# -*- coding: utf-8 -*-
"""
Shortest path algorithms for weighed graphs.
"""
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""
# Copyright (C) 2004-2010 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__all__ = ['dijkstra_path',
'dijkstra_path_length',
'bidirectional_dijkstra',
'single_source_dijkstra_path',
'single_source_dijkstra_path_length',
'all_pairs_dijkstra_path',
'all_pairs_dijkstra_path_length',
'single_source_dijkstra',
'dijkstra_predecessor_and_distance']
import heapq
import networkx as nx
def dijkstra_path(G,source,target):
"""Returns the shortest path from source to target in a weighted
graph G.
Parameters
----------
G : NetworkX graph
source : node
Starting node
target : node
Ending node
Returns
-------
path : list
List of nodes in a shortest path.
Examples
--------
>>> G=nx.path_graph(5)
>>> print nx.dijkstra_path(G,0,4)
[0, 1, 2, 3, 4]
Notes
------
Uses a bidirectional version of Dijkstra's algorithm.
Edge weight attributes must be numerical.
See Also
--------
bidirectional_dijkstra()
"""
# (length,path)=bidirectional_dijkstra(G,source,target) # faster, needs test
# return path
(length,path)=single_source_dijkstra(G,source)
try:
return path[target]
except KeyError:
raise nx.NetworkXError, \
"node %s not reachable from %s"%(source,target)
def dijkstra_path_length(G,source,target):
"""Returns the shortest path length from source to target in a weighted
graph G.
Parameters
----------
G : NetworkX graph, weighted
source : node label
starting node for path
target : node label
ending node for path
Returns
-------
length : number
Shortest path length.
Raises
------
NetworkXError
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5) # a weighted graph by default
>>> print nx.dijkstra_path_length(G,0,4)
4
Notes
-----
Edge weight attributes must be numerical.
See Also
--------
bidirectional_dijkstra()
"""
# (length,path)=bidirectional_dijkstra(G,source,target) # faster, needs test
# return length
(length,path)=single_source_dijkstra(G,source)
try:
return length[target]
except KeyError:
raise nx.NetworkXError, \
"node %s not reachable from %s"%(source,target)
def bidirectional_dijkstra(G, source, target):
"""Dijkstra's algorithm for shortest paths using bidirectional search.
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
Returns
-------
length : number
Shortest path length.
Returns a tuple of two dictionaries keyed by node.
The first dicdtionary stores distance from the source.
The second stores the path from the source to that node.
Raise an exception if no path exists.
Raises
------
NetworkXError
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> length,path=nx.bidirectional_dijkstra(G,0,4)
>>> print length
4
>>> print path
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
In practice bidirectional Dijkstra is much more than twice as fast as
ordinary Dijkstra.
Ordinary Dijkstra expands nodes in a sphere-like manner from the
source. The radius of this sphere will eventually be the length
of the shortest path. Bidirectional Dijkstra will expand nodes
from both the source and the target, making two spheres of half
this radius. Volume of the first sphere is pi*r*r while the
others are 2*pi*r/2*r/2, making up half the volume.
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
shortest_path
shortest_path_length
"""
if source is None or target is None:
raise NetworkXException(
"Bidirectional Dijkstra called with no source or target")
if source == target: return (0, [source])
#Init: Forward Backward
dists = [{}, {}]# dictionary of final distances
paths = [{source:[source]}, {target:[target]}] # dictionary of paths
fringe = [[], []] #heap of (distance, node) tuples for extracting next node to expand
seen = [{source:0}, {target:0} ]#dictionary of distances to nodes seen
#initialize fringe heap
heapq.heappush(fringe[0], (0, source))
heapq.heappush(fringe[1], (0, target))
#neighs for extracting correct neighbor information
if G.is_directed():
neighs = [G.successors_iter, G.predecessors_iter]
else:
neighs = [G.neighbors_iter, G.neighbors_iter]
#variables to hold shortest discovered path
#finaldist = 1e30000
finalpath = []
dir = 1
while fringe[0] and fringe[1]:
# choose direction
# dir == 0 is forward direction and dir == 1 is back
dir = 1-dir
# extract closest to expand
(dist, v )= heapq.heappop(fringe[dir])
if v in dists[dir]:
# Shortest path to v has already been found
continue
# update distance
dists[dir][v] = dist #equal to seen[dir][v]
if v in dists[1-dir]:
# if we have scanned v in both directions we are done
# we have now discovered the shortest path
return (finaldist,finalpath)
for w in neighs[dir](v):
if(dir==0): #forward
vwLength = dists[dir][v] + G[v][w].get('weight',1)
else: #back, must remember to change v,w->w,v
vwLength = dists[dir][v] + G[w][v].get('weight',1)
if w in dists[dir]:
if vwLength < dists[dir][w]:
raise ValueError,\
"Contradictory paths found: negative weights?"
elif w not in seen[dir] or vwLength < seen[dir][w]:
# relaxing
seen[dir][w] = vwLength
heapq.heappush(fringe[dir], (vwLength,w))
paths[dir][w] = paths[dir][v]+[w]
if w in seen[0] and w in seen[1]:
#see if this path is better than than the already
#discovered shortest path
totaldist = seen[0][w] + seen[1][w]
if finalpath == [] or finaldist > totaldist:
finaldist = totaldist
revpath = paths[1][w][:]
revpath.reverse()
finalpath = paths[0][w] + revpath[1:]
return False
#def dijkstra(G,source,target):
# return bidirectional_dijkstra(G,source,target)
def single_source_dijkstra_path(G,source):
"""Compute shortest path between source
and all other reachable nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path.
