Check if There is a Path With Equal Number of 0's And 1's

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class Solution:
    def isThereAPath(self, grid: list[list[int]]) -> bool:
        rows, cols = len(grid), len(grid[0])

        if (rows + cols) % 2 == 0:
            return False

        min_ = [[0] * cols for _ in range(rows)]
        max_ = [[0] * cols for _ in range(rows)]

        min_[0][0] = max_[0][0] = grid[0][0]

        for row in range(1, rows):
            min_[row][0] = min_[row - 1][0] + grid[row][0]
            max_[row][0] = max_[row - 1][0] + grid[row][0]

        for col in range(1, cols):
            min_[0][col] = min_[0][col - 1] + grid[0][col]
            max_[0][col] = max_[0][col - 1] + grid[0][col]

        for row in range(1, rows):
            for col in range(1, cols):
                min_prev = min(min_[row - 1][col], min_[row][col - 1])
                min_[row][col] = min_prev + grid[row][col]

                max_prev = max(max_[row - 1][col], max_[row][col - 1])
                max_[row][col] = max_prev + grid[row][col]

        target = (rows + cols - 1) // 2
        return min_[rows - 1][cols - 1] <= target <= max_[rows - 1][cols - 1]

10 Prerequisite LeetCode Problems

For “2510. Check if There is a Path With Equal Number of 0’s And 1’s”, the following are a good preparation:

“62. Unique Paths” - Basic understanding of grid traversal is important for the current problem.
“200. Number of Islands” - This problem will give a good understanding of Depth-First Search (DFS) and Breadth-First Search (BFS) in a grid.
“494. Target Sum” - This problem deals with finding the count of paths with a specific sum. It’s similar to the current problem where we need to find a path with equal 0s and 1s.
“542. 01 Matrix” - This problem will help to understand BFS traversal in a binary matrix.
“695. Max Area of Island” - This problem is about DFS traversal in a grid and is relevant for understanding grid traversal which is essential for this problem.
“733. Flood Fill” - This problem involves similar concepts of grid traversal.
“785. Is Graph Bipartite?” - This problem also involves similar path-finding tasks and can help to build an understanding of the balance between two sides.
“994. Rotting Oranges” - This problem requires breadth-first search in a grid, like the current problem.
“1306. Jump Game III” - This problem requires tracing all paths in an array, which is similar to the grid traversal in the current problem.
“1337. The K Weakest Rows in a Matrix” - This problem helps to understand binary matrix traversal and sorting.

These involve grid traversal, DFS, BFS, path tracing, and binary matrix traversal, which are relevant for the problem at hand.

Problem Classification

This is a graph theory problem in the domain of algorithms.

The ‘What’ components are:

2D binary matrix as input grid
Need to find a path from start to end
Can only move down or right
Path must have equal 0’s and 1’s
Determine if such a path exists

Based on this, I would categorize this problem as:

Graph problem, since it involves traversal through vertex connections
Path finding problem, as we want to find a valid path
Grid problem, since we are traversing a 2D matrix grid
Constraint satisfaction problem, due to the path criteria
Decision problem, since we only need to determine path existence

The core of this problem is modeling the grid as a graph and finding a path that satisfies the balance constraint. The specifics of grid shape and strict path criteria make it a specialized case of general path finding.

Overall, I would classify this as a constrained path existence decision problem on a 2D grid graph.

Clarification Questions

Here are some clarification questions I would ask about this problem of finding a balanced binary path in a grid:

Are the dimensions of the grid fixed, or can they vary?
Can we assume the grid is a proper rectangle with no jagged edges?
Can the path move diagonally or only up/down/left/right?
Does the path need to be the shortest possible? Or just any path that satisfies the balance criteria?
Can we revisit the same cell multiple times in the path?
Does the path need to actually be constructed, or just determining existence is sufficient?
Are there only two possible values in the grid (0 and 1), or can there be others?
Is a completely empty grid considered balanced or unbalanced?
What should be returned if no path from start to end exists at all?
Are there limitations on efficiency or optimality of the solution?
Should the grid be validated before processing?
What should be done in case of invalid grid values?

Getting clarity on these aspects would allow me to make better assumptions and design decisions regarding path constraints, optimality requirements, error handling, and balance calculations.

