Convert an Array Into a 2D Array With Conditions

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class Solution:
    def findMatrix(self, nums: List[int]) -> List[List[int]]:
        occurrences = defaultdict(int)
        max_occurrence = 0

        # Count the occurrences of each number in nums
        for num in nums:
            occurrences[num] += 1
            max_occurrence = max(max_occurrence, occurrences[num])

        # Initialize the result with max_occurrence number of rows
        result = [[] for _ in range(max_occurrence)]

        # Place the numbers in the rows according to their occurrences
        for num, count in occurrences.items():
            for i in range(count):
                result[i].append(num)

        return result

Identifying Problem Isomorphism

“Convert an Array Into a 2D Array With Conditions” can be mapped to “Group the People Given the Group Size They Belong To”.

The problems share similarities in the grouping elements based on certain conditions. In “Group the People Given the Group Size They Belong To”, people are grouped by the size they belong to, whereas in “Convert an Array Into a 2D Array With Conditions”, integers are grouped in a way that each row contains distinct integers, and the number of rows should be minimal.

They are approximately isomorphic. The grouping condition in “Convert an Array Into a 2D Array With Conditions” is more flexible, allowing different numbers of elements in each row, as long as each row contains distinct integers. Meanwhile, “Group the People Given the Group Size They Belong To” has a stricter condition where each group must contain exactly n members, where n is any given number.

They share similarities in the concept of grouping, but their conditions and requirements are different. Therefore, if you understand the solution to “Group the People Given the Group Size They Belong To”, it can be beneficial as a starting point to approach “Convert an Array Into a 2D Array With Conditions”. However, additional strategies or adjustments may be required to fully address the conditions in “Convert an Array Into a 2D Array With Conditions”.

“Convert an Array Into a 2D Array With Conditions” is more complex due to its additional requirement of minimizing the number of rows and allowing each row to have different numbers of elements.

10 Prerequisite LeetCode Problems

For “2610. Convert an Array Into a 2D Array With Conditions”, the following are a good preparation:

“442. Find All Duplicates in an Array” - This problem deals with finding duplicates in an array which is helpful for understanding the current problem.
“448. Find All Numbers Disappeared in an Array” - This problem introduces finding missing numbers in an array. It can enhance the understanding of working with array indices and their values.
“349. Intersection of Two Arrays” - This problem is about finding distinct common elements of two arrays, and understanding it can help you to grasp the concept of distinctness required in the current problem.
“217. Contains Duplicate” - This problem introduces the concept of checking for duplicates in an array, which is relevant for creating rows with distinct integers.
“566. Reshape the Matrix” - This problem deals with reshaping a 2D array, and it’s directly related to the current problem which involves creating a 2D array.
“485. Max Consecutive Ones” - This problem is about finding consecutive ones in an array. It can help in understanding how to track adjacent similar values, even though the current problem doesn’t require this, it’s good for enhancing array manipulation skills.
“283. Move Zeroes” - This problem involves moving certain values in an array to a particular position. It can enhance your skills in manipulating array values, which is useful in the current problem.
“268. Missing Number” - This problem is about finding the missing number in an array, understanding this can help in manipulating and examining array values.
“645. Set Mismatch” - This problem involves finding the duplicate and missing number in an array. It could be helpful in understanding more about working with array values and their indices.
“1287. Element Appearing More Than 25% In Sorted Array” - This problem introduces the concept of finding elements with certain conditions in an array, which can be useful in understanding how to divide array based on conditions.

These cover array manipulation, handling duplicates, and working with 2D arrays, which are key to solving “2610. Convert an Array Into a 2D Array With Conditions”.

Problem Classification

The problem falls under the “Arrays and 2D Arrays” domain. It involves manipulation of a 1D array to form a 2D array under given constraints.

What Components

Input: An integer array nums of length 1 to 200.
Constraints:

The 2D array should contain only elements from nums.
Each row in the 2D array must contain distinct integers.
The number of rows should be minimal.

Output: A 2D array that satisfies the constraints.

Problem Type: This is an optimization problem, aiming to minimize the number of rows in the resulting 2D array.
Sub-Problems: Breaking down the array to remove duplicates per row, and figuring out how to construct rows with distinct integers.
Algorithmic Concepts: Array manipulation, data structures like sets for quick lookup, and possibly sorting.

In summary, the problem involves array manipulation with an optimization goal, constrained by distinctness in each row and minimizing the number of rows.

Clarification Questions

Element Repetition: Can elements be repeated across different rows of the 2D array?
Row Order: Does the order of the rows in the output 2D array matter?
Element Order: Does the order of elements within each row matter?
Multiple Answers: Are multiple correct answers acceptable, or is there a preferred answer among them?
Empty Rows: Are empty rows allowed in the 2D array?
Input Constraints: Are there any specific constraints on the elements of the input array nums, such as them being positive integers only?
Row Length Variability: Can rows have varying lengths or should they aim to be uniform?
Performance Constraints: What is the expected size of the input array, and are there any performance requirements?
Element Coverage: Do all elements from the original array nums need to be used in the 2D array?
Output Format: In what format should the 2D array be returned? As a list of lists, or is another format acceptable?

Clarification questions like these can help eliminate ambiguities and make the problem statement more precise.

Problem Analysis and Key Insights

Minimize Rows: The problem aims to minimize the number of rows in the 2D array.
Distinct Elements in Rows: Each row must contain distinct integers.
Element Utilization: All elements from the original array must be used.
Multiple Valid Outputs: More than one correct answer is possible.
Variable Row Length: The 2D array can have varying row lengths.
Element Range: The elements in the array have an explicit range, which is between 1 and the length of nums.
Constraints: The constraints on the array length are lenient, suggesting that a relatively straightforward solution should suffice.

These insights help inform the strategy for solving the problem. For instance, aiming to minimize the number of rows might push us towards maximizing the use of repeated numbers in the same row where possible. The flexibility in row lengths also provides leeway in the arrangement of numbers.

