Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

# Source code for networkx.algorithms.shortest_paths.weighted

"""
Shortest path algorithms for weighed graphs.
"""

from collections import deque
from heapq import heappush, heappop
from itertools import count
import networkx as nx
from networkx.utils import generate_unique_node
from networkx.algorithms.shortest_paths.generic import _build_paths_from_predecessors

__all__ = [
"dijkstra_path",
"dijkstra_path_length",
"bidirectional_dijkstra",
"single_source_dijkstra",
"single_source_dijkstra_path",
"single_source_dijkstra_path_length",
"multi_source_dijkstra",
"multi_source_dijkstra_path",
"multi_source_dijkstra_path_length",
"all_pairs_dijkstra",
"all_pairs_dijkstra_path",
"all_pairs_dijkstra_path_length",
"dijkstra_predecessor_and_distance",
"bellman_ford_path",
"bellman_ford_path_length",
"single_source_bellman_ford",
"single_source_bellman_ford_path",
"single_source_bellman_ford_path_length",
"all_pairs_bellman_ford_path",
"all_pairs_bellman_ford_path_length",
"bellman_ford_predecessor_and_distance",
"negative_edge_cycle",
"johnson",
]

def _weight_function(G, weight):
"""Returns a function that returns the weight of an edge.

The returned function is specifically suitable for input to
functions :func:_dijkstra and :func:_bellman_ford_relaxation.

Parameters
----------
G : NetworkX graph.

weight : string or function
If it is callable, weight itself is returned. If it is a string,
it is assumed to be the name of the edge attribute that represents
the weight of an edge. In that case, a function is returned that
gets the edge weight according to the specified edge attribute.

Returns
-------
function
This function returns a callable that accepts exactly three inputs:
a node, an node adjacent to the first one, and the edge attribute
dictionary for the eedge joining those nodes. That function returns
a number representing the weight of an edge.

If G is a multigraph, and weight is not callable, the
minimum edge weight over all parallel edges is returned. If any edge
does not have an attribute with key weight, it is assumed to
have weight one.

"""
if callable(weight):
return weight
# If the weight keyword argument is not callable, we assume it is a
# string representing the edge attribute containing the weight of
# the edge.
if G.is_multigraph():
return lambda u, v, d: min(attr.get(weight, 1) for attr in d.values())
return lambda u, v, data: data.get(weight, 1)

[docs]def dijkstra_path(G, source, target, weight="weight"):
"""Returns the shortest weighted path from source to target in G.

Uses Dijkstra's Method to compute the shortest weighted path
between two nodes in a graph.

Parameters
----------
G : NetworkX graph

source : node
Starting node

target : node
Ending node

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
path : list
List of nodes in a shortest path.

Raises
------
NodeNotFound
If source is not in G.

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.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

The weight function can be used to include node weights.

>>> def func(u, v, d):
...     node_u_wt = G.nodes[u].get("node_weight", 1)
...     node_v_wt = G.nodes[v].get("node_weight", 1)
...     edge_wt = d.get("weight", 1)
...     return node_u_wt / 2 + node_v_wt / 2 + edge_wt

In this example we take the average of start and end node
weights of an edge and add it to the weight of the edge.

The function :func:single_source_dijkstra computes both
path and length-of-path if you need both, use that.

--------
bidirectional_dijkstra(), bellman_ford_path()
single_source_dijkstra()
"""
(length, path) = single_source_dijkstra(G, source, target=target, weight=weight)
return path

[docs]def dijkstra_path_length(G, source, target, weight="weight"):
"""Returns the shortest weighted path length in G from source to target.

Uses Dijkstra's Method to compute the shortest weighted path length
between two nodes in a graph.

Parameters
----------
G : NetworkX graph

source : node label
starting node for path

target : node label
ending node for path

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
length : number
Shortest path length.

Raises
------
NodeNotFound
If source is not in G.

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.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

The function :func:single_source_dijkstra computes both
path and length-of-path if you need both, use that.

