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. See also -------- :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. See also -------- :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): # Adapted from https://docs.python.org/3/library/itertools.html examples "List unique elements, preserving order. Remember all elements ever seen." # unique_everseen('AAAABBBCCDAABBB') --> A B C D seen = set() seen_add = seen.add for element in _filterfalse(seen.__contains__, iterable): seen_add(element) yield element