Problem Boundary

The scope of the problem is well-defined:

Input Scope: You’re given an integer array nums with a length constraint of 1 to 200, and each element’s value is between 1 and the length of the array.
Output Scope: The output is a 2D array that must meet the criteria laid out in the problem: minimal number of rows, distinct integers in each row, and utilization of all elements from the input array.
Operational Scope: The problem requires you to create the 2D array based on specific conditions but leaves room for multiple valid outputs.
Constraint Scope: Given the constraints, the problem needs to be solvable within reasonable time and space complexities, suggesting that a polynomial-time algorithm is expected.
Functional Scope: The problem doesn’t specify side-effects or extra functionalities like validation of inputs or error handling. It’s a pure transformation problem: given an array, transform it into a 2D array based on the rules provided.
Application Scope: While the problem itself is abstract, the techniques to solve it could be applicable in situations where data needs to be partitioned into distinct groups while minimizing the number of such groups.

The scope is restricted to solving this transformation problem based on the rules given, without any additional features or functionalities.

To establish the boundary of the problem, consider the following aspects:

Input Limits: The input array nums has a constrained length between 1 and 200, and elements are integers between 1 and the length of the array. This sets a boundary for the types and range of acceptable inputs.
Output Characteristics: The output must be a 2D array with the minimum number of rows, containing only elements from the input array. Each row must have distinct integers, and all input integers must be used. The boundary is established by these requirements; outputs that do not meet them are considered invalid.
Multiple Solutions: The problem allows for multiple valid answers, so the solution space is not singular. Any 2D array meeting the conditions is considered within the boundary of valid solutions.
Time Complexity: Though not explicitly stated, given the constraint on the length of the array, a solution with polynomial time complexity is expected, setting a computational boundary.
Edge Cases: Think of smallest and largest inputs, or unique scenarios like all elements being the same or all distinct. These edge cases define the boundaries for which your solution must be valid.
Functional Scope: The problem does not require additional functionalities like input validation or error handling. The boundary is thus purely algorithmic; transform the given array according to the rules.
Environment: There’s no mention of where the code will run, network latency, user interaction, or database access, narrowing the boundary to the algorithm itself.

By clearly defining these aspects, you set the boundary conditions within which the problem must be solved.

Distilling the Problem to Its Core Elements

Fundamental Concept: This problem is essentially about partitioning a set of numbers with duplicates into smaller sets with distinct numbers, while minimizing the number of smaller sets.
Simplest Description: Imagine you have a bag of numbered balls, some of which are the same number. You need to divide these balls into the smallest number of boxes possible, but each box can only have one ball of each number.
Core Problem: The core problem is to find the minimum number of sets (or rows in a 2D array) that can contain all the numbers from the given list without any duplicates in each set.
Key Components:
- Identify distinct elements in the array.
- Count occurrences of each distinct element.
- Create sets (rows in 2D array) using these counts to ensure each set has distinct elements.
- Minimize the number of sets.
Minimal Set of Operations:
- Count the occurrences of each distinct element in the input array.
- Loop through the counts, distributing elements into sets to ensure distinctness.
- Append each filled set to the final 2D array.
- Return the 2D array.

Understanding these aspects will give you a solid foundation to begin solving the problem.

Visual Model of the Problem

To visualize this problem, you can think of it as a process of sorting colored balls into boxes. Each color represents a unique integer from the array, and the number of balls of each color represents the frequency of that integer in the array.

Initial State: Imagine all the colored balls (integers) are in a large bag.
Task: Your job is to sort these balls into as few boxes as possible, with the constraint that each box can only have one ball of each color.
Actions:
- Pick a ball (integer) from the bag.
- Check if a box already exists that can accommodate this color without breaking the rule.
- If yes, put the ball in that box. If not, start a new box.
Final State: Once the bag is empty, you will have a minimal number of boxes, each with no duplicate colors. This maps to the 2D array, where each box is a row and the balls in it are the integers in that row.

Using this visualization, you can understand the problem in a more tangible way and it might also make explaining the solution easier.

Problem Restatement

You have a list of integers, some of which may be repeated. Your task is to organize these integers into a 2D array. The catch is that each row of this 2D array should only contain unique integers. Moreover, you need to use up all the integers from the original list while also making sure that the number of rows in the 2D array is as small as possible. You can return any valid 2D array that meets these conditions.

Requirements:

Use all integers from the original list.
Each row in the 2D array must have distinct integers.
Minimize the number of rows in the 2D array.

Constraints:

The original list has at least 1 and at most 200 integers.
Each integer in the list is between 1 and the length of the list.

Abstract Representation of the Problem

This problem is about partitioning a set of elements with repetitions into minimal disjoint sub-sets such that each sub-set has unique elements.

Key Elements:

Set of Elements: A list of integers, some of which might be repeated.
Partition: Dividing the set into multiple sub-sets.
Disjoint Sub-sets: Each sub-set should contain unique elements.
Minimal: Reduce the number of sub-sets as much as possible while satisfying all other conditions.

The objective is to find the minimal partition that satisfies these conditions.

Terminology

Partitioning: In mathematics and computer science, partitioning refers to dividing a set into non-overlapping subsets. Here, it means creating 2D arrays from a 1D array, adhering to specific rules.
Disjoint Sub-sets: Sub-sets that do not have any elements in common. In the context of this problem, each row in the 2D array must contain distinct integers.
Minimal: In the context of this problem, minimal refers to the requirement that the number of rows in the 2D array should be as few as possible while satisfying the other conditions.
Constraints: Limitations or conditions that a solution must adhere to. In this problem, the constraint is the range of array length and the range of elements.

Understanding these terms is crucial for defining the problem’s boundary and crafting a solution that adheres to the problem statement.

