conversely, if the bags have similar cookie counts, the unfairness would be less as the distribution would be more balanced.

Remember, this approach leverages a “greedy” strategy where at each step, we are making the choice that seems best at that moment, i.e., giving the next smallest bag to the child with the fewest cookies so far.

Thought Process

The key in the problem statement is the requirement to minimize the maximum total number of cookies a child gets. This points towards an optimization problem. We can interpret the “unfairness” as a measure of imbalance in the distribution of cookies. Hence, we are looking for a way to balance the distribution of cookies across children.

Another cue is that all cookies in a bag must go to one child, so we can’t split the content of a bag between children. This suggests that we should consider each bag as a single unit when thinking about distributing cookies.

A useful insight is that distributing the smallest number of cookies available to the child who has the least amount of cookies so far would likely minimize the maximum total cookies any child would get. This points to a potential “greedy” algorithm, where we make the optimal choice at each step with the hope that these local optimums will lead to a global optimum.

Given these insights, we can follow this approach:

1. Sort the bags of cookies in ascending order.
2. Initialize an array (or priority queue) to keep track of the total cookies each child has received.
3. Distribute the bags one by one, starting from the smallest bag, to the child with the least total cookies so far.

From Brute Force to Optimal Solution

Let’s start by examining the brute force solution.

Brute Force Solution

The most straightforward, or brute force, approach to this problem would be to generate all possible distributions of bags to the children, and then check each one to find the distribution with the minimum maximum total cookies. This could be done using recursive backtracking to generate all possible distributions.

However, the time complexity of this approach would be prohibitively high, likely on the order of O(n^k), where n is the number of bags and k is the number of children. This is because for each bag, we need to consider distributing it to each of the k children, and we have to do this for all n bags.

Clearly, this approach would be far too inefficient to be practical for even moderately sized inputs.

Moving Towards an Optimized Solution

Recognizing the inefficiency of the brute force solution, we need to find a way to avoid checking all possible distributions. As mentioned before, a greedy strategy seems promising here.

A key observation is that to minimize the maximum total cookies, we should aim to balance the distribution as much as possible. This suggests that at each step, we should distribute the next bag of cookies to the child who currently has the least total cookies. This can be done efficiently with a priority queue or heap, which can keep track of which child has the least cookies and allow us to update this in logarithmic time.

Optimized Solution

Here’s the Python code for the optimized solution, using the heapq module’s heap functionality to implement a priority queue:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14

import heapq

def minUnfairness(cookies, k):
    cookies.sort(reverse=True)
    totals = [0]*k
    heap = [(val, idx) for idx, val in enumerate(totals)]
    heapq.heapify(heap)

    for cookie in cookies:
        val, idx = heapq.heappop(heap)
        totals[idx] += cookie
        heapq.heappush(heap, (totals[idx], idx))

    return max(totals)

Time Complexity

The time complexity of this solution is O(n log k), where n is the number of bags and k is the number of children. The log k term comes from the heap operations (pushing and popping), which we do for each of the n bags. This is a significant improvement over the brute force approach.

Space Complexity

The space complexity of this solution is O(n + k). The n term is for the space needed to store the cookies array, and the k term is for the space needed to store the totals array and heap. If the input is not counted towards the space complexity, then the space complexity is O(k), because the space required for the totals and heap is proportional to the number of children, not the number of bags. This is also quite efficient.

This illustrates how we can optimize a brute force solution by applying observations and insights about the problem to design a more efficient algorithm. It’s important to always consider the time and space complexity of your solution and look for ways to improve it.

Coding Constructs

1. High-Level Problem-Solving Strategies: This code utilizes a Greedy algorithm, where at each step it makes the decision that seems best at the moment, aiming to balance the distribution of cookies among children. It also uses a Priority Queue (implemented with a heap) to efficiently select the child with the least cookies at each step.

2. Explaining to a Non-Programmer: Imagine you’re a teacher distributing cookies to children in your class. You want to make sure the child with the least cookies always gets the next batch, so no one child ends up with way more cookies than the others. This is what the program does - it always gives the next batch of cookies to the child with the fewest cookies at that moment.

3. Logical Elements: The main logical elements in this code include sorting, iteration (looping over the array of cookies), and the use of a priority queue (heap) to efficiently determine which child should get the next batch of cookies.

4. Algorithmic Approach in Plain English: First, the code sorts the cookies in descending order. Then, it creates a “heap” structure, which is just a clever way to keep track of which child currently has the least cookies. The code then goes through each batch of cookies (from largest to smallest) and gives it to the child who currently has the least cookies.

5. Key Steps or Operations: The key steps include sorting the cookies array, initializing the heap with the current totals of cookies for each child (starting with zero), and then iterating over the cookies array. In each iteration, the code removes (pops) the child with the least cookies from the heap, adds the next batch of cookies to their total, and then puts (pushes) them back into the heap. This ensures that the child with the least cookies is always at the front of the heap.

