Approach

To answer the question "How do you write a function to count the number of unique paths in a grid that includes obstacles?", you should follow a structured framework that covers problem understanding, solution formulation, and code implementation. Here’s how to break down your thought process:

1. Understand the Problem: Clarify the grid's dimensions, the start and end points, and the nature of obstacles.

2. Define the Constraints: Identify how obstacles affect movement. For instance, can the path go around them?

3. Choose a Suitable Algorithm: Consider using Dynamic Programming (DP) or Depth-First Search (DFS) for optimal pathfinding.

4. Implement the Solution: Write clean and efficient code, ensuring to handle edge cases.

5. Test Your Solution: Validate with different grid configurations to ensure accuracy.

Key Points

• Clarity on Grid Representation: Understand how the grid is represented (e.g., a 2D array) and how obstacles are defined (e.g., a value of 1 for obstacles and 0 for free spaces).

• Path Count Logic: Be clear on calculating paths by considering movements from one cell to another and how obstacles block these movements.

• Efficiency: Highlight the importance of time and space complexity in your solution, especially for larger grids.

• Use Cases: Mention practical applications of this problem in real-world scenarios like robotics and game development.

Standard Response

Here's a sample answer that follows best practices:

To solve the problem of counting unique paths in a grid that includes obstacles, we can employ a Dynamic Programming approach. Below is a detailed breakdown of the solution.

Problem Definition

• 0 indicates a free cell

• 1 indicates an obstacle

• We have a grid represented as a 2D array where:

The goal is to find the number of unique paths from the top-left corner (0,0) to the bottom-right corner (m-1,n-1) of the grid.

Step-by-Step Solution

• Initialize the DP Table: Create a 2D array dp of the same size as the grid where each cell will store the number of unique paths to that cell.

• Base Cases:

• If the starting cell or the ending cell is an obstacle, return 0 as no path exists.

• Set dp[0][0] = 1 if the start cell is free.

• Fill the DP Table:

• Loop through each cell in the grid:

• If the cell is an obstacle, set dp[i][j] = 0.

• Otherwise, set dp[i][j] as the sum of paths from the top cell (dp[i-1][j]) and the left cell (dp[i][j-1]).

• Make sure to check boundaries to avoid index errors.

• Return the Result: The value at dp[m-1][n-1] will give the total unique paths to the bottom-right corner.

Sample Code

Here’s the implementation in Python:

def uniquePathsWithObstacles(obstacleGrid):
 if not obstacleGrid or obstacleGrid[0][0] == 1 or obstacleGrid[-1][-1] == 1:
 return 0

 m, n = len(obstacleGrid), len(obstacleGrid[0])
 dp = [[0] * n for _ in range(m)]
 dp[0][0] = 1 # start point

 for i in range(m):
 for j in range(n):
 if obstacleGrid[i][j] == 1:
 dp[i][j] = 0 # obstacle
 else:
 if i > 0:
 dp[i][j] += dp[i - 1][j]
 if j > 0:
 dp[i][j] += dp[i][j - 1]

 return dp[-1][-1]

Tips & Variations

Common Mistakes to Avoid:

• Ignoring Edge Cases: Always check if the start or end cells are obstacles before proceeding.

• Incorrect DP Table Initialization: Ensure the DP table is correctly initialized and updated based on previous cell values.

Alternative Ways to Answer:

• Recursive Approach: Though less efficient, you can solve it using recursion with memoization. This would involve exploring paths from the start to the end recursively, storing results of subproblems to avoid redundant calculations.

Role-Specific Variations:

• Technical Roles: Focus on performance and time complexity, discussing the implications of using DP versus a brute-force approach.

• Creative Roles: Emphasize designing user-friendly visualizations for pathfinding, like showing the paths on