Note

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

# Source code for networkx.algorithms.connectivity.disjoint_paths

"""Flow based node and edge disjoint paths."""
import networkx as nx
from networkx.exception import NetworkXNoPath

# Define the default maximum flow function to use for the undelying
# maximum flow computations
from networkx.algorithms.flow import edmonds_karp
from networkx.algorithms.flow import preflow_push
from networkx.algorithms.flow import shortest_augmenting_path

default_flow_func = edmonds_karp
# Functions to build auxiliary data structures.
from .utils import build_auxiliary_node_connectivity
from .utils import build_auxiliary_edge_connectivity

from itertools import filterfalse as _filterfalse

__all__ = ["edge_disjoint_paths", "node_disjoint_paths"]

[docs]def edge_disjoint_paths(
G, s, t, flow_func=None, cutoff=None, auxiliary=None, residual=None
):
"""Returns the edges disjoint paths between source and target.

Edge disjoint paths are paths that do not share any edge. The
number of edge disjoint paths between source and target is equal
to their edge connectivity.

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

s : node
Source node for the flow.

t : node
Sink node for the flow.

flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:maximum_flow for
details). If flow_func is None, the default maximum flow function
(:meth:edmonds_karp) is used. The choice of the default function
may change from version to version and should not be relied on.
Default value: None.

cutoff : int
Maximum number of paths to yield. Some of the maximum flow
algorithms, such as :meth:edmonds_karp (the default) and
:meth:shortest_augmenting_path support the cutoff parameter,
and will terminate when the flow value reaches or exceeds the
cutoff. Other algorithms will ignore this parameter.
Default value: None.

auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based edge connectivity. It has
to have a graph attribute called mapping with a dictionary mapping
node names in G and in the auxiliary digraph. If provided
it will be reused instead of recreated. Default value: None.

residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be
reused instead of recreated. Default value: None.

Returns
-------
paths : generator
A generator of edge independent paths.

Raises
------
NetworkXNoPath
If there is no path between source and target.

NetworkXError
If source or target are not in the graph G.

--------
:meth:node_disjoint_paths
:meth:edge_connectivity
:meth:maximum_flow
:meth:edmonds_karp
:meth:preflow_push
:meth:shortest_augmenting_path

Examples
--------
We use in this example the platonic icosahedral graph, which has node
edge connectivity 5, thus there are 5 edge disjoint paths between any
pair of nodes.

>>> G = nx.icosahedral_graph()
>>> len(list(nx.edge_disjoint_paths(G, 0, 6)))
5

If you need to compute edge disjoint paths on several pairs of
nodes in the same graph, it is recommended that you reuse the
data structures that NetworkX uses in the computation: the
auxiliary digraph for edge connectivity, and the residual
network for the underlying maximum flow computation.

Example of how to compute edge disjoint paths among all pairs of
nodes of the platonic icosahedral graph reusing the data
structures.

>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, "capacity")
>>> result = {n: {} for n in G}
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as arguments
>>> for u, v in itertools.combinations(G, 2):
...     k = len(list(nx.edge_disjoint_paths(G, u, v, auxiliary=H, residual=R)))
...     result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True

You can also use alternative flow algorithms for computing edge disjoint
paths. For instance, in dense networks the algorithm
:meth:shortest_augmenting_path will usually perform better than
the default :meth:edmonds_karp which is faster for sparse
networks with highly skewed degree distributions. Alternative flow
functions have to be explicitly imported from the flow package.

>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(list(nx.edge_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path)))
5

Notes
-----
This is a flow based implementation of edge disjoint paths. We compute
the maximum flow between source and target on an auxiliary directed
network. The saturated edges in the residual network after running the
maximum flow algorithm correspond to edge disjoint paths between source
and target in the original network. This function handles both directed
and undirected graphs, and can use all flow algorithms from NetworkX flow
package.

"""
if s not in G:
raise nx.NetworkXError(f"node {s} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {t} not in graph")

if flow_func is None:
flow_func = default_flow_func

if auxiliary is None:
H = build_auxiliary_edge_connectivity(G)
else:
H = auxiliary

# Maximum possible edge disjoint paths
possible = min(H.out_degree(s), H.in_degree(t))
if not possible:
raise NetworkXNoPath

if cutoff is None:
cutoff = possible
else:
cutoff = min(cutoff, possible)

# Compute maximum flow between source and target. Flow functions in
# NetworkX return a residual network.
kwargs = dict(
capacity="capacity", residual=residual, cutoff=cutoff, value_only=True
)
if flow_func is preflow_push:
del kwargs["cutoff"]
if flow_func is shortest_augmenting_path:
kwargs["two_phase"] = True
R = flow_func(H, s, t, **kwargs)

if R.graph["flow_value"] == 0:
raise NetworkXNoPath

# Saturated edges in the residual network form the edge disjoint paths
# between source and target
cutset = [
(u, v)
for u, v, d in R.edges(data=True)
if d["capacity"] == d["flow"] and d["flow"] > 0
]
# This is equivalent of what flow.utils.build_flow_dict returns, but
# only for the nodes with saturated edges and without reporting 0 flows.
flow_dict = {n: {} for edge in cutset for n in edge}
for u, v in cutset:
flow_dict[u][v] = 1