--------
bidirectional_dijkstra(), bellman_ford_path_length()
single_source_dijkstra()

"""
if source == target:
return 0
weight = _weight_function(G, weight)
length = _dijkstra(G, source, weight, target=target)
try:
return length[target]
except KeyError as e:
raise nx.NetworkXNoPath(f"Node {target} not reachable from {source}") from e

[docs]def single_source_dijkstra_path(G, source, cutoff=None, weight="weight"):
"""Find shortest weighted paths in G from a source node.

Compute shortest path between source and all other reachable
nodes for a weighted graph.

Parameters
----------
G : NetworkX graph

source : node
Starting node for path.

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
paths : dictionary
Dictionary of shortest path lengths keyed by target.

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> path = nx.single_source_dijkstra_path(G, 0)
>>> path
[0, 1, 2, 3, 4]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

--------
single_source_dijkstra(), single_source_bellman_ford()

"""
return multi_source_dijkstra_path(G, {source}, cutoff=cutoff, weight=weight)

[docs]def single_source_dijkstra_path_length(G, source, cutoff=None, weight="weight"):
"""Find shortest weighted path lengths in G from a source node.

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

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
length : dict
Dict keyed by node to shortest path length from source.

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> length = nx.single_source_dijkstra_path_length(G, 0)
>>> length
4
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
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.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

--------
single_source_dijkstra(), single_source_bellman_ford_path_length()

"""
return multi_source_dijkstra_path_length(G, {source}, cutoff=cutoff, weight=weight)

[docs]def single_source_dijkstra(G, source, target=None, cutoff=None, weight="weight"):
"""Find shortest weighted paths and lengths from a source node.

Compute the shortest path length between source and all other
reachable nodes for a weighted graph.

Uses Dijkstra's algorithm to compute shortest paths and lengths
between a source and all other reachable nodes in a weighted graph.

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 return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
distance, path : pair of dictionaries, or numeric and list.
If target is None, paths and lengths to all nodes are computed.
The return value is a tuple of two dictionaries keyed by target nodes.
The first dictionary stores distance to each target node.
The second stores the path to each target node.
If target is not None, returns a tuple (distance, path), where
distance is the distance from source to target and path is a list
representing the path from source to target.

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> length, path = nx.single_source_dijkstra(G, 0)
>>> print(length)
4
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 3
4: 4
>>> path
[0, 1, 2, 3, 4]
>>> length, path = nx.single_source_dijkstra(G, 0, 1)
>>> length
1
>>> path
[0, 1]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

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).

--------
single_source_dijkstra_path()
single_source_dijkstra_path_length()
single_source_bellman_ford()
"""
return multi_source_dijkstra(
G, {source}, cutoff=cutoff, target=target, weight=weight
)

[docs]def multi_source_dijkstra_path(G, sources, cutoff=None, weight="weight"):
"""Find shortest weighted paths in G from a given set of source
nodes.

Compute shortest path between any of the source nodes and all other
reachable nodes for a weighted graph.

Parameters
----------
G : NetworkX graph

sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
paths : dictionary
Dictionary of shortest paths keyed by target.

Examples
--------
>>> G = nx.path_graph(5)
>>> path = nx.multi_source_dijkstra_path(G, {0, 4})
>>> path
[0, 1]
>>> path
[4, 3]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

Raises
------
ValueError
If sources is empty.
NodeNotFound
If any of sources is not in G.

--------
multi_source_dijkstra(), multi_source_bellman_ford()

"""
length, path = multi_source_dijkstra(G, sources, cutoff=cutoff, weight=weight)
return path

[docs]def multi_source_dijkstra_path_length(G, sources, cutoff=None, weight="weight"):
"""Find shortest weighted path lengths in G from a given set of
source nodes.

Compute the shortest path length between any of the source nodes and
all other reachable nodes for a weighted graph.

Parameters
----------
G : NetworkX graph

sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
length : dict
Dict keyed by node to shortest path length to nearest source.

Examples
--------
>>> G = nx.path_graph(5)
>>> length = nx.multi_source_dijkstra_path_length(G, {0, 4})
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 1
4: 0

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

Raises
------
ValueError
If sources is empty.
NodeNotFound
If any of sources is not in G.

