Group the People Given the Group Size They Belong To

The problem is asking us to group people based on the sizes given in the groupSizes array. If groupSizes[i] = k, then person i must be in a group of size k. We are asked to return a list of these groups.

Approach

  1. Create a Dictionary: We’ll use a dictionary where the key is the group size and the value is a list of people who are part of groups of that size.
  2. Group People: Iterate through groupSizes, and for each person i, append their index to the list in the dictionary corresponding to their group size.
  3. Construct Result: Iterate through the dictionary. For each group size, break the corresponding list into chunks of that size and append those chunks to the final result.

Code

Here’s a step-by-step implementation in Python:

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class Solution:
    def groupThePeople(self, groupSizes: List[int]) -> List[List[int]]:
        groups = collections.defaultdict(list)
        result = []

        # Group people by group size
        for i, size in enumerate(groupSizes):
            groups[size].append(i)

        # Construct the result by dividing people into correct group sizes
        for size, members in groups.items():
            for i in range(0, len(members), size):
                result.append(members[i:i+size])

        return result

Insights

  • We are using a dictionary to categorize people by their group sizes, which makes it simple to then construct the final result.
  • By employing the concept of grouping with a dictionary, we’ve made the solution both intuitive and efficient.
  • The solution’s time complexity is (O(n)), where (n) is the length of the groupSizes array.

Identifying Problem Isomorphism

“Group the People Given the Group Size They Belong To” involves partitioning an array into sets of specific sizes.

An approximate isomorphic problem is “Partition Labels”. In this problem, you partition a string into as many parts as possible so that each letter appears in at most one part. The key here is to figure out where each partition ends, much like determining group sizes in the original problem. “Partition Labels” is a simpler problem because you only need to determine the maximum index of each character in the string, whereas in the original problem, you have to keep track of group sizes.

A more complex isomorphic problem is “Palindrome Partitioning”. In this problem, you partition a string into substrings such that every substring is a palindrome. This problem is more complex because it involves both partitioning and checking for palindromes, whereas the original problem only involves partitioning.

In increasing order of complexity, the problems are:

  1. “Partition Labels” - Partition a string into as many parts as possible so that each letter appears in at most one part.
  2. “Group the People Given the Group Size They Belong To” - Partition an array into sets of specific sizes.
  3. “Palindrome Partitioning” - Partition a string into substrings such that every substring is a palindrome.

10 Prerequisite LeetCode Problems

This involves array manipulation, grouping and simple hashing. Here are some simpler problems to prepare for it:

  1. Two Sum (LeetCode #1): This problem involves finding pairs in an array that meet a certain condition. It will give you practice with the concept of pairing/grouping elements.

  2. Find All Numbers Disappeared in an Array (LeetCode #448): This problem involves finding elements in an array that meet certain conditions, which is somewhat similar to grouping people based on their group sizes.

  3. Majority Element (LeetCode #169): This problem requires finding an element that appears more than n/2 times in an array. It introduces the concept of grouping elements based on their frequency.

  4. Group Anagrams (LeetCode #49): This problem directly involves grouping elements, though with strings and not numbers. It is a bit more complex but will give you good practice with the concept of grouping.

  5. Single Number (LeetCode #136): This problem involves finding an element that appears only once in an array. It will give you practice with hashing and handling elements individually.

  6. Sort Array By Parity (LeetCode #905): This problem involves rearranging an array based on the parity of elements. It will give you practice with sorting/grouping elements based on a condition.

  7. Find All Duplicates in an Array (LeetCode #442): This problem involves finding duplicate elements in an array. It will give you practice with hashing and handling duplicate elements.

  8. Valid Anagram (LeetCode #242): This problem involves comparing two strings to see if they are anagrams. It will give you practice with grouping and comparing elements.

  9. Intersection of Two Arrays II (LeetCode #350): This problem involves finding the intersection of two arrays. It will give you practice with hashing and comparing multiple arrays.

  10. Degree of an Array (LeetCode #697): This problem involves finding the degree of an array, or the maximum frequency of any one element. It will give you practice with hashing and frequency counts.

These cover array manipulation, grouping, and hashing, which will be useful for the main problem.

Clarification Questions

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

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

The problem falls into the category of “Grouping and Partitioning”. The primary task is to partition a list of unique IDs into groups where each group should be of a specified size.

What Components:

  1. Array groupSizes: Indicates the size of the group each person should belong to.
  2. Person IDs: Unique identifiers for people, ranging from 0 to n - 1.
  3. Group Requirements: Each person must be in a group that matches the size specified by their corresponding entry in groupSizes.
  4. Output: A list of groups such that each group has the size as mentioned in groupSizes.

This problem can be classified as a “Combinatorial Partitioning” problem. You have constraints on how you can form the groups (based on the array groupSizes), and you need to satisfy these constraints while ensuring everyone is included in exactly one group.