Problem Simplification and Explanation

Breaking it Down:

You have a list of numbers, some of which might be repeated.
You need to organize these numbers into groups (rows in a 2D array).
Each group should have unique numbers, meaning no duplicates within the same group.
You should make as few groups as possible.

Key Concepts:

Array: Think of it as a list of items or a row of numbered boxes containing numbers.
2D Array: An array of arrays. Imagine it as a collection of rows stacked vertically.
Unique: No repeating numbers within the same group.
Minimal Groups: As few rows as possible in the 2D array.

How They Interact:

You pull numbers from the original list (array) to create new groups (rows) making sure the numbers in each group are unique. You continue doing this with the aim of creating as few groups as possible.

Metaphor:

Imagine you are sorting colored balls into different boxes. Each box can have balls of different colors, but no two balls in the same box can be of the same color. You want to use as few boxes as possible to store all the balls.

Constraints

Specific Characteristics for Efficient Solution:

Unique Elements: The problem specifically states that each row in the 2D array should contain unique elements. This can be exploited to speed up the row population. For instance, once a unique element is placed in a row, we know it won’t appear in the same row again.
Minimal Rows: We want the number of rows to be as minimal as possible. This means we should aim to fill up each row as much as we can before creating a new one.
Array Length Constraint: The length of the array is limited to 200, and each element value is between 1 and the length of the array. This fixed range can be useful for pre-allocating data structures like dictionaries or arrays to store frequency counts, thereby reducing the need for dynamic resizing.
Multiple Solutions Allowed: The problem statement is forgiving in that it allows for multiple correct answers. This gives us more flexibility in how we construct the 2D array.
No Ordering Requirement: The problem does not require the rows or the numbers within each row to be sorted. This means we can insert numbers into rows in the order we encounter them, without worrying about rearrangement.

By recognizing these characteristics, we can tailor our approach to make the most of these ’exploitable’ aspects. For example, knowing that we want to minimize the number of rows can guide us to try to fill up existing rows before creating new ones.

Key Insights from Analyzing Constraints:

Size Limitation: The input size is limited to 200 integers. This suggests that a solution with a time complexity of O(n^2) or better would likely be acceptable, though we should aim for better if possible.
Element Range: The value of each element in the array is between 1 and the length of the array. This helps in pre-allocating memory and reduces the need for handling out-of-bounds conditions. It also suggests that techniques like counting sort or frequency mapping could be useful.
Uniqueness Not Required: There is no requirement for the elements in the input array to be unique. This insight means we need a strategy to handle duplicates in a way that minimizes the number of rows.
Flexibility in Answers: Multiple correct answers are acceptable. This flexibility means we don’t need to find the “best” solution, just a valid one. This could make the problem easier to solve as we can stop as soon as we find any solution that meets the conditions.
No Row Length Constraints: Each row in the 2D array can have a different number of elements. This relaxation in constraints allows for a more dynamic approach to filling in the rows.
No Sorting Requirement: The 2D array output does not need to be sorted. This simplifies the problem as we can add elements to each row without worrying about maintaining any specific order.

Understanding these constraints can help focus the problem-solving approach by highlighting what we don’t have to worry about (e.g., sorting, finding the “best” answer) and what we do need to be concerned about (e.g., handling duplicates, minimizing row count).

Case Analysis

Additional Examples and Test Cases

1. Single Element (Edge Case)

Input: [1]
Output: [[1]]
Reasoning: The array contains only one element. It can be represented as a single row in the 2D array.

2. All Unique Elements

Input: [1, 2, 3, 4]
Output: [[1, 2, 3, 4]]
Reasoning: All elements are unique, so they can be placed in a single row.

3. Two Unique, Two Duplicates

Input: [1, 1, 2, 2]
Output: [[1, 2], [1, 2]]
Reasoning: Here, we have duplicates for each unique element. The minimal number of rows we can have is two.

4. Frequent Duplicates (Boundary Condition)

Input: [1, 1, 1, 1, 1, 1, 2]
Output: [[1, 2], [1], [1], [1], [1], [1]]
Reasoning: We have many duplicates of 1 and just one of 2. We should aim to minimize rows by placing 2 with one of the 1s, and the remaining 1s in individual rows.

5. Multiple Options (Flexibility in Answer)

Input: [1, 2, 2, 3, 3, 3]
Output: [[1, 2, 3], [2, 3], [3]] or [[1, 2], [2, 3], [3, 3]]
Reasoning: The problem allows multiple valid answers. Both outputs meet the conditions.

6. Descending Order

Input: [4, 3, 2, 1]
Output: [[4, 3, 2, 1]]
Reasoning: Even if the input is in descending order, it doesn’t affect the solution because sorting is not required.

7. Max Length Array (Edge Case)

Input: [1, 1, 1, ..., 1] (200 times)
Output: [[1], [1], [1], ..., [1]] (200 times)
Reasoning: When all numbers are the same and the array is of maximum length, the 2D array will have the maximum number of rows, each containing one element.

Edge Cases

Single Element: The input array has only one element.
Max Length Array: The input array has 200 elements, hitting the constraint’s upper boundary.

These test cases cover various aspects like duplicate handling, flexibility in output, and element order, and they provide a thorough understanding of the problem constraints and key insights.