--------
multi_source_dijkstra()

"""
if not sources:
raise ValueError("sources must not be empty")
weight = _weight_function(G, weight)
return _dijkstra_multisource(G, sources, weight, cutoff=cutoff)

[docs]def multi_source_dijkstra(G, sources, target=None, cutoff=None, weight="weight"):
"""Find shortest weighted paths and lengths from a given set of
source nodes.

Uses Dijkstra's algorithm to compute the shortest paths and lengths
between one of the source nodes and the given target, or all other
reachable nodes if not specified, for a weighted graph.

Parameters
----------
G : NetworkX graph

sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.

target : node label, optional
Ending node for path

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
distance, path : pair of dictionaries, or numeric and list
If target is None, returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from one of the source nodes.
The second stores the path from one of the sources to that node.
If target is not None, returns a tuple of (distance, path) where
distance is the distance from source to target and path is a list
representing the path from source to target.

Examples
--------
>>> G = nx.path_graph(5)
>>> length, path = nx.multi_source_dijkstra(G, {0, 4})
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 1
4: 0
>>> path
[0, 1]
>>> path
[4, 3]

>>> length, path = nx.multi_source_dijkstra(G, {0, 4}, 1)
>>> length
1
>>> path
[0, 1]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The weight function can be used to hide edges by returning None.
So weight = lambda u, v, d: 1 if d['color']=="red" else None
will find the shortest red path.

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).

Raises
------
ValueError
If sources is empty.
NodeNotFound
If any of sources is not in G.

--------
multi_source_dijkstra_path()
multi_source_dijkstra_path_length()

"""
if not sources:
raise ValueError("sources must not be empty")
if target in sources:
return (0, [target])
weight = _weight_function(G, weight)
paths = {source: [source] for source in sources}  # dictionary of paths
dist = _dijkstra_multisource(
G, sources, weight, paths=paths, cutoff=cutoff, target=target
)
if target is None:
return (dist, paths)
try:
return (dist[target], paths[target])
except KeyError as e:
raise nx.NetworkXNoPath(f"No path to {target}.") from e

def _dijkstra(G, source, weight, pred=None, paths=None, cutoff=None, target=None):
"""Uses Dijkstra's algorithm to find shortest weighted paths from a
single source.

This is a convenience function for :func:_dijkstra_multisource
with all the arguments the same, except the keyword argument
sources set to [source].

"""
return _dijkstra_multisource(
G, [source], weight, pred=pred, paths=paths, cutoff=cutoff, target=target
)

def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths

Parameters
----------
G : NetworkX graph

sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.

weight: function
Function with (u, v, data) input that returns that edges weight

pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.

paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.

target : node label, optional
Ending node for path. Search is halted when target is found.

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.

Raises
------
NodeNotFound
If any of sources is not in G.

Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.

"""
G_succ = G._succ if G.is_directed() else G._adj

push = heappush
pop = heappop
dist = {}  # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
if source not in G:
raise nx.NodeNotFound(f"Source {source} not in G")
seen[source] = 0
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 u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)

# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist

[docs]def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight="weight"):
"""Compute weighted shortest path length and predecessors.

Uses Dijkstra's Method to obtain the shortest weighted paths
and return dictionaries of predecessors for each node and
distance for each node from the source.

Parameters
----------
G : NetworkX graph

source : node label
Starting node for path

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
pred, distance : dictionaries
Returns two dictionaries representing a list of predecessors
of a node and the distance to each node.
Warning: If target is specified, the dicts are incomplete as they
only contain information for the nodes along a path to target.

Raises
------
NodeNotFound
If source is not in G.

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.

Examples
--------
>>> G = nx.path_graph(5, create_using=nx.DiGraph())
>>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0)
>>> sorted(pred.items())
[(0, []), (1, ), (2, ), (3, ), (4, )]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

>>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0, 1)
>>> sorted(pred.items())
[(0, []), (1, )]
>>> sorted(dist.items())
[(0, 0), (1, 1)]
"""

weight = _weight_function(G, weight)
pred = {source: []}  # dictionary of predecessors
return (pred, _dijkstra(G, source, weight, pred=pred, cutoff=cutoff))

[docs]def all_pairs_dijkstra(G, cutoff=None, weight="weight"):
"""Find shortest weighted paths and lengths between all nodes.

