求解图的单源最短路径分支界限法python

单源最短路径问题可以使用分支界限法求解，以下是Python实现：

import heapq


class Node:
    def __init__(self, v, w):
        self.v = v
        self.w = w


class State:
    def __init__(self, cost, path, visited):
        self.cost = cost
        self.path = path
        self.visited = visited


def shortest_path_bound(graph, s):
    n = len(graph)
    visited = [False] * n
    visited[s] = True
    pq = [Node(v, w) for v, w in graph[s]]
    heapq.heapify(pq)
    best_path = None
    best_cost = float('inf')
    while pq:
        node = heapq.heappop(pq)
        v, w = node.v, node.w
        if visited[v]:
            continue
        path = [s] + list(node.path)
        cost = w + sum(graph[u][v] for u, v in zip(path, path[1:]))
        if cost >= best_cost:
            continue
        visited[v] = True
        if all(visited):
            if cost < best_cost:
                best_path = path
                best_cost = cost
        else:
            for u, w in graph[v]:
                if not visited[u]:
                    heapq.heappush(pq, Node(u, w))
    return best_path, best_cost

其中，`graph`是邻接表形式的图表示，`s`是起点。函数返回最优路径和最优路径的长度。算法的时间复杂度为$O(b^d)$，其中$b$是分支因子，$d$是最短路径的长度。