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 _detect_unboundedness(R):
"""Detect infinite-capacity negative cycles."""
G = nx.DiGraph()
G.add_nodes_from(R)
# Value simulating infinity.
inf = R.graph["inf"]
# True infinity.
f_inf = float("inf")
for u in R:
for v, e in R[u].items():
# Compute the minimum weight of infinite-capacity (u, v) edges.
w = f_inf
for k, e in e.items():
if e["capacity"] == inf:
w = min(w, e["weight"])
if w != f_inf:
G.add_edge(u, v, weight=w)
if nx.negative_edge_cycle(G):
raise nx.NetworkXUnbounded(
"Negative cost cycle of infinite capacity found. "
"Min cost flow may be unbounded below.")
def bellman_ford(G, source, weight='weight', return_negative_cycle=False):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
return_negative_cycle: boolean, optional (default=False)
If a negative cycle is detected, return the path instead of
thowing an Exception.
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> pred
{0: None, 1: 0, 2: 1, 3: 2, 4: 3}
>>> dist
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
numb_nodes = len(G)
dist = {source: 0}
pred = {source: None}
if numb_nodes == 1:
return pred, dist
if G.is_multigraph():
def get_weight(edge_dict):
return min([eattr.get(weight, 1) for eattr in edge_dict.values()])
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
for i in range(numb_nodes):
no_changes = True
# Only need edges from nodes in dist b/c all others have dist==inf
for u, dist_u in list(
dist.items()): # get all edges from nodes in dist
for v, edict in G[u].items(): # double loop handles undirected too
dist_v = dist_u + get_weight(edict)
if v not in dist or dist[v] > dist_v:
dist[v] = dist_v
pred[v] = u
no_changes = False
if no_changes:
break
else:
if return_negative_cycle:
return pred, dist
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
return pred, dist
def dinitz_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
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
# 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 breath_first_search():
parents = {}
queue = deque([s])
while queue:
if t in parents:
break
u = queue.popleft()
for v in R_succ[u]:
attr = R_succ[u][v]
if v not in parents and attr['capacity'] - attr['flow'] > 0:
parents[v] = u
queue.append(v)
return parents
def depth_first_search(parents):
"""Build a path using DFS starting from the sink"""
path = []
u = t
flow = INF
while u != s:
path.append(u)
v = parents[u]
flow = min(flow, R_pred[u][v]['capacity'] - R_pred[u][v]['flow'])
u = v
path.append(s)
# Augment the flow along the path found
if flow > 0:
for u, v in pairwise(path):
R_pred[u][v]['flow'] += flow
R_pred[v][u]['flow'] -= flow
return flow
flow_value = 0
while flow_value < cutoff:
parents = breath_first_search()
if t not in parents:
break
this_flow = depth_first_search(parents)
if this_flow * 2 > INF:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
flow_value += this_flow
R.graph['flow_value'] = flow_value
return R
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
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, 'flow', 0)
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
def bellman_ford(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
dist = {source: 0}
pred = {source: None}
if len(G) == 1:
return pred, dist
if G.is_multigraph():
def get_weight(edge_dict):
return min(eattr.get(weight, 1) for eattr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
if G.is_directed():
G_succ = G.succ
else:
G_succ = G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque([source])
in_q = set([source])
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if the predecessor of u is in the queue.
if pred[u] not in in_q:
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + get_weight(e)
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = u
return pred, dist
def bellman_ford_path():
"""
Retorna o tamanho do menor caminho de um vértice para todos os outros
"""
source = get_user_input("Escolha o vértice de origem: ")
if G.is_multigraph():
weight = lambda u, v, d: min(attr.get("weight", 1) for attr in d.values())
weight = lambda u, v, data: data.get("weight", 1)
pred=None
paths=None
dist=None
target=None
iterations=0
startTime=time.time()
for s in source:
if s not in G:
raise nx.NodeNotFound("Source {} not in G".format(s))
if pred is None:
pred = {v: [] for v in source}
if dist is None:
dist = {v: 0 for v in source}
G_succ = G.succ if G.is_directed() else G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque(source)
in_q = set(source)
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if any of the predecessors of u is in the queue.
if all(pred_u not in in_q for pred_u in pred[u]):
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + weight(v, u, e)
iterations += 1
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = [u]
elif dist.get(v) is not None and dist_v == dist.get(v):
pred[v].append(u)
if paths is not None:
dsts = [target] if target is not None else pred
for dst in dsts:
path = [dst]
cur = dst
while pred[cur]:
cur = pred[cur][0]
path.append(cur)
path.reverse()
paths[dst] = path
length = dict(dist)
for node in G.nodes:
print('{}: {}'.format(node, length[node]))
print("Tempo de Execução: " + str(time.time()-startTime))
print("Iterações: " + str(iterations))
get_user_input("Aperte enter para continuar...")
def _bellman_ford_relaxation(G, pred, dist, source, weight):
"""Relaxation loop for Bellman–Ford algorithm
Parameters
----------
G : NetworkX graph
pred: dict
Keyed by node to predecessor in the path
dist: dict
Keyed by node to the distance from the source
source: list
List of source nodes
weight: string
Edge data key corresponding to the edge weight
Returns
-------
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle
"""
if G.is_multigraph():
def get_weight(edge_dict):
return min(eattr.get(weight, 1) for eattr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
G_succ = G.succ if G.is_directed() else G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque(source)
in_q = set(source)
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if the predecessor of u is in the queue.
if pred[u] not in in_q:
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + get_weight(e)
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = u
return pred, dist
def network_simplex(G, demand="demand", capacity="capacity", weight="weight"):
r"""Find a minimum cost flow satisfying all demands in digraph G.
