Assign Cookies

In this problem, the idea is to give each child the smallest cookie that will satisfy their greed factor. This way, you can maximize the number of content children. Here’s a step-by-step approach to solve this problem:

Sort both the g (greed factors of children) and s (sizes of cookies) arrays. Sorting helps us to match the smallest greed factor with the smallest available cookie.
Initialize two pointers, let’s say i and j, to point to the beginning of g and s respectively.
If g[i] <= s[j], it means the cookie size satisfies the child’s greed factor. Hence, we increment both i and j.
If g[i] > s[j], it means the cookie does not satisfy the child’s greed factor. Hence, we move on to the next size by incrementing j.
Continue this until we have exhausted either the greed factor array or the cookie size array. The count of content children will be equal to the pointer i.

Python solution:

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class Solution:
    def findContentChildren(self, g: List[int], s: List[int]) -> int:
        g.sort() # Sort the greed factor array
        s.sort() # Sort the cookie sizes array
        i = j = 0 # Initialize two pointers

        while i < len(g) and j < len(s): # While there are still children and cookies
            if g[i] <= s[j]: # If the current cookie satisfies the current child
                i += 1 # Move to the next child
            j += 1 # Move to the next cookie regardless

        return i # Return the count of content children

This algorithm runs in O(n log n) time due to the sorting, where n is the length of the longer list between g and s. The space complexity is O(1) because we only use a constant amount of extra space.

Identifying Problem Isomorphism

“Assign Cookies” requires you to maximize the number of children you can make content by assigning each child a cookie with size at least as large as the child’s greed factor. You are required to sort the greed factors and sizes, then iterate through them to make assignments.

An isomorphic problem is “455. Boat to Save People”. In this problem, you are given two arrays, one represents the weight of each person and the other represents the maximum weight each boat can carry. Each boat can carry at most two people at the same time, provided the sum of the weight of those people is at most the limit of the boat. Your goal is to minimize the number of boats. Although the conditions are different, the core logic of sorting and iterating is the same.

“Boat to Save People” is more complex than “Assign Cookies” because in “Boat to Save People” a boat can carry up to two people whereas in “Assign Cookies” each cookie can only be assigned to one child.

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class Solution:
    def findContentChildren(self, g: List[int], s: List[int]) -> int:
        g.sort()
        s.sort()
        i, j = 0, 0
        while i < len(g) and j < len(s):
            if s[j] >= g[i]:
                i += 1
            j += 1
        return i

Problem Classification

This problem is a resource allocation problem, falling under the optimization domain. We have a set of resources (cookies), a set of consumers (children), and an objective to satisfy (maximizing the number of content children). Let’s breakdown the ‘What’ components:

Resources (Cookies): Each cookie has a size s[j]. We have an array of these resources.
Consumers (Children): Each child has a greed factor g[i], which represents the minimum size of a cookie that will satisfy them. We have an array of these consumers.
Constraints: The number of cookies and children, and the sizes of the cookies and the greed factors of the children, have specific limitations.
Objective: We need to maximize the number of children who can be given a cookie that satisfies their minimum greed factor.

The key challenge here is to allocate the cookies in such a way that maximizes the satisfaction of the children. It can be further classified as a “Greedy Algorithm” problem, since at each step we try to make the decision that seems most promising at that time. It leads to the optimal solution under the problem’s constraints. The problem has a decision-making process involved, to match cookies to children based on their size and greed factor, respectively, hence it also falls under the category of Decision Making problems.

We are asked to identify the maximum possible number of satisfied children given the constraints, which makes this a Maximization Problem. It could also be seen as a Matching Problem, as we’re essentially matching cookies to children based on a specific rule.

This problem belongs to the domain of Greedy Algorithms. The reason is that we are asked to make a locally optimal choice at each step (i.e., assign the smallest possible cookie to each child) with the goal of maximizing the total number of satisfied children.

Additionally, it also involves elements of Sorting, since we need to sort both the children’s greed factors and the cookies’ sizes to match them effectively.