Visualizing these test cases can be helpful in understanding the nuances of the problem. Here’s how you might visualize them:

Single Element: Picture a single-cell in a 2D grid. There’s only one place for the number to go, so the visualization is straightforward.
```
Grid: 1
```
All Unique Elements: Imagine a single row with all unique numbers. Think of it as a straight line where each point represents a unique number.
```
Grid: 1 -> 2 -> 3 -> 4
```
Two Unique, Two Duplicates: Picture two rows, where each row contains the two unique numbers.
```
Grid:
1 -> 2
1 -> 2
```
Frequent Duplicates: Here, imagine a row with a mix of 1s and the solitary 2, followed by rows containing individual 1s.
```
Grid:
1 -> 2
1
1
1
1
1
```
Multiple Options: Two possible configurations can exist. Imagine these as rows where you are distributing the most frequent numbers first.
```
Option 1:
1 -> 2 -> 3
2 -> 3
3

Option 2:
1 -> 2
2 -> 3
3 -> 3
```
Descending Order: Like the ‘All Unique Elements’ but with the numbers in descending order, represented in a single row.
```
Grid: 4 -> 3 -> 2 -> 1
```
Max Length Array: Here, think of a column full of the same number, stretching from top to bottom.
```
Grid:
1
1
.
.
.
1 (200th time)
```

These visualizations help anchor key aspects of the problem like handling duplicates, unique elements, and distribution, making it easier to formulate an efficient solution.

Analyzing these different cases reveals several key insights:

Handling Duplicates: Duplicates should ideally be grouped together. The more frequent a number is, the more rows it will appear in. This is essential to minimize the number of rows.
Unique Elements: These are the easiest to handle; they can all be placed in a single row. This scenario points to the best-case efficiency of the algorithm.
Ordering Doesn’t Matter: Whether the numbers in a row are in ascending or descending order doesn’t affect the problem’s constraints. This means that sorting the array isn’t strictly necessary for the solution, but it could simplify the implementation.
Distribution: The objective is not just to separate distinct numbers but to do so in a way that minimizes the number of rows. The frequency of each number will guide its distribution across rows.
Flexibility in Output: The problem allows for multiple correct outputs. This means the problem has a degree of flexibility and we don’t have to find a unique solution, just a valid one.
Edge Cases: When the array has only one element or when the array has maximal length (200), the problem’s constraints still have to be met. These edge cases tell us that the solution should not only be efficient but also robust to handle these scenarios.
Constraints: The array length and values within it are modest (1 <= nums.length <= 200 and 1 <= nums[i] <= nums.length), which means we can afford algorithms that are not necessarily the most optimal in time complexity.

By recognizing these insights, we can strategize our approach to solving the problem more effectively.

Identification of Applicable Theoretical Concepts

Here are some mathematical or algorithmic concepts that can help simplify this problem:

Set Theory: The condition that each row must contain distinct integers hints at using a set for each row to enforce this constraint.
Frequency Counting: Given that the same integer can appear multiple times in the array but only once per row, counting the frequency of each number could be useful.
Greedy Algorithms: The problem asks for a minimal number of rows, so a greedy approach that tries to fill each row as much as possible before creating a new one could be effective.
Dynamic Programming: We are essentially partitioning the set of numbers into subsets (rows). Though it might be overkill for this problem, DP could be useful for similar problems with tighter constraints.
Partition Problem: This problem bears a resemblance to the partition problem, which is a known NP-hard problem. Here, however, the constraints are more relaxed, allowing for heuristic or greedy methods to solve it.
Bucketing: Considering the constraint that 1 <= nums[i] <= nums.length, you could use bucketing to store integers, making the algorithm more efficient.
Sorting: Though not strictly necessary, sorting the numbers can make it easier to organize them into rows, especially if combined with frequency counting.
Time Complexity: Given the problem’s constraints (array length <= 200), we don’t necessarily need to aim for the most optimized time complexity, giving us the flexibility to implement a simpler, perhaps less efficient, solution if it’s easier to understand or code.
Mathematical Functions: Functions like min, max, or sum may not directly apply but understanding the minimum and maximum possible values can help in boundary cases.

By leveraging these concepts, the problem becomes more manageable and can lead to a more efficient and effective solution.

Simple Explanation

Imagine you have a bunch of numbered balls, and some numbers repeat. You also have several buckets in front of you. You’re asked to put these balls into the buckets with two rules:

No two balls in the same bucket can have the same number.
You should use as few buckets as possible.

So, you’ll try to fill each bucket with different numbers before moving on to a new bucket. Your task is to figure out the best way to organize these balls into the buckets following these rules.

Problem Breakdown and Solution Methodology

Certainly. To solve this problem, we would employ a technique similar to sorting and bucketing. Here’s how the steps contribute to the overall solution:

Counting the Balls: First, count how many balls you have of each number. Imagine each number as a different type of fruit. Knowing the quantity helps you plan the arrangement.
- Metaphor: This is like separating a fruit basket into piles of apples, oranges, bananas, etc., and counting how many you have of each.
Prepare the Buckets: Next, create empty buckets where you’ll place the balls.
- Metaphor: These are your fruit bowls, waiting to be filled.
First Pass - Unique Fruits: Fill the first bucket with one of each type of ball (number). This ensures that each bucket has unique numbers.
- Metaphor: Take one of each type of fruit and make a fruit salad in the first bowl.
Second Pass - Repeat Fruits: Go back to your counting list. For any ball type where you had more than one, start filling them into a new bucket, again ensuring that the numbers are unique within that bucket.
- Metaphor: If you have more apples, make another fruit salad but only include the fruits you have left over.
Rinse and Repeat: Continue this process until you’ve placed all the balls into buckets.
- Metaphor: Continue making fruit salads until you run out of fruits.
Review Buckets: At this point, you should have all the balls sorted into buckets with unique numbers in each and using the minimum number of buckets possible.
- Metaphor: You end up with a few fruit bowls, each with a unique mix of fruits, and you’ve used the least number of bowls necessary.

How Parameters Affect the Solution

More Repeated Numbers: The more times a number repeats, the more buckets you’ll need, as each bucket can hold only one ball of each number.
Larger Range of Numbers: If the numbers range from 1 to a very large number, but you don’t have many balls, you’ll still likely end up with a single bucket.