Parameters
----------
G : NetworkX graph

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edge[u][v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Yields
------
(node, (distance, path)) : (node obj, (dict, dict))
Each source node has two associated dicts. The first holds distance
keyed by target and the second holds paths keyed by target.
(See single_source_dijkstra for the source/target node terminology.)
If desired you can apply dict() to this function to create a dict
keyed by source node to the two dicts.

Examples
--------
>>> G = nx.path_graph(5)
>>> len_path = dict(nx.all_pairs_dijkstra(G))
>>> print(len_path)
2
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"3 - {node}: {len_path[node]}")
3 - 0: 3
3 - 1: 2
3 - 2: 1
3 - 3: 0
3 - 4: 1
>>> len_path
[3, 2, 1]
>>> for n, (dist, path) in nx.all_pairs_dijkstra(G):
...     print(path)
[0, 1]

[2, 1]
[3, 2, 1]
[4, 3, 2, 1]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

The yielded dicts only have keys for reachable nodes.
"""
for n in G:
dist, path = single_source_dijkstra(G, n, cutoff=cutoff, weight=weight)
yield (n, (dist, path))

[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

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
distance : iterator
(source, dictionary) iterator with dictionary keyed by target and
shortest path length as the key value.

Examples
--------
>>> G = nx.path_graph(5)
>>> length = dict(nx.all_pairs_dijkstra_path_length(G))
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"1 - {node}: {length[node]}")
1 - 0: 1
1 - 1: 0
1 - 2: 1
1 - 3: 2
1 - 4: 3
>>> length
1
>>> length
0

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.
"""
length = single_source_dijkstra_path_length
for n in G:
yield (n, length(G, n, cutoff=cutoff, weight=weight))

[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

cutoff : integer or float, optional
Depth to stop the search. Only return paths with length <= cutoff.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.

Examples
--------
>>> G = nx.path_graph(5)
>>> path = dict(nx.all_pairs_dijkstra_path(G))
>>> print(path)
[0, 1, 2, 3, 4]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

--------
floyd_warshall(), all_pairs_bellman_ford_path()

"""
path = single_source_dijkstra_path
# TODO This can be trivially parallelized.
for n in G:
yield (n, path(G, n, cutoff=cutoff, weight=weight))

[docs]def bellman_ford_predecessor_and_distance(
G, source, target=None, weight="weight", heuristic=False
):
"""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 or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

heuristic : bool
Determines whether to use a heuristic to early detect negative
cycles at a hopefully negligible cost.

Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.

Raises
------
NodeNotFound
If source is not in G.

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
--------
>>> G = nx.path_graph(5, create_using=nx.DiGraph())
>>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0)
>>> sorted(pred.items())
[(0, []), (1, ), (2, ), (3, ), (4, )]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

>>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0, 1)
>>> sorted(pred.items())
[(0, []), (1, ), (2, ), (3, ), (4, )]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

>>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
>>> G["weight"] = -7
>>> nx.bellman_ford_predecessor_and_distance(G, 0)
Traceback (most recent call last):
...
networkx.exception.NetworkXUnbounded: Negative cost cycle detected.

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.

In NetworkX v2.1 and prior, the source node had predecessor [None].
In NetworkX v2.2 this changed to the source node having predecessor []
"""
if source not in G:
weight = _weight_function(G, weight)
if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)):
raise nx.NetworkXUnbounded("Negative cost cycle detected.")

dist = {source: 0}
pred = {source: []}

if len(G) == 1:
return pred, dist

weight = _weight_function(G, weight)

dist = _bellman_ford(
G, [source], weight, pred=pred, dist=dist, target=target, heuristic=heuristic
)
return (pred, dist)

def _bellman_ford(
G, source, weight, pred=None, paths=None, dist=None, target=None, heuristic=True
):
"""Relaxation loop for Bellman–Ford algorithm.

This is an implementation of the SPFA variant.
See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm

Parameters
----------
G : NetworkX graph

source: list
List of source nodes. The shortest path from any of the source
nodes will be found if multiple sources are provided.

weight : 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.

pred: dict of lists, optional (default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored

paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node
If None, paths are not stored

dist: dict, optional (default=None)
dict to store distance from source to the keyed node
If None, returned dist dict contents default to 0 for every node in the
source list

target: node label, optional
Ending node for path. Path lengths to other destinations may (and
probably will) be incorrect.

heuristic : bool
Determines whether to use a heuristic to early detect negative
cycles at a hopefully negligible cost.