The task can be further classified under Array Manipulation as we are given the data in the form of arrays and we need to manipulate and iterate through these arrays to find our solution.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

Constraints

Here are a few characteristics that can be exploited for an efficient solution to this problem:

Sorting: The size of cookies and the greed factor of children are numerical values that can be sorted. Once sorted, we can use a greedy approach to assign the smallest possible cookie that would satisfy a child, ensuring the maximum number of children are satisfied. This is because sorting brings a certain order to our data, allowing us to make efficient decisions about allocations.
Greedy Strategy: The objective is to maximize the number of content children. This means we want to satisfy as many children as possible, even if some get just enough to meet their greed factor. A child will be content as long as they receive a cookie of size greater than or equal to their greed factor. So, it makes sense to start with the least greedy child and give them the smallest possible cookie that would satisfy them. This ensures that larger cookies are available for children with larger greed factors.
Multiple Valid Outputs: It’s important to note that there might be multiple correct outputs. As long as the number of content children is maximized, the specific allocation of cookies to children doesn’t matter. This gives us some flexibility in our approach.
Range of Values: The problem constraints indicate that the sizes of cookies and the greed factors of the children can be quite large (up to 2^31 - 1). This suggests that simply iterating through all possible combinations could be very inefficient. Therefore, we need a solution that doesn’t rely on brute force.
Single Allocation: Each child can receive at most one cookie. This prevents us from splitting cookies to satisfy more children and simplifies the problem to making single choices for each child.
Infinite Greed Not Possible: As the greed factors are finite numbers, there won’t be a situation where a child’s greed cannot be satisfied by any cookie, unless the greed factor is larger than all available cookies, of course.

By recognizing these characteristics, we can make more informed decisions about how to approach this problem, and design an efficient algorithm to solve it.

Solution Approach and Analysis

Step 1: Sorting: First, we sort both arrays in ascending order. The greed factor array is sorted so that we can satisfy the least greedy children first. The cookie size array is sorted so we can efficiently pick the smallest possible cookie that will satisfy a child. For example, if we have greed factors [1,2,3] and cookie sizes [1,2], after sorting they remain the same. But if we have [2,1,3] and [2,1], after sorting we have [1,2,3] and [1,2].
Step 2: Pairing: Starting from the least greedy child and the smallest cookie, we check if the cookie can satisfy the child (i.e., the cookie size is greater than or equal to the child’s greed factor). If it does, we move to the next child and the next cookie. If it doesn’t, we keep the child but move to the next cookie (essentially discarding the current cookie as it can’t satisfy even the least greedy child). We continue this process until we run out of either cookies or children.

To visualize this, imagine you’re lining up the children and the cookies based on their size and greed factor. You start from the smallest child and the smallest cookie. If the cookie is big enough, you give it to the child, and both move to the next in line. If not, the child stays, but you move to the next bigger cookie.

Step 3: Counting: As we make successful pairings, we keep a count of the satisfied children. The aim is to maximize this count, which will be our final answer.

For example, if we have greed factors [1,2,3] and cookie sizes [1,2], we would pair the cookie of size 1 with the child with a greed factor of 1, and the cookie of size 2 with the child with a greed factor of 2. The final count of satisfied children would be 2.

If we change the parameters of the problem, such as the number of cookies or the greed factors of the children, it will affect the final count. For example, if we increase the size of all cookies, more children could potentially be satisfied. On the other hand, if we increase the greed factor of all children while keeping the same cookies, fewer children could be satisfied.

A significant change in the parameters might make a different algorithm more efficient. For example, if we have many more cookies than children, a more sophisticated algorithm that uses a priority queue instead of sorting might be faster. But for most cases, the simple sort-and-pair strategy would work well.

Thought Process

Understand the problem statement: The problem is essentially about satisfying the maximum number of children, each having a specific greed factor, with the available cookies that have certain sizes. A child is satisfied if the size of the cookie is greater than or equal to his/her greed factor.
Identify cues in the problem: The term ‘maximum number’ signals that it’s an optimization problem. The part ’s[j] >= g[i]’ implies a relationship between the elements of two different sets which needs to be satisfied.
Approach direction: We need to create pairs of children and cookies in a way that maximizes the number of satisfied children. The strategy that comes to mind is to sort both arrays, and try to satisfy the least greedy children first with the smallest possible cookie.
Generate insights: Sorting the arrays is crucial to the solution as it enables us to make the most out of the available cookies. By attempting to satisfy the least greedy children first, we leave the larger cookies for children with higher greed factors.

Let’s code the solution:

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def findContentChildren(g, s):
    g.sort()
    s.sort()

    child_i = cookie_j = 0
    # Start from the least greedy child and the smallest cookie.
    while child_i < len(g) and cookie_j < len(s):
        if s[cookie_j] >= g[child_i]:  # If the cookie can satisfy the child,
            child_i += 1  # move to the next child.
        cookie_j += 1  # Always move to the next cookie.

    return child_i  # The number of satisfied children.

