Python中常见路径算法的原理与实现详解

import heapq

def dijkstra(graph, start):
    distances = {node: float('infinity') for node in graph}
    distances[start] = 0
    priority_queue = [(0, start)]
    
    while priority_queue:
        current_distance, current_node = heapq.heappop(priority_queue)
        
        if current_distance > distances[current_node]:
            continue
        
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))
    
    return distances

# 示例图
graph = {
    'A': {'B': 5, 'C': 2},
    'B': {'D': 4, 'E': 2},
    'C': {'B': 1, 'F': 4},
    'D': {'E': 1},
    'E': {'F': 1},
    'F': {}
}

start_node = 'A'
shortest_distances = dijkstra(graph, start_node)
print(f"从节点 {start_node} 到其他节点的最短距离: {shortest_distances}")

import heapq

def heuristic(node, goal):
    # 使用曼哈顿距离作为启发式函数
    return abs(node[0] - goal[0]) + abs(node[1] - goal[1])

def a_star(graph, start, goal):
    open_set = [(0, start)]
    came_from = {}
    g_score = {node: float('infinity') for node in graph}
    g_score[start] = 0
    f_score = {node: float('infinity') for node in graph}
    f_score[start] = heuristic(start, goal)
    
    while open_set:
        current = heapq.heappop(open_set)[1]
        
        if current == goal:
            path = []
            while current in came_from:
                path.append(current)
                current = came_from[current]
            return path[::-1]
        
        for neighbor, weight in graph[current].items():
            tentative_g_score = g_score[current] + weight
            if tentative_g_score < g_score[neighbor]:
                came_from[neighbor] = current
                g_score[neighbor] = tentative_g_score
                f_score[neighbor] = g_score[neighbor] + heuristic(neighbor, goal)
                heapq.heappush(open_set, (f_score[neighbor], neighbor))
    
    return None

# 示例图（网格地图）
graph = {
    (0, 0): {(0, 1): 1, (1, 0): 1},
    (0, 1): {(0, 0): 1, (1, 1): 1},
    (1, 0): {(0, 0): 1, (1, 1): 1},
    (1, 1): {(0, 1): 1, (1, 0): 1}
}

start_node = (0, 0)
goal_node = (1, 1)
path = a_star(graph, start_node, goal_node)
print(f"从节点 {start_node} 到节点 {goal_node} 的路径: {path}")

def bellman_ford(graph, start):
    distances = {node: float('infinity') for node in graph}
    distances[start] = 0
    
    for _ in range(len(graph) - 1):
        for node in graph:
            for neighbor, weight in graph[node].items():
                if distances[node] + weight < distances[neighbor]:
                    distances[neighbor] = distances[node] + weight
    
    return distances

# 示例图（包含负权边）
graph = {
    'A': {'B': -1, 'C': 4},
    'B': {'C': 3, 'D': 2, 'E': 2},
    'C': {},
    'D': {'B': 1, 'C': 5},
    'E': {'D': -3}
}

start_node = 'A'
shortest_distances = bellman_ford(graph, start_node)
print(f"从节点 {start_node} 到其他节点的最短距离: {shortest_distances}")