This is a primal network simplex algorithm that uses the leaving
arc rule to prevent cycling.
G is a digraph with edge costs and capacities and in which nodes
have demand, i.e., they want to send or receive some amount of
flow. A negative demand means that the node wants to send flow, a
positive demand means that the node want to receive flow. A flow on
the digraph G satisfies all demand if the net flow into each node
is equal to the demand of that node.
Parameters
----------
G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is
to be found.
demand : string
Nodes of the graph G are expected to have an attribute demand
that indicates how much flow a node wants to send (negative
demand) or receive (positive demand). Note that the sum of the
demands should be 0 otherwise the problem in not feasible. If
this attribute is not present, a node is considered to have 0
demand. Default value: 'demand'.
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'.
weight : string
Edges of the graph G are expected to have an attribute weight
that indicates the cost incurred by sending one unit of flow on
that edge. If not present, the weight is considered to be 0.
Default value: 'weight'.
Returns
-------
flowCost : integer, float
Cost of a minimum cost flow satisfying all demands.
flowDict : dictionary
Dictionary of dictionaries keyed by nodes such that
flowDict[u][v] is the flow edge (u, v).
Raises
------
NetworkXError
This exception is raised if the input graph is not directed,
not connected or is a multigraph.
NetworkXUnfeasible
This exception is raised in the following situations:
* The sum of the demands is not zero. Then, there is no
flow satisfying all demands.
* There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of
negative cost and infinite capacity. Then, the cost of a flow
satisfying all demands is unbounded below.
Notes
-----
This algorithm is not guaranteed to work if edge weights or demands
are floating point numbers (overflows and roundoff errors can
cause problems). As a workaround you can use integer numbers by
multiplying the relevant edge attributes by a convenient
constant factor (eg 100).
See also
--------
cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost
Examples
--------
A simple example of a min cost flow problem.
>>> G = nx.DiGraph()
>>> G.add_node("a", demand=-5)
>>> G.add_node("d", demand=5)
>>> G.add_edge("a", "b", weight=3, capacity=4)
>>> G.add_edge("a", "c", weight=6, capacity=10)
>>> G.add_edge("b", "d", weight=1, capacity=9)
>>> G.add_edge("c", "d", weight=2, capacity=5)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost
24
>>> flowDict
{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
The mincost flow algorithm can also be used to solve shortest path
problems. To find the shortest path between two nodes u and v,
give all edges an infinite capacity, give node u a demand of -1 and
node v a demand a 1. Then run the network simplex. The value of a
min cost flow will be the distance between u and v and edges
carrying positive flow will indicate the path.
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from(
... [
... ("s", "u", 10),
... ("s", "x", 5),
... ("u", "v", 1),
... ("u", "x", 2),
... ("v", "y", 1),
... ("x", "u", 3),
... ("x", "v", 5),
... ("x", "y", 2),
... ("y", "s", 7),
... ("y", "v", 6),
... ]
... )
>>> G.add_node("s", demand=-1)
>>> G.add_node("v", demand=1)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight")
True
>>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0])
[('s', 'x'), ('u', 'v'), ('x', 'u')]
>>> nx.shortest_path(G, "s", "v", weight="weight")
['s', 'x', 'u', 'v']
It is possible to change the name of the attributes used for the
algorithm.
>>> G = nx.DiGraph()
>>> G.add_node("p", spam=-4)
>>> G.add_node("q", spam=2)
>>> G.add_node("a", spam=-2)
>>> G.add_node("d", spam=-1)
>>> G.add_node("t", spam=2)
>>> G.add_node("w", spam=3)
>>> G.add_edge("p", "q", cost=7, vacancies=5)
>>> G.add_edge("p", "a", cost=1, vacancies=4)
>>> G.add_edge("q", "d", cost=2, vacancies=3)
>>> G.add_edge("t", "q", cost=1, vacancies=2)
>>> G.add_edge("a", "t", cost=2, vacancies=4)
>>> G.add_edge("d", "w", cost=3, vacancies=4)
>>> G.add_edge("t", "w", cost=4, vacancies=1)
>>> flowCost, flowDict = nx.network_simplex(
... G, demand="spam", capacity="vacancies", weight="cost"
... )
>>> flowCost
37
>>> flowDict
{'p': {'q': 2, 'a': 2}, 'q': {'d': 1}, 'a': {'t': 4}, 'd': {'w': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}
References
----------
.. [1] Z. Kiraly, P. Kovacs.