Returns
-------
Returns a dict keyed by node to the distance from the source.
Dicts for paths and pred are in the mutated input dicts by those names.

Raises
------
NodeNotFound
If any of source is not in G.

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
"""
for s in source:
if s not in G:
raise nx.NodeNotFound(f"Source {s} not in G")

if pred is None:
pred = {v: [] for v in source}

if dist is None:
dist = {v: 0 for v in source}

# Heuristic Storage setup. Note: use None because nodes cannot be None
nonexistent_edge = (None, None)
pred_edge = {v: None for v in source}
recent_update = {v: nonexistent_edge for v in source}

G_succ = G.succ if G.is_directed() else 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 any of the predecessors of u is in the queue.
if all(pred_u not in in_q for pred_u in pred[u]):
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + weight(u, v, e)

if dist_v < dist.get(v, inf):
# In this conditional branch we are updating the path with v.
# If it happens that some earlier update also added node v
# that implies the existence of a negative cycle since
# after the update node v would lie on the update path twice.
# The update path is stored up to one of the source nodes,
# therefore u is always in the dict recent_update
if heuristic:
if v in recent_update[u]:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
# Transfer the recent update info from u to v if the
# same source node is the head of the update path.
# If the source node is responsible for the cost update,
# then clear the history and use it instead.
if v in pred_edge and pred_edge[v] == u:
recent_update[v] = recent_update[u]
else:
recent_update[v] = (u, v)

if v not in in_q:
q.append(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]
pred_edge[v] = u

elif dist.get(v) is not None and dist_v == dist.get(v):
pred[v].append(u)

if paths is not None:
sources = set(source)
dsts = [target] if target is not None else pred
for dst in dsts:
gen = _build_paths_from_predecessors(sources, dst, pred)
paths[dst] = next(gen)

return dist

[docs]def bellman_ford_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
------
NodeNotFound
If source is not in G.

NetworkXNoPath
If no path exists between source and target.

Examples
--------
>>> G = nx.path_graph(5)
>>> print(nx.bellman_ford_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.

--------
dijkstra_path(), bellman_ford_path_length()
"""
length, path = single_source_bellman_ford(G, source, target=target, weight=weight)
return path

[docs]def bellman_ford_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
------
NodeNotFound
If source is not in G.

NetworkXNoPath
If no path exists between source and target.

Examples
--------
>>> G = nx.path_graph(5)
>>> print(nx.bellman_ford_path_length(G, 0, 4))
4

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

--------
dijkstra_path_length(), bellman_ford_path()
"""
if source == target:
return 0

weight = _weight_function(G, weight)

length = _bellman_ford(G, [source], weight, target=target)

try:
return length[target]
except KeyError as e:
raise nx.NetworkXNoPath(f"node {target} not reachable from {source}") from e

[docs]def single_source_bellman_ford_path(G, source, 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

Returns
-------
paths : dictionary
Dictionary of shortest path lengths keyed by target.

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> path = nx.single_source_bellman_ford_path(G, 0)
>>> path
[0, 1, 2, 3, 4]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

--------
single_source_dijkstra(), single_source_bellman_ford()

"""
(length, path) = single_source_bellman_ford(G, source, weight=weight)
return path

[docs]def single_source_bellman_ford_path_length(G, source, 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.

Returns
-------
length : iterator
(target, shortest path length) iterator

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> length = dict(nx.single_source_bellman_ford_path_length(G, 0))
>>> length
4
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
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.

--------
single_source_dijkstra(), single_source_bellman_ford()

"""
weight = _weight_function(G, weight)
return _bellman_ford(G, [source], weight)

[docs]def single_source_bellman_ford(G, source, target=None, weight="weight"):
"""Compute shortest paths and lengths in a weighted graph G.

Uses Bellman-Ford algorithm for shortest paths.