Example Cases

Case with Repeats - [1, 3, 4, 1, 2, 3, 1]
- Count: 1’s = 3, 2’s = 1, 3’s = 2, 4’s = 1
- First Bucket: [1, 3, 4, 2]
- Second Bucket: [1, 3]
- Third Bucket: [1]
Case with No Repeats - [1, 2, 3, 4]
- Count: 1’s = 1, 2’s = 1, 3’s = 1, 4’s = 1
- First Bucket: [1, 2, 3, 4]

By understanding these key steps and parameters, we develop a strong grasp of how to solve this problem efficiently.

Identification of Applicable Theoretical Concepts

Here are the key terms in the problem and how they guide the problem-solving approach:

Integer Array (nums): This is our input. Knowing that we’re dealing with integers helps us immediately rule out concerns related to floating-point precision. Strategy: Basic array manipulation techniques will be useful.
2D Array: The problem specifies that our output must be a 2D array. Strategy: Knowing that we must output a 2D array suggests that some form of nesting or grouping will be involved in the solution.
Distinct Integers in Each Row: Each row in the 2D array must contain distinct integers. Strategy: A counting or mapping technique will help ensure that each row has unique integers. We may have to sort or iterate to enforce this.
Minimal Number of Rows: The 2D array should have the least number of rows possible. Strategy: This pushes us towards an optimization approach, possibly using greedy methods to fill as many unique elements into a single row as possible before moving to the next one.
Constraints (1 <= nums.length <= 200): Knowing the constraints gives us information about the scope of the problem. Strategy: Since the constraints are relatively small, an approach with a time complexity of O(n^2) or even slightly higher would be acceptable.

By identifying these key terms and their implications, we can tailor our approach to specifically address the requirements and constraints of the problem.

Integer Array (nums): A simple table with one row can represent the array. Each column in that row would represent an array element. This visualization can help you get an immediate grasp of what numbers you are dealing with.
2D Array: To visualize the final 2D array, you can use a table where each row of the table corresponds to a row in the 2D array. This helps in tracking how you’re allocating numbers from the input array to each row of the 2D array.
Distinct Integers in Each Row: Within each row of your 2D array table, mark or color-code duplicate numbers to highlight the requirement for distinct integers. This will give you a visual clue on where adjustments need to be made.
Minimal Number of Rows: You could start by putting all unique elements from the input array into the first row of your 2D array table. From there, observe how many rows you need as you add the rest of the numbers. The goal is to keep this table as short (few rows) as possible.
Constraints: You don’t necessarily visualize constraints like “1 <= nums.length <= 200” in a table, but being aware of them helps you know the boundaries of your solution.

By drawing these tables or diagrams, you can more easily identify patterns, opportunities for optimization, or potential pitfalls. It becomes a useful tool for problem-solving and helps you grasp the “big picture” of what you’re trying to achieve.

Inference of Problem-Solving Approach from the Problem Statement

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 of Approach:
1.1. Initialize an Empty 2D Array: We start by creating a 2D array that will hold the final output.
1.2. Sort Input Array: Sort the given integer array to handle duplicate numbers efficiently.
1.3. Populate First Row: Iterate through the sorted array to add unique numbers to the first row of the 2D array.
1.4. Handle Duplicates: For duplicate numbers, add them to a new row, ensuring that each row only has distinct numbers.
1.5. Check for Minimization: After fully populating the 2D array, check if the number of rows is minimal. If not, refine the 2D array.
1.6. Return 2D Array: Once all conditions are met, return the 2D array.
Granular, Actionable Steps:
2.1. Initialize: Create a 2D array, let’s call it result.
2.2. Sort: Use any sorting algorithm to sort the input array nums.
2.3. First Row: Use a loop to traverse nums. For each unique number, append it to the first row of result.
2.4. New Rows for Duplicates: If you encounter a duplicate, create a new row in result and add it there, while ensuring no duplicate exists in that new row.
2.5. Minimization Check: Observe the 2D array to ensure it has the least number of rows possible. Make adjustments as needed.
2.6. Output: Return the result array.
Independent Parts:
- Sorting the input array is independent of creating the 2D array.
- Populating unique numbers in the first row can be done independently of handling the duplicates.
Repeatable Patterns:
- The action of adding a number to a row in the 2D array is a repeatable pattern.
- The action of creating a new row for duplicates and filling it with unique numbers is another repeatable pattern.

By following this approach, you should be able to solve the problem effectively and ensure that each part contributes to the final, optimized solution.

Solution Approach and Analysis

Detailed Approach

Initialize Empty 2D Array (Bucket): Imagine you have an empty bucket. This bucket will hold smaller containers (arrays). Each container will hold a unique set of numbers from the original list. Let’s call this bucket result.
Sort the Input (Order the Numbers): Sort the input array nums. Think of it as ordering a messy pile of numbered balls.
Populate the First Container: Take the smallest ball (smallest number) from the pile and place it in the first container inside the bucket. Continue placing unique balls (numbers) into this first container.
Handle Duplicates (Additional Containers): When you pick up a ball that’s already in the first container, start a new container for it. Make sure each container holds unique numbers.
Check for Minimization (Optimize Containers): Look at all the containers and see if you can reduce their number by rearranging the balls, while keeping each container’s numbers unique.
Return the Bucket: Your bucket filled with containers is the 2D array that you need to return.

Parameters and Their Effects

Array Length: If the array is longer, the 2D array will likely have more rows (containers). The sorting step would also take longer.
Number Range: If the range of numbers is smaller, you’ll have more duplicates, and therefore, more containers in the bucket.
Duplicates: More duplicates mean more containers. However, more duplicates might also offer opportunities to minimize the number of containers by rearranging.

Example Case

Let’s use the first example from the problem statement: nums = [1,3,4,1,2,3,1]

Initialize: result = []
Sort: nums = [1,1,1,2,3,3,4]
First Container: The first container will hold [1,2,3,4]
Handle Duplicates:
- Second container will hold [1, 3]
- Third container will hold [1]
Check for Minimization: No need in this case, as it’s already minimized.
Return: result = [[1,2,3,4], [1,3], [1]]

By following this approach, the problem can be solved effectively while meeting all the constraints and conditions.