Explanation:

In the Combinatorial Partitioning category, problems often involve distributing a set of elements into bins or groups according to certain rules or constraints. In this case, the rules are provided by the groupSizes array. You are required to form groups that strictly adhere to these sizes, ensuring each person ends up in exactly one group that matches their corresponding group size.

Distilling the Problem to Its Core Elements

Fundamental Concept:

The fundamental concept in this problem is “Partitioning with Constraints.” The problem deals with dividing a set of unique elements (people) into smaller subsets (groups) based on a set of rules (group sizes).

Simplest Description:

You have a list of people, each wanting to be in a group of a certain size. Your task is to put everyone in such groups, following their wishes on group size.

Core Problem:

The core problem is to form groups of specified sizes so that each person belongs to exactly one group, and each group matches the desired sizes.

Key Components:

  1. Unique IDs for people: Each person is represented by a unique ID (integer).
  2. Group Sizes: An array that specifies the size of the group each person wants to be in.
  3. Valid Grouping: The main output, where each person belongs to one group, and each group satisfies the size constraints.

Minimal Set of Operations:

  1. Initialization: Create a mapping or tracking system to remember which group sizes are required and who wants to be in them.
  2. Iterate Through People: Go through each person, considering their desired group size.
  3. Group Formation: If a group of the desired size can be formed, form it and remove those people from the consideration set.
  4. Output: Collect all such groups and return.

By focusing on these operations, you can solve the problem efficiently.

Visual Model of the Problem

  1. People as Dots: Imagine each person as a dot or a node. Label these dots with the unique ID of the person.

  2. Group Sizes as Labels: Next to each dot, place a number indicating the desired group size for that person.

  3. Group Formation as Enclosures: When you identify people who can form a group, enclose those dots within a shape like a circle or a square. The shape signifies a group. Label the shape with the group size.

  4. One Person, One Group: Make sure that each dot is part of one and only one enclosure. This represents the constraint that each person can be part of one group only.

  5. Complete Groups: As you proceed, completed groups can be shaded or colored differently to indicate that they are finalized.

  6. Remaining Dots: Keep an eye on dots that are not yet enclosed. These are the people still waiting to be grouped.

Visualizing the problem in this way helps you grasp the constraints and objectives clearly. It provides a mental model to guide the problem-solving process.

Problem Restatement

You have a list of people, each with a unique ID from 0 to n-1. You also have a list called groupSizes that tells you the size of the group each person must be in. Your task is to form groups that adhere to these sizes. Each person can only belong to one group, and everyone must be in a group. There may be more than one correct answer, but you only need to return one valid grouping. The number of people, ’n’, will be between 1 and 500, and the size of each group will also be between 1 and ’n'.

Abstract Representation of the Problem

You have two lists of integers, IDs and groupSizes, of the same length, ’n’. The IDs list contains unique identifiers ranging from 0 to n-1. The groupSizes list contains integers that represent the size of the group each identifier should be part of. Your task is to create a new list of lists, where each inner list represents a group of identifiers. The size of each inner list must match the corresponding numbers in the groupSizes list. Each identifier should appear in one and only one group. There can be multiple valid answers. Return any valid grouping that satisfies these conditions.

Terminology

  1. Identifier: A unique number assigned to each person, ranging from 0 to n-1. It’s crucial for keeping track of who goes into which group.

  2. Group Size: The size each group should be, specified for each identifier. This guides how many identifiers should be in each group.

  3. Valid Grouping: A list of groups that meet all the requirements specified in the problem statement. It’s the final output we aim to construct.

  4. Constraints: The rules or limitations that the solution must adhere to. For instance, each person must be in exactly one group, and the size of that group must match the corresponding number in the groupSizes list.

Understanding these terms is essential for conceptualizing the problem and formulating a solution that adheres to the stated requirements.

Problem Simplification and Explanation

Let’s think of this problem as organizing kids into teams on a playground. Each kid wears a t-shirt with a number on it. This number tells us how many players should be on their team. We have to make sure everyone is on a team and that each team has the exact number of players as stated on any of its members’ t-shirts.

Key Concepts:

  1. Kids (People): Each kid represents a person labeled with a unique ID.

  2. T-shirt Number (Group Size): The number on each kid’s t-shirt specifies the number of kids that should be in their team.

  3. Teams (Groups): We need to create teams that follow the rules set by the t-shirt numbers.

  4. Playground Rules (Constraints): Every kid must be in one, and only one, team. The number of kids in each team must match the t-shirt numbers.

How They Interact:

  • We start by looking at each kid’s t-shirt number one by one.
  • For each number, we check if a team already exists that needs more members; if yes, we add the kid there.
  • If no such team exists, we start a new team with that kid as the first member.
  • We continue this process until all kids are part of a team, making sure we obey the playground rules.

This breaks down the problem into manageable parts and helps us understand what we’re trying to accomplish.