# Rebuild the edge disjoint paths from the flow dictionary.
paths_found = 0
for v in list(flow_dict[s]):
if paths_found >= cutoff:
# preflow_push does not support cutoff: we have to
# keep track of the paths founds and stop at cutoff.
break
path = [s]
if v == t:
path.append(v)
yield path
continue
u = v
while u != t:
path.append(u)
try:
u, _ = flow_dict[u].popitem()
except KeyError:
break
else:
path.append(t)
yield path
paths_found += 1

[docs]def node_disjoint_paths(
G, s, t, flow_func=None, cutoff=None, auxiliary=None, residual=None
):
r"""Computes node disjoint paths between source and target.

Node disjoint paths are paths that only share their first and last
nodes. The number of node independent paths between two nodes is
equal to their local node connectivity.

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

s : node
Source node.

t : node
Target node.

flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:maximum_flow for
details). If flow_func is None, the default maximum flow function
(:meth:edmonds_karp) is used. See below for details. The choice
of the default function may change from version to version and
should not be relied on. Default value: None.

cutoff : int
Maximum number of paths to yield. Some of the maximum flow
algorithms, such as :meth:edmonds_karp (the default) and
:meth:shortest_augmenting_path support the cutoff parameter,
and will terminate when the flow value reaches or exceeds the
cutoff. Other algorithms will ignore this parameter.
Default value: None.

auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based node connectivity. It has
to have a graph attribute called mapping with a dictionary mapping
node names in G and in the auxiliary digraph. If provided
it will be reused instead of recreated. Default value: None.

residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be
reused instead of recreated. Default value: None.

Returns
-------
paths : generator
Generator of node disjoint paths.

Raises
------
NetworkXNoPath
If there is no path between source and target.

NetworkXError
If source or target are not in the graph G.

Examples
--------
We use in this example the platonic icosahedral graph, which has node
node connectivity 5, thus there are 5 node disjoint paths between any
pair of non neighbor nodes.

>>> G = nx.icosahedral_graph()
>>> len(list(nx.node_disjoint_paths(G, 0, 6)))
5

If you need to compute node disjoint paths between several pairs of
nodes in the same graph, it is recommended that you reuse the
data structures that NetworkX uses in the computation: the
auxiliary digraph for node connectivity and node cuts, and the
residual network for the underlying maximum flow computation.

Example of how to compute node disjoint paths reusing the data
structures:

>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity
>>> H = build_auxiliary_node_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, "capacity")
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as arguments
>>> len(list(nx.node_disjoint_paths(G, 0, 6, auxiliary=H, residual=R)))
5

You can also use alternative flow algorithms for computing node disjoint
paths. For instance, in dense networks the algorithm
:meth:shortest_augmenting_path will usually perform better than
the default :meth:edmonds_karp which is faster for sparse
networks with highly skewed degree distributions. Alternative flow
functions have to be explicitly imported from the flow package.

>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(list(nx.node_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path)))
5

Notes
-----
This is a flow based implementation of node disjoint paths. We compute
the maximum flow between source and target on an auxiliary directed
network. The saturated edges in the residual network after running the
maximum flow algorithm correspond to node disjoint paths between source
and target in the original network. This function handles both directed
and undirected graphs, and can use all flow algorithms from NetworkX flow
package.

--------
:meth:edge_disjoint_paths
:meth:node_connectivity
:meth:maximum_flow
:meth:edmonds_karp
:meth:preflow_push
:meth:shortest_augmenting_path

"""
if s not in G:
raise nx.NetworkXError(f"node {s} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {t} not in graph")

if auxiliary is None:
H = build_auxiliary_node_connectivity(G)
else:
H = auxiliary

mapping = H.graph.get("mapping", None)
if mapping is None:
raise nx.NetworkXError("Invalid auxiliary digraph.")

# Maximum possible edge disjoint paths
possible = min(H.out_degree(f"{mapping[s]}B"), H.in_degree(f"{mapping[t]}A"))
if not possible:
raise NetworkXNoPath

if cutoff is None:
cutoff = possible
else:
cutoff = min(cutoff, possible)

kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H, cutoff=cutoff)

# The edge disjoint paths in the auxiliary digraph correspond to the node
# disjoint paths in the original graph.
paths_edges = edge_disjoint_paths(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs)
for path in paths_edges:
# Each node in the original graph maps to two nodes in auxiliary graph
yield list(_unique_everseen(H.nodes[node]["id"] for node in path))

def _unique_everseen(iterable):