.venv/Lib/site-packages/networkx/algorithms/connectivity/disjoint_paths.py

"""Flow based node and edge disjoint paths."""

import networkx as nx

# Define the default maximum flow function to use for the underlying
# maximum flow computations
from networkx.algorithms.flow import (
    edmonds_karp,
    preflow_push,
    shortest_augmenting_path,
)
from networkx.exception import NetworkXNoPath

default_flow_func = edmonds_karp
from itertools import filterfalse as _filterfalse

# Functions to build auxiliary data structures.
from .utils import build_auxiliary_edge_connectivity, build_auxiliary_node_connectivity

__all__ = ["edge_disjoint_paths", "node_disjoint_paths"]


@nx._dispatchable(
    graphs={"G": 0, "auxiliary?": 5},
    preserve_edge_attrs={"auxiliary": {"capacity": float("inf")}},
)
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 : integer or None (default: None)
        Maximum number of paths to yield. If specified, the maximum flow
        algorithm will terminate when the flow value reaches or exceeds the
        cutoff. This only works for flows that support the cutoff parameter
        (most do) and is ignored otherwise.

    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 = {
        "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


@nx._dispatchable(
    graphs={"G": 0, "auxiliary?": 5},
    preserve_node_attrs={"auxiliary": {"id": None}},
    preserve_graph_attrs={"auxiliary"},
)
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 : integer or None (default: None)
        Maximum number of paths to yield. If specified, the maximum flow
        algorithm will terminate when the flow value reaches or exceeds the
        cutoff. This only works for flows that support the cutoff parameter
        (most do) and is ignored otherwise.

    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
    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 = {
        "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