What is the optimal time complexity of solving N Queens?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for N Queens?

The space complexity is O(n) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in N Queens?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for N Queens?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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N Queens

Learn how to solve the 'N Queens' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Hard

The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other.

Given an integer n, return all distinct solutions to the n-queens puzzle. You may return the answer in any order.

Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' both indicate a queen and an empty space, respectively.

Implement a function solveNQueens(n: int) -> list.

Constraints

•1 <= n <= 9

Examples

Example 1

Input

Output

[[".Q..","...Q","Q...","..Q."],["..Q.","Q...",".Q..","...Q"]]

Explanation

There are exactly 2 distinct solutions to the 4-queens puzzle.

Example 2

Input

Output

[["Q"]]

Explanation

A single queen on a 1x1 board is the only solution.

Need a Hint?

Analyze the input constraints. Try sorting first (O(n log n)) or using a hash map/set to track seen elements in O(n) time.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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