Identify Invariant

In computer science, an invariant is a condition that holds true before and after some piece of code is executed. For this problem, the invariant is that “each row in the 2D array must contain distinct integers.”

Before and after each operation, whether it’s sorting the array or placing numbers into containers, this condition must hold true. Every time you add a number to a container (a row in the 2D array), you have to make sure that this invariant is maintained; that is, no two numbers in the same container are the same.

Maintaining this invariant leads us towards a correct solution, ensuring that each container has unique numbers and therefore abides by the problem constraints.

Identify Loop Invariant

A loop invariant in the context of this problem could be: “At the start of each iteration, all existing rows in the 2D array contain distinct integers.”

Assuming a loop is used to populate the 2D array, this invariant must hold true before entering the loop for the first time, remain true before each subsequent iteration, and still be true after the loop terminates. This ensures that the requirement of “each row in the 2D array contains distinct integers” is met throughout the entire operation of the loop, ultimately fulfilling the problem’s constraints.

In this problem, the terms “invariant” and “loop invariant” could refer to the same condition, namely: “Each row in the 2D array contains distinct integers.”

The term “invariant” broadly describes a condition that remains unaltered throughout the execution of a program or a specific block of code.

The “loop invariant” is a special case of an invariant. It specifically refers to a condition that holds true before and after each iteration of a loop.

So, in the context of this problem, the invariant condition is important in a broader sense for ensuring the integrity of the 2D array. If you’re using a loop to construct the 2D array, then this condition would also serve as your loop invariant.

Thought Process

Step 1: Understand the Problem

The first step is to understand the problem requirements. We need to form a 2D array using elements from a given list. The elements in each row should be distinct, and the number of rows should be as minimal as possible.

Step 2: Analyze Constraints

The problem tells us that the size of the list can be up to 200 elements, so we can aim for an approach that runs in O(n) or O(n log n) time.

Step 3: Identify Cues

The cues in the problem statement are:

We need to minimize the number of rows.
Each row should contain distinct integers.

Step 4: Formulate Approach

Use a hashmap to store how many times each element appears in the given list.
Initialize an empty list to serve as our 2D array.
Iteratively fill rows with unique elements, reducing the count in our hashmap as we go.

Step 5: Code

Here’s how to implement this in Python: PENDING TEST

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from collections import Counter

def create_2D_array(nums):
    # Step 1: Count occurrences of each number in nums
    num_count = Counter(nums)

    # Step 2: Initialize empty 2D array
    two_d_array = []

    # Step 3: Fill rows
    while num_count:
        new_row = []

        for num in list(num_count.keys()):
            new_row.append(num)
            num_count[num] -= 1

            if num_count[num] == 0:
                del num_count[num]

        two_d_array.append(new_row)

    return two_d_array

# Test cases
print(create_2D_array([1,3,4,1,2,3,1]))  # Should output something like [[1, 3, 4, 2], [1, 3], [1]]
print(create_2D_array([1,2,3,4]))  # Should output [[1, 2, 3, 4]]

Insights about the Problem Statement

The problem has a constraint about minimizing rows, which suggests that we should aim to fill each row as much as possible before creating a new one.
The problem allows for multiple valid solutions, which hints that a greedy approach might be suitable here.

With these steps and insights, you should have a clearer path to solve similar problems.

Establishing Preconditions and Postconditions

1. Parameters:

Inputs: The method takes one input parameter called nums.
Types: nums is a list of integers.
Representation: In the context of the problem, nums represents the given integer array from which a 2D array needs to be created.

2. Preconditions:

State: No specific state of the program is required before the method is called.
Constraints: The list nums must satisfy the constraints: 1 <= nums.length <= 200, 1 <= nums[i] <= nums.length.

3. Method Functionality:

Expected Function: The method is expected to return a 2D array satisfying the conditions stated in the problem.
Interaction: It only interacts with the input nums to produce the output. The current state of the program is irrelevant for this method.

4. Postconditions:

State: There is no change in the state of the program or values of parameters after the method returns.
Return Value: The return value is a 2D list of integers that represent a valid transformation of nums based on problem constraints.
Side Effects: The method has no side effects.

5. Error Handling:

Preconditions Not Met: If the preconditions on the input parameters are not met (although in this specific problem the constraints are straightforward), the behavior is undefined.
Response: Since Python is a dynamically typed language, type errors or constraint violations could lead to runtime errors, but they are not explicitly handled in this method. No special return value or exception is defined for such cases.

Problem Decomposition

1. Problem Understanding:

The problem is to transform a given list of integers into a 2D list. The transformed 2D list should use all elements from the original list exactly once. Each row of this 2D list should have distinct integers, and the number of rows should be minimal.

2. Initial Breakdown:

Major parts of the problem:

Count the occurrences of each integer in the given list.
Create a 2D list where each row has distinct integers.
Populate each row based on the occurrences of integers from step 1.

Count Occurrences:
- Iterate over the list to generate a count for each integer.
Initialize 2D List:
- Create an empty 2D list to start filling it up.
Populate 2D List:
- Add the most frequent numbers first in new or existing rows.
- Reduce the count for each added integer.
- Repeat until all integers are placed.

4. Task Identification:

Counting occurrences is a common task that can be reused.
Initializing a 2D list is a straightforward task.
Placing an integer in a row and reducing its count is a repeatable task.

5. Task Abstraction:

countOccurrences(nums: List[int]) -> Dict[int, int]
initialize2DList() -> List[List[int]]
placeIntegerInRow(row: List[int], integer: int, counts: Dict[int, int]) -> None

Each task is abstract enough to be clear and reusable but still specific to the problem’s context.

6. Method Naming:

countOccurrences: To count the occurrences of each integer in the list.
initialize2DList: To initialize an empty 2D list.
placeIntegerInRow: To place an integer in a row and update its count.

