"""
Compute the shortest paths and path lengths between nodes in the graph.
These algorithms work with undirected and directed graphs.
"""
import networkx as nx
__all__ = [
"shortest_path",
"all_shortest_paths",
"shortest_path_length",
"average_shortest_path_length",
"has_path",
]
[docs]def has_path(G, source, target):
"""Returns *True* if *G* has a path from *source* to *target*.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path
target : node
Ending node for path
"""
try:
nx.shortest_path(G, source, target)
except nx.NetworkXNoPath:
return False
return True
[docs]def shortest_path(G, source=None, target=None, weight=None, method="dijkstra"):
"""Compute shortest paths in the graph.
Parameters
----------
G : NetworkX graph
source : node, optional
Starting node for path. If not specified, compute shortest
paths for each possible starting node.
target : node, optional
Ending node for path. If not specified, compute shortest
paths to all possible nodes.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
method : string, optional (default = 'dijkstra')
The algorithm to use to compute the path.
Supported options: 'dijkstra', 'bellman-ford'.
Other inputs produce a ValueError.
If `weight` is None, unweighted graph methods are used, and this
suggestion is ignored.
Returns
-------
path: list or dictionary
All returned paths include both the source and target in the path.
If the source and target are both specified, return a single list
of nodes in a shortest path from the source to the target.
If only the source is specified, return a dictionary keyed by
targets with a list of nodes in a shortest path from the source
to one of the targets.
If only the target is specified, return a dictionary keyed by
sources with a list of nodes in a shortest path from one of the
sources to the target.
If neither the source nor target are specified return a dictionary
of dictionaries with path[source][target]=[list of nodes in path].
Raises
------
NodeNotFound
If `source` is not in `G`.
ValueError
If `method` is not among the supported options.
Examples
--------
>>> G = nx.path_graph(5)
>>> print(nx.shortest_path(G, source=0, target=4))
[0, 1, 2, 3, 4]
>>> p = nx.shortest_path(G, source=0) # target not specified
>>> p[4]
[0, 1, 2, 3, 4]
>>> p = nx.shortest_path(G, target=4) # source not specified
>>> p[0]
[0, 1, 2, 3, 4]
>>> p = nx.shortest_path(G) # source, target not specified
>>> p[0][4]
[0, 1, 2, 3, 4]
Notes
-----
There may be more than one shortest path between a source and target.
This returns only one of them.
See Also
--------
all_pairs_shortest_path()
all_pairs_dijkstra_path()
all_pairs_bellman_ford_path()
single_source_shortest_path()
single_source_dijkstra_path()
single_source_bellman_ford_path()
"""
if method not in ("dijkstra", "bellman-ford"):
# so we don't need to check in each branch later
raise ValueError(f"method not supported: {method}")
method = "unweighted" if weight is None else method
if source is None:
if target is None:
# Find paths between all pairs.
if method == "unweighted":
paths = dict(nx.all_pairs_shortest_path(G))
elif method == "dijkstra":
paths = dict(nx.all_pairs_dijkstra_path(G, weight=weight))
else: # method == 'bellman-ford':
paths = dict(nx.all_pairs_bellman_ford_path(G, weight=weight))
else:
# Find paths from all nodes co-accessible to the target.
if G.is_directed():
G = G.reverse(copy=False)
if method == "unweighted":
paths = nx.single_source_shortest_path(G, target)
elif method == "dijkstra":
paths = nx.single_source_dijkstra_path(G, target, weight=weight)
else: # method == 'bellman-ford':
paths = nx.single_source_bellman_ford_path(G, target, weight=weight)
# Now flip the paths so they go from a source to the target.
for target in paths:
paths[target] = list(reversed(paths[target]))
else:
if target is None:
# Find paths to all nodes accessible from the source.
if method == "unweighted":
paths = nx.single_source_shortest_path(G, source)
elif method == "dijkstra":
paths = nx.single_source_dijkstra_path(G, source, weight=weight)
else: # method == 'bellman-ford':
paths = nx.single_source_bellman_ford_path(G, source, weight=weight)
else:
# Find shortest source-target path.
if method == "unweighted":
paths = nx.bidirectional_shortest_path(G, source, target)
elif method == "dijkstra":
paths = nx.dijkstra_path(G, source, target, weight)
else: # method == 'bellman-ford':
paths = nx.bellman_ford_path(G, source, target, weight)
return paths
[docs]def shortest_path_length(G, source=None, target=None, weight=None, method="dijkstra"):
"""Compute shortest path lengths in the graph.
Parameters
----------
G : NetworkX graph
source : node, optional
Starting node for path.
If not specified, compute shortest path lengths using all nodes as
source nodes.
target : node, optional
Ending node for path.
If not specified, compute shortest path lengths using all nodes as
target nodes.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
method : string, optional (default = 'dijkstra')
The algorithm to use to compute the path length.
Supported options: 'dijkstra', 'bellman-ford'.
Other inputs produce a ValueError.
If `weight` is None, unweighted graph methods are used, and this
suggestion is ignored.
