Minimum Path Sum
Given an $m \times n$ grid filled with non-negative numbers, find a path from top left to bottom right, which minimizes the sum of all numbers along its path. Use DP.
Why Interviewers Ask This
Salesforce asks this to evaluate a candidate's ability to optimize recursive solutions into efficient dynamic programming algorithms. They specifically test if you can recognize overlapping subproblems and the optimal substructure property within grid-based pathfinding scenarios, ensuring you can balance time complexity against space constraints in real-world data processing tasks.
How to Answer This Question
1. Clarify constraints: Ask about grid dimensions and number ranges to determine if standard DP or optimized space is needed.
2. Define state explicitly: State that dp[i][j] represents the minimum path sum to reach cell (i, j).
3. Establish recurrence relation: Explain that the value at any cell equals its own weight plus the minimum of the top or left neighbor.
4. Handle base cases: Detail how to initialize the first row and column as cumulative sums since there is only one way to reach them.
5. Optimize space: Mention transforming the 2D array into a 1D array to reduce space complexity from O(m*n) to O(n), a key optimization Salesforce values for large datasets.
6. Verify logic: Walk through a small 2x2 example to validate the transition logic before coding.
Key Points to Cover
- Explicitly identifying the optimal substructure where the solution to the whole problem relies on optimal solutions to sub-problems
- Demonstrating awareness of boundary conditions for the first row and column initialization
- Proposing a space optimization strategy to reduce auxiliary memory usage from O(m*n) to O(n)
- Clearly articulating the recurrence relation involving min(top, left) + current_value
- Validating the logic with a concrete walkthrough of a small grid example
Sample Answer
To solve the Minimum Path Sum problem efficiently, I would approach it using Dynamic Programming. First, I'd clarify that we are looking for the shortest path sum moving only right or down. The core insight here is that the minimum cost to reach any cell (i, j) depends entirely on the minimum costs of the cells directly above it and to its left. Therefore, I define our state as dp[i][j], representing the minimum path sum to reach that specific coordinate. The recurrence relation becomes straightforward: the current cell's value plus the minimum of the two possible previous steps.
I would start by initializing the grid. For the first row, each cell must be the sum of all preceding cells because we can only move right. Similarly, the first column accumulates sums moving down. Once these boundaries are set, I iterate through the rest of the grid, updating each cell based on the recurrence relation. To align with Salesforce's focus on scalability, I would then optimize space complexity. Instead of maintaining a full 2D matrix, I can use a single 1D array to store the current row's values, updating it in place as we traverse. This reduces space usage from O(m*n) to O(n). Finally, I would return the value in the bottom-right corner, which now holds the global minimum sum. This approach ensures an O(m*n) time complexity while significantly reducing memory overhead.
Common Mistakes to Avoid
- Attempting a brute-force recursion without memoization, leading to exponential time complexity TLE
- Forgetting to handle the base cases for the first row and column, causing index out of bounds errors
- Confusing the direction of movement, such as allowing diagonal moves or up/left traversal instead of just right/down
- Ignoring the non-negative constraint assumption which simplifies the greedy vs DP decision making process
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