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This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.shortest_paths.weighted

```
# -*- coding: utf-8 -*-
"""
Shortest path algorithms for weighed graphs.
"""
__author__ = """\n""".join(['Aric Hagberg <hagberg@lanl.gov>',
'Loïc Séguin-C. <loicseguin@gmail.com>',
'Dan Schult <dschult@colgate.edu>'])
# Copyright (C) 2004-2011 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',
'single_source_dijkstra_path',
'single_source_dijkstra_path_length',
'all_pairs_dijkstra_path',
'all_pairs_dijkstra_path_length',
'dijkstra_predecessor_and_distance',
'bellman_ford',
'negative_edge_cycle',
'goldberg_radzik']
from collections import deque
from heapq import heappush, heappop
from itertools import count
import networkx as nx
from networkx.utils import generate_unique_node
[docs]def dijkstra_path(G, source, target, weight='weight'):
"""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
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
path : list
List of nodes in a shortest path.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path(G,0,4))
[0, 1, 2, 3, 4]
Notes
------
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
bidirectional_dijkstra()
"""
(length, path) = single_source_dijkstra(G, source, target=target,
weight=weight)
try:
return path[target]
except KeyError:
raise nx.NetworkXNoPath(
"node %s not reachable from %s" % (source, target))
[docs]def dijkstra_path_length(G, source, target, weight='weight'):
"""Returns the shortest path length from source to target
in a weighted graph.
Parameters
----------
G : NetworkX graph
source : node label
starting node for path
target : node label
ending node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
length : number
Shortest path length.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G=nx.path_graph(5)
>>> print(nx.dijkstra_path_length(G,0,4))
4
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
bidirectional_dijkstra()
"""
length = single_source_dijkstra_path_length(G, source, weight=weight)
try:
return length[target]
except KeyError:
raise nx.NetworkXNoPath(
"node %s not reachable from %s" % (source, target))
[docs]def single_source_dijkstra_path(G, source, cutoff=None, weight='weight'):
"""Compute shortest path between source and all other reachable
nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
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.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
single_source_dijkstra()
"""
(length, path) = single_source_dijkstra(
G, source, cutoff=cutoff, weight=weight)
return path
[docs]def single_source_dijkstra_path_length(G, source, cutoff=None,
weight='weight'):
"""Compute the 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
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
length : dictionary
Dictionary of shortest lengths 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 weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
single_source_dijkstra()
"""
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {source: 0}
c = count()
fringe = [] # use heapq with (distance,label) tuples
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
# 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():
minweight = min((dd.get(weight, 1)
for k, dd in keydata.items()))
edata.append((w, {weight: minweight}))
else:
edata = iter(G[v].items())
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
push(fringe, (vw_dist, next(c), w))
return dist
[docs]def single_source_dijkstra(G, source, target=None, cutoff=None, weight='weight'):
"""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 dictionary 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
---------
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
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 ({source: 0}, {source: [source]})
push = heappush
pop = heappop
dist = {} # dictionary of final distances
paths = {source: [source]} # dictionary of paths
seen = {source: 0}
c = count()
fringe = [] # use heapq with (distance,label) tuples
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(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():
minweight = min((dd.get(weight, 1)
for k, dd in keydata.items()))
edata.append((w, {weight: minweight}))
else:
edata = iter(G[v].items())
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
push(fringe, (vw_dist, next(c), w))
paths[w] = paths[v] + [w]
return (dist, paths)
[docs]def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight='weight'):
"""Compute shortest path length and predecessors on shortest paths
in weighted graphs.
Parameters
----------
G : NetworkX graph
source : node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
pred,distance : dictionaries
Returns two dictionaries representing a list of predecessors
of a node and the distance to each node.
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The list of predecessors contains more than one element only when
there are more than one shortest paths to the key node.
"""
push = heappush
pop = heappop
dist = {} # dictionary of final distances
pred = {source: []} # dictionary of predecessors
seen = {source: 0}
c = count()
fringe = [] # use heapq with (distance,label) tuples
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if G.is_multigraph():
edata = []
for w, keydata in G[v].items():
minweight = min((dd.get(weight, 1)
for k, dd in keydata.items()))
edata.append((w, {weight: minweight}))
else:
edata = iter(G[v].items())
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
push(fringe, (vw_dist, next(c), w))
pred[w] = [v]
elif vw_dist == seen[w]:
pred[w].append(v)
return (pred, dist)
[docs]def all_pairs_dijkstra_path_length(G, cutoff=None, weight='weight'):
""" Compute shortest path lengths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, 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
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionary returned only has keys for reachable node pairs.