Returns
-------
length: int or iterator
If the source and target are both specified, return the length of
the shortest path from the source to the target.
If only the source is specified, return a dict keyed by target
to the shortest path length from the source to that target.
If only the target is specified, return a dict keyed by source
to the shortest path length from that source to the target.
If neither the source nor target are specified, return an iterator
over (source, dictionary) where dictionary is keyed by target to
shortest path length from source to that target.
Raises
------
NodeNotFound
If `source` is not in `G`.
NetworkXNoPath
If no path exists between source and target.
ValueError
If `method` is not among the supported options.
Examples
--------
>>> G = nx.path_graph(5)
>>> nx.shortest_path_length(G, source=0, target=4)
4
>>> p = nx.shortest_path_length(G, source=0) # target not specified
>>> p[4]
4
>>> p = nx.shortest_path_length(G, target=4) # source not specified
>>> p[0]
4
>>> p = dict(nx.shortest_path_length(G)) # source,target not specified
>>> p[0][4]
4
Notes
-----
The length of the path is always 1 less than the number of nodes involved
in the path since the length measures the number of edges followed.
For digraphs this returns the shortest directed path length. To find path
lengths in the reverse direction use G.reverse(copy=False) first to flip
the edge orientation.
See Also
--------
all_pairs_shortest_path_length()
all_pairs_dijkstra_path_length()
all_pairs_bellman_ford_path_length()
single_source_shortest_path_length()
single_source_dijkstra_path_length()
single_source_bellman_ford_path_length()
"""
if method not in ("dijkstra", "bellman-ford"):
# so we don't need to check in each branch later
raise ValueError(f"method not supported: {method}")
method = "unweighted" if weight is None else method
if source is None:
if target is None:
# Find paths between all pairs.
if method == "unweighted":
paths = nx.all_pairs_shortest_path_length(G)
elif method == "dijkstra":
paths = nx.all_pairs_dijkstra_path_length(G, weight=weight)
else: # method == 'bellman-ford':
paths = nx.all_pairs_bellman_ford_path_length(G, weight=weight)
else:
# Find paths from all nodes co-accessible to the target.
if G.is_directed():
G = G.reverse(copy=False)
if method == "unweighted":
path_length = nx.single_source_shortest_path_length
paths = path_length(G, target)
elif method == "dijkstra":
path_length = nx.single_source_dijkstra_path_length
paths = path_length(G, target, weight=weight)
else: # method == 'bellman-ford':
path_length = nx.single_source_bellman_ford_path_length
paths = path_length(G, target, weight=weight)
else:
if target is None:
# Find paths to all nodes accessible from the source.
if method == "unweighted":
paths = nx.single_source_shortest_path_length(G, source)
elif method == "dijkstra":
path_length = nx.single_source_dijkstra_path_length
paths = path_length(G, source, weight=weight)
else: # method == 'bellman-ford':
path_length = nx.single_source_bellman_ford_path_length
paths = path_length(G, source, weight=weight)
else:
# Find shortest source-target path.
if method == "unweighted":
p = nx.bidirectional_shortest_path(G, source, target)
paths = len(p) - 1
elif method == "dijkstra":
paths = nx.dijkstra_path_length(G, source, target, weight)
else: # method == 'bellman-ford':
paths = nx.bellman_ford_path_length(G, source, target, weight)
return paths
[docs]def average_shortest_path_length(G, weight=None, method=None):
r"""Returns the average shortest path length.
The average shortest path length is
.. math::
a =\sum_{s,t \in V} \frac{d(s, t)}{n(n-1)}
where `V` is the set of nodes in `G`,
`d(s, t)` is the shortest path from `s` to `t`,
and `n` is the number of nodes in `G`.
Parameters
----------
G : NetworkX graph
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
method : string, optional (default = 'unweighted' or 'djikstra')
The algorithm to use to compute the path lengths.
Supported options are 'unweighted', 'dijkstra', 'bellman-ford',
'floyd-warshall' and 'floyd-warshall-numpy'.
Other method values produce a ValueError.
The default method is 'unweighted' if `weight` is None,
otherwise the default method is 'dijkstra'.
Raises
------
NetworkXPointlessConcept
If `G` is the null graph (that is, the graph on zero nodes).
NetworkXError
If `G` is not connected (or not weakly connected, in the case
of a directed graph).
ValueError
If `method` is not among the supported options.