Efficient implementation of minimum-cost flow algorithms.
Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012.
.. [2] R. Barr, F. Glover, D. Klingman.
Enhancement of spanning tree labeling procedures for network
optimization.
INFOR 17(1):16--34. 1979.
"""
###########################################################################
# Problem essentials extraction and sanity check
###########################################################################
if len(G) == 0:
raise nx.NetworkXError("graph has no nodes")
# Number all nodes and edges and hereafter reference them using ONLY their
# numbers
N = list(G) # nodes
I = {u: i for i, u in enumerate(N)} # node indices
D = [G.nodes[u].get(demand, 0) for u in N] # node demands
inf = float("inf")
for p, b in zip(N, D):
if abs(b) == inf:
raise nx.NetworkXError(f"node {p!r} has infinite demand")
multigraph = G.is_multigraph()
S = [] # edge sources
T = [] # edge targets
if multigraph:
K = [] # edge keys
E = {} # edge indices
U = [] # edge capacities
C = [] # edge weights
if not multigraph:
edges = G.edges(data=True)
else:
edges = G.edges(data=True, keys=True)
edges = (e for e in edges if e[0] != e[1] and e[-1].get(capacity, inf) != 0)
for i, e in enumerate(edges):
S.append(I[e[0]])
T.append(I[e[1]])
if multigraph:
K.append(e[2])
E[e[:-1]] = i
U.append(e[-1].get(capacity, inf))
C.append(e[-1].get(weight, 0))
for e, c in zip(E, C):
if abs(c) == inf:
raise nx.NetworkXError(f"edge {e!r} has infinite weight")
if not multigraph:
edges = nx.selfloop_edges(G, data=True)
else:
edges = nx.selfloop_edges(G, data=True, keys=True)
for e in edges:
if abs(e[-1].get(weight, 0)) == inf:
raise nx.NetworkXError(f"edge {e[:-1]!r} has infinite weight")
###########################################################################
# Quick infeasibility detection
###########################################################################
if sum(D) != 0:
raise nx.NetworkXUnfeasible("total node demand is not zero")
for e, u in zip(E, U):
if u < 0:
raise nx.NetworkXUnfeasible(f"edge {e!r} has negative capacity")
if not multigraph:
edges = nx.selfloop_edges(G, data=True)
else:
edges = nx.selfloop_edges(G, data=True, keys=True)
for e in edges:
if e[-1].get(capacity, inf) < 0:
raise nx.NetworkXUnfeasible(f"edge {e[:-1]!r} has negative capacity")
###########################################################################
# Initialization
###########################################################################
# Add a dummy node -1 and connect all existing nodes to it with infinite-
# capacity dummy edges. Node -1 will serve as the root of the
# spanning tree of the network simplex method. The new edges will used to
# trivially satisfy the node demands and create an initial strongly
# feasible spanning tree.
n = len(N) # number of nodes
for p, d in enumerate(D):
# Must be greater-than here. Zero-demand nodes must have
# edges pointing towards the root to ensure strong
# feasibility.
if d > 0:
S.append(-1)
T.append(p)
else:
S.append(p)
T.append(-1)
faux_inf = (
3
* max(
chain(
[sum(u for u in U if u < inf), sum(abs(c) for c in C)],
(abs(d) for d in D),
)
)
or 1
)
C.extend(repeat(faux_inf, n))
U.extend(repeat(faux_inf, n))
# Construct the initial spanning tree.
e = len(E) # number of edges
x = list(chain(repeat(0, e), (abs(d) for d in D))) # edge flows
pi = [faux_inf if d <= 0 else -faux_inf for d in D] # node potentials
parent = list(chain(repeat(-1, n), [None])) # parent nodes
edge = list(range(e, e + n)) # edges to parents
size = list(chain(repeat(1, n), [n + 1])) # subtree sizes
next = list(chain(range(1, n), [-1, 0])) # next nodes in depth-first thread
prev = list(range(-1, n)) # previous nodes in depth-first thread
last = list(chain(range(n), [n - 1])) # last descendants in depth-first thread
###########################################################################
# Pivot loop
###########################################################################
def reduced_cost(i):
"""Returns the reduced cost of an edge i."""
c = C[i] - pi[S[i]] + pi[T[i]]
return c if x[i] == 0 else -c
def find_entering_edges():
"""Yield entering edges until none can be found."""
if e == 0:
return
# Entering edges are found by combining Dantzig's rule and Bland's
# rule. The edges are cyclically grouped into blocks of size B. Within
# each block, Dantzig's rule is applied to find an entering edge. The
# blocks to search is determined following Bland's rule.
B = int(ceil(sqrt(e))) # pivot block size
M = (e + B - 1) // B # number of blocks needed to cover all edges
m = 0 # number of consecutive blocks without eligible
# entering edges
f = 0 # first edge in block
while m < M:
# Determine the next block of edges.
l = f + B
if l <= e:
edges = range(f, l)
else:
l -= e
edges = chain(range(f, e), range(l))
f = l
# Find the first edge with the lowest reduced cost.
i = min(edges, key=reduced_cost)
c = reduced_cost(i)
if c >= 0:
# No entering edge found in the current block.
m += 1
else:
# Entering edge found.
if x[i] == 0:
p = S[i]
q = T[i]
else:
p = T[i]
q = S[i]
yield i, p, q
m = 0
# All edges have nonnegative reduced costs. The current flow is
# optimal.
def find_apex(p, q):
"""Find the lowest common ancestor of nodes p and q in the spanning
tree.