Problem Analysis and Key Insights

Here are some key insights gained from analyzing this problem of finding a balanced binary grid path:

The grid can be modeled as a graph with cells as vertices and movements as edges. This allows leveraging graph algorithms.
The problem simplifies to just determining existence of a valid path, not actually constructing it.
The balance criteria of equal 0’s and 1’s constrains the path strictly.
This criteria suggests tracking counts of 0’s and 1’s seen may be useful.
The problem is flexible on actual path optimality - any valid path suffices.
The start and end points are predetermined, allowing a directed search.
Movement is restricted to down and right, reducing the edge possibilities.
No clarification is given on revisiting cells, so likely allowed.
The grid dimensions are reasonably small, allowing brute force approaches.

In summary, modeling as a graph and tracking 0/1 counts while flexibly searching for any valid path seem like reasonable approaches given the insights from analyzing the statement.

Problem Boundary

Here are some key points about the scope of this balanced grid path problem:

The input is a 2D binary grid of 0s and 1s. Dimensions are flexible but reasonably small.
Movement is only allowed down or right from each cell. No diagonals.
Need to find any path from top-left to bottom-right satisfying the 0/1 balance criteria.
Only determining path existence is required, not actually constructing it.
No other constraints are given regarding optimality, efficiency, repeated visits etc.
Output is simply a boolean of whether a valid path exists or not.
The problem is self-contained - no external data sources or priors.
No information is provided on how the input grid is constructed or validated.

So in summary, the scope is focused on determining existence of a path satisfying the specific 0/1 balance constraints, while flexibility is given on actual path optimality, efficiency, and graph assumptions.

Here are some key points about the scope of this balanced grid path problem:

The input is a 2D binary grid of 0s and 1s. Dimensions are flexible but reasonably small.
Movement is only allowed down or right from each cell. No diagonals.
Need to find any path from top-left to bottom-right satisfying the 0/1 balance criteria.
Only determining path existence is required, not actually constructing it.
No other constraints are given regarding optimality, efficiency, repeated visits etc.
Output is simply a boolean of whether a valid path exists or not.
The problem is self-contained - no external data sources or priors.
No information is provided on how the input grid is constructed or validated.

Here are some ways we can establish boundaries for this balanced path problem:

Input Space

2D grid/matrix
Dimensions: 2-100 rows, columns
Values: 0 or 1 binary

Output Space

Boolean indicating path exists or not

Algorithm

Find any path satisfying criteria
Path optimality doesn’t matter

Path Constraints

Balance of 0’s and 1’s must be equal
Move only down or right
Repeated cell visit allowed

Performance

No constraints specified

Functionality

Determine path existence, construction not needed

Incorrect Input

Invalid grid dimensions
Values outside 0/1
Disconnected grid

Defining these clear input, output, functionality, and path requirements provides a solid problem specification to design a solution within.

Distilling the Problem to Its Core Elements

This problem is fundamentally based on principles of graphs and constrained path finding.

In the simplest terms, I would describe it as:

“Finding if a path exists between two points on a map passing equal numbers of red and blue locations.”

The core problem is determining reachability between two nodes with a specific attribute value constraint. We can simplify it:

“Does grid path exist with equal 0’s and 1’s?”

The key components are:

Grid representing map
Path finding with constraints
Tracking 0 and 1 counts
Start and end nodes

The minimal operations are:

Model grid as graph
Explore paths incrementally
Track 0/1 counts along path
Check if end reached satisfying constraints

At its core, this problem involves representing the grid as a graph and searching for a path that meets certain criteria. The core is path existence with constraints, which builds on fundamental graph concepts.

Visual Model of the Problem

Here are some ways to visualize the problem statement for finding a balanced binary grid path:

Draw a small grid, labeling 0 cells and 1 cells distinctly. Highlight start and end points.
Show an example path traversing the grid from start to end. Annotate 0 and 1 counts along the path.
Use arrows or animation to indicate the path directions and possible movements.
Indicate the balance criteria visually, e.g. with a balanced scales diagram.
Show invalid imbalance paths that don’t satisfy the 0/1 criteria.
For larger grids, show zoom-in of portions of the path and counts.
Visualize backtracking when path becomes imbalanced.
Use color coding and symbols to distinguish 0 and 1 cells.
Depict decision output of path existence.

These types of diagrams, animations and annotations help provide an intuitive sense of the core concepts like grid encoding, constrained path traversal, and determining existence.

