Dijkstra 算法是一种用于计算单源最短路径的经典算法，适用于加权有向图，其中边的权重为非负数。以下是 Dijkstra 算法的 Python 实现：

pythonCopy Codeimport heapqdef dijkstra(graph, start):    """
    计算从起点 start 到图中所有其他节点的最短路径。
    :param graph: 字典表示的图，格式为 {节点: [(邻居, 权重), ...]}
    :param start: 起点节点
    :return: 从起点到每个节点的最短路径长度字典，以及前驱节点字典
    """
    # 初始化最短路径字典，所有节点的初始距离为无穷大
    shortest_paths = {node: float('inf') for node in graph}
    shortest_paths[start] = 0  # 起点到自身的距离为 0
    # 前驱节点字典，用于重建路径
    predecessors = {node: None for node in graph}    # 优先队列，存储 (距离, 节点) 元组
    priority_queue = [(0, start)]    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)        # 如果弹出的距离大于已知的最短距离，则跳过
        if current_distance > shortest_paths[current_node]:            continue
        # 遍历当前节点的邻居
        for neighbor, weight in graph[current_node]:
            distance = current_distance + weight            # 如果找到更短的路径，则更新最短路径和前驱节点
            if distance < shortest_paths[neighbor]:
                shortest_paths[neighbor] = distance
                predecessors[neighbor] = current_node
                heapq.heappush(priority_queue, (distance, neighbor))    return shortest_paths, predecessorsdef reconstruct_path(predecessors, start, target):    """
    根据前驱节点字典重建从起点到目标节点的路径。
    :param predecessors: 前驱节点字典
    :param start: 起点节点
    :param target: 目标节点
    :return: 从起点到目标节点的路径列表，如果不存在路径则返回空列表
    """
    path = []
    current = target    while current is not None:
        path.append(current)
        current = predecessors[current]
    path.reverse()    # 如果路径以起点开始，则返回路径，否则返回空列表
    return path if path[0] == start else []# 示例图graph = {    'A': [('B', 1), ('C', 4)],    'B': [('A', 1), ('C', 2), ('D', 5)],    'C': [('A', 4), ('B', 2), ('D', 1)],    'D': [('B', 5), ('C', 1)]
}
start_node = 'A'shortest_paths, predecessors = dijkstra(graph, start_node)print("最短路径长度:", shortest_paths)
target_node = 'D'path = reconstruct_path(predecessors, start_node, target_node)print(f"从 {start_node} 到 {target_node} 的路径:", path)

代码说明：

1. ‌ 图的表示‌：图使用一个字典表示，其中每个键是一个节点，值是一个列表，列表中的每个元素是一个元组，表示邻居节点和到该邻居的边的权重。

2. ‌ 优先队列‌：使用  heapq 模块实现优先队列，以便始终扩展当前已知距离最短的节点。

3. ‌ 最短路径计算‌：

   • 初始化所有节点的最短路径为无穷大，起点的最短路径为 0。
   • 使用优先队列存储待处理的节点，初始时只包含起点。
   • 每次从队列中取出距离最短的节点，更新其邻居的最短路径。

4. ‌ 路径重建‌：通过前驱节点字典，从目标节点回溯到起点，重建最短路径。

这个实现可以有效地计算单源最短路径，并重建从起点到任意目标节点的路径。

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