八皇后问题是一个经典的回溯算法问题,其中需要在一个 8x8 的棋盘上放置 8 个皇后,使得它们互相之间不能攻击到对方(即不能在同一行、同一列或同一对角线上)。
以下是一个使用递归回溯的示例代码来解决八皇后问题:
def solveNQueens(n):
def backtrack(row, cols, diagonals, anti_diagonals, board, solutions):
if row == n:
solutions.append(["".join(row) for row in board])
return
for col in range(n):
curr_diagonal = row - col
curr_anti_diagonal = row + col
if col not in cols and curr_diagonal not in diagonals and curr_anti_diagonal not in anti_diagonals:
board[row][col] = "Q"
cols.add(col)
diagonals.add(curr_diagonal)
anti_diagonals.add(curr_anti_diagonal)
backtrack(row + 1, cols, diagonals, anti_diagonals, board, solutions)
board[row][col] = "."
cols.remove(col)
diagonals.remove(curr_diagonal)
anti_diagonals.remove(curr_anti_diagonal)
board = [["."] * n for _ in range(n)]
solutions = []
cols = set()
diagonals = set()
anti_diagonals = set()
backtrack(0, cols, diagonals, anti_diagonals, board, solutions)
return solutions
使用该函数可以得到所有的解决方案:
solutions = solveNQueens(8)
for solution in solutions:
for row in solution:
print(row)
print()
除了八皇后问题外,还可以类似地解决其他棋子问题,例如 N 皇后问题,N 皇后问题的变种,以及其他棋子(如车、马、象、士、将等)在不同尺寸棋盘上的放置问题。其中,具体的解决方法和代码可能会有所不同,但基本的回溯算法思想是相似的。
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