This function sorts the greed factor array g and the size array s, then tries to pair them starting from the smallest values. It counts the number of successful pairings and returns this count as the result. The runtime complexity is O(n log n) due to the sorting operations, and the space complexity is O(1) as no additional data structures are used.

Separate the What from How

What:

Maximizing the number of satisfied children: The problem involves maximizing a quantity - the number of children who receive a cookie that meets or exceeds their “greed factor”.
Array Manipulation: The problem deals with manipulating arrays - the children’s greed factors and the cookies’ sizes are given in array format.

How:

Greedy Algorithms: A greedy algorithm is a simple, intuitive algorithm used in optimization problems, which makes the locally optimal choice at each step with the hope that these local optimizations will lead to a global optimum. Here, it refers to the approach of assigning the smallest possible cookie to each child to satisfy as many children as possible.
Sorting: Sorting is a technique of arranging data in a particular format or sequence, here it refers to sorting the arrays of children’s greed factors and cookie sizes so that we can assign cookies effectively.

If we focus on the What (needs to be done from the problem statement without considering how it might be implemented), how would you classify this problem?

When focusing on the “What” of the problem, it can be classified as an Optimization problem. Specifically, this is a Resource Allocation problem, where the resources are the cookies and the task is to distribute them among the children to maximize the number of satisfied children. The children can be thought of as tasks, each with a “cost” of satisfaction, and the goal is to allocate cookies in a way that maximizes the number of tasks completed.

Language Agnostic Coding Drills

Understanding the problem: The problem describes a scenario where you need to distribute cookies (resources) to children (tasks), each with a different greed factor (task cost). You can only assign one cookie to each child, and a cookie can satisfy a child if its size is greater or equal to the child’s greed factor. The goal is to maximize the number of satisfied children.
Sorting: Sorting is a key part of this problem’s solution. Both the children (by their greed factors) and the cookies (by their sizes) are sorted in increasing order.
Two-pointer approach: This problem uses the two-pointer approach to traverse two sorted arrays. One pointer i is for the array g (children’s greed factors) and another pointer j is for the array s (cookies’ sizes).
Conditional incrementation: Depending on the condition that if a cookie can satisfy a child’s greed, increment the respective pointers.
Comparison and decision-making: You have to compare the greed factor of the child with the size of the cookie. If the size of the cookie is greater than or equal to the greed factor of the child, you can assign the cookie to the child, thus increasing the count of content children.
Loop termination: The loop continues as long as there are children and cookies left to consider, and it stops when either the cookies or the children are exhausted.
Result return: The total number of content children is returned, which is tracked by the index i.

By learning these concepts separately, you’ll build a strong understanding of each piece. When combined, they’ll form a cohesive problem-solving strategy for this problem.

Targeted Drills in Python

Understanding the problem: For this drill, the task will be to create two lists, one representing the greed factor of children and the other the sizes of cookies. This will help in understanding the basic setup of the problem.
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g = [1,2,3] # Greed factors of children s = [1,1] # Sizes of cookies

Sorting: Here, the task will be to sort both lists in increasing order.

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g.sort()  # Sort greed factors
s.sort()  # Sort cookie sizes

Two-pointer approach: Initialize two pointers, each for traversing the sorted lists.
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i, j = 0, 0 # Initialize two pointers

Conditional incrementation: Write a condition to check if a cookie can satisfy a child’s greed, if so, increment the respective pointer.

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if s[j] >= g[i]:  # If the cookie can satisfy the child's greed
    i += 1  # Increment the pointer for the children

Comparison and decision-making: Write an if statement to compare the size of the cookie with the greed factor of the child.

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if s[j] >= g[i]:  # Compare cookie size with greed factor
    i += 1  # If the condition is met, increment the pointer for children
j += 1  # Always increment the pointer for cookies

Loop termination: Write a while loop that continues until either all the cookies or all the children have been considered.

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while i < len(g) and j < len(s):  # Loop until we exhaust children or cookies
    if s[j] >= g[i]:
        i += 1
    j += 1

Result return: Finally, return the count of content children.
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return i # Return the count of content children

By practicing these drills, you will become familiar with each of these individual coding concepts. Once you’re comfortable with them, you can then combine them all together to solve the complete problem.

