# -*- coding: utf-8 -*-
# Copyright (C) 2004-2018 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
# Authors: Aric Hagberg <aric.hagberg@gmail.com>
# Sérgio Nery Simões <sergionery@gmail.com>
"""
Compute the shortest paths and path lengths between nodes in the graph.
These algorithms work with undirected and directed graphs.
"""
from __future__ import division
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):
"""Return *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:
sp = nx.shortest_path(G, source, target)
except nx.NetworkXNoPath:
return False
return True
[docs]def shortest_path(G, source=None, target=None, weight=None):
"""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.
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].
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()
single_source_shortest_path()
single_source_dijkstra_path()
"""
if source is None:
if target is None:
# Find paths between all pairs.
if weight is None:
paths = dict(nx.all_pairs_shortest_path(G))
else:
paths = dict(nx.all_pairs_dijkstra_path(G, weight=weight))
else:
# Find paths from all nodes co-accessible to the target.
with nx.utils.reversed(G):
if weight is None:
paths = nx.single_source_shortest_path(G, target)
else:
paths = nx.single_source_dijkstra_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 weight is None:
paths = nx.single_source_shortest_path(G, source)
else:
paths = nx.single_source_dijkstra_path(G, source,
weight=weight)
else:
# Find shortest source-target path.
if weight is None:
paths = nx.bidirectional_shortest_path(G, source, target)
else:
paths = nx.dijkstra_path(G, source, target, weight)
return paths
[docs]def shortest_path_length(G, source=None, target=None, weight=None):
"""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.
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
------
NetworkXNoPath
If no path exists between source and target.
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()
single_source_shortest_path_length()
single_source_dijkstra_path_length()
"""
if source is None:
if target is None:
# Find paths between all pairs.
if weight is None:
paths = nx.all_pairs_shortest_path_length(G)
else:
paths = nx.all_pairs_dijkstra_path_length(G, weight=weight)
else:
# Find paths from all nodes co-accessible to the target.
with nx.utils.reversed(G):
if weight is None:
# We need to exhaust the iterator as Graph needs
# to be reversed.
path_length = nx.single_source_shortest_path_length
paths = path_length(G, target)
else:
path_length = nx.single_source_dijkstra_path_length
paths = path_length(G, target, weight=weight)
else:
if source not in G:
raise nx.NodeNotFound("Source {} not in G".format(source))
if target is None:
# Find paths to all nodes accessible from the source.
if weight is None:
paths = nx.single_source_shortest_path_length(G, source)
else:
paths = nx.single_source_dijkstra_path_length(G, source,
weight=weight)
else:
# Find shortest source-target path.
if weight is None:
p = nx.bidirectional_shortest_path(G, source, target)
paths = len(p) - 1
else:
paths = nx.dijkstra_path_length(G, source, target, weight)
return paths
[docs]def average_shortest_path_length(G, weight=None):
r"""Return 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.
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).
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 nx.connected_component_subgraphs(G):
... print(nx.average_shortest_path_length(C))
1.0
1.0
"""
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.
if weight is None:
def path_length(v): return nx.single_source_shortest_path_length(G, v)
else:
ssdpl = nx.single_source_dijkstra_path_length
def path_length(v): return ssdpl(G, v, weight=weight)
# 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())
return s / (n * (n - 1))
[docs]def all_shortest_paths(G, source, target, weight=None):
"""Compute all shortest 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.
Returns
-------
paths : generator of lists
A generator of all paths between source and target.
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.
See Also
--------
shortest_path()
single_source_shortest_path()
all_pairs_shortest_path()
"""
if weight is not None:
pred, dist = nx.dijkstra_predecessor_and_distance(G, source,
weight=weight)
else:
pred = nx.predecessor(G, source)
if source not in G:
raise nx.NodeNotFound('Source {} is not in G'.format(source))
if target not in pred:
raise nx.NetworkXNoPath()
stack = [[target, 0]]
top = 0
while top >= 0:
node, i = stack[top]
if node == source:
yield [p for p, n in reversed(stack[:top + 1])]
if len(pred[node]) > i:
top += 1
if top == len(stack):
stack.append([pred[node][i], 0])
else:
stack[top] = [pred[node][i], 0]
else:
stack[top - 1][1] += 1
top -= 1