N-Queens
Time O(n^2 * n!) · Space O(n) · Official statement on LeetCode
Solutions
# Time: O(n^2 * n!)
# Space: O(n)
class Solution(object):
def solveNQueens(self, n):
"""
:type n: int
:rtype: List[List[str]]
"""
def dfs(row):
if row == n:
result.append(map(lambda x: '.'*x + "Q" + '.'*(n-x-1), curr))
return
for i in xrange(n):
if cols[i] or main_diag[row+i] or anti_diag[row-i+(n-1)]:
continue
cols[i] = main_diag[row+i] = anti_diag[row-i+(n-1)] = True
curr.append(i)
dfs(row+1)
curr.pop()
cols[i] = main_diag[row+i] = anti_diag[row-i+(n-1)] = False
result, curr = [], []
cols, main_diag, anti_diag = [False]*n, [False]*(2*n-1), [False]*(2*n-1)
dfs(0)
return result
# For any point (x,y), if we want the new point (p,q) don't share the same row, column, or diagonal.
# then there must have ```p+q != x+y``` and ```p-q!= x-y```
# the former focus on eliminate 'left bottom right top' diagonal
# the latter focus on eliminate 'left top right bottom' diagonal
# - col_per_row: the list of column index per row
# - cur_row：current row we are seraching for valid column
# - xy_diff：the list of x-y
# - xy_sum：the list of x+y
class Solution2(object):
def solveNQueens(self, n):
"""
:type n: int
:rtype: List[List[str]]
"""
def dfs(col_per_row, xy_diff, xy_sum):
cur_row = len(col_per_row)
if cur_row == n:
ress.append(col_per_row)
for col in range(n):
if col not in col_per_row and cur_row-col not in xy_diff and cur_row+col not in xy_sum:
dfs(col_per_row+[col], xy_diff+[cur_row-col], xy_sum+[cur_row+col])
ress = []
dfs([], [], [])
return [['.'*i + 'Q' + '.'*(n-i-1) for i in res] for res in ress]
Beginner Explanation
What is N-Queens?
N-Queens (LeetCode #51) is a Hard problem that primarily trains backtracking.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with dfs backtracking.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for N-Queens
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to dfs backtracking.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(n^2 * n!)) and space (O(n)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(n^2 * n!) time and O(n) space.
Pattern focus: dfs backtracking
Use the pattern as a checklist:
- dfs backtracking — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(n^2 * n!) |
| Space | O(n) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on N-Queens
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dfs backtracking — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to dfs backtracking:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: backtracking.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve N-Queens in a second language (python).
- Drill 3–5 more problems tagged backtracking.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the dfs backtracking approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
N-Queens (#51) — Hard. Pattern: dfs backtracking. Complexity: O(n^2 * n!) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of N-Queens?+
The reference solutions aim for O(n^2 * n!) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does N-Queens use?+
It primarily maps to dfs backtracking, within the broader topic of backtracking.
Is N-Queens good for interviews?+
Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/n-queens/