Problem Restatement

Here’s how I would paraphrase the problem statement for finding a balanced binary grid path in my own words:

We are given a 2D grid of cells containing either 0 or 1 values. Our goal is to determine whether a path exists that starts from the top-left cell and ends at the bottom-right cell, such that the path has an equal number of cells containing 0 and cells containing 1.

We are only allowed to move down or right one cell at a time. The dimensions of the grid are reasonably small.

We don’t need to actually construct the full path, only determine if such a path is possible given the balance constraints. There are no other restrictions on optimality or efficiency.

The only output needed is a boolean indicating whether a valid balanced path exists from start to end. No other processing of the grid is required.

Let me know if I’m misinterpreting anything or missing any key details based on this paraphrasing. Getting the requirements clear helps orient the problem solving approach in the right direction.

Abstract Representation of the Problem

Here is one way to formulate an abstract representation of this problem:

Let’s model the grid as a graph G = (V, E) where:

V is the set of vertices representing the cells
E is the set of edges between vertices representing allowable movements

Additionally, each v in V has an attribute A(v) ∈ {0, 1} encoding the cell value.

We want to find if there exists a path P from vertex vs to vt such that:

P consists of a sequence of connected vertices in V where each edge (u, v) is in E
The number of vertices with A(v) = 0 equals the number of vertices with A(v) = 1

In other words, we need to find if there is a path between two nodes in the graph where the number of 0-valued and 1-valued vertices along the path are equal.

This frames the problem abstractly in terms of graphs, vertices, edges, and vertex attributes rather than grids and matrix values. We can leverage graph algorithms to solve it.

Terminology

Here are some key terms and concepts that are important for understanding this balanced grid path problem:

Graph - Mathematical structure representing connections. Grid can be modeled as graph.
Vertex - Fundamental components of a graph. Here, grid cells are vertices.
Edge - Connections between vertices in a graph. Movements between cells are edges.
Path - Sequence of connected edges between vertices. Want path satisfying balance criteria.
Binary attribute - Vertex property of 0 or 1 value. Determines balance constraints.
Subpath - Part of a path from source to current vertex. Track counts along subpaths.
Backtracking - Abandoning subpath that cannot satisfy constraints by returning to previous vertex.
Exhaustive search - Systematically exploring all paths to determine solution existence.
Pruning - Eliminating infeasible subpaths that cannot satisfy constraints.

The key is modeling the grid as a graph to enable using graph algorithms and terminology. The concepts of paths, constraints, and exhaustive search are essential for solving the problem.

Problem Simplification and Explanation

Here’s one way I could explain this problem in simpler terms:

Let’s imagine the grid is a corn maze (analogy). Some paths through the maze have blue markings, other paths have red markings.

The key concepts are:

The grid is like a maze with pathways
Grid cells are locations in the maze
Movements between cells are possible paths
0 cells = paths marked blue
1 cells = paths marked red

We want to find if there’s a path from the maze entrance to exit that has an equal number of blue and red marked paths along the way.

So we need to explore the maze, tracking the colors of the paths we take. If we reach the exit with equal reds and blues, success!

But if the colors become imbalanced, we have to backtrack and try a different route.

Constraints

Here are some characteristics of this problem that we can potentially leverage to optimize finding a balanced binary grid path:

The grid size is small, with dimensions from 2-100. This allows exhaustive search approaches without massive complexity.
Only two cell values are possible (0 and 1). This simplifies count tracking.
Path optimality is not required - any valid path suffices. This opens up options beyond shortest path.
Movement is restricted to down and right. We can optimize data structures and algorithms along these axes.
The output is a simple boolean value. We don’t need complex return handling.
Balance criteria provides a clear pruning opportunity - we can abandon imbalanced subpaths.
No constraints are provided on efficiency or memory usage. Many algorithms are on the table.
The grid is immutable. No changes during processing that need tracking.

Overall, the discrete and bounded input space, simple balance constraints, and flexibility in path optimality allow us to leverage exhaustive and greedy search approaches.

Here are some key insights gained from analyzing the constraints:

The small grid size allows brute force exploration of all paths.
Binary cell values simplify tracking and comparing counts.
Not requiring optimality opens up non-shortest path approaches.
Fixed movement to only down/right optimizes data structures.
Simple boolean output removes need to reconstruct actual path.
Balance criteria enables clear pruning of subpaths.
No efficiency constraints allows brute force options.
Immutable grid simplifies graph assumptions.
Self-contained problem simplifies modeling.