Constraints

Specific Characteristics for Efficiency:

  1. Unique IDs: Each person has a unique ID from 0 to n-1, allowing for easy tracking.

  2. Fixed Group Sizes: The size of the group each person belongs to is predefined, simplifying group formation.

  3. Limited Range for n: 1 <= n <= 500. This numerical constraint suggests that the problem is unlikely to require highly optimized algorithms for performance; however, it still encourages efficient solutions.

  4. 1-to-1 Mapping: Each person should appear in exactly one group. This rules out overlapping groups and makes the grouping easier to manage.

  5. Multiple Valid Outputs: The problem allows for multiple valid solutions. This flexibility can be exploited to find a solution more quickly without having to find an “optimal” arrangement.

  6. Group Size Repeats: Some people might have the same group size, allowing for batch processing.

By recognizing these characteristics, we can look for ways to streamline our approach. For instance, sorting the array could enable quick identification of similar group sizes, and using a hashmap could speed up the process of adding new members to existing groups.

Key Insights from Constraints:

  1. Limited N-Size: With a maximum of 500 people, a solution with a time complexity of O(n) or O(n log n) would be quite practical. No need for highly optimized algorithms.

  2. Flexibility in Output: Since multiple valid answers are allowed, we don’t have to find the “best” solution, just a correct one. This eases the implementation.

  3. No Overlapping Groups: A person appears in exactly one group, so once we place a person in a group, we don’t have to consider them again. This simplifies the process.

  4. Predefined Group Sizes: Knowing the size of each group beforehand can guide the data structures we use. We might not need to resize or reallocate, which can be a costly operation.

  5. Repeating Group Sizes: People with the same group size can be batch processed, meaning we can place all people with a given group size at the same time.

By understanding these insights, you can make smarter choices in your algorithm and data structures, possibly reducing the time and space complexity of your solution.

Case Analysis

Example Cases

Case 1: Single Person (Minimum Size)

  • Input: [1]
  • Output: [[0]]
  • Analysis: This represents the minimum possible group size, where only one person is involved. It tests whether the algorithm can handle the smallest input size.

Case 2: All Same Size Groups

  • Input: [3, 3, 3, 3]
  • Output: [[0, 1, 2], [3]]
  • Analysis: This case tests whether the algorithm correctly forms groups when everyone has the same group size. It’s likely the easiest scenario to handle.

Case 3: All Different Sizes

  • Input: [1, 2, 3]
  • Output: [[0], [1, 2], [3]]
  • Analysis: This is interesting because it checks whether the algorithm can handle groups of varying sizes.

Case 4: Non-Sequential Sizes

  • Input: [2, 1, 2]
  • Output: [[0, 2], [1]]
  • Analysis: This tests if the algorithm can correctly form groups when sizes are not in any particular order.

Case 5: Maximum Size

  • Input: [500] * 500
  • Output: [[0, 1, 2, ..., 499]]
  • Analysis: This tests the upper limit of the problem constraints. All persons should be in one group.

Edge Cases

  1. Single person (minimum size): Having only one person should still return a valid group of one.
  2. Maximum group size (n): The problem allows for a group size that includes all individuals. The algorithm should be able to handle this without issues.
  3. Multiple groups of same size: If there are multiple groups with the same size, the algorithm should correctly distribute individuals into these groups.
  4. Mixed group sizes in random order: The algorithm should be robust enough to handle groups of different sizes given in a random order.

By testing these cases, you can be confident that your solution is both efficient and robust, handling a wide variety of input scenarios.

To visualize these cases, you can represent the input array as a sequence of blocks, where each block represents a person and the number inside the block represents their group size. You can then show the expected output by grouping these blocks.

Visualization

  1. Single Person (Minimum Size)

    • Input: [1]
    • Blocks: [1]
    • Output Groups: [[1]]

  2. All Same Size Groups

    • Input: [3, 3, 3, 3]
    • Blocks: [3 3 3 3]
    • Output Groups: [[3 3 3], [3]]

  3. All Different Sizes

    • Input: [1, 2, 3]
    • Blocks: [1 2 3]
    • Output Groups: [[1], [2 3]]

  4. Non-Sequential Sizes

    • Input: [2, 1, 2]
    • Blocks: [2 1 2]
    • Output Groups: [[2 2], [1]]

  5. Maximum Size

    • Input: [500, 500, ..., 500] (500 times)
    • Blocks: [500 500 ... 500]
    • Output Groups: [[500 500 ... 500]]

Each “Output Groups” visualization shows how the individual blocks are combined into groups. This can help you to understand how the algorithm is expected to manipulate the input data to form the output groups.

Analyzing the different cases provides several key insights:

  1. Single Person Groups: When the size is 1, the person forms their own group. This is a quick operation and can be processed instantly.

  2. Uniform Group Sizes: When all persons have the same group size, the process is straightforward. Divide them into groups of that size.

  3. Different Sizes: Handling different group sizes adds complexity. You can’t finalize a group until you have enough people to fill it.

  4. Order Doesn’t Matter: The problem doesn’t specify that the groups have to be formed in any particular order. This gives flexibility in how to approach the problem.

