N-Queens
Solve the N-Queens puzzle, where $N$ queens must be placed on an $N \times N$ chessboard such that no two queens attack each other. Return all distinct solutions. Use Backtracking.
Why Interviewers Ask This
Stripe asks this to evaluate a candidate's mastery of recursive backtracking and constraint satisfaction under pressure. They specifically look for the ability to visualize board state changes, manage complex indexing without off-by-one errors, and optimize pruning strategies to avoid unnecessary computation on large N values.
How to Answer This Question
1. Clarify Constraints: Immediately ask about input size limits (e.g., N=8 vs N=12) and expected output format to gauge performance requirements.
2. Define State: Explain your representation of the board, such as using three boolean arrays for columns and diagonals instead of a full matrix to achieve O(1) conflict checks.
3. Outline Backtracking Logic: Describe the recursive function where you iterate through rows, attempting to place a queen in valid columns, then recursing to the next row.
4. Pruning Strategy: Detail how you skip invalid placements immediately based on column or diagonal conflicts before making the recursive call.
5. Optimization Discussion: Propose bit manipulation techniques if N is large, demonstrating awareness of space-time trade-offs relevant to Stripe's high-throughput systems.
Key Points to Cover
- Demonstrating O(N^2) space optimization using sets instead of a 2D array
- Explaining the logic for calculating unique diagonal indices (row - col and row + col)
- Clearly articulating the base case and the recursive step structure
- Discussing pruning techniques to eliminate invalid branches early
- Mentioning bit manipulation as an advanced optimization for performance
Sample Answer
To solve the N-Queens problem efficiently, I would first clarify the constraints with the interviewer, as the solution complexity grows exponentially. My approach uses depth-first search with backtracking. Instead of maintaining a full N x N grid, which is memory-intensive, I will use three sets to track occupied columns, left diagonals, and right diagonals. This allows us to check validity in constant time.
The core algorithm iterates row by row. For each row, we attempt to place a queen in every column. Before placing, we check if the column or either diagonal is already occupied. If safe, we mark these coordinates in our sets, add the position to our current path, and recursively move to the next row. If we reach row N, we have found a valid solution and add it to our results list. Upon returning from recursion, we backtrack by unmarking the column and diagonals to explore other possibilities.
For optimization, especially given Stripe's focus on efficiency, I would discuss bit manipulation. We can represent the board state using integers where bits indicate occupied positions, allowing bitwise operations to find valid spots instantly. This reduces both time complexity and overhead, ensuring the solution scales better for larger inputs while remaining clean and readable.
Common Mistakes to Avoid
- Using a full 2D grid for state tracking, leading to inefficient O(N^2) lookups per placement
- Failing to handle diagonal index calculations correctly, causing false positives in conflict detection
- Not resetting the board state after recursion returns, resulting in incorrect subsequent solutions
- Ignoring edge cases like N=1 or N=2 where no solution exists or is trivial
Practice This Question with AI
Answer this question orally or via text and get instant AI-powered feedback on your response quality, structure, and delivery.