Parameters
----------
G : NetworkX graph

source : node label
Starting node for path

target : node label, optional
Ending node for path

Returns
-------
distance, path : pair of dictionaries, or numeric and list
If target is None, returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from one of the source nodes.
The second stores the path from one of the sources to that node.
If target is not None, returns a tuple of (distance, path) where
distance is the distance from source to target and path is a list
representing the path from source to target.

Raises
------
NodeNotFound
If source is not in G.

Examples
--------
>>> G = nx.path_graph(5)
>>> length, path = nx.single_source_bellman_ford(G, 0)
>>> print(length)
4
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 3
4: 4
>>> path
[0, 1, 2, 3, 4]
>>> length, path = nx.single_source_bellman_ford(G, 0, 1)
>>> length
1
>>> path
[0, 1]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

--------
single_source_dijkstra()
single_source_bellman_ford_path()
single_source_bellman_ford_path_length()
"""
if source == target:
return (0, [source])

weight = _weight_function(G, weight)

paths = {source: [source]}  # dictionary of paths
dist = _bellman_ford(G, [source], weight, paths=paths, target=target)
if target is None:
return (dist, paths)
try:
return (dist[target], paths[target])
except KeyError as e:
msg = f"Node {target} not reachable from {source}"
raise nx.NetworkXNoPath(msg) from e

[docs]def all_pairs_bellman_ford_path_length(G, 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

Returns
-------
distance : iterator
(source, dictionary) iterator with dictionary keyed by target and
shortest path length as the key value.

Examples
--------
>>> G = nx.path_graph(5)
>>> length = dict(nx.all_pairs_bellman_ford_path_length(G))
>>> for node in [0, 1, 2, 3, 4]:
...     print(f"1 - {node}: {length[node]}")
1 - 0: 1
1 - 1: 0
1 - 2: 1
1 - 3: 2
1 - 4: 3
>>> length
1
>>> length
0

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.
"""
length = single_source_bellman_ford_path_length
for n in G:
yield (n, dict(length(G, n, weight=weight)))

[docs]def all_pairs_bellman_ford_path(G, 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

Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.

Examples
--------
>>> G = nx.path_graph(5)
>>> path = dict(nx.all_pairs_bellman_ford_path(G))
>>> print(path)
[0, 1, 2, 3, 4]

Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.

--------
floyd_warshall(), all_pairs_dijkstra_path()

"""
path = single_source_bellman_ford_path
# TODO This can be trivially parallelized.
for n in G:
yield (n, path(G, n, 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 or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.

Raises
------
NodeNotFound
If source is not in G.

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
--------
>>> 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)]

>>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
>>> G["weight"] = -7
Traceback (most recent call last):
...
networkx.exception.NetworkXUnbounded: Negative cost cycle detected.

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:
weight = _weight_function(G, weight)
if any(weight(u, v, d) < 0 for u, v, d in nx.selfloop_edges(G, data=True)):
raise nx.NetworkXUnbounded("Negative cost cycle detected.")

if len(G) == 1:
return {source: None}, {source: 0}

if G.is_directed():
G_succ = G.succ
else:

inf = float("inf")
d = {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
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 + weight(u, v, 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 = {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] + weight(u, v, 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())))
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()

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 = weight(u, v, e)
if d_u + w_e < d[v]:
d[v] = d_u + w_e
pred[v] = u
return relabeled

# Set of nodes relabled in the last round of scan operations. Denoted by B
# in Goldberg and Radzik's paper.
relabeled = {source}

while relabeled:
to_scan = topo_sort(relabeled)
relabeled = relax(to_scan)

d = {u: d[u] for u in pred}
return pred, d

[docs]def negative_edge_cycle(G, weight="weight", heuristic=True):
"""Returns True if there exists a negative edge cycle anywhere in G.

Parameters
----------
G : NetworkX graph

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

heuristic : bool
Determines whether to use a heuristic to early detect negative
cycles at a negligible cost. In case of graphs with a negative cycle,
the performance of detection increases by at least an order of magnitude.

Returns
-------
negative_cycle : bool
True if a negative edge cycle exists, otherwise False.