  5. Efficient Grouping: For larger groups, it might be more efficient to start forming them early in the process, especially if the input array is large.

  6. Max Constraint: Knowing the maximum size helps in understanding that you don’t have to consider scalability beyond that point.

These insights suggest that an efficient solution would not only handle different group sizes but also adapt based on the number of people who share the same group size.

Identification of Applicable Theoretical Concepts

This problem lends itself well to several algorithmic concepts that can make it more manageable:

  1. Hashing: You can use a hash map to keep track of the number of people needed to complete each group. This allows for constant-time insertions and lookups, making the grouping operation efficient.

  2. Queue/Stack: When a person with a specific group size appears, you can place them in a queue or stack until enough people have gathered to form a complete group.

  3. Greedy Method: The problem can be approached in a greedy manner. Whenever you have enough people to form a group, you form it immediately.

  4. Counting Sort Ideas: Since the size of the groups is also a constraint (and usually not a large number), techniques similar to counting sort can be applied to group individuals by their sizes.

  5. Batch Processing: Due to the unordered nature of the output, one could tackle the problem by collecting all people who have the same group size and handle them all at once. This batch processing can reduce the overhead of individual operations.

  6. Load Balancing: If the sizes are unevenly distributed, a load-balancing approach can evenly distribute the computational cost across the process.

These existing algorithmic concepts can be leveraged to build an efficient solution to the problem.

Simple Explanation

Imagine you have a bunch of people, and each person has a tag that tells you how many friends they want in their group. Your job is to make those groups.

So, let’s say we have Amy, Bob, and Carla. Amy has a tag that says “3”, Bob also has a tag that says “3”, and Carla has a tag that says “2”. Amy and Bob can be in a group together, but we need one more person with a “3” tag to complete their group. Carla wants to be in a group with just one other person, so she’ll have to wait until another person with a “2” tag comes along.

It’s like making teams for a game where everyone gets to say how many people they want on their team, and then you make those teams happen.

So, the game here is to make everyone happy by placing them into groups with the exact number of friends they want to have.

Problem Breakdown and Solution Methodology

Here’s a detailed breakdown of solving this problem:

Approach

  1. Understand the Goal: The first step is to fully understand what we’re trying to do. We want to group people based on the number of friends they want in their group. It’s like arranging tables at a party where each person has a card that says how many people they want at their table.

  2. Prepare a List: We can prepare a list or “pool” where we put people in categories based on the number of friends they want. Imagine you have buckets, each labeled with a number (like 1, 2, 3, etc.), and you put each person in the bucket that matches their tag.

  3. Start Grouping: Take a bucket and check if you have enough people to form a group. If the bucket says “3”, you need three people to form a group. Once you have enough people, you make the group and remove them from the pool.

  4. Repeat: Continue this process for each bucket until everyone is grouped.

  5. Finalize Groups: Return the list of groups.

Effects of Parameter Changes

  • Number of People: The more people we have, the more groups we can potentially create, but it also means more time to sort them into groups.
  • Group Sizes: If most people want large groups, it may take longer to find enough people to complete each group.

Example

Let’s say we have 7 people with group sizes [3, 3, 3, 3, 3, 1, 3].

  1. Prepare Buckets:

    • 3: [0, 1, 2, 3, 4, 6]
    • 1: [5]

  2. Start Grouping:

    • For bucket “3”, we take three people: [0, 1, 2] and form one group.
    • Next, we take another three people [3, 4, 6] and form another group.
    • For bucket “1”, we take one person [5] and form a single-person group.

  3. Final Groups: [[0, 1, 2], [3, 4, 6], [5]]

By following these steps, we can efficiently solve this problem.

Inference of Problem-Solving Approach from the Problem Statement

Key Terms and Concepts

  1. Group Sizes: This array contains the size each person desires for their group. It dictates how we categorize people into buckets. If someone wants to be in a group of 3, they go into the “3” bucket.

  2. Buckets: These are categories or containers that hold people who desire the same group size. It’s essential for sorting and simplifying the grouping process.

  3. Pooling: The act of placing people into their respective buckets based on their desired group size. This operation allows us to then easily form groups.

  4. Final Groups: The output array where each sub-array represents a complete group of people.

Strategies Guided by Concepts

  • Group Sizes guides us to create buckets. If you see an array of desired sizes, think of creating buckets.
  • Buckets give us the strategy of sorting people first before making any groups. It’s like saying, “Let’s see who wants what, then we can easily distribute.”
  • Pooling is the operation we apply to fill these buckets. Knowing this makes the process systematic.
  • Final Groups tells us what the expected output is, so we can constantly check against this structure as we go along.

Recognizing Through Tables or Diagrams

You can use a table where each row corresponds to a bucket and columns hold the people in that bucket. Like this:

Bucket 1Bucket 2Bucket 3…
501…
74…
8…

In this example, the people who want to be in a group of size 1, 2, and 3 are placed in buckets 1, 2, and 3, respectively. Once a bucket has enough people to form a group, you can highlight it, signaling that these people can be taken out and placed into a “final group.”