6. Algorithmic Patterns: This code follows the Greedy algorithmic pattern, where the locally optimal choice is made at each step in the hope that these local choices will lead to a global optimum. It also uses a heap to implement a Priority Queue, which is a common data structure for efficiently selecting an item with the highest (or lowest) priority.

Language Agnostic Coding Drills

Let’s deconstruct the solution for the “Minimum Unfairness in Cookie Distribution” problem and identify the underlying coding concepts.

1. Array or List Manipulation: Difficulty - Easy. This concept is about understanding how to work with lists (or arrays), such as accessing elements, iterating over them, or manipulating them. It’s a fundamental concept in any programming language.

2. Sorting: Difficulty - Easy. Sorting is an algorithmic concept that orders the elements of a list in a specific sequence - either ascending or descending. In this problem, we sorted the cookies list in descending order.

3. Heap Data Structure: Difficulty - Medium. Heaps are a type of binary tree data structure that satisfies the heap property. They’re used to efficiently get and remove the smallest element in the heap, and in our case, we used a min-heap. Understanding heaps involves understanding binary trees and the operations that can be performed on them, making it a medium-difficulty concept.

4. Heap Manipulation: Difficulty - Medium. This involves understanding how to use the heap data structure provided by the programming language’s standard library, which includes methods for creating a heap, pushing elements, and popping elements. In our solution, we converted a list into a heap and used push and pop operations.

5. Problem Solving and Logic Building: Difficulty - Hard. This involves understanding the problem, mapping the problem to known data structures, algorithms, or techniques, and devising a step-by-step approach to reach the solution. Here, the challenge was realizing that a heap could efficiently track the child with the least cookies, enabling an optimal distribution of the cookies.

Problem-solving approach:

1. We start with the problem statement and understand that we have to distribute the cookies such that the difference between the child who gets the most and the least is minimized. This involves understanding the requirements and constraints of the problem.

2. Next, we think about what data structures and techniques could help us achieve the distribution efficiently. Here, we realize that a heap data structure can be used to efficiently track the child with the least cookies.

3. Then, we think about how to distribute the cookies. We realize that giving the biggest cookie to the child with the least total number of cookies so far would minimize the unfairness. This is where we use the heap to efficiently get and update the child with the least cookies.

4. To implement this approach, we first initialize a heap with zeros, each representing a child. Then, we sort the cookies in descending order.

5. Next, we distribute the cookies. For each cookie, we pop the child with the least cookies from the heap, give the cookie to the child, and push the child back into the heap.

6. Finally, after all cookies are distributed, the maximum total cookies a child has received (which is the last element in our heap) is our answer. This involves understanding that the heap structure will have the maximum element at the end after all the distributions are done.

Remember, the actual coding part is a translation of this thought process into a specific programming language syntax.

Targeted Drills in Python

Sure, I can create coding drills based on the concepts I identified earlier.

1. Array or List Manipulation: In Python, arrays are represented as lists. Here’s a basic drill to illustrate list manipulation in Python:

   1 2 3 4 5 6 7 8 9 10 11 12 13

   # Define a list nums = [1, 2, 3, 4, 5] # Access elements first_num = nums[0] last_num = nums[-1] # Iterating over a list for num in nums: print(num) # List slicing first_three = nums[:3]

   This concept is fundamental to our problem as we need to iterate over the list of cookies and manipulate it.

2. Sorting: Sorting a list in Python is straightforward using the built-in sort() function.

   1 2 3

   nums = [5, 3, 1, 2, 4] nums.sort() # sorts in ascending order nums.sort(reverse=True) # sorts in descending order

   We need to sort the cookies in descending order before we distribute them.

3. Heap Data Structure and Heap Manipulation: Python provides the heapq module to use heaps.

   1 2 3 4 5 6 7 8 9 10 11 12

   import heapq # Define a heap heap = [] # Push elements into the heap heapq.heappush(heap, 1) heapq.heappush(heap, 2) heapq.heappush(heap, 3) # Pop the smallest element smallest = heapq.heappop(heap)

   In our problem, we use a heap to keep track of the child with the least cookies, and we manipulate the heap using the heappush and heappop functions.

Problem-specific Drills:

1. Using a Heap to Track Minimum Element: A key concept in our problem is using a heap to efficiently track and update the minimum element. This drill illustrates that concept:

   1 2 3 4 5 6 7 8 9 10 11 12 13

   import heapq # Define a heap heap = [3, 1, 2] # Transform list into a heap heapq.heapify(heap) # Pop the smallest element smallest = heapq.heappop(heap) # Push a new element into the heap heapq.heappush(heap, 0)

   In our problem, we need to continually update the heap by popping the minimum element and pushing the updated value back in.

Integrating the Drills:

These drills cover the individual concepts used in our solution. To solve the problem, we integrate these drills as follows:

1. First, we represent the cookies and children using lists.

2. We sort the cookies in descending order.

3. We initialize a heap with zeros to represent the children.

4. We then iterate over the sorted cookies. For each cookie, we pop the child with the least cookies from the heap, give the cookie to the child, and push the child back into the heap.

5. Finally, after all the cookies are distributed, the last element in our heap is the maximum total cookies a child has received, which is our answer.