Q&A

What if each child can get more than 1 cookies? Still each cookie can only be assigned to 1 child.

If each child can receive more than one cookie, the problem becomes a bit more complex, but the basic approach of trying to satisfy the least greedy children first remains the same. However, now we can’t move on to the next child as soon as we satisfy the current one. Instead, we have to keep giving them cookies until we either run out of cookies or we hit a cookie that is too small for the current child.

Here’s the adjusted Python code:

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def findContentChildren(g, s):
    g.sort()
    s.sort()

    child_i = 0
    content_count = 0  # The number of satisfied children.
    
    for cookie_j in range(len(s)):  # Go through all the cookies.
        if s[cookie_j] >= g[child_i]:  # If the cookie can satisfy the child,
            content_count += 1  # Count this as a satisfied child.
            child_i += 1  # Move to the next child.
            if child_i == len(g):  # If we have no more children,
                break  # exit the loop.
    
    return content_count

Now, instead of iterating over the cookies and children simultaneously, we iterate over the cookies in a loop and only move on to the next child when we hit a cookie that can’t satisfy the current one. The time complexity of this approach remains O(n log n) due to sorting, and the space complexity is still O(1).

blah blah blah

What you’ve presented here is an argument for the correctness of the greedy approach in the cookie problem using a proof by contradiction. In simpler terms, you’re arguing that:

Suppose there is an optimal solution A that’s different from the solution B produced by the greedy algorithm.
By the way the greedy algorithm works, every child in B is at least as greedy as the child in the same position in A. This is because the greedy algorithm always satisfies the least greedy child first.
Therefore, any child beyond the end of B could be added to B without making it any worse than A, creating a contradiction, since we assumed A was optimal and thus no better solution could exist.
So, our assumption that A and B were different must be wrong, and the greedy solution must be optimal.

It’s a rigorous argument and holds up mathematically. This proof firmly establishes that the greedy algorithm is indeed an optimal solution for this problem. It’s a great demonstration of how careful reasoning can often prove the optimality of certain approaches to problems.

What if two children have same greed factor?

If two children have the same greed factor, it does not impact the solution or the approach to the problem. The greedy algorithm can still be used, and it will still yield the optimal result.

The algorithm focuses on satisfying the least greedy child first. If there are two children with the same greed factor, you can satisfy either of them (assuming you have a cookie of suitable size), and it won’t affect the overall maximum number of children that can be satisfied.

To put it simply, the order in which you satisfy children with the same greed factor doesn’t impact the final outcome. This makes sense intuitively; if you have two children who are equally satisfied by a cookie of the same size, then it doesn’t matter which of them you give the cookie to first.

Is it possible without sorting ?

The problem could technically be solved without sorting, but it would be much less efficient. If we don’t sort, we would need to scan the entire list of cookies for each child to find an adequate cookie, which would result in a time complexity of O(n^2) for n children and cookies.

The reason why sorting works well here is because it allows us to use a greedy algorithm - we can start with the least greedy child and the smallest cookie and try to match them. If the smallest cookie is too small for the least greedy child, it is also too small for all other children, and we can discard it. If it’s large enough, we can give it to the least greedy child, satisfying one child and reducing the problem size. This is a much more efficient approach, with a time complexity of O(n log n), primarily due to the sorting operation. The pairing operation after the sort is linear, O(n).

In short, while it’s possible to solve the problem without sorting, doing so would likely result in a less efficient solution.

Are cookies and children’s load factors sorted in all test cases

The problem statement does not specify that the greed factors of the children or the sizes of the cookies are sorted. Therefore, we should not assume that they are sorted in the input.

In real-world scenarios or competitive programming, we usually have to handle the sorting part ourselves if it is needed for the problem solution. Hence, in your algorithm, you should include the step of sorting both arrays if it is necessary for your approach.

Also, it’s a good practice to not make assumptions that are not stated in the problem. Always carefully read the problem statement and understand the constraints thoroughly before proceeding with your solution.

