Warning

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

# Source code for networkx.algorithms.simple_paths

# -*- coding: utf-8 -*-
#    Sergio Nery Simoes <sergionery@gmail.com>
from heapq import heappush, heappop
from itertools import count

import networkx as nx
from networkx.utils import not_implemented_for

__author__ = """\n""".join(['Sérgio Nery Simões <sergionery@gmail.com>',
'Aric Hagberg <aric.hagberg@gmail.com>',
'Andrey Paramonov',
'Jordi Torrents <jordi.t21@gmail.com>'])

__all__ = [
'all_simple_paths',
'shortest_simple_paths',
]

[docs]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 : node
Ending node for 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.

Examples
--------
>>> 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]
>>> paths = nx.all_simple_paths(G, source=0, target=3, cutoff=2)
>>> print(list(paths))
[[0, 1, 3], [0, 2, 3], [0, 3]]

Notes
-----
This algorithm uses a modified depth-first search to generate the
paths _.  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
----------
..  R. Sedgewick, "Algorithms in C, Part 5: Graph Algorithms",
Addison Wesley Professional, 3rd ed., 2001.

--------
all_shortest_paths, shortest_path
"""
if source not in G:
raise nx.NetworkXError('source node %s not in graph'%source)
if target not in G:
raise nx.NetworkXError('target node %s not in graph'%target)
if cutoff is None:
cutoff = len(G)-1
if G.is_multigraph():
return _all_simple_paths_multigraph(G, source, target, cutoff=cutoff)
else:
return _all_simple_paths_graph(G, source, target, cutoff=cutoff)

def _all_simple_paths_graph(G, source, target, cutoff=None):
if cutoff < 1:
return
visited = [source]
stack = [iter(G[source])]
while stack:
children = stack[-1]
child = next(children, None)
if child is None:
stack.pop()
visited.pop()
elif len(visited) < cutoff:
if child == target:
yield visited + [target]
elif child not in visited:
visited.append(child)
stack.append(iter(G[child]))
else: #len(visited) == cutoff:
if child == target or target in children:
yield visited + [target]
stack.pop()
visited.pop()

def _all_simple_paths_multigraph(G, source, target, cutoff=None):
if cutoff < 1:
return
visited = [source]
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.pop()
elif len(visited) < cutoff:
if child == target:
yield visited + [target]
elif child not in visited:
visited.append(child)
stack.append((v for u,v in G.edges(child)))
else: #len(visited) == cutoff:
count = ([child]+list(children)).count(target)
for i in range(count):
yield visited + [target]
stack.pop()
visited.pop()

@not_implemented_for('multigraph')
[docs]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 allawed.

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

source : node
Starting node for path

target : node
Ending node for path

weight : string
Name of the edge attribute to be used as a weight. 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 _.  Finding
the first K paths requires O(KN^3) operations.

--------
all_shortest_paths
shortest_path
all_simple_paths

References
----------
..  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.NetworkXError('source node %s not in graph' % source)

if target not in G:
raise nx.NetworkXError('target node %s not in graph' % target)

if weight is None:
length_func = len
shortest_path_func = _bidirectional_shortest_path
else:
def length_func(path):
return sum(G.edge[u][v][weight] for (u, v) in zip(path, path[1:]))
shortest_path_func = _bidirectional_dijkstra

listA = list()
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:
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

if listB:
path = listB.pop()
yield path
listA.append(path)
prev_path = path
else:
break

class PathBuffer(object):

def __init__(self):
self.paths = set()
self.sortedpaths = list()
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))

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):
"""Return 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 convinience 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.

--------
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]
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 target == source:
return ({target:None},{source:None},source)

# handle either directed or undirected
if G.is_directed():
Gpred=G.predecessors_iter
Gsucc=G.successors_iter
else:
Gpred=G.neighbors_iter
Gsucc=G.neighbors_iter

# support optional nodes filter
if ignore_nodes:
def filter_iter(nodes_iter):
def iterate(v):
for w in nodes_iter(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_iter):
def iterate(v):
for w in nodes_iter(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)

# predecesssor and successors in search
pred={source:None}
succ={target:None}

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("No path between %s and %s." % (source, 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, optional (default='weight')
Edge data key 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).

--------
shortest_path
shortest_path_length
"""
if source == target:
return (0, [source])

# handle either directed or undirected
if G.is_directed():
Gpred=G.predecessors_iter
Gsucc=G.successors_iter
else:
Gpred=G.neighbors_iter
Gsucc=G.neighbors_iter

# support optional nodes filter
if ignore_nodes:
def filter_iter(nodes_iter):
def iterate(v):
for w in nodes_iter(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_iter):
def iterate(v):
for w in nodes_iter(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, next(c), source))
push(fringe, (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 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 in neighs[dir](v):
if(dir == 0):  # forward
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[v][w].items()))
else:
minweight = G[v][w].get(weight, 1)
vwLength = dists[dir][v] + minweight  # G[v][w].get(weight,1)
else:  # back, must remember to change v,w->w,v
if G.is_multigraph():
minweight = min((dd.get(weight, 1)
for k, dd in G[w][v].items()))
else:
minweight = G[w][v].get(weight, 1)
vwLength = dists[dir][v] + minweight  # G[w][v].get(weight,1)

if w in dists[dir]:
if vwLength < dists[dir][w]:
raise ValueError(
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("No path between %s and %s." % (source, target))