"""
size_p = size[p]
size_q = size[q]
while True:
while size_p < size_q:
p = parent[p]
size_p = size[p]
while size_p > size_q:
q = parent[q]
size_q = size[q]
if size_p == size_q:
if p != q:
p = parent[p]
size_p = size[p]
q = parent[q]
size_q = size[q]
else:
return p
def trace_path(p, w):
"""Returns the nodes and edges on the path from node p to its ancestor
w.
"""
Wn = [p]
We = []
while p != w:
We.append(edge[p])
p = parent[p]
Wn.append(p)
return Wn, We
def find_cycle(i, p, q):
"""Returns the nodes and edges on the cycle containing edge i == (p, q)
when the latter is added to the spanning tree.
The cycle is oriented in the direction from p to q.
"""
w = find_apex(p, q)
Wn, We = trace_path(p, w)
Wn.reverse()
We.reverse()
if We != [i]:
We.append(i)
WnR, WeR = trace_path(q, w)
del WnR[-1]
Wn += WnR
We += WeR
return Wn, We
def residual_capacity(i, p):
"""Returns the residual capacity of an edge i in the direction away
from its endpoint p.
"""
return U[i] - x[i] if S[i] == p else x[i]
def find_leaving_edge(Wn, We):
"""Returns the leaving edge in a cycle represented by Wn and We."""
j, s = min(
zip(reversed(We), reversed(Wn)), key=lambda i_p: residual_capacity(*i_p)
)
t = T[j] if S[j] == s else S[j]
return j, s, t
def augment_flow(Wn, We, f):
"""Augment f units of flow along a cycle represented by Wn and We."""
for i, p in zip(We, Wn):
if S[i] == p:
x[i] += f
else:
x[i] -= f
def trace_subtree(p):
"""Yield the nodes in the subtree rooted at a node p."""
yield p
l = last[p]
while p != l:
p = next[p]
yield p
def remove_edge(s, t):
"""Remove an edge (s, t) where parent[t] == s from the spanning tree."""
size_t = size[t]
prev_t = prev[t]
last_t = last[t]
next_last_t = next[last_t]
# Remove (s, t).
parent[t] = None
edge[t] = None
# Remove the subtree rooted at t from the depth-first thread.
next[prev_t] = next_last_t
prev[next_last_t] = prev_t
next[last_t] = t
prev[t] = last_t
# Update the subtree sizes and last descendants of the (old) acenstors
# of t.
while s is not None:
size[s] -= size_t
if last[s] == last_t:
last[s] = prev_t
s = parent[s]
def make_root(q):
"""Make a node q the root of its containing subtree."""
ancestors = []
while q is not None:
ancestors.append(q)
q = parent[q]
ancestors.reverse()
for p, q in zip(ancestors, islice(ancestors, 1, None)):
size_p = size[p]
last_p = last[p]
prev_q = prev[q]
last_q = last[q]
next_last_q = next[last_q]
# Make p a child of q.
parent[p] = q
parent[q] = None
edge[p] = edge[q]
edge[q] = None
size[p] = size_p - size[q]
size[q] = size_p
# Remove the subtree rooted at q from the depth-first thread.
next[prev_q] = next_last_q
prev[next_last_q] = prev_q
next[last_q] = q
prev[q] = last_q
if last_p == last_q:
last[p] = prev_q
last_p = prev_q
# Add the remaining parts of the subtree rooted at p as a subtree
# of q in the depth-first thread.
prev[p] = last_q
next[last_q] = p
next[last_p] = q
prev[q] = last_p
last[q] = last_p
def add_edge(i, p, q):
"""Add an edge (p, q) to the spanning tree where q is the root of a
subtree.
"""
last_p = last[p]
next_last_p = next[last_p]
size_q = size[q]
last_q = last[q]
# Make q a child of p.
parent[q] = p
edge[q] = i
# Insert the subtree rooted at q into the depth-first thread.
next[last_p] = q
prev[q] = last_p
prev[next_last_p] = last_q
next[last_q] = next_last_p
# Update the subtree sizes and last descendants of the (new) ancestors
# of q.
while p is not None:
size[p] += size_q
if last[p] == last_p:
last[p] = last_q
p = parent[p]
def update_potentials(i, p, q):
"""Update the potentials of the nodes in the subtree rooted at a node
q connected to its parent p by an edge i.
