What is the optimal dynamic programming approach to maximize rocks collected in a grid?
This problem challenges candidates to find the path from the bottom-left to the top-right corner of a grid that maximizes the sum of collected values.
Why Interviewers Ask This
This question tests advanced problem-solving skills, specifically the application of dynamic programming to grid-based optimization problems. It evaluates whether the candidate can define appropriate state transitions and handle dependencies between cells. Additionally, reconstructing the actual path from the DP table demonstrates a deeper understanding beyond just computing the maximum value.
How to Answer This Question
Define the DP state clearly, such as dp[i][j] representing the maximum rocks collected to reach cell (i, j). Initialize the base cases for the first row and column. Iterate through the grid to fill the DP table based on the recurrence relation involving the maximum of the previous valid cells. Finally, backtrack from the destination to the source to reconstruct the optimal path.
Key Points to Cover
- Define DP state for max rocks
- Initialize boundary conditions
- Use recurrence relation with max()
- Backtrack to reconstruct path
Sample Answer
I would use dynamic programming where dp[i][j] stores the maximum rocks collected reaching cell (i, j). I initialize the bottom-left corner and then fill the first row and column by accumulating values. For other cells, I take the current cell's value plus the maximum of the rock counts from the cell below or to the left. After filling the table, I backtrack from the top-right to the bottom-left to construct the path, ensuring I always choose the direction that contributed to the maximum sum.
Common Mistakes to Avoid
- Incorrect initialization of boundary rows/columns
- Failing to track the path during DP computation
- Using recursion without memoization leading to TLE
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