Note

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

Source code for networkx.algorithms.flow.boykovkolmogorov

"""
Boykov-Kolmogorov algorithm for maximum flow problems.
"""
from collections import deque
from operator import itemgetter

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

__all__ = ["boykov_kolmogorov"]


[docs]def boykov_kolmogorov( G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None ): r"""Find a maximum single-commodity flow using Boykov-Kolmogorov 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 worse case complexity $O(n^2 m |C|)$ for $n$ nodes, $m$ edges, and $|C|$ the cost of the minimum cut [1]_. This implementation uses the marking heuristic defined in [2]_ which improves its running time in many practical problems. 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 boykov_kolmogorov 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 = boykov_kolmogorov(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value 3.0 >>> flow_value == R.graph["flow_value"] True A nice feature of the Boykov-Kolmogorov algorithm is that a partition of the nodes that defines a minimum cut can be easily computed based on the search trees used during the algorithm. These trees are stored in the graph attribute `trees` of the residual network. >>> source_tree, target_tree = R.graph["trees"] >>> partition = (set(source_tree), set(G) - set(source_tree)) Or equivalently: >>> partition = (set(G) - set(target_tree), set(target_tree)) References ---------- .. [1] Boykov, Y., & Kolmogorov, V. (2004). An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26(9), 1124-1137. http://www.csd.uwo.ca/~yuri/Papers/pami04.pdf .. [2] Vladimir Kolmogorov. Graph-based Algorithms for Multi-camera Reconstruction Problem. PhD thesis, Cornell University, CS Department, 2003. pp. 109-114. https://pub.ist.ac.at/~vnk/papers/thesis.pdf """ R = boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff) R.graph["algorithm"] = "boykov_kolmogorov" return R
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff): 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. # This is way too slow # nx.set_edge_attributes(R, 0, 'flow') for u in R: for e in R[u].values(): e["flow"] = 0 # Use an arbitrary high value as infinite. It is computed # when building the residual network. INF = R.graph["inf"] if cutoff is None: cutoff = INF R_succ = R.succ R_pred = R.pred def grow(): """Bidirectional breadth-first search for the growth stage. Returns a connecting edge, that is and edge that connects a node from the source search tree with a node from the target search tree. The first node in the connecting edge is always from the source tree and the last node from the target tree. """ while active: u = active[0] if u in source_tree: this_tree = source_tree other_tree = target_tree neighbors = R_succ else: this_tree = target_tree other_tree = source_tree neighbors = R_pred for v, attr in neighbors[u].items(): if attr["capacity"] - attr["flow"] > 0: if v not in this_tree: if v in other_tree: return (u, v) if this_tree is source_tree else (v, u) this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] active.append(v) elif v in this_tree and _is_closer(u, v): this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] _ = active.popleft() return None, None def augment(u, v): """Augmentation stage. Reconstruct path and determine its residual capacity. We start from a connecting edge, which links a node from the source tree to a node from the target tree. The connecting edge is the output of the grow function and the input of this function. """ attr = R_succ[u][v] flow = min(INF, attr["capacity"] - attr["flow"]) path = [u] # Trace a path from u to s in source_tree. w = u while w != s: n = w w = source_tree[n] attr = R_pred[n][w] flow = min(flow, attr["capacity"] - attr["flow"]) path.append(w) path.reverse() # Trace a path from v to t in target_tree. path.append(v) w = v while w != t: n = w w = target_tree[n] attr = R_succ[n][w] flow = min(flow, attr["capacity"] - attr["flow"]) path.append(w) # Augment flow along the path and check for saturated edges. it = iter(path) u = next(it) these_orphans = [] for v in it: R_succ[u][v]["flow"] += flow R_succ[v][u]["flow"] -= flow if R_succ[u][v]["flow"] == R_succ[u][v]["capacity"]: if v in source_tree: source_tree[v] = None these_orphans.append(v) if u in target_tree: target_tree[u] = None these_orphans.append(u) u = v orphans.extend(sorted(these_orphans, key=dist.get)) return flow def adopt(): """Adoption stage. Reconstruct search trees by adopting or discarding orphans. During augmentation stage some edges got saturated and thus the source and target search trees broke down to forests, with orphans as roots of some of its trees. We have to reconstruct the search trees rooted to source and target before we can grow them again. """ while orphans: u = orphans.popleft() if u in source_tree: tree = source_tree neighbors = R_pred else: tree = target_tree neighbors = R_succ nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr["capacity"] - attr["flow"] > 0: if _has_valid_root(v, tree): tree[u] = v dist[u] = dist[v] + 1 timestamp[u] = time break else: nbrs = ( (n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree ) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr["capacity"] - attr["flow"] > 0: if v not in active: active.append(v) if tree[v] == u: tree[v] = None orphans.appendleft(v) if u in active: active.remove(u) del tree[u] def _has_valid_root(n, tree): path = [] v = n while v is not None: path.append(v) if v == s or v == t: base_dist = 0 break elif timestamp[v] == time: base_dist = dist[v] break v = tree[v] else: return False length = len(path) for i, u in enumerate(path, 1): dist[u] = base_dist + length - i timestamp[u] = time return True def _is_closer(u, v): return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1 source_tree = {s: None} target_tree = {t: None} active = deque([s, t]) orphans = deque() flow_value = 0 # data structures for the marking heuristic time = 1 timestamp = {s: time, t: time} dist = {s: 0, t: 0} while flow_value < cutoff: # Growth stage u, v = grow() if u is None: break time += 1 # Augmentation stage flow_value += augment(u, v) # Adoption stage adopt() if flow_value * 2 > INF: raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.") # Add source and target tree in a graph attribute. # A partition that defines a minimum cut can be directly # computed from the search trees as explained in the docstrings. R.graph["trees"] = (source_tree, target_tree) # Add the standard flow_value graph attribute. R.graph["flow_value"] = flow_value return R