Note

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

Source code for networkx.algorithms.flow.edmondskarp

"""
Edmonds-Karp algorithm for maximum flow problems.
"""

import networkx as nx
from networkx.algorithms.flow.utils import build_residual_network

__all__ = ["edmonds_karp"]


def edmonds_karp_core(R, s, t, cutoff):
    """Implementation of the Edmonds-Karp algorithm.
    """
    R_nodes = R.nodes
    R_pred = R.pred
    R_succ = R.succ

    inf = R.graph["inf"]

    def augment(path):
        """Augment flow along a path from s to t.
        """
        # Determine the path residual capacity.
        flow = inf
        it = iter(path)
        u = next(it)
        for v in it:
            attr = R_succ[u][v]
            flow = min(flow, attr["capacity"] - attr["flow"])
            u = v
        if flow * 2 > inf:
            raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.")
        # Augment flow along the path.
        it = iter(path)
        u = next(it)
        for v in it:
            R_succ[u][v]["flow"] += flow
            R_succ[v][u]["flow"] -= flow
            u = v
        return flow

    def bidirectional_bfs():
        """Bidirectional breadth-first search for an augmenting path.
        """
        pred = {s: None}
        q_s = [s]
        succ = {t: None}
        q_t = [t]
        while True:
            q = []
            if len(q_s) <= len(q_t):
                for u in q_s:
                    for v, attr in R_succ[u].items():
                        if v not in pred and attr["flow"] < attr["capacity"]:
                            pred[v] = u
                            if v in succ:
                                return v, pred, succ
                            q.append(v)
                if not q:
                    return None, None, None
                q_s = q
            else:
                for u in q_t:
                    for v, attr in R_pred[u].items():
                        if v not in succ and attr["flow"] < attr["capacity"]:
                            succ[v] = u
                            if v in pred:
                                return v, pred, succ
                            q.append(v)
                if not q:
                    return None, None, None
                q_t = q

    # Look for shortest augmenting paths using breadth-first search.
    flow_value = 0
    while flow_value < cutoff:
        v, pred, succ = bidirectional_bfs()
        if pred is None:
            break
        path = [v]
        # Trace a path from s to v.
        u = v
        while u != s:
            u = pred[u]
            path.append(u)
        path.reverse()
        # Trace a path from v to t.
        u = v
        while u != t:
            u = succ[u]
            path.append(u)
        flow_value += augment(path)

    return flow_value


def edmonds_karp_impl(G, s, t, capacity, residual, cutoff):
    """Implementation of the Edmonds-Karp algorithm.
    """
    if s not in G:
        raise nx.NetworkXError(f"node {str(s)} not in graph")
    if t not in G:
        raise nx.NetworkXError(f"node {str(t)} not in graph")
    if s == t:
        raise nx.NetworkXError("source and sink are the same node")

    if residual is None:
        R = build_residual_network(G, capacity)
    else:
        R = residual

    # Initialize/reset the residual network.
    for u in R:
        for e in R[u].values():
            e["flow"] = 0

    if cutoff is None:
        cutoff = float("inf")
    R.graph["flow_value"] = edmonds_karp_core(R, s, t, cutoff)

    return R


[docs]def edmonds_karp( G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None ): """Find a maximum single-commodity flow using the Edmonds-Karp algorithm. This function returns the residual network resulting after computing the maximum flow. See below for details about the conventions NetworkX uses for defining residual networks. This algorithm has a running time of $O(n m^2)$ for $n$ nodes and $m$ edges. Parameters ---------- G : NetworkX graph Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity. s : node Source node for the flow. t : node Sink node for the flow. capacity : string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. residual : NetworkX graph Residual network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None. value_only : bool If True compute only the value of the maximum flow. This parameter will be ignored by this algorithm because it is not applicable. cutoff : integer, float If specified, the algorithm will terminate when the flow value reaches or exceeds the cutoff. In this case, it may be unable to immediately determine a minimum cut. Default value: None. Returns ------- R : NetworkX DiGraph Residual network after computing the maximum flow. Raises ------ NetworkXError The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded If the graph has a path of infinite capacity, the value of a feasible flow on the graph is unbounded above and the function raises a NetworkXUnbounded. See also -------- :meth:`maximum_flow` :meth:`minimum_cut` :meth:`preflow_push` :meth:`shortest_augmenting_path` Notes ----- The residual network :samp:`R` from an input graph :samp:`G` has the same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists in :samp:`G`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists in :samp:`G` or zero otherwise. If the capacity is infinite, :samp:`R[u][v]['capacity']` will have a high arbitrary finite value that does not affect the solution of the problem. This value is stored in :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. The flow value, defined as the total flow into :samp:`t`, the sink, is stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum :samp:`s`-:samp:`t` cut. Examples -------- >>> from networkx.algorithms.flow import edmonds_karp The functions that implement flow algorithms and output a residual network, such as this one, are not imported to the base NetworkX namespace, so you have to explicitly import them from the flow package. >>> G = nx.DiGraph() >>> G.add_edge("x", "a", capacity=3.0) >>> G.add_edge("x", "b", capacity=1.0) >>> G.add_edge("a", "c", capacity=3.0) >>> G.add_edge("b", "c", capacity=5.0) >>> G.add_edge("b", "d", capacity=4.0) >>> G.add_edge("d", "e", capacity=2.0) >>> G.add_edge("c", "y", capacity=2.0) >>> G.add_edge("e", "y", capacity=3.0) >>> R = edmonds_karp(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value 3.0 >>> flow_value == R.graph["flow_value"] True """ R = edmonds_karp_impl(G, s, t, capacity, residual, cutoff) R.graph["algorithm"] = "edmonds_karp" return R