# Source code for networkx.algorithms.simple_paths

```
from heapq import heappop, heappush
from itertools import count
import networkx as nx
from networkx.algorithms.shortest_paths.weighted import _weight_function
from networkx.utils import not_implemented_for, pairwise
__all__ = [
"all_simple_paths",
"is_simple_path",
"shortest_simple_paths",
"all_simple_edge_paths",
]
[docs]
@nx._dispatch
def is_simple_path(G, nodes):
"""Returns True if and only if `nodes` form a simple path in `G`.
A *simple path* in a graph is a nonempty sequence of nodes in which
no node appears more than once in the sequence, and each adjacent
pair of nodes in the sequence is adjacent in the graph.
Parameters
----------
G : graph
A NetworkX graph.
nodes : list
A list of one or more nodes in the graph `G`.
Returns
-------
bool
Whether the given list of nodes represents a simple path in `G`.
Notes
-----
An empty list of nodes is not a path but a list of one node is a
path. Here's an explanation why.
This function operates on *node paths*. One could also consider
*edge paths*. There is a bijection between node paths and edge
paths.
The *length of a path* is the number of edges in the path, so a list
of nodes of length *n* corresponds to a path of length *n* - 1.
Thus the smallest edge path would be a list of zero edges, the empty
path. This corresponds to a list of one node.
To convert between a node path and an edge path, you can use code
like the following::
>>> from networkx.utils import pairwise
>>> nodes = [0, 1, 2, 3]
>>> edges = list(pairwise(nodes))
>>> edges
[(0, 1), (1, 2), (2, 3)]
>>> nodes = [edges[0][0]] + [v for u, v in edges]
>>> nodes
[0, 1, 2, 3]
Examples
--------
>>> G = nx.cycle_graph(4)
>>> nx.is_simple_path(G, [2, 3, 0])
True
>>> nx.is_simple_path(G, [0, 2])
False
"""
# The empty list is not a valid path. Could also return
# NetworkXPointlessConcept here.
if len(nodes) == 0:
return False
# If the list is a single node, just check that the node is actually
# in the graph.
if len(nodes) == 1:
return nodes[0] in G
# check that all nodes in the list are in the graph, if at least one
# is not in the graph, then this is not a simple path
if not all(n in G for n in nodes):
return False
# If the list contains repeated nodes, then it's not a simple path
if len(set(nodes)) != len(nodes):
return False
# Test that each adjacent pair of nodes is adjacent.
return all(v in G[u] for u, v in pairwise(nodes))
[docs]
@nx._dispatch
def all_simple_paths(G, source, target, cutoff=None):
"""Generate all simple paths in the graph G from source to target.
A simple path is a path with no repeated nodes.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
target : nodes
Single node or iterable of nodes at which to end path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
path_generator: generator
A generator that produces lists of simple paths. If there are no paths
between the source and target within the given cutoff the generator
produces no output. If it is possible to traverse the same sequence of
nodes in multiple ways, namely through parallel edges, then it will be
returned multiple times (once for each viable edge combination).
Examples
--------
This iterator generates lists of nodes::
>>> G = nx.complete_graph(4)
>>> for path in nx.all_simple_paths(G, source=0, target=3):
... print(path)
...
[0, 1, 2, 3]
[0, 1, 3]
[0, 2, 1, 3]
[0, 2, 3]
[0, 3]
You can generate only those paths that are shorter than a certain
length by using the `cutoff` keyword argument::
>>> paths = nx.all_simple_paths(G, source=0, target=3, cutoff=2)
>>> print(list(paths))
[[0, 1, 3], [0, 2, 3], [0, 3]]
To get each path as the corresponding list of edges, you can use the
:func:`networkx.utils.pairwise` helper function::
>>> paths = nx.all_simple_paths(G, source=0, target=3)
>>> for path in map(nx.utils.pairwise, paths):
... print(list(path))
[(0, 1), (1, 2), (2, 3)]
[(0, 1), (1, 3)]
[(0, 2), (2, 1), (1, 3)]
[(0, 2), (2, 3)]
[(0, 3)]
Pass an iterable of nodes as target to generate all paths ending in any of several nodes::
>>> G = nx.complete_graph(4)
>>> for path in nx.all_simple_paths(G, source=0, target=[3, 2]):
... print(path)
...
