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 steps is always (m + n - 2). The core challenge is determining how many ways we can arrange these specific moves.
Initially, I might consider Dynamic Programming. We could create a 2D array where each cell represents the number of ways to reach it, calculated as the sum of the cell above and the cell to the left. While effective, this uses O(m*n) space. We can optimize this to O(n) space since we only need the previous row's data at any moment.
However, for a more elegant and efficient solution often preferred in high-scale environments like Netflix, I would apply Combinatorics. Since the sequence of moves is fixed in length, the problem reduces to choosing which (m-1) positions out of the total (m+n-2) steps are 'down' moves. The answer is simply the binomial coefficient C(m+n-2, m-1).
I would implement this using an iterative approach to calculate the combination, avoiding large factorials that could cause integer overflow. For example, if m=3 and n=7, we calculate C(8, 2), which is 28. This approach runs in O(min(m,n)) time and O(1) space, offering the best performance profile.
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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