分支限界法求n皇后问题效果图

### 使用分支限界法求解N皇后问题的可视化 在探讨如何通过分支限界法(branch and bound)来解决N皇后问题并实现其可视化效果时，可以借助图形化界面展示算法执行过程中的状态变化。此方法能够有效地帮助理解该类组合优化问题的求解机制。 #### 构建基本框架 为了创建一个简单的可视化工具，Python编程语言及其库matplotlib和pygame是非常合适的选择。下面是一个简化版的例子： ```python import numpy as np import matplotlib.pyplot as plt from time import sleep def draw_board(board): """绘制棋盘""" fig, ax = plt.subplots() n = board.shape[0] ax.imshow(-board, cmap='binary') for i in range(n + 1): ax.axhline(y=i - .5, color='.7', lw=2) ax.axvline(x=i - .5, color='.7', lw=2) plt.xticks([]) plt.yticks([]) def branch_and_bound_n_queens(N, delay=.5): solutions = [] col_free = [True] * N up_free = [True] * (2*N - 1) down_free = [True] * (2*N - 1) queens = {} row_Qs = [] def placequeen(row_num): global solution_count for col_index in range(N): if col_free[col_index] and \ up_free[row_num + col_index] and \ down_free[N - 1 + col_index - row_num]: queens[row_num] = col_index col_free[col_index] = False up_free[row_num + col_index] = False down_free[N - 1 + col_index - row_num] = False if row_num == N-1: sol = ['.'*i+'Q'+'.'*(N-i-1) for i in list(queens.values())] solutions.append(sol.copy()) # 绘制当前解决方案 board = np.zeros((N,N)) for r,c in enumerate(list(queens.values())): board[r][c]=1 draw_board(board) plt.show(block=False) plt.pause(delay) plt.close() else: placequeen(row_num+1) col_free[col_index] = True up_free[row_num + col_index] = True down_free[N - 1 + col_index - row_num] = True placequeen(0) return solutions if __name__ == "__main__": N = int(input("Enter number of Queens: ")) branch_and_bound_n_queens(N) ``` 上述代码实现了基于分支限界的N皇后问题求解器，并利用`matplotlib`动态展示了每一步骤的结果。每次找到新的有效布局后都会暂停一段时间(`delay`)以便观察者能清楚看到变化[^1]。