[0, 1, 2]
[0, 1, 2, 3]
[0, 1, 3]
[0, 1, 3, 2]
[0, 2]
[0, 2, 1, 3]
[0, 2, 3]
[0, 3]
[0, 3, 1, 2]
[0, 3, 2]
Iterate over each path from the root nodes to the leaf nodes in a
directed acyclic graph using a functional programming approach::
>>> from itertools import chain
>>> from itertools import product
>>> from itertools import starmap
>>> from functools import partial
>>>
>>> chaini = chain.from_iterable
>>>
>>> G = nx.DiGraph([(0, 1), (1, 2), (0, 3), (3, 2)])
>>> roots = (v for v, d in G.in_degree() if d == 0)
>>> leaves = (v for v, d in G.out_degree() if d == 0)
>>> all_paths = partial(nx.all_simple_paths, G)
>>> list(chaini(starmap(all_paths, product(roots, leaves))))
[[0, 1, 2], [0, 3, 2]]
The same list computed using an iterative approach::
>>> G = nx.DiGraph([(0, 1), (1, 2), (0, 3), (3, 2)])
>>> roots = (v for v, d in G.in_degree() if d == 0)
>>> leaves = (v for v, d in G.out_degree() if d == 0)
>>> all_paths = []
>>> for root in roots:
... for leaf in leaves:
... paths = nx.all_simple_paths(G, root, leaf)
... all_paths.extend(paths)
>>> all_paths
[[0, 1, 2], [0, 3, 2]]
Iterate over each path from the root nodes to the leaf nodes in a
directed acyclic graph passing all leaves together to avoid unnecessary
compute::
>>> G = nx.DiGraph([(0, 1), (2, 1), (1, 3), (1, 4)])
>>> roots = (v for v, d in G.in_degree() if d == 0)
>>> leaves = [v for v, d in G.out_degree() if d == 0]
>>> all_paths = []
>>> for root in roots:
... paths = nx.all_simple_paths(G, root, leaves)
... all_paths.extend(paths)
>>> all_paths
[[0, 1, 3], [0, 1, 4], [2, 1, 3], [2, 1, 4]]
If parallel edges offer multiple ways to traverse a given sequence of
nodes, this sequence of nodes will be returned multiple times:
>>> G = nx.MultiDiGraph([(0, 1), (0, 1), (1, 2)])
>>> list(nx.all_simple_paths(G, 0, 2))
[[0, 1, 2], [0, 1, 2]]
Notes
-----
This algorithm uses a modified depth-first search to generate the
paths [1]_. A single path can be found in $O(V+E)$ time but the
number of simple paths in a graph can be very large, e.g. $O(n!)$ in
the complete graph of order $n$.
This function does not check that a path exists between `source` and
`target`. For large graphs, this may result in very long runtimes.
Consider using `has_path` to check that a path exists between `source` and
`target` before calling this function on large graphs.
References
----------
.. [1] R. Sedgewick, "Algorithms in C, Part 5: Graph Algorithms",
Addison Wesley Professional, 3rd ed., 2001.
See Also
--------
all_shortest_paths, shortest_path, has_path
"""
if source not in G:
raise nx.NodeNotFound(f"source node {source} not in graph")
if target in G:
targets = {target}
else:
try:
targets = set(target)
except TypeError as err:
raise nx.NodeNotFound(f"target node {target} not in graph") from err
if source in targets:
return _empty_generator()
if cutoff is None:
cutoff = len(G) - 1
if cutoff < 1:
return _empty_generator()
if G.is_multigraph():
return _all_simple_paths_multigraph(G, source, targets, cutoff)
else:
return _all_simple_paths_graph(G, source, targets, cutoff)
def _empty_generator():
yield from ()
def _all_simple_paths_graph(G, source, targets, cutoff):
visited = {source: True}
stack = [iter(G[source])]
while stack:
children = stack[-1]
child = next(children, None)
if child is None:
stack.pop()
visited.popitem()
elif len(visited) < cutoff:
if child in visited:
continue
if child in targets:
yield list(visited) + [child]
visited[child] = True
if targets - set(visited.keys()): # expand stack until find all targets
stack.append(iter(G[child]))
else:
visited.popitem() # maybe other ways to child
else: # len(visited) == cutoff:
for target in (targets & (set(children) | {child})) - set(visited.keys()):
yield list(visited) + [target]
stack.pop()
visited.popitem()
def _all_simple_paths_multigraph(G, source, targets, cutoff):
visited = {source: True}
stack = [(v for u, v in G.edges(source))]
while stack:
children = stack[-1]
child = next(children, None)
if child is None:
stack.pop()
visited.popitem()
elif len(visited) < cutoff:
if child in visited:
continue
if child in targets:
yield list(visited) + [child]
visited[child] = True
if targets - set(visited.keys()):
stack.append((v for u, v in G.edges(child)))
else:
visited.popitem()
else: # len(visited) == cutoff:
for target in targets - set(visited.keys()):
count = ([child] + list(children)).count(target)
for i in range(count):
yield list(visited) + [target]
stack.pop()
visited.popitem()
[docs]
@nx._dispatch
def all_simple_edge_paths(G, source, target, cutoff=None):
"""Generate lists of edges for all simple paths in G from source to target.
