"""
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.
https://doi.org/10.1109/TPAMI.2004.60
.. [2] Vladimir Kolmogorov. Graph-based Algorithms for Multi-camera
Reconstruction Problem. PhD thesis, Cornell University, CS Department,
2003. pp. 109-114.
https://web.archive.org/web/20170809091249/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