"""
paths = {}
for n in G:
paths[n] = single_source_dijkstra_path_length(G, n, cutoff=cutoff,
weight=weight)
return paths
[docs]def all_pairs_dijkstra_path(G, cutoff=None, weight='weight'):
""" Compute shortest paths between all nodes in a weighted graph.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
cutoff : integer or float, 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]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
See Also
--------
floyd_warshall()
"""
paths = {}
for n in G:
paths[n] = single_source_dijkstra_path(G, n, cutoff=cutoff,
weight=weight)
return paths
[docs]def bellman_ford(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
dist = {source: 0}
pred = {source: None}
if len(G) == 1:
return pred, dist
if G.is_multigraph():
def get_weight(edge_dict):
return min(eattr.get(weight, 1) for eattr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
if G.is_directed():
G_succ = G.succ
else:
G_succ = G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque([source])
in_q = set([source])
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if the predecessor of u is in the queue.
if pred[u] not in in_q:
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + get_weight(e)
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = u
return pred, dist
def goldberg_radzik(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.goldberg_radzik(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.goldberg_radzik, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
if len(G) == 1:
return {source: None}, {source: 0}
if G.is_multigraph():
def get_weight(edge_dict):
return min(attr.get(weight, 1) for attr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
if G.is_directed():
G_succ = G.succ
else:
G_succ = G.adj
inf = float('inf')
d = dict((u, inf) for u in G)
d[source] = 0
pred = {source: None}
def topo_sort(relabeled):
"""Topologically sort nodes relabeled in the previous round and detect
negative cycles.
"""
# List of nodes to scan in this round. Denoted by A in Goldberg and
# Radzik's paper.
to_scan = []
# In the DFS in the loop below, neg_count records for each node the
# number of edges of negative reduced costs on the path from a DFS root
# to the node in the DFS forest. The reduced cost of an edge (u, v) is
# defined as d[u] + weight[u][v] - d[v].
#
# neg_count also doubles as the DFS visit marker array.
neg_count = {}
for u in relabeled:
# Skip visited nodes.
if u in neg_count:
continue
d_u = d[u]
# Skip nodes without out-edges of negative reduced costs.
if all(d_u + get_weight(e) >= d[v] for v, e in G_succ[u].items()):
continue
# Nonrecursive DFS that inserts nodes reachable from u via edges of
# nonpositive reduced costs into to_scan in (reverse) topological
# order.
stack = [(u, iter(G_succ[u].items()))]
in_stack = set([u])
neg_count[u] = 0
while stack:
u, it = stack[-1]
try:
v, e = next(it)
except StopIteration:
to_scan.append(u)
stack.pop()
in_stack.remove(u)
continue
t = d[u] + get_weight(e)
d_v = d[v]
if t <= d_v:
is_neg = t < d_v
d[v] = t
pred[v] = u
if v not in neg_count:
neg_count[v] = neg_count[u] + int(is_neg)
stack.append((v, iter(G_succ[v].items())))
in_stack.add(v)
elif (v in in_stack and
neg_count[u] + int(is_neg) > neg_count[v]):
# (u, v) is a back edge, and the cycle formed by the
# path v to u and (u, v) contains at least one edge of
# negative reduced cost. The cycle must be of negative
# cost.
raise nx.NetworkXUnbounded(
'Negative cost cycle detected.')
to_scan.reverse()
return to_scan
def relax(to_scan):
"""Relax out-edges of relabeled nodes.
"""
relabeled = set()
# Scan nodes in to_scan in topological order and relax incident
# out-edges. Add the relabled nodes to labeled.
for u in to_scan:
d_u = d[u]
for v, e in G_succ[u].items():
w_e = get_weight(e)
if d_u + w_e < d[v]:
d[v] = d_u + w_e
pred[v] = u
relabeled.add(v)
return relabeled
# Set of nodes relabled in the last round of scan operations. Denoted by B
# in Goldberg and Radzik's paper.
relabeled = set([source])
while relabeled:
to_scan = topo_sort(relabeled)
relabeled = relax(to_scan)
d = dict((u, d[u]) for u in pred)
return pred, d
[docs]def negative_edge_cycle(G, weight='weight'):
"""Return True if there exists a negative edge cycle anywhere in G.
Parameters
----------
G : NetworkX graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
negative_cycle : bool
True if a negative edge cycle exists, otherwise False.
Examples
--------
>>> import networkx as nx
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> print(nx.negative_edge_cycle(G))
False
>>> G[1][2]['weight'] = -7
>>> print(nx.negative_edge_cycle(G))
True
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
This algorithm uses bellman_ford() but finds negative cycles
on any component by first adding a new node connected to
every node, and starting bellman_ford on that node. It then
removes that extra node.
"""
newnode = generate_unique_node()
G.add_edges_from([(newnode, n) for n in G])
try:
bellman_ford(G, newnode, weight)
except nx.NetworkXUnbounded:
return True
finally:
G.remove_node(newnode)
return False
[docs]def bidirectional_dijkstra(G, source, target, weight='weight'):
"""Dijkstra's algorithm for shortest paths using bidirectional search.
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
length : number
Shortest path length.
Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
Raises
------
NetworkXNoPath
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 == target:
return (0, [source])
push = heappush
pop = heappop
# 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
c = count()
# initialize fringe heap
push(fringe[0], (0, next(c), source))
push(fringe[1], (0, next(c), 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) = pop(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
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[v][w].items()))
else:
minweight = G[v][w].get(weight, 1)
vwLength = dists[dir][v] + minweight # G[v][w].get(weight,1)
else: # back, must remember to change v,w->w,v
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[w][v].items()))
else:
minweight = G[w][v].get(weight, 1)
vwLength = dists[dir][v] + minweight # 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
push(fringe[dir], (vwLength, next(c), 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:]
raise nx.NetworkXNoPath("No path between %s and %s." % (source, target))
```