
Dijkstra(G,s) finds all shortest paths from s to each other vertex in the graph, and shortestPath(G,s,t) uses Dijkstra to find the shortest path from s to t. Uses the priorityDictionary data structure (Recipe 117228) to keep track of estimated distances to each vertex.
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87  # 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 neighbors,
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 by edge
insertion and removal. Such modifications are not
needed here but are important in other graph algorithms.
Since dictionaries obey iterator protocol, a graph
represented as described here could be handed without
modification to an algorithm using Guido's representation.
Of course, G and G[v] need not be Python dict objects;
they can be any other object that obeys dict protocol,
for instance a wrapper in which vertices are URLs
and a call to G[v] loads the web page and finds its 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 seen
before 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() # est.dist. of nonfinal vert.
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 alreadyfinal 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

As an example of the input format, here is the graph from Cormen, Leiserson, and Rivest (Introduction to Algorithms, 1st edition), page 528:
<pre>
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}}
</pre>
The shortest path from s to v is ['s', 'x', 'u', 'v'] and has length 9.

Can be simplified (with tradeoffs in time and memory). Dijkstra's algorithm can be simplified by allowing a (cost, vertex)
pair to be present multiple times in the priority queue:
If there are n vertices and m edges, the modifiedDijkstra algorithm
takes O(n+m) space and O(m*log(m)) time. In comparison, the Dijkstra
algorithm takes O(n) space and O((n+m)*log(n)) time. All time bounds
assume no hash collisions.
Using Eppstein's (excellent) dictionary graph representation, it takes O(n+m) space
to store a graph in memory, thus the memory overhead of the
modifiedDijkstra algorithm is reasonable.
I tested running times on a Pentium 3, and for complete graphs of ~2000
vertices, this modified Dijkstra function is several times slower than
Eppstein's function, and for sparse graphs with ~50000 vertices and
~50000*3 edges, the modified Dijkstra function is several times faster
than Eppstein's function.
P.S.: Eppstein has also implemented the modified algorithm in Python (see pythondev). This implementation is faster.
GraphViz Output. I have prepared a little script to visualize the network in graphViz just insert the output file in Dot or in this webpage http://ashitani.jp/gv/
Variable Naming. Would 'vwLength' be better named 'startwLength' or maybe even 'pathDistance'?
Thanks for the recipe, but I think it would be ten times more readable with names such as "graph", "estimated_distances", "final_distances", etc.
Thanks for posting this. I had a stab at an implementation on my blog: http://rebrained.com/?p=392
It's a little pedestrian compared to the examples above as it follows the algorithm fairly literally.
I took the modified version from Connelly Barnes, made it a bit simpler (adding some extra cost for the path/list operations), and uploaded the code and visualization to http://pyalgoviz.appspot.com. It shows a nice stepbystep progression of the nodes being visited. My version of the algorithm:
Hi David, I test this code by:
graph_dict = { "s1":{"s1": 0, "s2": 2, "s10": 1, "s12": 4, "s5":3}, "s2":{"s1": 1, "s2": 0, "s10": 4, "s12": 2, "s5":2}, "s10":{"s1": 2, "s2": 1, "s10": 0, "s12":1, "s5":4}, "s12":{"s1": 3, "s2": 5, "s10": 2, "s12":0,"s5":1}, "s5":{"s1": 3, "s2": 5, "s10": 2, "s12":4,"s5":0}, }
the distance turns out correctly, but the path doesn't. I also use function shortestPath to calculate path between s1 and s12, the result shows below:
({'s12': 2, 's1': 0, 's10': 1, 's5': 3, 's2': 2}, {'s2': 's1', 's10': 's1', 's5': 's1', 's12': 's1'}) ['s1', 's12']
Actually, the best path is s1>s10>s12.
I wirte the dijkstra too, but I think its time complexity is too high. How to make it faster?
```python
```