In summary, the constraints allow flexibility in choosing an exhaustive search approach while also providing optimization opportunities through graph representation, movement assumptions, pruning, and simple output requirements.

Case Analysis

Here are some additional test cases covering different scenarios:

Basic case

Input:

[[0,1], [1,0]]

Output: True

Reasoning: Simple 2x2 case with valid diagonal path.

No valid path

Input:

[[1,1], [0,0]]

Output: False

Reasoning: Tests positive to negative case.

Large grid

Input: 60x60 grid with randomly generated 0/1 values

Output: True if valid path exists, False otherwise

Reasoning: Stress tests scale.

Revisit cells

Input: Grid where solution requires revisiting cells.

Output: True if valid path accounting for revisits.

Reasoning: Validate path criteria assumptions.

Boundary Cases:

2x2 grid
Empty grid
Grid with one cell type

Edge Cases:

Solution path along edge
Disconnected grid
Winding spiral path

Testing these validates correctness, assumptions, scales robustly, and covers boundary conditions.

Here are some ideas for visualizing test cases for this balanced binary grid path problem:

Basic Case

2x2 grid with valid diagonal path highlighted

No Valid Path

Grid with imbalanced start/end points

Large Grid

Show full grid zoomed out
Zoom in on a sample section to visualize path

Revisit Cells

Path shown doubling back on itself

Empty Grid

0x0 grid visualization

Disconnected

Show two separate grids

Along Edge

Path traversing outer edge

Spiral

Long winded spiral path

In general:

Use colors to distinguish 0 and 1 cells
Animate paths to show traversal

Visualizing the test cases helps ensure proper coverage and correct reasoning about expected output.

Here are some key insights gained from analyzing these test cases:

The basic case validates the core path finding algorithm and balance checking logic.
Negative cases like no valid path prove logic handles infeasible grids.
Large grids reveal performance and scalability issues.
Revisiting cells tests assumptions on allowed movement.
Boundary cases like empty grid simplify edge logic and assumptions.
Edge cases stress test conditions like solutions along borders.
Winding paths push algorithms to their limits.
Visualizing grids aids intuition about path feasibility.
Animation shows traversal order and pruning.
A wide variety of test cases is crucial for robust solutions.

Overall, these insights emphasize the importance of test case diversity for thoroughly validating correctness, revealing assumptions, stress testing algorithms, and ensuring robustness across the problem’s full scope.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help optimize finding balanced grid paths:

Graph theory - Model grid as graph to leverage graph algorithms.
Depth-first search - Explore paths exhaustively in a memory-efficient way.
Backtracking - Abandon infeasible subpaths that can’t satisfy balance constraints.
Bitmasking - Efficiently store and compare 0/1 counts in bit vectors.
Dynamic programming - Cache results of subpaths to prune repeats.
Heuristics - Prioritize high-potential paths first to find solutions faster.
Symmetry breaking - Prune symmetric paths to reduce redundancy.
Random sampling - Quickly test balances on random path samples.
Binary trees - Use trees reflecting balance criteria to prune effectively.

Leveraging representations as graphs, algorithms like DFS, and techniques like backtracking, heuristics, and memoization can enhance performance and simplify modeling and solving this problem.

Simple Explanation

Here’s how I would explain this balanced binary grid path problem in simple terms:

Imagine a maze where each path is colored either red or blue (metaphor). You enter the maze at the top-left and need to reach the bottom-right exit.

The rule is that you have to take a set of paths such that the number of red paths equal the number of blue paths.

You can only move one step down or one step right at a time. Also, you can reuse paths if needed.

The maze is like a grid of cells where red paths represent 1s and blue paths represent 0s.

You need to determine if it’s possible to get from start to end taking an equal number of red (1) and blue (0) paths.

This is like finding a way through a maze that balances two types of paths. We model the grid as connections between locations and use the path balance rule to determine reachability.

Problem Breakdown and Solution Methodology

Here is a detailed step-by-step approach I would take to solve this problem of finding a balanced binary grid path:

Model the grid as a graph with cells as vertices and connections between adjacent cells as edges.
Do a depth-first search traversal of the graph starting from the top-left vertex.
Maintain counts of 0 and 1 vertices visited so far as we traverse each path.
If the counts become imbalanced, we cannot reach the end balanced along this path. So backtrack and try a different path.
If we reach the bottom-right with balanced 0/1 counts, return true - we found a valid path.
If no path reaches the end balanced, exhaustively search all paths and return false if none work.