Examples
--------
>>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
>>> print(nx.negative_edge_cycle(G))
False
>>> G["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_predecessor_and_distance() but finds
negative cycles on any component by first adding a new node connected to
every node, and starting bellman_ford_predecessor_and_distance 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_predecessor_and_distance(G, newnode, weight, heuristic=heuristic)
except nx.NetworkXUnbounded:
return True
finally:
G.remove_node(newnode)
return False

[docs]def bidirectional_dijkstra(G, source, target, weight="weight"):
r"""Dijkstra's algorithm for shortest paths using bidirectional search.

Parameters
----------
G : NetworkX graph

source : node
Starting node.

target : node
Ending node.

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
length, path : number and list
length is the distance from source to target.
path is a list of nodes on a path from source to target.

Raises
------
NodeNotFound
If either source or target is not in G.

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).

--------
shortest_path
shortest_path_length
"""
if source not in G or target not in G:
msg = f"Either source {source} or target {target} is not in G"
raise nx.NodeNotFound(msg)

if source == target:
return (0, [source])

weight = _weight_function(G, weight)
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) for choosing node to expand
seen = [{source: 0}, {target: 0}]  # dict of distances to seen nodes
c = count()
# initialize fringe heap
push(fringe, (0, next(c), source))
push(fringe, (0, next(c), target))
# neighs for extracting correct neighbor information
if G.is_directed():
neighs = [G._succ, G._pred]
else:
# variables to hold shortest discovered path
# finaldist = 1e30000
finalpath = []
dir = 1
while fringe and fringe:
# 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, d in neighs[dir][v].items():
if dir == 0:  # forward
vwLength = dists[dir][v] + weight(v, w, d)
else:  # back, must remember to change v,w->w,v
vwLength = dists[dir][v] + weight(w, v, d)
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 and w in seen:
# see if this path is better than than the already
# discovered shortest path
totaldist = seen[w] + seen[w]
if finalpath == [] or finaldist > totaldist:
finaldist = totaldist
revpath = paths[w][:]
revpath.reverse()
finalpath = paths[w] + revpath[1:]
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")

[docs]def johnson(G, weight="weight"):
r"""Uses Johnson's Algorithm to compute shortest paths.

Johnson's Algorithm finds a shortest path between each pair of
nodes in a weighted graph even if negative weights are present.

Parameters
----------
G : NetworkX graph

weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining u to v will be G.edges[u, v][weight]). If no
such edge attribute exists, the weight of the edge is assumed to
be one.

If this 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.

Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.

Raises
------
NetworkXError
If given graph is not weighted.

Examples
--------
>>> graph = nx.DiGraph()
...     [("0", "3", 3), ("0", "1", -5), ("0", "2", 2), ("1", "2", 4), ("2", "3", 1)]
... )
>>> paths = nx.johnson(graph, weight="weight")
>>> paths["0"]["2"]
['0', '1', '2']

Notes
-----
Johnson's algorithm is suitable even for graphs with negative weights. It
works by using the Bellman–Ford algorithm to compute a transformation of
the input graph that removes all negative weights, allowing Dijkstra's
algorithm to be used on the transformed graph.

The time complexity of this algorithm is $O(n^2 \log n + n m)$,
where $n$ is the number of nodes and $m$ the number of edges in the
graph. For dense graphs, this may be faster than the Floyd–Warshall
algorithm.

--------
floyd_warshall_predecessor_and_distance
floyd_warshall_numpy
all_pairs_shortest_path
all_pairs_shortest_path_length
all_pairs_dijkstra_path
bellman_ford_predecessor_and_distance
all_pairs_bellman_ford_path
all_pairs_bellman_ford_path_length

"""
if not nx.is_weighted(G, weight=weight):
raise nx.NetworkXError("Graph is not weighted.")

dist = {v: 0 for v in G}
pred = {v: [] for v in G}
weight = _weight_function(G, weight)

# Calculate distance of shortest paths
dist_bellman = _bellman_ford(G, list(G), weight, pred=pred, dist=dist)

# Update the weight function to take into account the Bellman--Ford
# relaxation distances.
def new_weight(u, v, d):
return weight(u, v, d) + dist_bellman[u] - dist_bellman[v]

def dist_path(v):
paths = {v: [v]}
_dijkstra(G, v, new_weight, paths=paths)
return paths

return {v: dist_path(v) for v in G}