最短路径问题的Dijkstra算法 -python
最短路径问题的Dijkstra算法
是由荷兰计算机科学家艾兹赫尔·戴克斯特拉提出。迪科斯彻算法使用了广度优先搜索解决非负权有向图的单源最短路径问题，算法最终得到一个最短路径树> 。该算法常用于路由算法或者作为其他图算法的一个子模块。
# Dijkstra's algorithm for shortest paths
# David Eppstein, UC Irvine, 4 April 2002
# http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/117228
from priodict import priorityDictionary
def Dijkstra(G,start,end=None):
"""
Find shortest paths from the start vertex to all vertices nearer than or equal to the end.
The input graph G is assumed to have the following representation:
A vertex can be any object that can be used as an index into a dictionary.
G is a dictionary, indexed by vertices. For any vertex v, G[v] is itself a dictionary,
indexed by the neighbors of v. For any edge v->w, G[v][w] is the length of the edge.
This is related to the representation in <http://www.python.org/doc/essays/graphs.html>
where Guido van Rossum suggests representing graphs as dictionaries mapping vertices
to lists of outgoing edges, however dictionaries of edges have many advantages over lists:
they can store extra information (here, the lengths), they support fast existence tests,
and they allow easy modification of the graph structure by edge insertion and removal.
Such modifications are not needed here but are important in many other graph algorithms.
Since dictionaries obey iterator protocol, a graph represented as described here could
be handed without modification to an algorithm expecting Guido's graph representation.
Of course, G and G[v] need not be actual Python dict objects, they can be any other
type of object that obeys dict protocol, for instance one could use a wrapper in which vertices
are URLs of web pages and a call to G[v] loads the web page and finds its outgoing links.
The output is a pair (D,P) where D[v] is the distance from start to v and P[v] is the
predecessor of v along the shortest path from s to v.
Dijkstra's algorithm is only guaranteed to work correctly when all edge lengths are positive.
This code does not verify this property for all edges (only the edges examined until the end
vertex is reached), but will correctly compute shortest paths even for some graphs with negative
edges, and will raise an exception if it discovers that a negative edge has caused it to make a mistake.
"""
D = {} # dictionary of final distances
P = {} # dictionary of predecessors
Q = priorityDictionary() # estimated distances of non-final vertices
Q[start] = 0
for v in Q:
D[v] = Q[v]
if v == end: break
for w in G[v]:
vwLength = D[v] + G[v][w]
if w in D:
if vwLength < D[w]:
raise ValueError, "Dijkstra: found better path to already-final vertex"
elif w not in Q or vwLength < Q[w]:
Q[w] = vwLength
P[w] = v
return (D,P)
def shortestPath(G,start,end):
"""
Find a single shortest path from the given start vertex to the given end vertex.
The input has the same conventions as Dijkstra().
The output is a list of the vertices in order along the shortest path.
"""
D,P = Dijkstra(G,start,end)
Path = []
while 1:
Path.append(end)
if end == start: break
end = P[end]
Path.reverse()
return Path
# example, CLR p.528
G = {'s': {'u':10, 'x':5},
'u': {'v':1, 'x':2},
'v': {'y':4},
'x':{'u':3,'v':9,'y':2},
'y':{'s':7,'v':6}}
print Dijkstra(G,'s')
print shortestPath(G,'s','v')