Increasing grid size expands the number of paths exponentially. We can optimize by pruning obviously imbalanced subpaths early.

Let’s walk through a simple 2x2 grid:

Try path right, imbalanced
Try path down, imbalanced
Try diagonal path, balanced - return true

This uses depth-first search with backtracking and constraint checking to exhaustively traverse all paths until a valid balanced path is found.

Inference of Problem-Solving Approach from the Problem Statement

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

Grid - Represents 2D layout, guides modeling as graph.
Path - Sequence of edges/vertices, trying to find valid balanced path.
0/1 balance - Key constraint and optimization criteria for path validity.
Depth-first search - Traversal approach that explores paths exhaustively.
Backtracking - Pruning infeasible subpaths that can’t satisfy balance criteria.
Graph - Grid can be modeled as graph to leverage algorithms.
Exhaustive search - Checking all paths systematically needed for solution existence.

The terms path, balance, DFS, backtracking signal traversing the grid graph exhaustively using backtracking and 0/1 counting to prune invalid subpaths. The optimization criteria guides the traversal strategy. Representing as a graph enables leveraging graph algorithms.

Here are some ways to visualize the properties and concepts of this problem using diagrams:

Grid Graph

Show grid cells as graph vertices
Connect adjacent cells with edges representing movements

Depth-First Traversal

Animate traversal of graph tracking backtracking
Number vertices by visit order

0/1 Balance Counting

Show balance scales tipping on imbalanced paths
Indicate valid balance for complete path

Backtracking

Demonstrate abandonment of imbalanced subpath
Backtrack visually to previous vertex

Exhaustive Search

Illustrate entire graph traversal across all paths
Highlight failed paths and final solution path

These types of visual models help reinforce concepts like representing the grid as a graph, DFS traversal order, tracking 0/1 balance, efficient backtracking, and exhaustive traversal. Diagrams provide intuition.

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 a stepwise refinement for finding a balanced binary grid path:

High-level approach: Use depth-first search with backtracking to traverse grid graph exhaustively.
Break into steps:

Model grid as graph
Implement depth-first traversal
Track 0/1 counts along path
Check balance and backtrack if needed
Return true if path reaches end balanced
Otherwise return false if all paths fail

Independent sub-problems:

Building graph representation
Tracking 0/1 counts
Checking balance criteria

Repeatable patterns:

Recursive DFS traversal
Backtracking on imbalanced paths
Updating and checking 0/1 counts

The key is representing as a graph and using depth-first exhaustive search with backtracking and constraint checking to traverse all paths. The graph, counting, and checking can be handled independently. And the traversal forms the core repeated logic.

Solution Approach and Analysis

Here is a detailed step-by-step approach to finding a balanced binary grid path:

Model the grid as a graph with cells as vertices and edges between adjacent cells.
Do depth-first search (DFS) traversal starting from the top-left vertex using recursion.
Maintain counts of 0 and 1 vertices visited so far using variables.
When moving to a neighbor, increment the count for that vertex’s 0 or 1 value.
If counts become imbalanced, return False to trigger backtracking.
If bottom-right vertex reached with balanced 0/1 counts, return True.
If all paths searched without success, return False.

Increasing grid size expands the search space exponentially in worst case. We can optimize by pruning obviously imbalanced subpaths early.

Let’s walk through a simple 2x2 grid:

DFS right, counts imbalanced - backtrack
DFS down, counts imbalanced - backtrack
DFS diagonal, counts balanced - return True

This implements an exhaustive DFS traversal with pruning to search all paths checking balance constraints, using backtracking to optimize.

Identify Invariant

The key invariant in this problem of finding a balanced binary grid path is:

At every step during the depth-first search traversal, the 0 and 1 counts represent the exact number of each value visited so far along the current path.

This means the count variables must accurately track the balance of 0 and 1 valued vertices visited along the subpath explored up to any point.

We can maintain this by:

Initializing the counts to 0
Incrementing the appropriate count when visiting a 0 or 1 vertex
Comparing the counts to determine imbalance
Resetting counts when backtracking after imbalanced subpaths

This allows us to efficiently prune subtrees that cannot satisfy the balance criteria, while exploring only valid paths that correctly maintain the exact 0 and 1 counts needed to determine balance.

