The Question
Coding

N-Queens

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 '.' indicate a queen and an empty space, respectively. Constraints: - 1 <= n <= 12
Java
Backtracking
Recursion
Questions & Insights

Clarifying Questions

What is the maximum value for N? (Usually 1 \le N \le 12 for standard backtracking, as complexity is O(N!)).
Should the output be a list of all possible board configurations or just the total count of solutions? (Assumption: Return all distinct board configurations as a list of strings).
How should the board be represented in the output? (Assumption: A list of strings where 'Q' represents a queen and '.' represents an empty space).
Are there any specific memory constraints we should be aware of? (Assumption: Standard heap limits apply; O(N^2) space for the board is acceptable).

Thinking Process

Incremental Placement: Since each queen must be in a unique row and a unique column, we can iterate through the board row by row and attempt to place a queen in each column of the current row.
Constraint Checking: A queen at (r, c) attacks others if they share the same column, the same major diagonal (row - col = constant), or the same minor diagonal (row + col = constant). We can use boolean arrays to track occupied columns and diagonals for O(1) validation.
Backtracking Mechanism: If a placement is valid, we mark the column and diagonals as occupied, move to the next row, and recursively continue. If we reach row N, a solution is found. After the recursive call, we must "unmark" (backtrack) to explore other configurations.
Efficiency: Using three boolean arrays (one for columns, two for diagonals) is more efficient than scanning the board for collisions in every step.
Implementation Breakdown

Problem Set

Functional Requirement: Find all distinct N \times N chessboard configurations such that N queens are placed without attacking each other.
Constraints:
Time Complexity: O(N!)
Space Complexity: O(N) (excluding the output storage)
N is a positive integer.

Approach

Algorithm: Backtracking
Data Structure: boolean[] arrays for constant-time lookups of occupied paths; char[][] for board state.
Complexity:
Time: O(N!) — though the upper bound is N^N, the pruning of columns and diagonals makes it closer to N!.
Space: O(N) — to store the recursion stack and the state of columns/diagonals. O(N^2) if including the board representation.

Implementation

Wrap Up

Advanced Topics

Bitmasking Optimization: Instead of boolean arrays, we can use three integers as bitmasks (cols, left_diag, right_diag) to track constraints. Bitwise operations (AND, OR, LSHIFT, RSHIFT) can significantly speed up the execution by reducing the constant factor in the O(N!) complexity.
Symmetry: An N \times N board is symmetric. One can solve for only half of the first row's columns and then reflect the solutions across the vertical axis to find the other half, potentially halving the computation time.
Parallelization: Since each branch of the first row's decisions is independent, the search can be distributed across multiple threads or nodes.