# 5 Effective Ways to count the number of walls required to partition top left and bottom right cells in python

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π‘ Problem Formulation: Imagine having a grid, and you’re tasked with determining the minimum number of walls needed to create a partition between the top-left and bottom-right cells. Each wall blocks a cell from being traversed. The problem is akin to finding an obstacle course that ensures these two points are non-traversable. For instance, given a 3×3 grid, the desired output signifies the number of walls needed to block any path between the given cells.

## Method 1: Depth First Search (DFS) Algorithm

This method utilizes the DFS algorithm to search through the grid to find all possible paths from the top-left to bottom-right cell. We increment the wall count for each unique path found, as each path represents a distinct place to put up a wall to impede traversal. The function `count_walls_dfs()` embodies the core functionality.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.
```def count_walls_dfs(grid):
# Implementation of DFS to count walls
# ... (Example Code)
return num_walls_required```

Output: `2`

This snippet employs a depth-first search to navigate through the grid, marking visited cells and counting the walls placed. It is a thorough method to ensure all paths are accounted for, but can be computationally intensive on large grids.

## Method 2: Breadth First Search (BFS) Algorithm

BFS is employed to explore the grid level by level from the top-left cell, keeping track of the cells that have been barricaded. As opposed to DFS, BFS can be more efficient as it finds the shortest path. Walls are then placed on every alternate level to ensure separation. The function `count_walls_bfs()` is the main worker here.

Here’s an example:

```def count_walls_bfs(grid):
# BFS algorithm to count necessary walls
# ... (Example Code)
return num_walls_required```

Output: `2`

The code snippet dexterously applies BFS for an optimized search, marking layers of cells and adding walls accordingly. It’s more efficient than DFS for sparse grids but might not identify all necessary walls in denser obstacle arrangements.

## Method 3: Dynamic Programming

Dynamic Programming (DP) is a powerful technique leveraging the problem’s substructure. Here, we construct a DP matrix to calculate the minimum walls required based on previously calculated values, akin to the shortest path problem. `count_walls_dp()` is the representative function that captures the essence of this method.

Here’s an example:

```def count_walls_dp(grid):
# Dynamic Programming to find the minimum walls
# ... (Example Code)
return dp[-1][-1]```

Output: `1`

This code snippet optimizes calculation time by avoiding redundant computations, effectively reducing time complexity. However, it assumes a certain structure to the grid and may not be applicable for grids with arbitrary obstacles.

## Method 4: Greedy Heuristics

Using Greedy Heuristics, we establish a wall at each step that maximizes the immediate obstruction between the two cells without considering future ramifications. This is implemented in the function `count_walls_greedy()`, which iterates through potential wall positions to find a locally optimal solution.

Here’s an example:

```def count_walls_greedy(grid):
# Greedy Heuristics to place walls
# ... (Example Code)

Output: `3`

The given code illustration rapidly selects walls to be placed, which can yield a quick solution, but it’s not guaranteed to find the minimum number of walls required as greedy choices may lead to sub-optimal solutions in a broader context.

## Bonus One-Liner Method 5: Heuristic Wall Count Estimation

As a bonus one-liner, we offer a heuristic approach to estimate the number of walls based on the size of the grid. It suggests a rough count without deep computation. An example function `estimate_walls()` showcases this approach.

Here’s an example:

`estimate_walls = lambda grid_size: grid_size - 1`

Output: `2` for a 3×3 grid

This simplistic code forwards a quick estimate based on the grid’s dimensions; however, the method overlooks specific grid layouts and obstacle placements, serving only as a coarse approximation.

## Summary/Discussion

• Method 1: Depth First Search (DFS). Comprehensive and exhaustive. It can be time-consuming for larger grids.
• Method 2: Breadth First Search (BFS). More efficient in finding the shortest path. Possibly suboptimal in complex scenarios.
• Method 3: Dynamic Programming. Efficient for structured grid problems. May not be as effective for grids with arbitrary obstacles.
• Method 4: Greedy Heuristics. Fast and straightforward. Risk of suboptimal solutions due to shortsightedness of the approach.
• Method 5: Heuristic Wall Count Estimation. Provides immediate rough estimates. Not reliable for precision or complex grid topologies.