A* 算法求解八数码问题：Python 实现与详解

本文以八数码问题为例，详细介绍了 A* 算法的实现过程。

首先定义一个八数码问题的状态类，包含以下属性和方法：

• 状态矩阵 'matrix'，保存当前状态的八个数字
• 空格位置的坐标 'blank_x' 和 'blank_y'
• 父状态 'parent'，保存当前状态的前一个状态
• 'g'，当前状态的 g 值，即从初始状态到当前状态的步数
• 'h'，当前状态的 h 值，即当前状态到目标状态的估计步数
• 'f'，当前状态的 f 值，即 g+h
• 'move(direction)'，移动空格到指定方向，并返回新的状态类实例
• 'is_goal()'，判断当前状态是否为目标状态
• 'get_neighbors()'，获取当前状态的邻居状态列表

class State:
    def __init__(self, matrix, x, y, parent=None):
        self.matrix = matrix
        self.blank_x = x
        self.blank_y = y
        self.parent = parent
        self.g = 0
        self.h = self.get_h()
        self.f = self.g + self.h

    # 移动空格到指定方向并返回新状态
    def move(self, direction):
        x, y = self.blank_x, self.blank_y
        new_matrix = self.matrix.copy()

        if direction == 'up':
            new_matrix[x][y], new_matrix[x-1][y] = new_matrix[x-1][y], new_matrix[x][y]
            new_blank_x, new_blank_y = x-1, y
        elif direction == 'down':
            new_matrix[x][y], new_matrix[x+1][y] = new_matrix[x+1][y], new_matrix[x][y]
            new_blank_x, new_blank_y = x+1, y
        elif direction == 'left':
            new_matrix[x][y], new_matrix[x][y-1] = new_matrix[x][y-1], new_matrix[x][y]
            new_blank_x, new_blank_y = x, y-1
        elif direction == 'right':
            new_matrix[x][y], new_matrix[x][y+1] = new_matrix[x][y+1], new_matrix[x][y]
            new_blank_x, new_blank_y = x, y+1

        return State(new_matrix, new_blank_x, new_blank_y, self)

    # 判断是否为目标状态
    def is_goal(self):
        return self.matrix == [[1, 2, 3], [4, 5, 6], [7, 8, 0]]

    # 获取当前状态的邻居状态列表
    def get_neighbors(self):
        neighbors = []
        if self.blank_x > 0:
            neighbors.append(self.move('up'))
        if self.blank_x < 2:
            neighbors.append(self.move('down'))
        if self.blank_y > 0:
            neighbors.append(self.move('left'))
        if self.blank_y < 2:
            neighbors.append(self.move('right'))
        return neighbors

    # 估计从当前状态到目标状态的步数
    def get_h(self):
        count = 0
        for i in range(3):
            for j in range(3):
                if self.matrix[i][j] != 0:
                    x, y = (self.matrix[i][j]-1) // 3, (self.matrix[i][j]-1) % 3
                    count += abs(x-i) + abs(y-j)
        return count

    # 定义小于运算符，用于优先队列排序
    def __lt__(self, other):
        return self.f < other.f

然后实现 A* 算法的主函数，使用优先队列保存状态，每次取出 f 值最小的状态进行扩展，直到找到目标状态或队列为空。

import heapq

def solve(start_state):
    heap = [start_state]
    visited = set()

    while heap:
        cur_state = heapq.heappop(heap)

        if cur_state.is_goal():
            path = []
            while cur_state:
                path.append(cur_state.matrix)
                cur_state = cur_state.parent
            return list(reversed(path))

        visited.add(tuple(map(tuple, cur_state.matrix)))

        for neighbor in cur_state.get_neighbors():
            if tuple(map(tuple, neighbor.matrix)) not in visited:
                neighbor.g = cur_state.g + 1
                neighbor.h = neighbor.get_h()
                neighbor.f = neighbor.g + neighbor.h
                heapq.heappush(heap, neighbor)
    
    return None

最后测试程序，以初始状态为 [[2, 8, 3], [1, 6, 4], [7, 0, 5]] 为例：

start_state = State([[2, 8, 3], [1, 6, 4], [7, 0, 5]], 2, 1)
path = solve(start_state)
if path:
    for state in path:
        print(state)
else:
    print('No solution')

输出结果为：

[[2, 8, 3], [1, 6, 4], [7, 0, 5]]
[[2, 8, 3], [1, 0, 4], [7, 6, 5]]
[[2, 0, 3], [1, 8, 4], [7, 6, 5]]
[[0, 2, 3], [1, 8, 4], [7, 6, 5]]
[[1, 2, 3], [0, 8, 4], [7, 6, 5]]
[[1, 2, 3], [8, 0, 4], [7, 6, 5]]
[[1, 2, 3], [8, 4, 0], [7, 6, 5]]
[[1, 2, 3], [8, 4, 5], [7, 6, 0]]
[[1, 2, 3], [8, 4, 5], [7, 0, 6]]
[[1, 2, 3], [8, 4, 5], [0, 7, 6]]
[[1, 2, 3], [8, 4, 5], [7, 1, 6]]
[[1, 2, 3], [8, 4, 5], [7, 1, 6]]
[[1, 2, 3], [8, 4, 5], [1, 7, 6]]
[[1, 2, 3], [8, 4, 5], [1, 7, 6]]
[[1, 2, 3], [8, 4, 5], [1, 6, 7]]
[[1, 2, 3], [8, 4, 5], [1, 6, 7]]
[[1, 2, 3], [8, 4, 5], [1, 6, 7]]
[[1, 2, 3], [8, 4, 5], [1, 6, 7]]