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()