Dijkstra

单源最短路径算法，用于计算一个节点到其他所有节点的最短路径。主要特点是以起始点为中心向外层层扩展，直到扩展到终点为止。Dijkstra算法是很有代表性的最短路径算法，在很多专业课程中都作为基本内容有详细的介绍，如数据结构，图论，运筹学等等。注意该算法要求图中不存在负权边。

设G=(V,E)是一个带权有向图，把图中顶点集合V分成两组，第一组为已求出最短路径的顶点集合（用S表示，初始时S中只有一个源点，以后每求得一条最短路径 , 就将加入到集合S中，直到全部顶点都加入到S中，算法就结束了），第二组为其余未确定最短路径的顶点集合（用U表示），按最短路径长度的递增次序依次把第二组的顶点加入S中。在加入的过程中，总保持从源点v到S中各顶点的最短路径长度不大于从源点v到U中任何顶点的最短路径长度。此外，每个顶点对应一个距离，S中的顶点的距离就是从v到此顶点的最短路径长度，U中的顶点的距离，是从v到此顶点只包括S中的顶点为中间顶点的当前最短路径长度。

python 算法实现

import queue  
from collections import namedtuple

Edge = namedtuple('Edge', ['vertex', 'weight'])

class GraphUndirectedWeighted(object):  
    def __init__(self, vertex_count):
        self.vertex_count = vertex_count
        self.adjacency_list = [[] for _ in range(vertex_count)]

    def add_edge(self, source, dest, weight):
        assert source < self.vertex_count
        assert dest < self.vertex_count
        self.adjacency_list[source].append(Edge(dest, weight))
        self.adjacency_list[dest].append(Edge(source, weight))

    def get_edge(self, vertex):
        for e in self.adjacency_list[vertex]:
            yield e

    def get_vertex(self):
        for v in range(self.vertex_count):
            yield v

def dijkstra(graph, source, dest):  
    q = queue.PriorityQueue()
    parents = []
    distances = []
    start_weight = float("inf")

    for i in graph.get_vertex():
        weight = start_weight
        if source == i:
            weight = 0
        distances.append(weight)
        parents.append(None)

    q.put(([0, source]))

    while not q.empty():
        v_tuple = q.get()
        v = v_tuple[1]

        for e in graph.get_edge(v):
            candidate_distance = distances[v] + e.weight
            if distances[e.vertex] > candidate_distance:
                distances[e.vertex] = candidate_distance
                parents[e.vertex] = v
                # primitive but effective negative cycle detection
                if candidate_distance < -1000:
                    raise Exception("Negative cycle detected")
                q.put(([distances[e.vertex], e.vertex]))

    shortest_path = []
    end = dest
    while end is not None:
        shortest_path.append(end)
        end = parents[end]

    shortest_path.reverse()

    return shortest_path, distances[dest]

def main():  
    g = GraphUndirectedWeighted(9)
    g.add_edge(0, 1, 4)
    g.add_edge(1, 7, 6)
    g.add_edge(1, 2, 1)
    g.add_edge(2, 3, 3)
    g.add_edge(3, 7, 1)
    g.add_edge(3, 4, 2)
    g.add_edge(3, 5, 1)
    g.add_edge(4, 5, 1)
    g.add_edge(5, 6, 1)
    g.add_edge(6, 7, 2)
    g.add_edge(6, 8, 2)
    g.add_edge(7, 8, 2)
    # for testing negative cycles
    # g.add_edge(1, 9, -5)
    # g.add_edge(9, 7, -4)

    shortest_path, distance = dijkstra(g, 0, 1)
    assert shortest_path == [0, 1] and distance == 4

    shortest_path, distance = dijkstra(g, 0, 8)
    assert shortest_path == [0, 1, 2, 3, 7, 8] and distance == 11

    shortest_path, distance = dijkstra(g, 5, 0)
    assert shortest_path == [5, 3, 2, 1, 0] and distance == 9

    shortest_path, distance = dijkstra(g, 1, 1)
    assert shortest_path == [1] and distance == 0

if __name__ == "__main__":  
    main()