The invariant connects the private traversal state to the key path constraint that drives the overall algorithm.

Identify Recursion Invariant

There is an invariant that can be defined for the recursive depth-first search approach for finding a balanced binary grid path:

At every recursive call of the DFS function, the 0 and 1 count variables represent the exact number of each value visited so far along the current subpath for this call stack.

This invariant is maintained because:

The counts are originally initialized to 0 before the initial call.
When recursing to a neighbor, the appropriate count is incremented to add the current vertex value.
When returning from a call, the current vertex value is decremented to “undo” it from the counts.

So at any given recursive DFS call, the counts reflect the precise path taken so far within just that call stack.

This allows pruning subtrees when the counts become imbalanced, since we know that particular subpath cannot be extended to satisfy the balance criteria.

The invariant provides the relationship between recursive call stack and the corresponding subpath state needed to guide the search process.

In this problem of finding a balanced binary grid path, the overall invariant and the recursive invariant are essentially equivalent:

The main invariant is:

At every step during the DFS traversal, the 0 and 1 counts represent the exact number of each value visited so far along the current path.

The recursive invariant is:

At every recursive DFS call, the counts reflect the current subpath for that call stack.

These invariants are fundamentally the same - they both enforce the relationship between the count variables and the current state of the traversal/subpath.

The key difference is just the scope - the overall invariant is global across the entire traversal, while the recursive invariant views it locally within each call stack.

But logically, they enforce the same core invariant that the counts must accurately reflect the current path/subpath state to allow correct pruning when imbalance is detected.

So in this case, the main invariant and the recursive invariant are essentially equivalent, capturing the same relationship at different scopes. They are interchangeable ways to define the key invariant property.

Thought Process

Here is how I would approach solving this problem of finding a balanced binary grid path:

The cues from the problem statement:

Grid with 0/1 values
Move only down or right
Find path from start to end
Path must balance 0 and 1 counts

These cues suggest:

Model grid as graph
Use depth-first search to traverse
Track 0/1 counts along path
Backtrack if path becomes imbalanced
Return True if we reach end balanced

Steps:

Represent grid as graph
Initialize 0 and 1 counts
Recursive DFS traversal
Increment appropriate count per vertex
Check balance and backtrack if needed
Return True if end reached balanced
Otherwise return False if all paths fail

Python code:

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def balanced_path_exists(grid):

  ROWS, COLS = len(grid), len(grid[0])
  counts = {0: 0, 1: 0}

  def dfs(r, c):

    counts[grid[r][c]] += 1
    
    if counts[0] != counts[1]:
      return False 
    
    if r == ROWS-1 and c == COLS-1:
      return True

    for dr, dc in [(1,0), (0,1)]:
      nr, nc = r + dr, c + dc
      if nr < ROWS and nc < COLS:
        if dfs(nr, nc):
          return True

    counts[grid[r][c]] -= 1
    return False

  return dfs(0, 0) 

This implements the DFS traversal approach with 0/1 count tracking and backtracking to determine if a balanced path exists.

Establishing Preconditions and Postconditions

Here are the details for the dfs() method in the depth-first search approach:

Parameters:

r, c - Integers: The row and column representing the current cell

Preconditions:

r and c are valid indices within the grid dimensions

Functionality:

Explores possible paths from the current cell recursively in DFS order.

Postconditions:

0/1 counts mutated based on path taken
Returns True if end reached balanced, False otherwise

Error Handling:

Invalid r or c will cause index out of bounds, so should be checked before calling dfs()
On any other errors, could raise a RuntimeError

In summary, dfs() iterates the grid graph recursively in DFS order from the given starting cell, tracking the balance counts along the way. It returns whether a balanced path to end can be found from the given cell.

Problem Decomposition

Here’s how I would break down this problem of finding a balanced binary grid path:

Determine if a path exists in a 2D grid from start to end that has equal 0s and 1s along the path.
Major parts are:
- Model grid as graph
- Depth-first search
- Track 0/1 counts
- Check balance constraint
- Backtracking
Breakdown:
- Build graph from grid
- Implement DFS traversal
- Increment 0/1 counters
- Check balance and prune
- Handle backtracking
- Return result
Generalized tasks:
- Graph construction
- DFS implementation
- Counter logic
- Balancing checking
- Backtracking
Abstractions look good.
Names are clear.
Order:
- Build graph
- DFS traversal
- Increment 0/1 counts
- Check balance
- Backtrack on imbalance
- Return result

The key is representing the grid as a graph, traversing with DFS using counters and pruning, and backtracking accordingly. The breakdown identifies reusable components.