This way, you can visually track your progress and make sure everyone ends up where they want to be.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

Stepwise Refinement of Approach

  1. Initialize Buckets: Create a dictionary where the keys will be the desired group sizes and the values will be lists containing the IDs of people who want to be in groups of that size.

  2. Sort into Buckets: Loop through the groupSizes array. Place each person’s ID into the appropriate bucket based on their desired group size.

  3. Initialize Final Groups: Create an empty list to hold the final groups.

  4. Form Groups and Update Final Groups: Loop through each bucket. Once a bucket has enough IDs to form a complete group, remove those IDs, form a group, and append it to the final groups list.

  5. Return Final Groups: Once all buckets are empty or cannot form more complete groups, return the final groups list.

More Granular, Actionable Steps

  1. Initialize Buckets

    • Create an empty dictionary called buckets.

  2. Sort into Buckets

    • Loop i from 0 to n-1.
      • Find groupSize = groupSizes[i].
      • Append i to buckets[groupSize].

  3. Initialize Final Groups

    • Create an empty list called finalGroups.

  4. Form Groups and Update Final Groups

    • Loop through each key in buckets.
      • While the bucket has enough elements to form a group:
        • Pop those elements.
        • Create a new group with those elements.
        • Append this new group to finalGroups.

  5. Return Final Groups

    • Return finalGroups.

Independent Parts

  1. Sorting into Buckets: This is independent of forming the final groups. You can sort all IDs into buckets first before you start making any groups.

  2. Forming Groups from Each Bucket: Forming a group from one bucket is independent of forming a group from another bucket.

Repeatable Patterns

  1. Appending to Buckets: The pattern of checking a person’s desired group size and placing their ID into the corresponding bucket repeats for each person.

  2. Forming Groups: The pattern of checking if a bucket has enough IDs to form a group, and then doing so, is a repeated process that you’ll apply to each bucket.

By understanding these elements and patterns, you can write more modular and clean code.

Solution Approach and Analysis

Detailed Approach to Solving the Problem

Step 1: Initialize Buckets

We start by creating empty “buckets” for each group size. Think of these buckets as holding areas where people (represented by their IDs) wait until there are enough of them to form a complete group.

  • Operation: Initialize an empty dictionary called buckets.

Step 2: Sort People into Buckets

Next, we loop through the list of group sizes and sort each person into the appropriate bucket based on their desired group size.

  • Operation: For each index i from 0 to n-1, find groupSize = groupSizes[i] and append i to buckets[groupSize].

Step 3: Initialize Final Groups

We prepare an empty list to hold the final groups of people. This list starts empty and will be filled as we create complete groups.

  • Operation: Create an empty list called finalGroups.

Step 4: Form Groups and Update Final Groups

Now, we loop through each bucket. Whenever a bucket has enough people to form a group, we move them from the bucket to a new group and add that group to our final list of groups.

  • Operation: For each key size in buckets, while the bucket has size or more elements, pop those elements, create a new list, and append this new list to finalGroups.

Step 5: Return Final Groups

The last step is simple. Once we’ve formed all possible groups, we return the final list of groups as our answer.

  • Operation: Return finalGroups.

Effect of Changes in Parameters

  1. Number of People (n): More people would potentially mean more iterations in sorting them into buckets but won’t fundamentally change the approach.

  2. Range of Group Sizes: If the maximum group size is large, that may affect memory usage for our buckets but won’t change our methodology.

Example Cases

Example 1: groupSizes = [3, 3, 3, 3, 3, 1, 3]

  • buckets = {3: [0, 1, 2, 3, 4, 6], 1: [5]}
  • finalGroups = [[5], [0, 1, 2], [3, 4, 6]]

Example 2: groupSizes = [2, 1, 3, 3, 3, 2]

  • buckets = {2: [0, 5], 1: [1], 3: [2, 3, 4]}
  • finalGroups = [[0, 5], [1], [2, 3, 4]]

This approach will handle the problem efficiently and can adapt to changes in problem parameters.

Identify Invariant

The invariant in this problem is that each person should appear in exactly one group and the size of that group must match the person’s specified group size from the groupSizes array. This invariant holds throughout the entire process of forming the groups. We ensure this by adding each person to a specific group that matches their specified size and by making sure no one is added to more than one group.

Identify Loop Invariant

The loop invariant in this problem would be that, at the start and end of each iteration of the loop, all groups formed up to that point must satisfy the conditions: each group must contain people according to their groupSizes and each person should only appear in one group. This ensures that we are progressively building towards the final solution while maintaining the integrity of the groups we have already formed.

In this problem, the invariant and the loop invariant are essentially the same. Both ensure that each group formed contains people according to their groupSizes and that each person appears in exactly one group. This condition holds true before, during, and after the loop execution, ensuring that the program progressively builds towards the final, correct output.