"""
if q == T[i]:
d = pi[p] - C[i] - pi[q]
else:
d = pi[p] + C[i] - pi[q]
for q in trace_subtree(q):
pi[q] += d
# Pivot loop
for i, p, q in find_entering_edges():
Wn, We = find_cycle(i, p, q)
j, s, t = find_leaving_edge(Wn, We)
augment_flow(Wn, We, residual_capacity(j, s))
# Do nothing more if the entering edge is the same as the leaving edge.
if i != j:
if parent[t] != s:
# Ensure that s is the parent of t.
s, t = t, s
if We.index(i) > We.index(j):
# Ensure that q is in the subtree rooted at t.
p, q = q, p
remove_edge(s, t)
make_root(q)
add_edge(i, p, q)
update_potentials(i, p, q)
###########################################################################
# Infeasibility and unboundedness detection
###########################################################################
if any(x[i] != 0 for i in range(-n, 0)):
raise nx.NetworkXUnfeasible("no flow satisfies all node demands")
if any(x[i] * 2 >= faux_inf for i in range(e)) or any(
e[-1].get(capacity, inf) == inf and e[-1].get(weight, 0) < 0
for e in nx.selfloop_edges(G, data=True)
):
raise nx.NetworkXUnbounded("negative cycle with infinite capacity found")
###########################################################################
# Flow cost calculation and flow dict construction
###########################################################################
del x[e:]
flow_cost = sum(c * x for c, x in zip(C, x))
flow_dict = {n: {} for n in N}
def add_entry(e):
"""Add a flow dict entry."""
d = flow_dict[e[0]]
for k in e[1:-2]:
try:
d = d[k]
except KeyError:
t = {}
d[k] = t
d = t
d[e[-2]] = e[-1]
S = (N[s] for s in S) # Use original nodes.
T = (N[t] for t in T) # Use original nodes.
if not multigraph:
for e in zip(S, T, x):
add_entry(e)
edges = G.edges(data=True)
else:
for e in zip(S, T, K, x):
add_entry(e)
edges = G.edges(data=True, keys=True)
for e in edges:
if e[0] != e[1]:
if e[-1].get(capacity, inf) == 0:
add_entry(e[:-1] + (0,))
else:
c = e[-1].get(weight, 0)
if c >= 0:
add_entry(e[:-1] + (0,))
else:
u = e[-1][capacity]
flow_cost += c * u
add_entry(e[:-1] + (u,))
return flow_cost, flow_dict
def _find_leaving_edge(H,
T,
cycle,
newEdge,
capacity='capacity',
reverse=False):
"""Find an edge that will leave the basis and the value by which we
can increase or decrease the flow on that edge.
The leaving arc rule is used to prevent cycling.
If cycle has no reverse edge and no forward edge of finite
capacity, it means that cycle is a negative cost infinite capacity
cycle. This implies that the cost of a flow satisfying all demands
is unbounded below. An exception is raised in this case.
"""
eps = False
leavingEdge = ()
# If cycle is a digon.
if len(cycle) == 3:
u, v = newEdge
if capacity not in H[u][v] and capacity not in H[v][u]:
raise nx.NetworkXUnbounded(
"Negative cost cycle of infinite capacity found. " +
"Min cost flow unbounded below.")
if reverse:
if H[u][v].get('flow', 0) > H[v][u].get('flow', 0):
return (v, u), H[v][u].get('flow', 0)
else:
return (u, v), H[u][v].get('flow', 0)
else:
uv_residual = H[u][v].get(capacity, 0) - H[u][v].get('flow', 0)
vu_residual = H[v][u].get(capacity, 0) - H[v][u].get('flow', 0)
if (uv_residual > vu_residual):
return (v, u), vu_residual
else:
return (u, v), uv_residual
# Find the forward edge with the minimum value for capacity - 'flow'
# and the reverse edge with the minimum value for 'flow'.
for index, u in enumerate(cycle[:-1]):
edgeCapacity = False
edge = ()
v = cycle[index + 1]
if (u, v) in T.edges() + [newEdge]: #forward edge
if capacity in H[u][v]: # edge (u, v) has finite capacity
edgeCapacity = H[u][v][capacity] - H[u][v].get('flow', 0)
edge = (u, v)
else: #reverse edge
edgeCapacity = H[v][u].get('flow', 0)
edge = (v, u)
# Determine if edge might be the leaving edge.
if edge:
if leavingEdge:
if edgeCapacity < eps:
eps = edgeCapacity
leavingEdge = edge
else:
eps = edgeCapacity
leavingEdge = edge
if not leavingEdge:
raise nx.NetworkXUnbounded(
"Negative cost cycle of infinite capacity found. " +
"Min cost flow unbounded below.")
return leavingEdge, eps
def minimum_st_edge_cut(G, s, t, capacity='capacity'):
"""Returns the edges of the cut-set of a minimum (s, t)-cut.
We use the max-flow min-cut theorem, i.e., the capacity of a minimum
capacity cut is equal to the flow value of a maximum flow.
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'.
Returns
-------
cutset : set
Set of edges that, if removed from the graph, will disconnect it
Raises
------
NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have
infinite capacity and the function raises a NetworkXError.
