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This approach involves using backtracking to place queens row by row, exploring valid positions recursively. Each queen placement is validated based on column and diagonal restrictions. If a valid placement is found, the algorithm progresses to the next row, otherwise it backtracks and attempts alternative placements.
Time Complexity: O(N!), because we are trying N possible positions for each row for N queens.
Space Complexity: O(N^2), due to the storage of the board states and recursion stack.
1from typing import List
2
3class Solution:
4 def solveNQueens(self, n: int) -> List[List[str]]:
5 def backtrack(row = 0, cols = 0, d1 = 0, d2 = 0, solution = []):
6 if row == n:
7 results.append(solution)
8 else:
9 for col in range(n):
10 if not (cols & (1 << col)) and not (d1 & (1 << (row - col + n - 1))) and not (d2 & (1 << (row + col))):
11 backtrack(row + 1,
12 cols | (1 << col),
13 d1 | (1 << (row - col + n - 1)),
14 d2 | (1 << (row + col)),
15 solution + [col])
16
17 results = []
18 backtrack()
19 return [["." * i + "Q" + "." * (n - i - 1) for i in solution] for solution in results]
20
21s = Solution()
22print(s.solveNQueens(4))This Python solution employs a bit manipulation technique for efficient bitmask checking of column and diagonal occupation. The goal is to reduce overhead by leveraging bitwise operations. The backtrack function tries possibilities recursively, assembling string results once a complete solution is identified.
This approach optimizes the placement verification using bit manipulation, reducing the auxiliary space by managing columns and diagonals in integer bitmasks. This permits efficient bitwise operations for placing queens and checking potential attacks.
Time Complexity: O(N!), each row considers N placements, simplified by binary check.
Space Complexity: O(N), due to condensed state tracking via differential lists.
1from typing import List
2
3def solveNQueens
This Python implementation enhances the typical backtracking with bit positioning. The current column and diagonals are represented with list-formed negative and positive differential tracking to accommodate queen positions and check constraints efficiently.