**π‘ Problem Formulation:** Imagine a grid where each cell may contain fire or be safe, and a person is trying to reach either the top-left or bottom-right cell without passing through fire. We aim to create programs in Python that can check whether such a path exists. For example, given a grid with fire (represented by ‘F’) and safe cells (represented by ‘S’), the program should return True if a safe path exists and False otherwise.

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

Depth-First Search (DFS) is a classic graph traversal algorithm that explores as far as possible along each branch before backtracking. In the context of our problem, DFS can be used to navigate through the grid, avoiding fire cells, and finding a path to the desired corner if one exists.

Here’s an example:

def dfs(grid, x, y, visited): if x<0 or y=len(grid) or y>=len(grid[0]) or grid[x][y] == 'F' or visited[x][y]: return False if (x, y) in ((0, 0), (len(grid) - 1, len(grid[0]) - 1)): return True visited[x][y] = True return (dfs(grid, x+1, y, visited) or dfs(grid, x-1, y, visited) or dfs(grid, x, y+1, visited) or dfs(grid, x, y-1, visited)) grid = [['S', 'F', 'S'], ['S', 'S', 'F'], ['F', 'S', 'S']] visited = [[False]*len(grid[0]) for _ in range(len(grid))] print(dfs(grid, 0, 0, visited))

Output: True

The code snippet performs DFS starting from the top-left corner of the grid. It recursively explores all four directions from the current cell, avoiding fire cells and cells that have already been visited. If the DFS reaches the top-left or bottom-right cell, the function returns True; otherwise, it backtracks.

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

Breadth-First Search (BFS) is another graph traversal algorithm that explores all neighbors of a node before moving on to their neighbors. BFS can be adapted to find the shortest path on a grid from one point to another, avoiding fire cells in our scenario.

Here’s an example:

from collections import deque def bfs(grid): if grid[0][0] == 'F' or grid[-1][-1] == 'F': return False queue = deque([(0, 0)]) visited = set([(0, 0)]) while queue: x, y = queue.popleft() if (x, y) == (len(grid)-1, len(grid[0])-1): return True for dx, dy in [(1,0), (-1,0), (0,1), (0,-1)]: nx, ny = x + dx, y + dy if 0 <= nx < len(grid) and 0 <= ny < len(grid[0]) and grid[nx][ny] != 'F' and (nx, ny) not in visited: queue.append((nx, ny)) visited.add((nx, ny)) return False grid = [['S', 'F', 'S'], ['S', 'S', 'F'], ['F', 'S', 'S']] print(bfs(grid))

Output: True

This snippet uses BFS to traverse the grid, starting from the top-left corner. It inserts the coordinates into a queue and checks each direction for safe cells while avoiding revisiting cells. If it reaches the bottom-right cell, it returns True, signaling a path exists.

## Method 3: Dynamic Programming

Dynamic Programming (DP) can optimize the pathfinding problem by storing the results of subproblems. We solve the grid navigation by dividing it into smaller problems, each representing a sub-grid, and solve each recursively with memorization to avoid the fire.

Here’s an example:

def can_reach(grid): def dp(x, y): if x<0 or y<0 or grid[x][y] == 'F': return False if (x, y) == (0, 0) or (x, y) == (len(grid) - 1, len(grid[0]) - 1): return True if (x, y) in memo: return memo[(x, y)] grid[x][y] = 'F' # Temporarily mark as fire to prevent revisits memo[(x, y)] = dp(x+1, y) or dp(x-1, y) or dp(x, y+1) or dp(x, y-1) grid[x][y] = 'S' # Reset back to safe return memo[(x, y)] memo = {} return dp(len(grid) - 1, len(grid[0]) - 1) or dp(0, 0) grid = [['S', 'F', 'S'], ['S', 'S', 'F'], ['F', 'S', 'S']] print(can_reach(grid))

Output: True

The DP approach checks if a path exists from either corner by recursively searching through the grid, while memorizing and reusing subproblem results. Temporarily marking cells as ‘F’ prevents revisiting, and ‘S’ resets them, ensuring all safe paths are explored.

## Method 4: A* Search Algorithm

The A* Search Algorithm combines features of both BFS and DFS, with added heuristics to improve the efficiency of the search. In our grid scenario, the heuristic could be the Manhattan distance to the target cell. A* is typically used for pathfinding and graph traversal.

Here’s an example:

import heapq def a_star_search(grid): target = (len(grid) - 1, len(grid[0]) - 1) if grid[0][0] == 'F' or grid[target[0]][target[1]] == 'F': return False open_set = [(0, 0, 0)] # cost, x, y came_from = {} cost_so_far = {(0, 0): 0} while open_set: current_cost, x, y = heapq.heappop(open_set) if (x, y) == target: return True for dx, dy in [(1,0), (-1,0), (0,1), (0,-1)]: next_x, next_y = x + dx, y + dy if 0 <= next_x < len(grid) and 0 <= next_y < len(grid[0]) and grid[next_x][next_y] != 'F': new_cost = current_cost + 1 if (next_x, next_y) not in cost_so_far or new_cost < cost_so_far[(next_x, next_y)]: cost_so_far[(next_x, next_y)] = new_cost priority = new_cost + abs(next_x - target[0]) + abs(next_y - target[1]) heapq.heappush(open_set, (priority, next_x, next_y)) came_from[(next_x, next_y)] = (x, y) return False grid = [['S', 'F', 'S'], ['S', 'S', 'F'], ['F', 'S', 'S']] print(a_star_search(grid))

Output: True

This A* Search snippet utilizes a priority queue and heuristics based on the Manhattan distance to the target cell to explore paths quickly and efficiently. By updating costs and maintaining a record of the path, A* guarantees that the first instance it reaches the target cell is along the shortest path.

## Bonus One-Liner Method 5: Brute Force with itertools

Utilizing Python’s itertools library, we can brute force all possible paths in the grid and check if any path avoids the fire. This method is not practical for large grids due to exponential time complexity, but it’s an interesting one-liner for smaller cases.

Here’s an example:

from itertools import product grid = [['S', 'F', 'S'], ['S', 'S', 'F'], ['F', 'S', 'S']] print(any(all(grid[x][y] == 'S' for x, y in path) for path in product(*[[(i, j) for i in range(len(grid))] for j in range(len(grid[0]))]) if (0, 0) in path and (len(grid)-1, len(grid[0])-1) in path)

Output: True

This one-liner creates all possible paths by creating Cartesian products of the grid coordinates and then checks if any path contains only safe cells and includes both corners.

## Summary/Discussion

**Method 1: Depth-First Search (DFS):**Allows comprehensive exploration of all paths. It is easy to implement and understand but may be inefficient if the grid is large, as it does not prioritize paths leading closer to the goal.**Method 2: Breadth-First Search (BFS):**Good for finding the shortest path and works well with the assumption of equally weighted steps. BFS can be more efficient than DFS but uses more memory for its queue, especially in wide grids.**Method 3: Dynamic Programming:**Effective for cases where the same subproblems are encountered multiple times, reducing redundant work. However, the memory overhead can be significant due to storing the result of each subproblem.**Method 4: A* Search Algorithm:**The main advantage is efficiency through use of heuristics. It’s faster than BFS and DFS in reaching a goal, but proper heuristic selection is essential and heuristics for complex scenarios may be hard to design.**Bonus One-Liner Method 5: Brute Force with itertools:**This method is simple and requires very little code, but its brute force nature leads to exponential time complexity, making it impractical for all but the smallest grids.