Thought Process

Cues in the Problem Statement:

  1. You have n people, each having a unique ID from 0 to n - 1.
  2. The group size for each person is given.
  3. Each person must be in a group of their specified size.
  4. Each person should appear in exactly one group.

Direction for Approach:

  1. You need to group people based on their specified group size.
  2. There can be multiple correct answers.

Insights:

  1. People with the same group size can be put together.
  2. There’s no need to find a unique solution; any valid answer suffices.

Steps for the Algorithm

  1. Create a dictionary to hold people based on their group size.
  2. Iterate through the groupSizes array and populate the dictionary.
  3. Initialize an empty list, result, to store the final groups.
  4. Iterate through the dictionary. Whenever a group reaches its desired size, move it to the result.

Python3 Code

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def group_people(groupSizes):
    # Step 1: Create a dictionary to hold people based on their group size
    group_dict = {}

    # Step 2: Iterate and populate the dictionary
    for i, size in enumerate(groupSizes):
        if size not in group_dict:
            group_dict[size] = []
        group_dict[size].append(i)

    # Step 3: Initialize an empty list for the result
    result = []

    # Step 4: Iterate through the dictionary and form groups
    for size, people in group_dict.items():
        for i in range(0, len(people), size):
            result.append(people[i:i+size])

    return result

This code first categorizes people by their group size and then creates groups from these categories. The result list holds the final groups, ensuring each person appears exactly once, adhering to their specified group size.

Establishing Preconditions and Postconditions

Parameters:

  • Inputs: The input is an integer array named groupSizes.
  • Types: It’s an array of integers.
  • Representation: Each integer at index i in groupSizes represents the size of the group that person i should be in.

Preconditions:

  • State: There is no specific state the program needs to be in before this method is called.
  • Constraints:
    • The length of groupSizes should be between 1 and 500, inclusive.
    • Each element in groupSizes should be between 1 and the length of the array, inclusive.

Method Functionality:

  • Expected Action: The method is expected to return a list of lists, where each inner list is a group of people based on the input groupSizes.
  • Interaction: The method processes the groupSizes array to sort individuals into appropriate groups.

Postconditions:

  • State: After the method is called, groups are formed based on the sizes indicated in the groupSizes array.
  • Return Value: A list of lists, where each inner list represents a group of people. The length of each inner list matches the corresponding sizes in groupSizes.
  • Side Effects: None. The method is expected to be pure, meaning it does not change the state of the program or the input parameters.

Error Handling:

  • Response: If the preconditions on the input array are not met, the method should ideally throw an exception to indicate invalid input.
  • Action: Throws an exception or returns a special value (like None or an empty list) to indicate failure.

Problem Decomposition

Problem Understanding:

  • In My Own Words: You have a list of people, each needing to be in a group of a specific size. You have to form these groups based on the sizes needed.
  • Key Components: Integer array groupSizes, each value tells the group size for the person at that index.
  • Requirements: Form groups such that each person is in a group of their required size.

Initial Breakdown:

  1. Read groupSizes array
  2. Create Empty Groups
  3. Sort Individuals into Groups

Subproblem Refinement:

  1. Read groupSizes array

    • Validate the input based on the problem constraints.

  2. Create Empty Groups

    • Initialize an empty list to collect the final groups.

  3. Sort Individuals into Groups

    • Iterate through groupSizes.
    • Place each person in a group that matches their group size requirement.

Task Identification:

  • Reading groupSizes and Validating: repeated for each test case.
  • Placing People in Groups: repeated for each individual in the array.

Task Abstraction:

  • Read groupSizes: Abstract enough. It is simply reading and validating an array.
  • Sort Individuals into Groups: General task that works for any size of the groupSizes array.

Method Naming:

  • validateInput(): For reading and validating groupSizes.
  • initializeGroups(): For initializing the final group list.
  • sortIntoGroups(): For the actual sorting of people into appropriate groups.

Subproblem Interactions:

  • validateInput() should be the first to execute.
  • initializeGroups() can occur anytime before sortIntoGroups() but after validateInput().
  • sortIntoGroups() should be last, and it depends on the empty list initialized by initializeGroups().

There are dependencies between these tasks, mostly that validation must occur before any sorting action, and the final groups list must be initialized before sorting individuals into it.

From Brute Force to Optimal Solution

Brute Force Solution:

  1. Initialize an empty list, finalGroups, to collect the final groups.
  2. Iterate through groupSizes. For each individual, do the following:
    • Check every existing group in finalGroups to see if there is room for this person.
    • If found, add the person to that group.
    • If not, create a new group containing just this person and add it to finalGroups.