Returns
-------
paths : dictionary
Dictionary of shortest path lengths keyed by target.
Examples
--------
>>> G=nx.path_graph(5)
>>> path=nx.single_source_dijkstra_path(G,0)
>>> path[4]
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
See Also
--------
single_source_dijkstra()
"""
(length,path)=single_source_dijkstra(G,source)
return path
def single_source_dijkstra_path_length(G,source):
"""Compute shortest path length between source
and all other reachable nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
Returns
-------
paths : dictionary
Dictionary of shortest paths keyed by target.
Examples
--------
>>> G=nx.path_graph(5)
>>> length=nx.single_source_dijkstra_path_length(G,0)
>>> length[4]
4
>>> print length
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
Notes
-----
Edge data must be numerical values for XGraph and XDiGraphs.
See Also
--------
single_source_dijkstra()
"""
(length,path)=single_source_dijkstra(G,source)
return length
def single_source_dijkstra(G,source,target=None,cutoff=None ):
"""Compute shortest paths and lengths in a weighted graph G.
Uses Dijkstra's algorithm for shortest paths.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
target : node label, optional
Ending node for path
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance,path : dictionaries
Returns a tuple of two dictionaries keyed by node.
The first dicdtionary stores distance from the source.
The second stores the path from the source to that node.
Examples
--------
>>> G=nx.path_graph(5)
>>> length,path=nx.single_source_dijkstra(G,0)
>>> print length[4]
4
>>> print length
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
>>> path[4]
[0, 1, 2, 3, 4]
Notes
---------
Distances are calculated as sums of weighted edges traversed.
Edges must hold numerical values for Graph and DiGraphs.
Based on the Python cookbook recipe (119466) at
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
single_source_dijkstra_path()
single_source_dijkstra_path_length()
"""
if source==target: return (0, [source])
dist = {} # dictionary of final distances
paths = {source:[source]} # dictionary of paths
seen = {source:0}
fringe=[] # use heapq with (distance,label) tuples
heapq.heappush(fringe,(0,source))
while fringe:
(d,v)=heapq.heappop(fringe)
if v in dist: continue # already searched this node.
dist[v] = d
if v == target: break
#for ignore,w,edgedata in G.edges_iter(v,data=True):
#is about 30% slower than the following
if G.is_multigraph():
edata=[]
for w,keydata in G[v].items():
edata.append((w,
{'weight':min((dd.get('weight',1)
for k,dd in keydata.iteritems()))}))
else:
edata=G[v].iteritems()
for w,edgedata in edata:
vw_dist = dist[v] + edgedata.get('weight',1)
if cutoff is not None:
if vw_dist>cutoff:
continue
if w in dist:
if vw_dist < dist[w]:
raise ValueError,\
"Contradictory paths found: negative weights?"
elif w not in seen or vw_dist < seen[w]:
seen[w] = vw_dist
heapq.heappush(fringe,(vw_dist,w))
paths[w] = paths[v]+[w]
return (dist,paths)
def dijkstra_predecessor_and_distance(G,source):
"""Compute shorest path length and predecessors on shortest paths
in weighted graphs.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
Returns
-------
pred,distance : dictionaries
Returns two dictionaries representing a list of predecessors
of a node and the distance to each node.
Notes
-----
The list of predecessors contains more than one element only when
there are more than one shortest paths to the key node.
"""
push=heapq.heappush
pop=heapq.heappop
dist = {} # dictionary of final distances
pred = {source:[]} # dictionary of predecessors
seen = {source:0}
fringe=[] # use heapq with (distance,label) tuples
push(fringe,(0,source))
while fringe:
(d,v)=pop(fringe)
if v in dist: continue # already searched this node.
dist[v] = d
if G.is_multigraph():
edata=( (w,min(edgedata.values()))
for w,edgedata in G[v].iteritems() )
else:
edata=G[v].iteritems()
for w,edgedata in edata:
vw_dist = dist[v] + edgedata.get('weight',1)
if w in dist:
if vw_dist < dist[w]:
raise ValueError,\
"Contradictory paths found: negative weights?"
elif w not in seen or vw_dist < seen[w]:
seen[w] = vw_dist
push(fringe,(vw_dist,w))
pred[w] = [v]
elif vw_dist==seen[w]:
pred[w].append(v)
return (pred,dist)
def all_pairs_dijkstra_path_length(G):
""" Compute shortest path lengths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest path lengths.
Examples
--------
>>> G=nx.path_graph(5)
>>> length=nx.all_pairs_dijkstra_path_length(G)
>>> print length[1][4]
3
>>> length[1]
{0: 1, 1: 0, 2: 1, 3: 2, 4: 3}
Notes
-----
The dictionary returned only has keys for reachable node pairs.
"""
paths={}
for n in G:
paths[n]=single_source_dijkstra_path_length(G,n)
return paths
def all_pairs_dijkstra_path(G):
""" Compute shortest paths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.
Examples
--------
>>> G=nx.path_graph(5)
>>> path=nx.all_pairs_dijkstra_path(G)
>>> print path[0][4]
[0, 1, 2, 3, 4]
See Also
--------
floyd_warshall()
"""
paths={}
for n in G:
paths[n]=single_source_dijkstra_path(G,n)
return paths
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