7. Subproblem Interactions:

Start by counting occurrences with countOccurrences.
Initialize the 2D list using initialize2DList.
Populate the 2D list iteratively using placeIntegerInRow.

The first task must be completed before the others, as its results are used in the following steps. After initialization, the population step can proceed iteratively until completion. There are no cyclic dependencies among tasks.

From Brute Force to Optimal Solution

Brute-Force Solution:

Approach:

Generate all possible permutations of the input list.
For each permutation, try to partition it into rows of distinct integers.
Keep track of the partition with the minimum number of rows.

Code (Python):

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from itertools import permutations

def brute_force_solution(nums):
    min_rows = float('inf')
    best_partition = None
    
    for perm in permutations(nums):
        rows = []
        current_row = []
        
        for num in perm:
            if num not in current_row:
                current_row.append(num)
            else:
                rows.append(current_row)
                current_row = [num]
        
        rows.append(current_row)
        
        if len(rows) < min_rows:
            min_rows = len(rows)
            best_partition = rows

    return best_partition

nums = [2, 1, 1, 1, 2, 2, 3]
print(brute_force_solution(nums))

Inefficiencies:

Generating all permutations has a time complexity of (O(N!)), where (N) is the length of the list. This is extremely inefficient.
We then check each permutation to find the minimum number of rows, adding another loop.

Optimized Solution:

Step 1: Count Occurrences

Instead of generating all permutations, we start by counting the occurrences of each number.

Step 2: Initialize 2D List

Initialize an empty list to represent the 2D array.

Step 3: Place Integers with High Frequency First

Start by placing the integers with higher occurrences in the first available row that doesn’t already contain that integer. We can use a priority queue for this.

Code (Python):

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from collections import Counter
from queue import PriorityQueue

def optimized_solution(nums):
    counts = Counter(nums)
    pq = PriorityQueue()
    
    for num, count in counts.items():
        pq.put((-count, num))
    
    rows = []
    
    while not pq.empty():
        count, num = pq.get()
        count = -count
        
        placed = False
        for row in rows:
            if num not in row:
                row.append(num)
                placed = True
                break
                
        if not placed:
            rows.append([num])
        
        if count > 1:
            pq.put((-(count - 1), num))

    return rows

nums = [2, 1, 1, 1, 2, 2, 3]
print(optimized_solution(nums))

PENDING TEST

Time Complexity:

Counting occurrences: (O(N))
Priority Queue operations: (O(N \log N))
Populating 2D list: (O(N))

Overall time complexity: (O(N \log N))

Space Complexity:

Counter Dictionary: (O(U)), where (U) is the number of unique integers.
Priority Queue: (O(U))
2D List: (O(N))

Overall space complexity: (O(N))

Summary:

The optimized approach is far more efficient compared to the brute-force method. It reduces the time complexity to (O(N \log N)) from an impractical (O(N!)) and uses linear space (O(N)).

Code Explanation and Design Decisions

1. Initial Parameters:

nums: This is a list of integers. It represents the original sequence of numbers that we need to partition into rows such that each row has distinct integers.

2. Primary Loop:

The primary loop is iterating over the Priority Queue (pq) which is populated based on the frequency of each unique number in nums.

Each iteration does the following:

Takes out the number with the highest remaining frequency (most urgent to place).
Places it into the first available row where it can fit (the row doesn’t already contain this number).

3. Conditions or Branches:

if num not in row: Checks whether the current number exists in the current row we are examining. The condition is there to ensure that each row only contains distinct integers.

4. Updates or Modifications:

row.append(num): We add the number to the row once we find a suitable row. This reflects that the number has been successfully placed.
pq.put((-(count - 1), num)): If more of the same number need to be placed, we decrement its frequency (count) by 1 and put it back in the priority queue.

5. Invariant:

An invariant here is that each row in the 2D array rows will always contain distinct integers. This is maintained by the condition if num not in row, ensuring the property stays true throughout each iteration.

6. Final Output:

The final output is a 2D list (rows) where each inner list is a row of distinct integers. This 2D list satisfies the problem’s requirement of partitioning the original sequence into the minimum number of such rows.

Coding Constructs

1. High-Level Strategies:

The code primarily uses a Priority Queue data structure to manage the frequency of each number in the list and Greedy Algorithm to place the numbers into rows, always selecting the most frequent yet-to-be-placed number first.

2. Purpose for a Non-Programmer:

The code organizes a list of numbers into separate groups. Each group should contain only unique numbers, and we want to use as few groups as possible. Imagine you have a bag of colored balls and you’re trying to sort them into boxes, but each box can have only one ball of each color.

3. Logical Elements:

Priority Queue: To manage numbers based on their frequency.
Loop: To iterate through the numbers.
Conditionals: To check whether a number can be placed in a particular row.
Arrays/List: To store the final rows and temporary rows for checks.

4. Algorithmic Approach:

Count how many times each number appears in the list.
Put these numbers in a Priority Queue based on how often they appear.
Go through the Priority Queue, always selecting the number that appears the most times but hasn’t yet been placed into a row.
Place this number into the first available row where it fits (no duplicate numbers in the row).
Repeat steps 3-4 until all numbers are placed.

5. Key Steps:

Count the frequency of each unique number.
Insert the unique numbers into a Priority Queue based on their frequency.
Loop to pick numbers from the Priority Queue and place them into rows such that each row has distinct numbers.

6. Algorithmic Patterns:

Greedy Algorithm: Always picking the most frequent yet-to-be-placed number to ensure minimal rows are used.
Priority Queue: Ensures the most frequent numbers are dealt with first.
Looping: Iterating over the Priority Queue to place each number.
Condition Checking: To make sure each row has distinct numbers.

