Unique Paths
A robot is located at the top-left corner of an $m \times n$ grid. It can only move down or right. How many unique paths are there to the bottom-right corner? Use DP or Combinatorics.
Why Interviewers Ask This
Netflix evaluates this problem to assess a candidate's ability to optimize solutions beyond brute force. They specifically look for the transition from an O(m*n) dynamic programming approach to an O(min(m,n)) combinatorial solution, testing mathematical insight and efficiency awareness crucial for their high-performance engineering culture.
How to Answer This Question
1. Clarify Constraints: Immediately ask about grid dimensions (m, n) and data types to determine if integer overflow is a concern, reflecting Netflix's focus on robust production code.
2. Define the Naive Approach: Briefly explain the recursive solution with memoization or a standard DP table, acknowledging it solves the problem but highlighting its space complexity.
3. Optimize Space: Propose reducing the DP space from O(m*n) to O(n) by observing that only the previous row is needed, demonstrating practical optimization skills.
4. Derive Combinatorics: Pivot to the mathematical insight that any path consists of exactly (m-1) downs and (n-1) rights. Explain the formula C((m+n-2), (m-1)).
5. Address Edge Cases: Explicitly mention handling m=1 or n=1, and discuss potential overflow when calculating factorials, proposing iterative multiplication to solve it safely.
Key Points to Cover
- Demonstrating the shift from DP to a Combinatorial solution shows deeper mathematical maturity
- Explicitly discussing space complexity reduction from O(m*n) to O(n) or O(1)
- Proactively addressing integer overflow risks when calculating combinations
- Clearly articulating the logic that total steps equal (m-1) downs plus (n-1) rights
- Aligning the solution style with Netflix's expectation for clean, optimal, and production-ready code
Sample Answer
To solve the Unique Paths problem, I would first visualize the grid. A robot needs to make exactly (m-1) moves down and (n-1) moves right to reach the destination, regardless of the order. This means the total number of…
Common Mistakes to Avoid
- Stopping at the DP solution without exploring the O(1) space or combinatorial optimization
- Attempting to compute full factorials directly, leading to unnecessary overflow errors
- Failing to handle edge cases where one dimension equals 1
- Not explaining the underlying logic of why the path count is a combination problem rather than just stating the formula
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