Examples
--------
>>> 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)
>>> sorted(nx.minimum_edge_cut(G, 'x', 'y'))
[('c', 'y'), ('x', 'b')]
>>> nx.minimum_cut_value(G, 'x', 'y')
3.0
"""
try:
H = nx.ford_fulkerson(G, s, t, capacity=capacity)
cutset = set()
# Compute reachable nodes from source in the residual network
reachable = set(nx.single_source_shortest_path(H,s))
# And unreachable nodes
others = set(H) - reachable # - set([s])
# Any edge in the original network linking these two partitions
# is part of the edge cutset
for u, nbrs in ((n, G[n]) for n in reachable):
cutset.update((u,v) for v in nbrs if v in others)
return cutset
except nx.NetworkXUnbounded:
# Should we raise any other exception or just let ford_fulkerson
# propagate nx.NetworkXUnbounded ?
raise nx.NetworkXUnbounded("Infinite capacity path, no minimum cut.")
def _bellman_ford_relaxation(G, pred, dist, source, weight):
"""Relaxation loop for Bellman–Ford algorithm
Parameters
----------
G : NetworkX graph
pred: dict
Keyed by node to predecessor in the path
dist: dict
Keyed by node to the distance from the source
source: list
List of source nodes
weight : function
The weight of an edge is the value returned by the function. The
function must accept exactly three positional arguments: the two
endpoints of an edge and the dictionary of edge attributes for
that edge. The function must return a number.
Returns
-------
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle
"""
G_succ = G.succ if G.is_directed() else G.adj
inf = float('inf')
n = len(G)
count = {}
q = deque(source)
in_q = set(source)
while q:
u = q.popleft()
in_q.remove(u)
# Skip relaxations if the predecessor of u is in the queue.
if pred[u] not in in_q:
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + weight(u, v, e)
if dist_v < dist.get(v, inf):
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
pred[v] = u
return pred, dist
def bellman_ford(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edge[u][v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
weight = _weight_function(G, weight)
if any(weight(u, v, d) < 0 for u, v, d in G.selfloop_edges(data=True)):
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
dist = {source: 0}
pred = {source: None}
if len(G) == 1:
return pred, dist
return _bellman_ford_relaxation(G, pred, dist, [source], weight)
def build_residual_network(G, s, t, capacity):
"""Build a residual network and initialize a zero flow.
"""
if G.is_multigraph():
raise nx.NetworkXError(
'MultiGraph and MultiDiGraph not supported (yet).')
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
if s == t:
raise nx.NetworkXError('source and sink are the same node')
R = nx.DiGraph()
R.add_nodes_from(G, excess=0)
inf = float('inf')
# Extract edges with positive capacities. Self loops excluded.
edge_list = [(u, v, attr) for u, v, attr in G.edges_iter(data=True)
if u != v and attr.get(capacity, inf) > 0]
# Simulate infinity with twice the sum of the finite edge capacities or any
# positive value if the sum is zero. This allows the infinite-capacity
# edges to be distinguished for unboundedness detection and directly
# participate in residual capacity calculation. If the maximum flow is
# finite, these edges cannot appear in the minimum cut and thus guarantee
# correctness.
inf = 2 * sum(attr[capacity] for u, v, attr in edge_list
if capacity in attr and attr[capacity] != inf) or 1
if G.is_directed():
for u, v, attr in edge_list:
r = min(attr.get(capacity, inf), inf)
if not R.has_edge(u, v):
# Both (u, v) and (v, u) must be present in the residual
# network.
R.add_edge(u, v, capacity=r, flow=0)
R.add_edge(v, u, capacity=0, flow=0)
else:
# The edge (u, v) was added when (v, u) was visited.
R[u][v]['capacity'] = r
else:
for u, v, attr in edge_list:
# Add a pair of edges with equal residual capacities.
r = min(attr.get(capacity, inf), inf)
R.add_edge(u, v, capacity=r, flow=0)
R.add_edge(v, u, capacity=r, flow=0)
# Record the value simulating infinity.
R.graph['inf'] = inf
# Detect unboundedness by determining reachability of t from s using only
# infinite-capacity edges.
q = deque([s])
seen = set([s])
while q:
u = q.popleft()
for v, attr in R[u].items():
if attr['capacity'] == inf and v not in seen:
if v == t:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
seen.add(v)
q.append(v)
return R
def ford_fulkerson(G, s, t, capacity='capacity'):
"""Find a maximum single-commodity flow using the Ford-Fulkerson
algorithm.
This algorithm uses Edmonds-Karp-Dinitz path selection rule which
guarantees a running time of O(nm^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'.
Returns
-------
flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
flow_dict : dictionary
Dictionary of dictionaries keyed by nodes such that
flow_dict[u][v] is the flow edge (u, v).
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.
Examples
--------
>>> import networkx as nx
>>> 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)
>>> flow, F = nx.ford_fulkerson(G, 'x', 'y')
>>> flow
3.0
"""
if G.is_multigraph():
raise nx.NetworkXError(
'MultiGraph and MultiDiGraph not supported (yet).')
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
auxiliary, inf_capacity_flows = _create_auxiliary_digraph(
G, capacity=capacity)
flow_value = 0 # Initial feasible flow.