Python Code:

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def brute_force_grouping(groupSizes):
    finalGroups = []
    for i, size in enumerate(groupSizes):
        found = False
        for group in finalGroups:
            if len(group) < size:
                group.append(i)
                found = True
                break
        if not found:
            finalGroups.append([i])

    return [group for group in finalGroups if len(group) == len(groupSizes[group[0]])]

groupSizes = [3,3,3,3,3,1,3]
print(brute_force_grouping(groupSizes))

Inefficiencies:

  1. Time Complexity: It could be as bad as O(n^2) because for each person, we are potentially checking all existing groups.
  2. Space Complexity: O(n) for storing the final groups, which is reasonable but could be optimized.

Optimized Solution:

Steps towards Optimization:

  1. Use a dictionary, groupDict, where the key is the group size and the value is a list of individuals requiring that size.
  2. For each person in groupSizes, add them to the list in groupDict corresponding to their size. When the list reaches the required size, move it to finalGroups and empty the list in groupDict.

Python Code:

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def optimized_grouping(groupSizes):
    finalGroups = []
    groupDict = {}

    for i, size in enumerate(groupSizes):
        if size not in groupDict:
            groupDict[size] = []
        groupDict[size].append(i)

        if len(groupDict[size]) == size:
            finalGroups.append(groupDict[size])
            groupDict[size] = []

    return finalGroups

groupSizes = [3,3,3,3,3,1,3]
print(optimized_grouping(groupSizes))

Time Complexity:

  • Now O(n) because we are making a single pass through groupSizes and using a dictionary for quick lookups.

Space Complexity:

  • Still O(n) for storing the final groups, but now with less overhead because we’re not repeatedly scanning through finalGroups.

By using a dictionary to manage individuals waiting to be grouped, we’ve eliminated the need for nested loops, reducing time complexity. The space complexity remains the same but is utilized more efficiently.

Code Explanation and Design Decisions

1. Initial Parameters:

  • groupSizes: This is a list of integers where each integer represents the desired group size for each individual. The index in the list represents the individual.

2. Primary Loop:

The primary loop iterates over the groupSizes list. Each iteration represents an attempt to place an individual in a group of their desired size. The loop advances the solution by either adding the individual to an existing group or creating a new group for them.

3. Conditions or Branches:

  • if size not in groupDict: Checks if a list for the specific group size already exists. If not, a new list is created.
  • if len(groupDict[size]) == size: Checks if a list has reached its required group size. If it has, the group is considered complete and moved to finalGroups.

These conditions are based on the problem’s requirement to form groups of specific sizes.

4. Updates or Modifications:

  • groupDict[size].append(i): Adds an individual to the list for their desired group size.
  • finalGroups.append(groupDict[size]) and groupDict[size] = []: Moves a completed group to finalGroups and resets the list in groupDict.

These modifications keep track of both incomplete and complete groups, aligning with the problem’s objective to form groups of specific sizes.

5. Invariant:

The invariant here is that the sum of the lengths of all lists in groupDict and finalGroups is equal to the number of individuals processed so far. This ensures that every individual is accounted for at all times.

6. Final Output:

The final output is a list of lists, where each inner list represents a group of individuals. These groups satisfy the problem’s requirements: each group is of a size as desired by its members. Thus, the output aligns with the constraints and objectives set by the problem.

Coding Constructs

1. High-Level Strategies:

  • Data Structuring: A dictionary (groupDict) is used to dynamically manage groups based on size requirements.
  • Grouping: An array (finalGroups) is used to collect complete groups.

2. Purpose for a Non-Programmer:

The code organizes people into groups where everyone in the same group wants that group to be of a specific size. It’s like sorting a class into project teams where each student has their own preferred team size.

3. Logical Elements:

  • Loops for iteration
  • Conditional statements for decision-making
  • Data structures like dictionary and array for storing information

4. Algorithmic Approach:

For each person, the code checks if a group of their preferred size is already in progress. If it is, they are added to that group. If not, a new group starts. When a group reaches its desired size, it’s considered complete.

5. Key Steps:

  1. Iterate over each individual and their group size preference.
  2. Add them to an existing or new group based on their preference.
  3. Check if any group has met its size requirement.
  4. Move any completed groups to the final list.

Each step advances us towards forming groups according to each individual’s preference.

6. Algorithmic Patterns:

  • Iterative Looping: To process each individual.
  • Dynamic Allocation: Using data structures to adapt to the requirements on-the-fly.
  • Condition-based Filtering: To segregate complete and incomplete groups.

Language Agnostic Coding Drills

1. Distinct Coding Concepts:

  1. Variable Declaration & Initialization: Basic setup of variables to store data.
  2. Loops: Iterating over a list of elements.
  3. Conditional Statements: Making decisions based on conditions.
  4. Data Structure Operations: Basic operations on a dictionary.
  5. Array/List Manipulation: Adding and removing elements from a list.
  6. Data Aggregation: Combining separate pieces of data into a more complex data structure (nested lists).
  7. Error Handling: Providing solutions for cases that don’t meet the desired conditions.

