The N-Queens II problem requires finding the number of distinct solutions to place n queens on an n x n chessboard, where no two queens threaten each other. Using backtracking with pruning allows for efficient solution exploration by pruning invalid branches early. This approach reduces unnecessary calculations and ensures correctness in counting the solutions.
Problem Statement
The N-Queens II problem involves placing n queens on an n x n chessboard so that no two queens threaten each other. You must return the number of distinct solutions where each solution consists of n queens positioned on the board.
The challenge is to find all possible ways to place the queens such that no two queens share the same row, column, or diagonal. For example, when n = 4, there are two distinct solutions. Your task is to determine the number of solutions for any value of n within the constraint 1 <= n <= 9.
Examples
Example 1
Input: n = 4
Output: 2
There are two distinct solutions to the 4-queens puzzle as shown.
Example 2
Input: n = 1
Output: 1
Example details omitted.
Constraints
- 1 <= n <= 9
Solution Approach
Backtracking with Pruning
To solve the problem, use backtracking to place queens row by row. At each step, check whether placing a queen in a particular position is safe by ensuring that no other queen is in the same column or diagonal. Prune invalid paths early by skipping placements that violate the constraints.
Column, Diagonal Tracking
Utilize additional data structures to track which columns and diagonals are under attack. This allows for constant-time checks when placing a queen, ensuring that you don't revisit invalid solutions. This tracking minimizes unnecessary re-evaluations, making the backtracking process more efficient.
Iterative Search and Count
As you place queens on the board, continue the search recursively. Each time a valid configuration is found (i.e., all queens are placed safely), increment the count of solutions. Once all rows are filled, a valid configuration is found, and the search backtracks to explore other possibilities.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the implementation and the pruning effectiveness, but it is generally exponential with respect to n. The space complexity depends on the data structures used for tracking columns and diagonals, but it can be managed to O(n). The pruning optimizations help reduce the number of recursive calls made during the backtracking process, improving the overall efficiency.
What Interviewers Usually Probe
- Does the candidate show an understanding of the backtracking approach and pruning?
- Can they efficiently explain the significance of using tracking arrays for columns and diagonals?
- Are they aware of the time complexity implications of backtracking solutions?
Common Pitfalls or Variants
Common pitfalls
- Failing to prune invalid solutions early, leading to unnecessary exploration of invalid configurations.
- Overcomplicating the tracking of columns and diagonals, which can slow down the backtracking process.
- Not accounting for all possible board configurations and missing edge cases, such as n = 1.
Follow-up variants
- Increasing the board size beyond n = 9 introduces more computational complexity.
- Changing the problem to return all solutions instead of just counting them requires storing all valid configurations.
- Allowing some queens to attack others (modified constraint) would change the approach significantly.
How GhostInterview Helps
- GhostInterview guides you through the backtracking solution and how to optimize it with pruning techniques to improve efficiency.
- We help you focus on the right approach by making sure your understanding of column and diagonal tracking is clear and applied correctly.
- GhostInterview tests your ability to reason about and optimize solutions by highlighting the time and space complexity aspects of backtracking.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
How do I solve N-Queens II efficiently?
Use backtracking combined with pruning to reduce unnecessary checks. Track columns and diagonals to ensure each queen placement is valid.
What is the time complexity of the N-Queens II problem?
The time complexity is exponential, typically O(n!), but can be improved with pruning and efficient tracking of attacked columns and diagonals.
What should I avoid when solving N-Queens II?
Avoid unnecessary recursive calls by pruning invalid configurations early. Also, ensure that you manage column and diagonal tracking properly.
What is the primary pattern for solving N-Queens II?
The primary pattern is backtracking with pruning, where invalid configurations are pruned early to minimize unnecessary computations.
Can I apply N-Queens II to larger board sizes?
While the problem is solvable for n up to 9, larger board sizes increase computational complexity significantly. Efficient backtracking and pruning are essential to managing this growth.
Need direct help with N-Queens II instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture N-Queens II from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Solve the N-Queens problem using backtracking with pruning, exploring all valid board placements while avoiding conflicts efficiently.
Open problem page#47 Permutations IIGenerate all unique permutations of an array containing duplicates using backtracking and pruning to avoid repeated sequences efficiently.
Open problem page#46 PermutationsGenerate all possible orderings of a distinct integer array using backtracking search with careful pruning to avoid duplicates.
Open problem page