# As long as there is an (s, t)-path in the auxiliary digraph, find
# the shortest (with respect to the number of arcs) such path and
# augment the flow on this path.
while True:
try:
path_nodes = nx.bidirectional_shortest_path(auxiliary, s, t)
except nx.NetworkXNoPath:
break
# Get the list of edges in the shortest path.
path_edges = list(zip(path_nodes[:-1], path_nodes[1:]))
# Find the minimum capacity of an edge in the path.
try:
path_capacity = min([
auxiliary[u][v][capacity] for u, v in path_edges
if capacity in auxiliary[u][v]
])
except ValueError:
# path of infinite capacity implies no max flow
raise nx.NetworkXUnbounded(
"Infinite capacity path, flow unbounded above.")
flow_value += path_capacity
# Augment the flow along the path.
for u, v in path_edges:
edge_attr = auxiliary[u][v]
if capacity in edge_attr:
edge_attr[capacity] -= path_capacity
if edge_attr[capacity] == 0:
auxiliary.remove_edge(u, v)
else:
inf_capacity_flows[(u, v)] += path_capacity
if auxiliary.has_edge(v, u):
if capacity in auxiliary[v][u]:
auxiliary[v][u][capacity] += path_capacity
else:
auxiliary.add_edge(v, u, {capacity: path_capacity})
flow_dict = _create_flow_dict(G,
auxiliary,
inf_capacity_flows,
capacity=capacity)
return flow_value, flow_dict
def _build_residual_network(G, demand, capacity, weight):
"""Build a residual network and initialize a zero flow.
"""
if sum(G.nodes[u].get(demand, 0) for u in G) != 0:
raise nx.NetworkXUnfeasible("Sum of the demands should be 0.")
R = nx.MultiDiGraph()
R.add_nodes_from(
(u, {"excess": -G.nodes[u].get(demand, 0), "potential": 0}) for u in G
)
inf = float("inf")
# Detect selfloops with infinite capacities and negative weights.
for u, v, e in nx.selfloop_edges(G, data=True):
if e.get(weight, 0) < 0 and e.get(capacity, inf) == inf:
raise nx.NetworkXUnbounded(
"Negative cost cycle of infinite capacity found. "
"Min cost flow may be unbounded below."
)
# Extract edges with positive capacities. Self loops excluded.
if G.is_multigraph():
edge_list = [
(u, v, k, e)
for u, v, k, e in G.edges(data=True, keys=True)
if u != v and e.get(capacity, inf) > 0
]
else:
edge_list = [
(u, v, 0, e)
for u, v, e in G.edges(data=True)
if u != v and e.get(capacity, inf) > 0
]
# Simulate infinity with the larger of the sum of absolute node imbalances
# the sum of finite edge capacities or any positive value if both sums are
# zero. This allows the infinite-capacity edges to be distinguished for
# unboundedness detection and directly participate in residual capacity
# calculation.
inf = (
max(
sum(abs(R.nodes[u]["excess"]) for u in R),
2
* sum(
e[capacity]
for u, v, k, e in edge_list
if capacity in e and e[capacity] != inf
),
)
or 1
)
for u, v, k, e in edge_list:
r = min(e.get(capacity, inf), inf)
w = e.get(weight, 0)
# Add both (u, v) and (v, u) into the residual network marked with the
# original key. (key[1] == True) indicates the (u, v) is in the
# original network.
R.add_edge(u, v, key=(k, True), capacity=r, weight=w, flow=0)
R.add_edge(v, u, key=(k, False), capacity=0, weight=-w, flow=0)
# Record the value simulating infinity.
R.graph["inf"] = inf
_detect_unboundedness(R)
return R
def network_simplex(G, demand="demand", capacity="capacity", weight="weight"):
r"""Find a minimum cost flow satisfying all demands in digraph G.
This is a primal network simplex algorithm that uses the leaving
arc rule to prevent cycling.
G is a digraph with edge costs and capacities and in which nodes
have demand, i.e., they want to send or receive some amount of
flow. A negative demand means that the node wants to send flow, a
positive demand means that the node want to receive flow. A flow on
the digraph G satisfies all demand if the net flow into each node
is equal to the demand of that node.
Parameters
----------
G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is
to be found.
demand : string
Nodes of the graph G are expected to have an attribute demand
that indicates how much flow a node wants to send (negative
demand) or receive (positive demand). Note that the sum of the
demands should be 0 otherwise the problem in not feasible. If
this attribute is not present, a node is considered to have 0
demand. Default value: 'demand'.
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'.
weight : string
Edges of the graph G are expected to have an attribute weight
that indicates the cost incurred by sending one unit of flow on
that edge. If not present, the weight is considered to be 0.
Default value: 'weight'.
Returns
-------
flowCost : integer, float
Cost of a minimum cost flow satisfying all demands.
flowDict : dictionary
Dictionary of dictionaries keyed by nodes such that
flowDict[u][v] is the flow edge (u, v).
Raises
------
NetworkXError
This exception is raised if the input graph is not directed or
not connected.
NetworkXUnfeasible
This exception is raised in the following situations:
* The sum of the demands is not zero. Then, there is no
flow satisfying all demands.