Language Agnostic Coding Drills

1. Dissect the Code into Concepts

Variable Declaration: Storing essential values for later use.
Array/List Manipulation: Creating and manipulating lists or arrays to store data.
Looping: Iterating over sets of elements.
Conditionals: Implementing if-else statements for decision making.
Data Structures: Understanding and using a Priority Queue.
Counting Frequency: Counting the occurrences of unique elements in a list.
Greedy Algorithm: Making the locally optimal choice to reach a global optimum.

2. Concepts in Order of Increasing Difficulty

Variable Declaration: Easiest. Almost every code uses variables to store data.
Array/List Manipulation: A fundamental concept but requires understanding how to create, access, and modify data.
Looping: Loops are easy but add a level of complexity because they involve repetition and often conditionals.
Conditionals: These add decision-making into your code but also introduce more potential points of failure or bugs.
Counting Frequency: Involves looping and conditionals together, along with storing the results, hence more complex.
Data Structures: Knowing how to use a Priority Queue involves understanding how it works, making this more difficult.
Greedy Algorithm: Most difficult as it requires understanding not just how to code the algorithm but why it works to solve this specific problem.

3. Problem-Solving Approach

Variable Declaration: First, declare variables that will hold the frequency count and final groups.
Array/List Manipulation: Initialize an empty list or array to hold the final rows.
Counting Frequency: Loop through the given list to count the frequency of each number. Store this data for later use.
Data Structures: Insert the unique numbers and their frequencies into a Priority Queue.
Looping and Conditionals: Begin a loop to go through the Priority Queue. In each iteration, use conditionals to check if the most frequent number can be placed into any of the existing rows.
Greedy Algorithm: If it can, place it in. If not, create a new row. Always choose the number with the highest frequency that hasn’t yet been placed.
Array/List Manipulation: As numbers are placed, update the list or array that holds the rows.

By sequentially applying these drills, you can piece together the final solution. Each concept, or “drill,” serves as a building block that contributes to solving the problem.

Targeted Drills in Python

1. Coding Drills for General Concepts

Variable Declaration

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# Variable Declaration
x = 10
name = "Alice"

Array/List Manipulation

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# Array/List Manipulation
my_list = []
my_list.append(1)
my_list.append(2)

Looping

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# Looping
for i in range(5):
    print(i)

Conditionals

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# Conditionals
x = 10
if x > 5:
    print("Greater")
else:
    print("Smaller or equal")

Counting Frequency

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# Counting Frequency
freq = {}
arr = [1, 1, 2, 2, 2, 3]
for num in arr:
    if num in freq:
        freq[num] += 1
    else:
        freq[num] = 1

Data Structures

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# Data Structures (Priority Queue)
import heapq

pq = []
heapq.heappush(pq, 1)
heapq.heappush(pq, 2)
heapq.heappop(pq)

Greedy Algorithm

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# Greedy Algorithm
coins = [1, 2, 5]
amount = 11
count = 0
for coin in reversed(coins):
    count += amount // coin
    amount %= coin

2. Problem-Specific Concepts

For our problem, the most critical specific concept is how to use Priority Queue along with Greedy Algorithms.

Problem-Specific Priority Queue with Greedy

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# Using Priority Queue and Greedy together
import heapq

pq = []
freq = {1: 2, 2: 3, 3: 1}
for key, value in freq.items():
    heapq.heappush(pq, (-value, key))

while pq:
    value, key = heapq.heappop(pq)
    # Apply the greedy algorithm logic here

3. Assembling the Pieces

To solve the problem, we’ll integrate these drills in the following order:

Variable Declaration: Declare variables to store frequency counts and final output.
Array/List Manipulation: Initialize arrays or lists to hold temporary and final data.
Counting Frequency: Use a loop to go through the input array and fill in the frequency dictionary.
Data Structures: Fill in the Priority Queue using the frequency dictionary.
Looping and Conditionals: Loop through the Priority Queue to place elements into the rows.
Greedy Algorithm: Inside the loop, make greedy choices based on the available frequency and constraints.
Array/List Manipulation: Update the final output list with each iteration, as you place elements into the rows.

By following this sequence and integrating each coding drill, you can assemble a comprehensive solution to the problem.

Convert an Array Into a 2D Array With Conditions

Identifying Problem Isomorphism

10 Prerequisite LeetCode Problems

Problem Classification

What Components

Clarification Questions

Problem Analysis and Key Insights

Problem Boundary

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Additional Examples and Test Cases

1. Single Element (Edge Case)

2. All Unique Elements

3. Two Unique, Two Duplicates

4. Frequent Duplicates (Boundary Condition)

5. Multiple Options (Flexibility in Answer)

6. Descending Order

7. Max Length Array (Edge Case)

Edge Cases

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

How Parameters Affect the Solution

Example Cases

Identification of Applicable Theoretical Concepts

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Detailed Approach

Parameters and Their Effects

Example Case

Identify Invariant

Identify Loop Invariant

Thought Process

Step 1: Understand the Problem

Step 2: Analyze Constraints

Step 3: Identify Cues

Step 4: Formulate Approach

Step 5: Code

Insights about the Problem Statement

Establishing Preconditions and Postconditions

1. Parameters:

2. Preconditions:

3. Method Functionality:

4. Postconditions:

5. Error Handling:

Problem Decomposition

1. Problem Understanding:

2. Initial Breakdown:

3. Subproblem Refinement:

4. Task Identification:

5. Task Abstraction:

6. Method Naming:

7. Subproblem Interactions:

From Brute Force to Optimal Solution

Brute-Force Solution:

Approach:

Code (Python):

Inefficiencies:

Optimized Solution:

Step 1: Count Occurrences

Step 2: Initialize 2D List

Step 3: Place Integers with High Frequency First

Code (Python):

Time Complexity:

Space Complexity:

Summary:

Code Explanation and Design Decisions

1. Initial Parameters:

2. Primary Loop:

3. Conditions or Branches:

4. Updates or Modifications:

5. Invariant: