Flatten 2D Vector

The given task is to design an iterator that can flatten a 2D vector and provide methods to retrieve the next element and check if there are more elements.

Here is the implementation:

__init__(self, vec: List[List[int]]): The constructor takes the 2D vector and initializes the object. We’ll keep two pointers, one to track the current row and the other for the current column.
next(self) -> int: Returns the next integer in the 2D vector and moves the pointers forward. If the current row ends, it moves to the next row.
hasNext(self) -> bool: Checks if there are more integers in the 2D vector.

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class Vector2D:

    def __init__(self, vec: List[List[int]]):
        self.vec = vec
        self.row = 0
        self.col = 0

    def next(self) -> int:
        # Move to the next valid row if the current row is empty
        while self.row < len(self.vec) and self.col == len(self.vec[self.row]):
            self.row += 1
            self.col = 0
        
        # Retrieve the value and increment the column pointer
        val = self.vec[self.row][self.col]
        self.col += 1
        return val

    def hasNext(self) -> bool:
        # Move to the next valid row if the current row is empty
        while self.row < len(self.vec) and self.col == len(self.vec[self.row]):
            self.row += 1
            self.col = 0
        
        return self.row < len(self.vec)

The above code first ensures that it is pointing to a valid row that has elements by incrementing the row pointer if the current row is empty.

The next method retrieves the next value and moves the pointers.
The hasNext method returns True if there are more elements, False otherwise.

The time complexity for both the next and hasNext methods is O(1) on average, as they only make a constant amount of work for each call over the entire sequence of operations. The space complexity is O(1) as we only use a constant extra space to store the pointers.

The Vector2D class is a Python class that represents a 2D iterator for a given 2D vector. The class should implement two methods: next, which returns the next element in the 2D vector, and hasNext, which checks if there are any more elements in the 2D vector.

We’ll maintain two pointers: outer to track the current list in the 2D vector, and inner to track the current element in that list. Here’s the implementation:

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class Vector2D:

    def __init__(self, vec: List[List[int]]):
        self.vec = vec
        self.outer = 0
        self.inner = 0
        
        # Move to the next valid position if the current one is invalid
        self.moveNext()

    def next(self) -> int:
        # Get the value at the current valid position
        result = self.vec[self.outer][self.inner]

        # Move to the next element
        self.inner += 1
        self.moveNext()
        
        return result

    def hasNext(self) -> bool:
        return self.outer < len(self.vec)

    def moveNext(self):
        # Move to the next valid position
        while self.outer < len(self.vec) and self.inner >= len(self.vec[self.outer]):
            self.outer += 1
            self.inner = 0

The moveNext method ensures that the current position is valid or moves to the next valid position if it is not. The next method retrieves the value at the current valid position and moves to the next one. The hasNext method checks if there are more elements in the vector.

The given code ensures that all operations are in constant time, so the time complexity for both next and hasNext is O(1), and the space complexity is also O(1).

10 Prerequisite LeetCode Problems

“Flatten 2D Vector” involves understanding 2D arrays, iterators, and how to effectively flatten a 2D structure into 1D. Here are 10 simple problems to prepare:

341. Flatten Nested List Iterator: This problem also involves creating an iterator for a complex data structure, in this case a nested list.
173. Binary Search Tree Iterator: Another iterator problem, this time with a binary search tree. It will help you understand how to deal with more complex data structures.
284. Peeking Iterator: This problem will help you understand the mechanism of an iterator and how to handle the situation when you need to peek the next element.
232. Implement Queue using Stacks: This problem requires you to understand how to use one data structure (stacks) to implement another (queue), which can be helpful for understanding how to manipulate data structures.
155. Min Stack: This problem involves designing a stack that supports retrieving the minimum element, which can help you understand how to maintain additional properties while manipulating data structures.
295. Find Median from Data Stream: This problem will help you understand how to maintain a dynamic data structure and find the median.
380. Insert Delete GetRandom O(1): This problem is about implementing a data structure with specific operations. The techniques used can be useful for implementing the iterator in the problem.
706. Design HashMap: This problem will help you understand how to design a basic data structure, which is essential for more complex ones.
348. Design Tic-Tac-Toe: This problem is a good practice for handling 2D arrays.
54. Spiral Matrix: This problem is a great way to practice traversing 2D arrays, which is the first step in solving the given problem.

These cover design of data structures and iterators, which will be helpful when tackling “Flatten 2D Vector”.

Clarification Questions

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

Problem Analysis and Key Insights

Here are some key insights gained from analyzing the problem statement:

Need to design a custom iterator instead of relying on native language constructs.
The iterator must present the 2D vector as a flat 1D sequence.
Tracking current position will be crucial to provide sequential access.
hasNext() likely needs to check if any elements left in current row or any remaining rows.
next() needs to navigate to the next element’s position before returning it.
Input constraints on vector size limits search space and operations.
Follow-up suggests solutions using only built-in iterators may exist.
next() and hasNext() imply a unidirectional forward-only iterator.
No constraint on how underlying data is traversed (row-wise, col-wise, etc).

The key insights are around tracking position across two dimensions, presenting as a flat sequence, and incrementally moving the position to enable sequential access. The constraints allow flexibility in implementation.

Problem Boundary

Based on analyzing the problem statement, here is how I would summarize the scope:

Inputs:

2D vector of integers

Outputs:

Next element in flattened sequence
Boolean indicating if more elements remain

Objective:

Design iterator class providing sequential access over 2D vector
Implement next() and hasNext() methods

Focus:

Track position across nested structure
Increment position to move to next element
Check remaining elements for hasNext()
Abstract 2D structure as flat sequence

Out of Scope:

Supporting previous(), remove(), etc - unidirectional forward iterator
Optimizing underlying traversal order
Space complexity unless excessive
Thread safety or concurrency

So in essence, the scope focuses on designing an iterator over a 2D structure that presents a flat sequential view via next(), hasNext() and position tracking. It gives flexibility in the implementation details.

Here are some ways we can establish clear boundaries for the scope of this 2D vector iterator problem:

The input is limited to just the 2D vector of integers. No other inputs.
The outputs are the next element and a boolean for hasNext(). We don’t need to return or yield anything else.
We only need to implement the next() and hasNext() behaviors. Other methods like previous(), remove(), etc. are out of scope.
No specific constraints are provided on time/space complexity, so we have flexibility to optimize as needed.
The iterator should provide a sequential, flat view of the 2D structure. The underlying traversal order doesn’t matter.
No need to handle concurrent modification or thread safety based on problem description.
No requirement to validate inputs upfront. Can assume next() calls are valid.
Integer vector provides bounded datatypes. Generic types don’t need to be handled.
Reasonable constraints on number of calls allows more complex implementations.

Keeping the scope narrowly focused on the essential next(), hasNext() behaviors and sequential access over the 2D input allows solving the core problem without getting distracted by unnecessary generalizations or embellishments.

Problem Classification

This is an iterator design problem in the domain of data structures and algorithms. The key ‘What’ components are:

2D vector - Nested array structure to iterate over
Iterator - Object providing sequential access to elements
next() - Method to return next element
hasNext() - Method to check for more elements

We can further classify this problem as:

Iterator design - Creating a custom iterator interface
2D/Nested iteration - Iterating over nested container
Sequential access - Providing sequential element access
Stateful traversal - Maintaining internal state to track position

The core challenge is designing an iterator class that provides sequential access over a 2D nested array structure by maintaining traversal state internally.

This requires skills in iterator design, manipulating multidimensional data structures, and stateful traversal - all focused on sequentially accessing elements in a complex 2D structure. The key is tracking state across two dimensions.

Distilling the Problem to Its Core Elements

Here are my thoughts on analyzing this 2D vector iterator problem:

Fundamental concept:

This problem boils down to providing sequential access over nested data. The core technique is tracking state across dimensions.

Simple explanation:

We want to take a 2D grid of numbers and iterate through it one element at a time.

Core problem:

Design an iterator class to provide sequential access over a nested 2D structure. We can simplify it to creating a flat iterator over 2D data.

Key components:

2D vector
Tracking position
next() to return element
hasNext() to check for more

Minimal operations:

Initialize position
Return element at current position
Increment position(s)
Check if reached end

So in essence, this problem involves keeping track of position across two dimensions internally to sequentially return elements in a flat way over a 2D structure.

Visual Model of the Problem

Here are some ways we could visualize this 2D vector iterator problem to provide more intuition:

Grid - Showing the nested 2D structure as a grid can help visualize the position tracking across rows and columns.
Path - Illustrate a path traversing the grid from the current position to the next element returned.
Overlay numbers - Put sequential numbers overlaid on the grid to represent the flat iterator order.
Animation - Animate a token moving from one element to the next to highlight order.
Navigation map - Show a map view with current location that moves over the 2D grid.
Interactive - Create an interactive grid that highlights elements as next() is clicked to showcase iteration order.
State machine - Visualize iterator state tracking with a state diagram.

Representing the nested 2D structure as a grid and illustrating traversal across it helps provide intuition about maintaining position. Interactive visuals could further enhance understanding.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We need to design an iterator class that provides sequential access to the elements of a 2D vector.

The 2D vector is an array of arrays containing integers.

The iterator should implement two methods:

next() - Returns the next element in the flattened 2D structure and internally advances the position.
hasNext() - Returns a boolean indicating if any more elements remain to be iterated through.

The iterator should maintain internal state to keep track of the current position as it sequentially walks through the 2D structure.

It needs to expose the nested 2D vector as a flat linear sequence via the next() method.

The iterator is read-only and only needs to support forward iteration.

There are constraints provided on the maximum 2D vector size and number of operations.

So in essence, we need to create an iterator class that internally tracks position over a 2D structure to provide the illusion of a flat sequential view. It must maintain state across calls to enable iterating through the nested arrays linearly.

Abstract Representation of the Problem

Here is one way we could formulate an abstract representation of this 2D vector iterator problem:

We want to design an iterator over a logical grid G containing R rows and C columns of elements.

The iterator should present the elements of G as a flat uni-directional sequence S of length R*C.

It should implement:

next() - Return next element from S
hasNext() - Return true if elements remain in S

The iterator should maintain internal state to keep track of current position within S as next() is called repeatedly.

More formally:

G - Logical grid of elements
S - Sequential ordering of G’s elements

Iterator:

next() - Return si where i is current position in S
hasNext() - Return true if i < |S|
Maintain index i as state through calls

By representing the problem in terms of an abstract grid G and sequential traversal order S, we can focus on the key logical requirements around ordering and state maintenance separately from the concrete 2D vector and integer details. This helps relate it to general iteration concepts.

Terminology

Here are some key terms and concepts that are important for this 2D vector iterator problem:

Iterator - Object providing sequential traversal over a data structure by encapsulating state and iteration logic. Central concept.

Next/HasNext - Methods to return next element and check for more elements. Core iterator API.

Stateful iteration - Maintaining state like current position across iteration calls. Needed to track across 2D structure.

Nested loops - Iterating over containers within containers, like 2D grid. Challenging to flatten.

Linear sequencing - Abstracting nested structure as a linear flat sequence. Goal of iterator.

Position - Current location within structure. Crucial for indexed access.

Multidimensional arrays - Grid-like structure containing elements at indexes along dimensions. Input data structure.

Amortized analysis - Analyzing cost aggregated over sequence of operations. Good approach for iterator cost model.

The key is modeling the problem around iterators, which manage stateful traversal and present linear sequencing over nested multidimensional arrays. The terms help frame the problem appropriately.

Problem Simplification and Explanation

Here’s one way I could break down this 2D iterator problem into simpler concepts using a metaphor:

Let’s compare traversing a 2D grid to navigating a neighborhood with North-South roads and East-West avenues.

The rows of the grid are like the avenues running North-South. The columns are like the roads running East-West.

We want to visit each house in the neighborhood by going street-by-street sequentially, as if the 2D grid of houses were flattened into a 1D list of houses.

Our location on the map represents the iterator’s position in the grid. We update the position after visiting each house.

Checking if any unvisited houses remain is like checking if there are more elements for the iterator.

The key concepts are:

Grid = Neighborhood map
Rows = Avenues
Columns = Roads
Houses = Elements
Location = Iterator position

This real-world traversal analogy provides intuition about navigating and tracking position on a 2D grid sequentially, which maps to the problem of iteratively flattening 2D array contents.

Constraints

Here are some characteristics and constraints from the problem statement that could help optimize finding an efficient solution:

2D grid structure - Can leverage nested loops and linear indexing formulas.
Integer elements - Allows direct storage and access without serialization.
Read-only access - No need to optimize for updates and mutations.
Uni-directional iteration - Simpler to implement compared to bidirectional.
Maximum array sizes defined - Can limit nesting levels and position variables.
Bounded ops count - Can focus optimization on key traversal logic over one-time ops.
No specified output order - Underlying traversal can use any row/column order.
Valid input assumed - No need for input validation checks.
Follow-up suggests native solutions - Good indicatornative language constructs may suffice.

These characteristics suggest nested loops with linear indexing may be suitable for traversing multidimensional arrays in a read-only unidirectional way. The bounds help limit scope allowing optimization on the key traversal logic.

Here are some key insights gained from analyzing the constraints:

Grid structure points to nested loops as a straightforward way to traverse 2D data.
Integer elements allow direct storage and access without serialization complexity.
Read-only unidirectional access simplifies logic over read-write or bidirectional.
Maximum sizes suggest position can be tracked with simple numeric variables rather than complex objects.
Bounded operations count allows focusing optimization on traversal rather than one-time initialization logic.
Output order flexibility allows traversal optimization - no set constraints.
No input validation needed means we can assume next() calls are valid.
Native solutions may be possible since constraints aren’t too restrictive.

Overall, the constraints suggest a simple nested loop solution may be reasonable here, provided we encapsulate it in an iterator class correctly. The bounds help limit scope and simplify assumptions. There appears to be flexibility allowing optimization of the traversal.

Case Analysis

Here are some additional test cases covering different scenarios:

Base cases:

1x1 grid - Single element
2x1 grid - Single row
1x2 grid - Single column

Grid varieties:

Sparse grid - Few elements
Dense grid - Lots of elements
Jagged rows - Varying row sizes

Constraints:

Maximum rows and columns
Maximum total elements

Ordering:

Row-major order
Column-major order
Diagonal order

Edge cases:

Empty grid
Grid with one element

The edge cases include empty and single element grids which should be handled gracefully. Other cases validate traversal order, sparse/dense data, input size constraints, and testing different traversal orders. This provides broad test coverage.

Here are some ways to visualize the different test cases for this 2D vector iterator:

Base cases - Show 1x1, 2x1, 1x2 grids. Highlight iteration order.
Sparse grid - Grid with many blank spots. Show traversal path.
Dense grid - Grid filled with elements. Illustrate order.
Jagged rows - Rows of different lengths. Annotate indexing logic.
Large grids - Zoomed out grids with many elements. Heatmap overlay of traversal.
Different orders - Show row-wise, column-wise, diagonal patterns.
Edge cases - Empty and single element grids. Loop conditions.

We can represent cases using:

Grid diagrams with traversal order highlighted
Heatmaps showing element access patterns
Animations stepping through traversal
Illustrations of grid indexing formulas

These visualizations provide intuition for how different grids and traversal orders interact with the iterator implementation.

Here are some key insights gained by analyzing the different test cases:

Empty and single element grids reveal need for defensive coding and boundary checks.
Different grid sizes and densities validate traversal logic handles varied shapes and sparseness.
Jagged rows highlight need for properly calculating row ends rather than assumptions.
Large inputs validate performance and scalability.
Varying traversal orders verify underlying implementation is decoupled from sequence.
Visualizing order provides intuition on position state needing tracking across indices.
Illustrating indexing formulas helps validate calculation logic.
Animations emphasize the incremental movement and access pattern.

Overall, the cases validate traversing grids of diverse shapes and sizes in multiple patterns. Visualizations build intuition. Defensive coding and decoupling of ordering are highlighted. Together they increase confidence in a robust iterator implementation.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help simplify and solve this 2D vector iterator problem:

Linear indexing - Treating 2D coordinates as 1D index using row-major or column-major formulas. Simplifies position tracking.
Modular arithmetic - For resetting position to 0 when hitting end of rows/columns.
Nested loops - Directly maps to iterating over nested 2D structure.
Amortized analysis - Aggregate cost over sequence of operations to bound performance.
State machines - Model iterator state transitions as state machine.
Matrix transformations - Techniques for operating on matrix data like rotations. Could reorder traversal.
Design patterns - Iterator pattern captures abstraction of traversal over containers.
Discrete math - Formal techniques for sequences, indices, transformations.

In particular, linear indexing, modular arithmetic, nested loops, amortized analysis, and the iterator design pattern seem like potentially useful techniques that could simplify modeling the problem and lead to an optimized solution.

Simple Explanation

Here is how I would explain this 2D vector iterator problem in simple non-technical terms:

Let’s imagine you have a big spreadsheet filled with numbers - some cells are filled, some are empty.

Normally, to read this spreadsheet you need to go left-to-right across each row. When you reach the end, you go down and start the next row from left.

We want to create something that lets you read each cell of the spreadsheet one-by-one in order, by automatically moving across rows and columns behind the scenes.

It’s like if you could read across the rows as if they were one long list of cells rather than a 2D grid.

The iterator is like a helper that tracks which cell you are currently on, gets the next cell value for you, and moves the position internally each time.

So in plain terms, we want to take a 2D grid of data like a spreadsheet and access it sequentially like a list by using an iterator object to abstract the navigation and position details away.

Problem Breakdown and Solution Methodology

Here is how I would approach solving this 2D vector iterator problem:

Create an Iterator class to represent the iterator.
Store the 2D vector of integers as a member variable.
Maintain index variables tracking current row and column position.
Implement next():
- Return vector[row][col]
- Increment col
- If col reaches end of row, reset col to 0 and increment row
Implement hasNext():
- Check if row is less than number of rows in vector
- Or if row is at end, check if col is less than number of cols in current row
Increment indices to point to next element after each next() call.

For example on a 3x2 grid:

Initially indices (0, 0)

next() returns grid[0][0], increments col to 1

next() returns grid[0][1], resets col to 0, increments row to 1

This uses nested indices to traverse the 2D structure and incrementally move the position.

Changes affecting solution:

Larger grid means larger index variables
More next() calls increases index math computations

The indices track position across the nested structure to enable sequentially accessing elements.

Inference of Problem-Solving Approach from the Problem Statement

Here are the key terms and how they guide the solution approach:

Iterator - Indicates implementing using iterator pattern with encapsulated state.

2D vector - Nested structure suggests need for tracking indices across dimensions.

Next/hasNext - Core iterator methods imply maintaining position state internally.

Stateful - Necessitates storing traversal position in iterator object.

Sequential access - Signals incrementally advancing position from one element to the next.

Forward-only - Uni-directional traversal simplifies logic rather than bidirectional.

Position - Index tracking needed to access elements in structured order.

Linear sequence - Flattening 2D structure requires abstractions using indices.

Nested arrays - 2D input structure points to using nested indices for position.

The core terms like iterator, indices, position, sequential, and nested arrays suggest using nested indices within an iterator object to sequentially traverse the 2D structure. Tracking stateful position enables presenting a linear sequence over the nested array.

We can visualize the key properties and structures of this problem using tables and diagrams:

Nested indices:

[(0, 0), (0, 1), (0, 2)]
[(1, 0), (1, 1), (1, 2)] 
[(2, 0), (2, 1), (2, 2)]

Showing the row and column index visually conveys the 2D coordinate positioning.

Iteration order:

[ (0,0), 
  (0,1),
  (0,2),
  (1,0),
  (1,1),
  (1,2),
  (2,0) ]

Illustrating the linear traversal order provides intuition.

Position state:

class Iterator:

  def __init__(grid):
    self.row = 0
    self.col = 0

This demonstrates encapsulating the position variables within the iterator state.

The core concepts around nested indices, linear order, and state encapsulation can be effectively captured visually. Diagrams provide insight into the array indexing and traversal order.

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?

Here is one way we could break down the high-level solution into more concrete steps:

Define Iterator class to represent iterator
Store 2D vector array as member variable
Initialize position indices row and col to 0
Implement next():
- Return array[row][col]
- Increment col
- If col reached end of row, increment row and reset col
Implement hasNext():

If row < num rows, return true
Else if col < num cols in current row, return true
Else return false

Increment indices after next() to advance position

This decomposes the solution into:

Iterator class setup
Index initialization
Core traversal logic
Check completion logic
Advancing indices

The traversal and completion check logic can be developed independently from the iterator class definition.

There is a repeated pattern of incrementing indices within the bounds of the data structure.

Solution Approach and Analysis

Here is how I would approach solving this 2D iterator problem:

Define an Iterator class to encapsulate traversal state
Store row and column indices r and c initialized to 0
Implement next():

Get element at vector[r][c]
Increment c
If c is past end of row:
- Increment r
- Reset c to 0

Implement hasNext():

If r < number of rows, return true
Else if c < number of cols in row r, return true
Else return false

Increment indices r, c after next() call

For example on 2x3 vector:

Initially r=0, c=0

next() returns vec[0][0], increments c to 1

next() returns vec[0][1], increments c to 2

next() returns vec[0][2], increments r to 1, resets c to 0

This traverses the 2D structure row-wise using nested indices, while abstracting away the traversal details behind the iterator interface.

Changes affecting solution:

Larger grid means larger index range
Different traversal orders impact index update logic

The indices act as state to enable sequential access over the 2D structure.

Identify Invariant

The invariant in this 2D iterator problem is:

At every point, the iterator’s indices r and c correctly reflect the position that corresponds to the next element in the flattened sequential order over the 2D vector.

This can be proven by induction:

Base case: Initially r and c are 0, corresponding to the first position.

Inductive step: Assume r and c represent the current position. After next() call, the indices are incremented and reset based on the 2D structure to point to the next position.

Therefore, by induction, the indices always correspond to the real sequential position in the traversal order, which is crucial for next() and hasNext() to work correctly.

This invariant ensures:

The next element can be accessed reliably using the indices.
hasNext() will correctly reflect if there are elements left.
The indices can be relied on for positioning.

By maintaining this key invariant after each index update, the iterator state remains synchronized with the real traversal progression.

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Here is how I would think through solving this type of iterator problem:

First, identify cues from the problem statement:

Need to provide sequential access over nested structure
Requires implementing next() and hasNext() methods
Must hide internal traversal from caller

This suggests:

Using an iterator class to encapsulate traversal state
Tracking position internally to iterate over elements
Incrementing position after each call

Some insights about the problem:

Can use indices to represent position
Indices must be updated properly based on structure
Index math navigates 2D coordinates

I would:

Define Iterator class
Store 2D array, initialize indices
Implement next():
- Return array at current indices
- Increment indices based on dimensions
Implement hasNext()
- Check indices against array bounds
Increment indices after next()

Here is Python code implementing this approach:

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class Iterator:

  def __init__(self, array):
    self.array = array
    self.r = 0
    self.c = 0
    
  def next(self):
    result = self.array[self.r][self.c]
    self.c += 1
    if self.c == len(self.array[0]):
      self.c = 0
      self.r += 1
    return result

  def hasNext(self):
    # Check indices against array size

This uses iterator state encapsulation and nested indices to provide sequential traversal over 2D arrays.

PENDING TEST

Establishing Preconditions and Postconditions

Here is how I would analyze the next() method:

Parameters: None
Preconditions:

Iterator object must be initialized
next() must be called in sequence

Functionality:

Gets element at current position based on indices
Increments column index
If column overflow, increments row, resets column
Returns element at computed position

Postconditions:

Current position advances to next element
Indices updated to reflect new position
Next element in flattened order is returned

Error handling:

If preconditions not met, throw InvalidOperationException
If indices out of bounds, throw IndexOutOfBoundsException

In summary, next() relies on and updates the iterator state to sequentially advance the position and return the next element in the flattened order. It requires being called in sequence on a properly initialized iterator.

Problem Decomposition

Here is how I would break down this problem into logical components:

Problem Understanding: We need to implement an iterator that provides sequential access over a 2D vector (array of arrays) abstraction.
Initial Breakdown:

Store 2D vector data
Maintain position state
Retrieve next element
Check if elements remain

Subproblem Refinement:

Initialize class with vector
Use indices for row and column position
Access vector at current indices
Increment indices based on dimensions
Check indices against vector bounds

Task Identification:

Storing grid
Position initialization
Index incrementing
Element accessing
Completion checking

Task Abstraction:

Grid storage
Position initialization
Position increment
Element fetching
Existence checking

Method Naming:

storeGrid
initPos
incrementPos
getElement
hasMore

Subproblem Interactions:

Store grid
Initialize starting position
Loop:
- Get element at current position
- Increment position
- Check if elements remain

This breaks the problem down into manageable subtasks focusing on key elements like position tracking and traversal logic.

From Brute Force to Optimal Solution

Here is one approach to optimize a brute force solution:

Brute force:

Nest 2 for loops iterating over 2D array rows and columns
Store elements in a flattened 1D array
Iterate over 1D array to provide access

This uses O(N^2) loops to flatten, plus O(N) space for 1D array.

Optimizations:

Replace 2D loops with linear index calculation:
- Map 2D coordinates to 1D index
- Provides element access in O(1)

Avoid nested iteration to flatten.

Store current index vs copying all elements:
- Track single index variable
- Compute next index on the fly
- Use index to retrieve element

Eliminate extra storage space.

Separate traversal from access:
- Iterator manages traversal
- Keeps grid storage intact
- Encapsulates index tracking

Clean separation of responsibilities.

Lazy element access:
- Lookup elements as needed
- Avoid upfront flattening work

Further improve efficiency.

Together these optimizations improve the complexity from O(N^2) time and O(N) space to O(1) for both by incrementally computing index and position.

Code Explanation and Design Decisions

Here is how I would analyze this code’s logic:

The 2D grid input establishes the data structure to traverse. The indices represent iteration state.
The outer loop iterates through the logical flattened sequence by incrementing the indices. This traverses the full data set.
The conditional checks if index overflowed the row, indicating the need to advance to the next row. This handles transitioning between rows.
The indices are updated to advance the position each iteration. This progresses the traversal through the 2D grid linearly.
The invariant is that the indices always correctly point to the next element in the flattened order. This ensures sequential access.
The return value delivers the next element in the 2D structure in linear order. This satisfies the API requirements.

In summary, the code loops through a linear traversal order by correctly updating the indices each iteration. The indices act as state allowing sequential access over the 2D grid in a flattened manner.

Coding Constructs

Here is how I would analyze this code:

The high-level techniques appear to be linear iteration and stateful traversal. The grid is iterated over sequentially while maintaining state.
I would explain the code is traversing a 2D grid row-by-row as if it were a flat list. The state keeps track of the position.
The logical constructs are:

A loop for iterating
Indices for position state
Index math to increment position
Conditional check for overflow
Returning next element

The algorithm iterates through the 2D grid linearly using indices to track position. The indices are incremented and checked to advance the position.
The key steps are initializing the indices, incrementing them sequentially, checking for overflow to next row, and returning the next element. This traverses the 2D structure.
The core strategies are maintaining state across iterations and using state to traverse a structure sequentially. These could be implemented in any language.

The analysis focuses on the high-level stateful traversal approach rather than language-specific syntax. Explaining it in plain language helps reinforce the concepts.

Language Agnostic Coding Drills

Here is how I would break down this code into coding concepts:

Level 1:

Variable initialization - Define starting state
Returning value - Retrieve element

Level 2:

Index increment - Advance position
Modular arithmetic - Reset within bounds

Level 3:

Looping - Iterate over structure
Conditional check - Detect overflow

Level 4:

Encapsulation - Hide traversal behind class
Stateful traversal - Maintain position across iterations

The overall problem solving approach would be:

Initialize iterator state variables
Implement position incrementing using modular arithmetic
Add loop to iterate over structure
Check for overflow to reset position
Encapsulate state and traversal in iterator class
Iterate, incrementing state, to traverse structure
Return next element based on current state

Each concept builds incrementally from initializing state to traversing structures by maintaining state. Combined appropriately, they map the problem constraints to an efficient solution.

Targeted Drills in Python

Here are some example Python coding drills for the key concepts:

Variable initialization

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row = 0 
col = 0

Index incrementing

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col += 1
if col == num_cols:
  col = 0
  row += 1

Looping

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while row < num_rows:

  # Increment indices

  row += 1

Encapsulation

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class Iterator:

  def __init__(grid):
    self.row = 0
    self.col = 0
    
  def next(self):
    # Traversal logic

Stateful traversal

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# Within Iterator

def next(self):
  val = grid[self.row][self.col]
  
  # Increment position
  
  return val

These pieces could be integrated sequentially:

Initialize state variables
Implement position incrementing
Encapsulate state in iterator class
Add looping
Increment state to traverse structure
Return next elements

Each drill builds up knowledge culminating in the full iterator solution.