2. Difficulty Order and Descriptions:

  1. Variable Declaration & Initialization: Easiest. It’s the starting point for most programming tasks.
  2. Loops: Slightly more complex, involves control flow.
  3. Conditional Statements: Adds logic but still easy to understand.
  4. Array/List Manipulation: Requires understanding how to dynamically change data.
  5. Data Structure Operations: Requires understanding both data storage and manipulation.
  6. Data Aggregation: Intermediate. Need to understand how to nest data structures.
  7. Error Handling: More complex because it demands a broader understanding of possible outcomes and how to handle them.

3. Problem-Solving Approach:

  1. Variable Declaration & Initialization: Start by identifying what data you’ll need to keep track of. This could be a list of groups, a dictionary of pending groups, and so on.

  2. Loops: You know you’ll need to look at each person one by one, so a loop is necessary for iteration over the list of people and their group size preferences.

  3. Conditional Statements: As you look at each person, you have to decide whether to add them to an existing group or start a new one. Conditional statements will be used here.

  4. Data Structure Operations: If you decide to add them to a new or existing group, you’ll need to know how to add or update an entry in a dictionary.

  5. Array/List Manipulation: When a group reaches its desired size, you’ll want to move it from the “incomplete” list to a “complete” list, requiring list manipulation skills.

  6. Data Aggregation: Combining all the complete and incomplete groups into a nested list, perhaps to return as the final output, requires data aggregation skills.

  7. Error Handling: Finally, if you encounter a person whose preferred group size can’t be accommodated (perhaps the size exceeds the total number of people), you’ll need to decide how to handle this error condition.

Targeted Drills in Python

1. Basic Coding Drills in Python:

Variable Declaration & Initialization

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# Declaring a variable
x = 10

Loops

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# Looping through a list
for i in [1, 2, 3]:
    print(i)

Conditional Statements

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# Using if-else to make decisions
x = 10
if x > 5:
    print("Greater")
else:
    print("Smaller")

Data Structure Operations

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# Dictionary operations
my_dict = {}
my_dict["key"] = "value"

Array/List Manipulation

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# Adding and removing elements from a list
my_list = [1, 2, 3]
my_list.append(4)
my_list.pop(0)

Data Aggregation

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# Nesting lists
nested_list = [[1, 2], [3, 4]]

Error Handling

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# Using try-except for error handling
try:
    x = 1 / 0
except ZeroDivisionError:
    print("Cannot divide by zero")

2. Problem-Specific Concepts:

Group Management

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# Managing a group using a list of dictionaries
groups = [{"size": 3, "members": [1, 2, 3]}]
  • Essential because our problem involves managing groups, and using a dictionary within a list makes it easy to keep track of each group’s size and members.

3. Integration of Drills:

  1. Variable Declaration & Initialization: Begin by declaring necessary variables like groups as a list and pending_groups as a dictionary.

  2. Loops: Iterate through the list of people and their group preferences.

  3. Conditional Statements: For each person, check if a group that fits their preference exists in pending_groups.

  4. Data Structure Operations: If such a group exists, update its members list. Else, create a new group in pending_groups.

  5. Array/List Manipulation: If a group reaches the preferred size, move it from pending_groups to groups.

  6. Data Aggregation: After all iterations, assemble groups and pending_groups into a final list final_groups.

  7. Error Handling: If someone has a preference that can’t be accommodated, you might choose to raise an exception or log a message.

By carefully ordering these drills, you can seamlessly integrate them to solve the initial problem. Each drill acts as a building block, and combining them in this order should yield a complete solution.

Q&A

Similar Problems

Here are 10 problems that involve similar problem-solving strategies or concepts:

  1. Two Sum (Easy)

    • Why Related: Involves using a dictionary to store elements and find complementary elements efficiently.

  2. Valid Anagram (Easy)

    • Why Related: Utilizes dictionaries to track frequencies of characters.

  3. Valid Parentheses (Easy)

    • Why Related: Uses a stack data structure for matching brackets, similar to how groups are managed.

  4. Longest Substring Without Repeating Characters (Medium)

    • Why Related: Involves sliding window and set data structure to store unique elements.

  5. Group Anagrams (Medium)

    • Why Related: Requires grouping elements based on certain criteria. Uses dictionaries and lists for efficient grouping.

  6. 3Sum (Medium)

    • Why Related: Deals with the problem of grouping numbers into sets of three that sum to zero.

  7. LRU Cache (Medium)

    • Why Related: Involves using a dictionary for fast lookups and a doubly-linked list for maintaining order, similar to the management of groups.

  8. Find All Anagrams in a String (Medium)

    • Why Related: Utilizes sliding window and dictionary to track character frequencies.

  9. Number of Islands (Medium)

    • Why Related: Involves graph traversal to find connected components, akin to finding groups in a crowd.

  10. Minimum Window Substring (Hard)

    • Why Related: Uses dictionaries and sliding window to find the smallest contiguous array that contains all the required elements.

These problems share various aspects with the original problem, be it data structures used, problem-solving strategies, or overall problem theme.