I do not understand why my soln is wrong. I know the TC is a bit high but please tell me the reason.

class Solution { public int findContentChildren(int[] g, int[] s) { int count = 0; for (int i = 0; i < g.length; i++) { for (int j = 0; j < s.length; j++) { if (s[j] >= g[i]) { count++; s[j] = Integer.MIN_VALUE; break; } } } return count; } }

Your solution is incorrect because you are not prioritizing the cookie allocation optimally. Your code simply checks each child and tries to allocate a cookie that can satisfy that child’s greed. However, without sorting the ‘g’ and ’s’ arrays, there’s no guarantee that the child with the smallest ‘g’ value gets considered first. This can lead to a situation where a bigger cookie is assigned to a child with a lesser ‘g’ value, which could have otherwise been given to a child with a higher ‘g’ value.

Consider this scenario:

Children’s greed factors: g = [1,2] Cookies sizes: s = [1,2]

Here, the child with ‘g’ value 1 can be satisfied with either of the cookies. However, the child with ‘g’ value 2 can only be satisfied by the cookie of size 2.

In your solution, it’s possible that the child with ‘g’ value 2 gets considered first and receives the cookie of size 1. This leaves the child with ‘g’ value 1 satisfied but the child with ‘g’ value 2 unsatisfied, even though it would’ve been possible to satisfy both.

If you sort both ‘g’ and ’s’ first, you ensure that you’re considering the children in increasing order of their greed and assigning the smallest possible cookie that can satisfy them. This approach guarantees that you’ll be able to satisfy the maximum number of children.

So, always remember that sometimes the order in which you process the data can be crucial to finding the optimal solution. In this case, sorting is a necessary step to find the correct solution.

I don’t understand this problem. Why in the example 3;, we have 3 cookies?

In the problem statement, each child i has a ‘greed factor’ g[i] and each cookie j has a size s[j]. A child will be content if they receive a cookie that is of a size greater than or equal to their greed factor. The goal is to maximize the number of children that are content.

If you’re referring to an example where the greed factors g = [1,2] and the cookie sizes s = [1,2,3], then we have 2 children and 3 cookies. Here, each child can be given a cookie that satisfies their greed factor. The child with a greed factor of 1 can be given a cookie of size 1, and the child with a greed factor of 2 can be given a cookie of size 2 (or 3). Therefore, in this case, both children can be made content, and the output should be 2.

The third cookie in this example is not needed, but the problem statement allows for there to be different numbers of children and cookies. If there’s any specific part of the problem you’re finding difficult to understand, please let me know so I can help clarify.

Why is this not working for a test case in which we need to output 960 but 955 is being output? If I sort the vector, then it works, but why is that? I am using bruteforce approach which should check all possible combinations, so why there is a difference of 5 in my output?

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int findContentChildren(vector<int> g, vector<int> s)
	{
		set<int> receivers;

		for (int i = 0; i < s.size(); i++)
		{
			//cout << "Cookie #" << i << " of size " << s[i] << endl;
			for (int j = 0; j < g.size(); j++)
			{
				//cout << "Checking child #" << j << " with greed " << g[j] << endl;
				
				if  (receivers.find(j) == receivers.end() && s[i] >= g[j])
				{
					//cout << "Added #" << j << endl;
					receivers.insert(j);
					break;
				}
			}
		}

		return receivers.size();
	}

The provided code doesn’t guarantee to satisfy the maximum number of children, because it doesn’t always satisfy the child with the least “greed” factor first. Specifically, it loops over the cookies, then for each cookie, it loops over the children and assigns the cookie to the first child it encounters whose “greed” is less than or equal to the cookie size. The issue arises from the fact that a cookie could be given to a child with a higher “greed” factor, while a child with a lower “greed” factor could remain unsatisfied.

Consider an example:

g = [1, 2, 3], s = [3, 1, 2] - greed factors and cookie sizes are the same but not in the same order.

With the existing code, here’s how the distribution might look:

Cookie of size 3 goes to child with greed 1.
Cookie of size 1 goes to child with greed 2.
Cookie of size 2 cannot satisfy any remaining child (with greed 3).

Although all children could be satisfied, only 2 children get cookies because of how the cookies are distributed.

Sorting both the greed factor and size arrays in increasing order ensures that each cookie goes to the child with the smallest possible “greed” factor that it can satisfy. In other words, each cookie goes to the “least greedy” child it can satisfy, thereby maximizing the number of satisfied children. This is why the code works when you sort the vectors.

In the sorted version of the example above:

g = [1, 2, 3], s = [1, 2, 3] (both sorted in increasing order).

The distribution now looks like:

Cookie of size 1 goes to child with greed 1.
Cookie of size 2 goes to child with greed 2.
Cookie of size 3 goes to child with greed 3.

Now all children get cookies.