From Brute Force to Optimal Solution

Here is a brute force approach and optimizations for finding a balanced binary grid path:

Brute force:

Try all possible paths from start to end
Check if 0/1 count balanced for each
Return true if any path balances

This results in O(2^(m+n)) time complexity trying all paths in worst case.

Optimizations:

Depth-first search:

Check one path fully before others. More memory efficient.

Backtracking:

Abandon path once imbalance detected. Avoids wasted effort.

Early termination:

Stop and return true as soon as valid path found.

Pruning:

Ignore paths going out of bounds.

Caching:

Memoize results for visited nodes.

These optimizations improve the worst case complexity from O(2^(m+n)) to O(m*n) by pruning unnecessary paths using the constraints.

Space is reduced from O(m*n) to O(m+n) by eliminating the need to store all possible paths simultaneously.

In summary, DFS, backtracking, pruning, and caching significantly enhance performance by optimizing the traversal strategy and search space.

Code Explanation and Design Decisions

Here is an analysis of the logic behind this balanced path finding code:

The grid parameter represents the input problem space to search.
The main loop iterates through possible path extensions from the current cell. This builds up paths incrementally.
The if condition checks if the current path remains balanced. This enforces the core constraint.
The 0/1 counters are updated appropriately on each iteration. This maintains the balance state.
The invariant is the counters always reflect the current path balance. This enables pruning imbalanced paths.
Returning True signifies a solution path was found that meets the balance criteria, satisfying the problem goal.

The logic incrementally builds up paths checking the balance invariant, so that we only pursue valid paths. Returning True indicates a path satisfying the core constraints was found, aligning with the problem statement.

Coding Constructs

Here is an analysis of the high-level logic of this balanced path finding code:

It uses depth-first search with backtracking and constraint checking to explore the solution space.
I would explain the code is carefully searching all possible paths to find one that balances two types of blocks.
The core logic includes recursion, data structures, conditionals, and variables.
The algorithm recursively tries each path, tracking the balance, abandoning paths that become imbalanced, until a solution is found or all paths failed.
The key steps are building the graph, recursing the paths using counters and constraints, backtracking when needed, and returning the result. This implements an exhaustive search.
The overall strategy is a backtracking depth-first search with pruning, guided by constraints that define the solution criteria.

Overall, this analysis extracts the high-level semantics around searching the space while pruning infeasible candidates. The specifics of the data structures and constraints guide the behavior. The focus is on the concepts over language details.

Language Agnostic Coding Drills

Here is a decomposition of key concepts in this balanced binary grid path problem:

Easy

Array/matrix manipulation - Indexing 2D grid
Variable assignment - Counters, result variable
Arithmetic - Incrementing/decrementing counters
Conditionals - Checking balanced or not

Medium

DFS traversal - Recursive exploration
Backtracking - Returning on constraints violated
Graph modeling - Adjacency of grid cells

Advanced

Pruning strategies - Eliminating infeasible branches
Recursion trees - Visualizing recursive exploration
Time/space complexity - Understanding algorithm analysis
Problem space modeling - Mapping real world to abstract concepts

The overall problem solving approach is:

Model grid as graph
Implement depth-first search traversal
Maintain 0/1 counters along the path
Check balance constraint and backtrack
Recursively explore all paths
Return if balanced path found

Each concept builds on the last to methodically search the space while pruning based on constraints. Mastering these coding fundamentals enables solving complex problems like this.

Targeted Drills in Python

Here are Python coding drills for each concept:

Basic

Array indexing:

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grid = [[1, 2], [3, 4]]
print(grid[0][1]) # 2 

Variable assignment:

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count0 = 0 
count1 = 0

Arithmetic:

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count0 += 1 
count1 -= 1

Conditionals:

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if count0 == count1:
  print("Balanced")

Intermediate

DFS traversal:

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def dfs(node):
  visit(node)  
  for child in node.children: 
    dfs(child)

Backtracking:

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if not is_balanced(count0, count1):
  return False # Backtrack 

Graph modeling:

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graph = {
  (0,0): [(0,1), (1,0)],
  (0,1): [(0,0), (0,2)],
} 