A simple path is a path with no repeated nodes.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
target : nodes
Single node or iterable of nodes at which to end path
cutoff : integer, optional
Depth to stop the search. Only paths of length <= cutoff are returned.
Returns
-------
path_generator: generator
A generator that produces lists of simple paths. If there are no paths
between the source and target within the given cutoff the generator
produces no output.
For multigraphs, the list of edges have elements of the form `(u,v,k)`.
Where `k` corresponds to the edge key.
Examples
--------
Print the simple path edges of a Graph::
>>> g = nx.Graph([(1, 2), (2, 4), (1, 3), (3, 4)])
>>> for path in sorted(nx.all_simple_edge_paths(g, 1, 4)):
... print(path)
[(1, 2), (2, 4)]
[(1, 3), (3, 4)]
Print the simple path edges of a MultiGraph. Returned edges come with
their associated keys::
>>> mg = nx.MultiGraph()
>>> mg.add_edge(1, 2, key="k0")
'k0'
>>> mg.add_edge(1, 2, key="k1")
'k1'
>>> mg.add_edge(2, 3, key="k0")
'k0'
>>> for path in sorted(nx.all_simple_edge_paths(mg, 1, 3)):
... print(path)
[(1, 2, 'k0'), (2, 3, 'k0')]
[(1, 2, 'k1'), (2, 3, 'k0')]
Notes
-----
This algorithm uses a modified depth-first search to generate the
paths [1]_. A single path can be found in $O(V+E)$ time but the
number of simple paths in a graph can be very large, e.g. $O(n!)$ in
the complete graph of order $n$.
References
----------
.. [1] R. Sedgewick, "Algorithms in C, Part 5: Graph Algorithms",
Addison Wesley Professional, 3rd ed., 2001.
See Also
--------
all_shortest_paths, shortest_path, all_simple_paths
"""
if source not in G:
raise nx.NodeNotFound("source node %s not in graph" % source)
if target in G:
targets = {target}
else:
try:
targets = set(target)
except TypeError:
raise nx.NodeNotFound("target node %s not in graph" % target)
if source in targets:
return []
if cutoff is None:
cutoff = len(G) - 1
if cutoff < 1:
return []
if G.is_multigraph():
for simp_path in _all_simple_edge_paths_multigraph(G, source, targets, cutoff):
yield simp_path
else:
for simp_path in _all_simple_paths_graph(G, source, targets, cutoff):
yield list(zip(simp_path[:-1], simp_path[1:]))
def _all_simple_edge_paths_multigraph(G, source, targets, cutoff):
if not cutoff or cutoff < 1:
return []
visited = [source]
stack = [iter(G.edges(source, keys=True))]
while stack:
children = stack[-1]
child = next(children, None)
if child is None:
stack.pop()
visited.pop()
elif len(visited) < cutoff:
if child[1] in targets:
yield visited[1:] + [child]
elif child[1] not in [v[0] for v in visited[1:]]:
visited.append(child)
stack.append(iter(G.edges(child[1], keys=True)))
else: # len(visited) == cutoff:
for u, v, k in [child] + list(children):
if v in targets:
yield visited[1:] + [(u, v, k)]
stack.pop()
visited.pop()
[docs]
@not_implemented_for("multigraph")
@nx._dispatch(edge_attrs="weight")
def shortest_simple_paths(G, source, target, weight=None):
"""Generate all simple paths in the graph G from source to target,
starting from shortest ones.
A simple path is a path with no repeated nodes.
If a weighted shortest path search is to be used, no negative weights
are allowed.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
weight : string or function
If it is a string, it is the name of the edge attribute to be
used as a weight.