Examples
--------
>>> G = nx.path_graph(5)
>>> nx.average_shortest_path_length(G)
2.0
For disconnected graphs, you can compute the average shortest path
length for each component
>>> G = nx.Graph([(1, 2), (3, 4)])
>>> for C in (G.subgraph(c).copy() for c in nx.connected_components(G)):
... print(nx.average_shortest_path_length(C))
1.0
1.0
"""
single_source_methods = ["unweighted", "dijkstra", "bellman-ford"]
all_pairs_methods = ["floyd-warshall", "floyd-warshall-numpy"]
supported_methods = single_source_methods + all_pairs_methods
if method is None:
method = "unweighted" if weight is None else "dijkstra"
if method not in supported_methods:
raise ValueError(f"method not supported: {method}")
n = len(G)
# For the special case of the null graph, raise an exception, since
# there are no paths in the null graph.
if n == 0:
msg = (
"the null graph has no paths, thus there is no average"
"shortest path length"
)
raise nx.NetworkXPointlessConcept(msg)
# For the special case of the trivial graph, return zero immediately.
if n == 1:
return 0
# Shortest path length is undefined if the graph is disconnected.
if G.is_directed() and not nx.is_weakly_connected(G):
raise nx.NetworkXError("Graph is not weakly connected.")
if not G.is_directed() and not nx.is_connected(G):
raise nx.NetworkXError("Graph is not connected.")
# Compute all-pairs shortest paths.
def path_length(v):
if method == "unweighted":
return nx.single_source_shortest_path_length(G, v)
elif method == "dijkstra":
return nx.single_source_dijkstra_path_length(G, v, weight=weight)
elif method == "bellman-ford":
return nx.single_source_bellman_ford_path_length(G, v, weight=weight)
if method in single_source_methods:
# Sum the distances for each (ordered) pair of source and target node.
s = sum(l for u in G for l in path_length(u).values())
else:
if method == "floyd-warshall":
all_pairs = nx.floyd_warshall(G, weight=weight)
s = sum([sum(t.values()) for t in all_pairs.values()])
elif method == "floyd-warshall-numpy":
s = nx.floyd_warshall_numpy(G, weight=weight).sum()
return s / (n * (n - 1))
[docs]def all_shortest_paths(G, source, target, weight=None, method="dijkstra"):
"""Compute all shortest simple paths in the graph.
Parameters
----------
G : NetworkX graph
source : node
Starting node for path.
target : node
Ending node for path.
weight : None or string, optional (default = None)
If None, every edge has weight/distance/cost 1.
If a string, use this edge attribute as the edge weight.
Any edge attribute not present defaults to 1.
method : string, optional (default = 'dijkstra')
The algorithm to use to compute the path lengths.
Supported options: 'dijkstra', 'bellman-ford'.
Other inputs produce a ValueError.
If `weight` is None, unweighted graph methods are used, and this
suggestion is ignored.
Returns
-------
paths : generator of lists
A generator of all paths between source and target.
Raises
------
ValueError
If `method` is not among the supported options.
NetworkXNoPath
If `target` cannot be reached from `source`.
Examples
--------
>>> G = nx.Graph()
>>> nx.add_path(G, [0, 1, 2])
>>> nx.add_path(G, [0, 10, 2])
>>> print([p for p in nx.all_shortest_paths(G, source=0, target=2)])
[[0, 1, 2], [0, 10, 2]]
Notes
-----
There may be many shortest paths between the source and target. If G
contains zero-weight cycles, this function will not produce all shortest
paths because doing so would produce infinitely many paths of unbounded
length -- instead, we only produce the shortest simple paths.
See Also
--------
shortest_path()
single_source_shortest_path()
all_pairs_shortest_path()
"""
method = "unweighted" if weight is None else method
if method == "unweighted":
pred = nx.predecessor(G, source)
elif method == "dijkstra":
pred, dist = nx.dijkstra_predecessor_and_distance(G, source, weight=weight)
elif method == "bellman-ford":
pred, dist = nx.bellman_ford_predecessor_and_distance(G, source, weight=weight)
else:
raise ValueError(f"method not supported: {method}")
return _build_paths_from_predecessors({source}, target, pred)
def _build_paths_from_predecessors(sources, target, pred):
"""Compute all simple paths to target, given the predecessors found in
pred, terminating when any source in sources is found.
Parameters
----------
sources : set
Starting nodes for path.
target : node
Ending node for path.
pred : dict
A dictionary of predecessor lists, keyed by node
Returns
-------
paths : generator of lists
A generator of all paths between source and target.
Raises
------
NetworkXNoPath
If `target` cannot be reached from `source`.
Notes
-----
There may be many paths between the sources and target. If there are
cycles among the predecessors, this function will not produce all
possible paths because doing so would produce infinitely many paths
of unbounded length -- instead, we only produce simple paths.
See Also
--------
shortest_path()
single_source_shortest_path()
all_pairs_shortest_path()
all_shortest_paths()
bellman_ford_path()
"""
if target not in pred:
raise nx.NetworkXNoPath(
f"Target {target} cannot be reached" f"from given sources"
)
seen = {target}
stack = [[target, 0]]
top = 0
while top >= 0:
node, i = stack[top]
if node in sources:
yield [p for p, n in reversed(stack[: top + 1])]
if len(pred[node]) > i:
stack[top][1] = i + 1
next = pred[node][i]
if next in seen:
continue
else:
seen.add(next)
top += 1
if top == len(stack):
stack.append([next, 0])
else:
stack[top][:] = [next, 0]
else:
seen.discard(node)
top -= 1