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Recipe 119466: Dijkstra's algorithm for shortest paths

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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# 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 non-final 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 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

Discussion

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.

Comments

  1. 1. At 6:08 p.m. on 2 oct 2005, Connelly Barnes said:

    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:

    def shortest_path(G, start, end):
   def flatten(L):       # Flatten linked list of form [0,[1,[2,[]]]]
      while len(L) > 0:
         yield L[0]
         L = L[1]

   q = [(0, start, ())]  # Heap of (cost, path_head, path_rest).
   visited = set()       # Visited vertices.
   while True:
      (cost, v1, path) = heapq.heappop(q)
      if v1 not in visited:
         visited.add(v1)
         if v1 == end:
            return list(flatten(path))[::-1] + [v1]
         path = (v1, path)
         for (v2, cost2) in G[v1].iteritems():
            if v2 not in visited:
               heapq.heappush(q, (cost + cost2, v2, path))

    If there are n vertices and m edges, the modified-Dijkstra 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

    modified-Dijkstra 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 python-dev). This implementation is faster.

  2. 2. At 3:25 a.m. on 6 mar 2008, Ciamac Faridani said:

    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/

    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}}

f = open('dotgraph.txt','w')
f.writelines('digraph G {\nnode [width=.3,height=.3,shape=octagon,style=filled,color=skyblue];\noverlap="false";\nrankdir="LR";\n')
f.writelines
for i in G:

    for j in G[i]:
        s= '      '+ i
        s +=  ' -> ' +  j + ' [label="' + str(G[i][j]) + '"]'
        s+=';\n'
        f.writelines(s)

f.writelines('}')
f.close()
print "done!"
  3. 3. At 7:18 a.m. on 14 mar 2008, Eoghan Murray said:

    Variable Naming. Would 'vwLength' be better named 'startwLength' or maybe even 'pathDistance'?

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