If it is a function, the weight of an edge is the value returned
by the function. The function must accept exactly three positional
arguments: the two endpoints of an edge and the dictionary of edge
attributes for that edge. The function must return a number.
If None all edges are considered to have unit weight. Default
value None.
Returns
-------
path_generator: generator
A generator that produces lists of simple paths, in order from
shortest to longest.
Raises
------
NetworkXNoPath
If no path exists between source and target.
NetworkXError
If source or target nodes are not in the input graph.
NetworkXNotImplemented
If the input graph is a Multi[Di]Graph.
Examples
--------
>>> G = nx.cycle_graph(7)
>>> paths = list(nx.shortest_simple_paths(G, 0, 3))
>>> print(paths)
[[0, 1, 2, 3], [0, 6, 5, 4, 3]]
You can use this function to efficiently compute the k shortest/best
paths between two nodes.
>>> from itertools import islice
>>> def k_shortest_paths(G, source, target, k, weight=None):
... return list(
... islice(nx.shortest_simple_paths(G, source, target, weight=weight), k)
... )
>>> for path in k_shortest_paths(G, 0, 3, 2):
... print(path)
[0, 1, 2, 3]
[0, 6, 5, 4, 3]
Notes
-----
This procedure is based on algorithm by Jin Y. Yen [1]_. Finding
the first $K$ paths requires $O(KN^3)$ operations.
See Also
--------
all_shortest_paths
shortest_path
all_simple_paths
References
----------
.. [1] Jin Y. Yen, "Finding the K Shortest Loopless Paths in a
Network", Management Science, Vol. 17, No. 11, Theory Series
(Jul., 1971), pp. 712-716.
"""
if source not in G:
raise nx.NodeNotFound(f"source node {source} not in graph")
if target not in G:
raise nx.NodeNotFound(f"target node {target} not in graph")
if weight is None:
length_func = len
shortest_path_func = _bidirectional_shortest_path
else:
wt = _weight_function(G, weight)
def length_func(path):
return sum(
wt(u, v, G.get_edge_data(u, v)) for (u, v) in zip(path, path[1:])
)
shortest_path_func = _bidirectional_dijkstra
listA = []
listB = PathBuffer()
prev_path = None
while True:
if not prev_path:
length, path = shortest_path_func(G, source, target, weight=weight)
listB.push(length, path)
else:
ignore_nodes = set()
ignore_edges = set()
for i in range(1, len(prev_path)):
root = prev_path[:i]
root_length = length_func(root)
for path in listA:
if path[:i] == root:
ignore_edges.add((path[i - 1], path[i]))
try:
length, spur = shortest_path_func(
G,
root[-1],
target,
ignore_nodes=ignore_nodes,
ignore_edges=ignore_edges,
weight=weight,
)
path = root[:-1] + spur
listB.push(root_length + length, path)
except nx.NetworkXNoPath:
pass
ignore_nodes.add(root[-1])
if listB:
path = listB.pop()
yield path
listA.append(path)
prev_path = path
else:
break
class PathBuffer:
def __init__(self):
self.paths = set()
self.sortedpaths = []
self.counter = count()
def __len__(self):
return len(self.sortedpaths)
def push(self, cost, path):
hashable_path = tuple(path)
if hashable_path not in self.paths:
heappush(self.sortedpaths, (cost, next(self.counter), path))
self.paths.add(hashable_path)
def pop(self):
(cost, num, path) = heappop(self.sortedpaths)
hashable_path = tuple(path)
self.paths.remove(hashable_path)
return path
def _bidirectional_shortest_path(
G, source, target, ignore_nodes=None, ignore_edges=None, weight=None
):
"""Returns the shortest path between source and target ignoring
nodes and edges in the containers ignore_nodes and ignore_edges.
This is a custom modification of the standard bidirectional shortest
path implementation at networkx.algorithms.unweighted
Parameters
----------
G : NetworkX graph
source : node
starting node for path
target : node
ending node for path
ignore_nodes : container of nodes
nodes to ignore, optional
ignore_edges : container of edges
edges to ignore, optional
weight : None
This function accepts a weight argument for convenience of
shortest_simple_paths function. It will be ignored.
Returns
-------
path: list
List of nodes in a path from source to target.
Raises
------
NetworkXNoPath
If no path exists between source and target.
See Also
--------
shortest_path
"""
# call helper to do the real work
results = _bidirectional_pred_succ(G, source, target, ignore_nodes, ignore_edges)
pred, succ, w = results
# build path from pred+w+succ
path = []
# from w to target
while w is not None:
path.append(w)
w = succ[w]
# from source to w
w = pred[path[0]]
while w is not None:
path.insert(0, w)
w = pred[w]
return len(path), path
def _bidirectional_pred_succ(G, source, target, ignore_nodes=None, ignore_edges=None):
"""Bidirectional shortest path helper.
Returns (pred,succ,w) where
pred is a dictionary of predecessors from w to the source, and
succ is a dictionary of successors from w to the target.
"""
# does BFS from both source and target and meets in the middle
if ignore_nodes and (source in ignore_nodes or target in ignore_nodes):
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
if target == source:
return ({target: None}, {source: None}, source)
# handle either directed or undirected
if G.is_directed():
Gpred = G.predecessors
Gsucc = G.successors
else:
Gpred = G.neighbors
Gsucc = G.neighbors
# support optional nodes filter
if ignore_nodes:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if w not in ignore_nodes:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
# support optional edges filter
if ignore_edges:
if G.is_directed():
def filter_pred_iter(pred_iter):
def iterate(v):
for w in pred_iter(v):
if (w, v) not in ignore_edges:
yield w
return iterate
def filter_succ_iter(succ_iter):
def iterate(v):
for w in succ_iter(v):
if (v, w) not in ignore_edges:
yield w
return iterate
Gpred = filter_pred_iter(Gpred)
Gsucc = filter_succ_iter(Gsucc)
else:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if (v, w) not in ignore_edges and (w, v) not in ignore_edges:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
# predecessor and successors in search
pred = {source: None}
succ = {target: None}
# initialize fringes, start with forward
forward_fringe = [source]
reverse_fringe = [target]
while forward_fringe and reverse_fringe:
if len(forward_fringe) <= len(reverse_fringe):
this_level = forward_fringe
forward_fringe = []
for v in this_level:
for w in Gsucc(v):
if w not in pred:
forward_fringe.append(w)
pred[w] = v
if w in succ:
# found path
return pred, succ, w
else:
this_level = reverse_fringe
reverse_fringe = []
for v in this_level:
for w in Gpred(v):
if w not in succ:
succ[w] = v
reverse_fringe.append(w)
if w in pred:
# found path
return pred, succ, w
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
def _bidirectional_dijkstra(
G, source, target, weight="weight", ignore_nodes=None, ignore_edges=None
):
"""Dijkstra's algorithm for shortest paths using bidirectional search.
This function returns the shortest path between source and target
ignoring nodes and edges in the containers ignore_nodes and
ignore_edges.
This is a custom modification of the standard Dijkstra bidirectional
shortest path implementation at networkx.algorithms.weighted
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight: string, function, optional (default='weight')
Edge data key or weight function corresponding to the edge weight
ignore_nodes : container of nodes
nodes to ignore, optional
ignore_edges : container of edges
edges to ignore, optional
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.
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 ignore_nodes and (source in ignore_nodes or target in ignore_nodes):
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
if source == target:
if source not in G:
raise nx.NodeNotFound(f"Node {source} not in graph")
return (0, [source])
# handle either directed or undirected
if G.is_directed():
Gpred = G.predecessors
Gsucc = G.successors
else:
Gpred = G.neighbors
Gsucc = G.neighbors
# support optional nodes filter
if ignore_nodes:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if w not in ignore_nodes:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
# support optional edges filter
if ignore_edges:
if G.is_directed():
def filter_pred_iter(pred_iter):
def iterate(v):
for w in pred_iter(v):
if (w, v) not in ignore_edges:
yield w
return iterate
def filter_succ_iter(succ_iter):
def iterate(v):
for w in succ_iter(v):
if (v, w) not in ignore_edges:
yield w
return iterate
Gpred = filter_pred_iter(Gpred)
Gsucc = filter_succ_iter(Gsucc)
else:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if (v, w) not in ignore_edges and (w, v) not in ignore_edges:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
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
neighs = [Gsucc, Gpred]
# 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)
wt = _weight_function(G, weight)
for w in neighs[dir](v):
if dir == 0: # forward
minweight = wt(v, w, G.get_edge_data(v, w))
vwLength = dists[dir][v] + minweight
else: # back, must remember to change v,w->w,v
minweight = wt(w, v, G.get_edge_data(w, v))
vwLength = dists[dir][v] + minweight
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 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(f"No path between {source} and {target}.")
```