* There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of
negative cost and infinite capacity. Then, the cost of a flow
satisfying all demands is unbounded below.
Notes
-----
This algorithm is not guaranteed to work if edge weights or demands
are floating point numbers (overflows and roundoff errors can
cause problems). As a workaround you can use integer numbers by
multiplying the relevant edge attributes by a convenient
constant factor (eg 100).
See also
--------
cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost
Examples
--------
A simple example of a min cost flow problem.
>>> G = nx.DiGraph()
>>> G.add_node("a", demand=-5)
>>> G.add_node("d", demand=5)
>>> G.add_edge("a", "b", weight=3, capacity=4)
>>> G.add_edge("a", "c", weight=6, capacity=10)
>>> G.add_edge("b", "d", weight=1, capacity=9)
>>> G.add_edge("c", "d", weight=2, capacity=5)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost
24
>>> flowDict
{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
The mincost flow algorithm can also be used to solve shortest path
problems. To find the shortest path between two nodes u and v,
give all edges an infinite capacity, give node u a demand of -1 and
node v a demand a 1. Then run the network simplex. The value of a
min cost flow will be the distance between u and v and edges
carrying positive flow will indicate the path.
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from(
... [
... ("s", "u", 10),
... ("s", "x", 5),
... ("u", "v", 1),
... ("u", "x", 2),
... ("v", "y", 1),
... ("x", "u", 3),
... ("x", "v", 5),
... ("x", "y", 2),
... ("y", "s", 7),
... ("y", "v", 6),
... ]
... )
>>> G.add_node("s", demand=-1)
>>> G.add_node("v", demand=1)
>>> flowCost, flowDict = nx.network_simplex(G)
>>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight")
True
>>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0])
[('s', 'x'), ('u', 'v'), ('x', 'u')]
>>> nx.shortest_path(G, "s", "v", weight="weight")
['s', 'x', 'u', 'v']
It is possible to change the name of the attributes used for the
algorithm.
>>> G = nx.DiGraph()
>>> G.add_node("p", spam=-4)
>>> G.add_node("q", spam=2)
>>> G.add_node("a", spam=-2)
>>> G.add_node("d", spam=-1)
>>> G.add_node("t", spam=2)
>>> G.add_node("w", spam=3)
>>> G.add_edge("p", "q", cost=7, vacancies=5)
>>> G.add_edge("p", "a", cost=1, vacancies=4)
>>> G.add_edge("q", "d", cost=2, vacancies=3)
>>> G.add_edge("t", "q", cost=1, vacancies=2)
>>> G.add_edge("a", "t", cost=2, vacancies=4)
>>> G.add_edge("d", "w", cost=3, vacancies=4)
>>> G.add_edge("t", "w", cost=4, vacancies=1)
>>> flowCost, flowDict = nx.network_simplex(
... G, demand="spam", capacity="vacancies", weight="cost"
... )
>>> flowCost
37
>>> flowDict
{'p': {'q': 2, 'a': 2}, 'q': {'d': 1}, 'a': {'t': 4}, 'd': {'w': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}
References
----------
.. [1] Z. Kiraly, P. Kovacs.
Efficient implementation of minimum-cost flow algorithms.
Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012.
.. [2] R. Barr, F. Glover, D. Klingman.
Enhancement of spanning tree labeling procedures for network
optimization.
INFOR 17(1):16--34. 1979.
"""
###########################################################################
# Problem essentials extraction and sanity check
###########################################################################
if len(G) == 0:
raise nx.NetworkXError("graph has no nodes")
multigraph = G.is_multigraph()
# extracting data essential to problem
DEAF = _DataEssentialsAndFunctions(
G, multigraph, demand=demand, capacity=capacity, weight=weight
)
###########################################################################
# Quick Error Detection
###########################################################################
inf = float("inf")
for u, d in zip(DEAF.node_list, DEAF.node_demands):
if abs(d) == inf:
raise nx.NetworkXError(f"node {u!r} has infinite demand")
for e, w in zip(DEAF.edge_indices, DEAF.edge_weights):
if abs(w) == inf:
raise nx.NetworkXError(f"edge {e!r} has infinite weight")
if not multigraph:
edges = nx.selfloop_edges(G, data=True)
else:
edges = nx.selfloop_edges(G, data=True, keys=True)
for e in edges:
if abs(e[-1].get(weight, 0)) == inf:
raise nx.NetworkXError(f"edge {e[:-1]!r} has infinite weight")
###########################################################################
# Quick Infeasibility Detection
###########################################################################
if sum(DEAF.node_demands) != 0:
raise nx.NetworkXUnfeasible("total node demand is not zero")
for e, c in zip(DEAF.edge_indices, DEAF.edge_capacities):
if c < 0:
raise nx.NetworkXUnfeasible(f"edge {e!r} has negative capacity")
if not multigraph:
edges = nx.selfloop_edges(G, data=True)
else:
edges = nx.selfloop_edges(G, data=True, keys=True)
for e in edges:
if e[-1].get(capacity, inf) < 0:
raise nx.NetworkXUnfeasible(f"edge {e[:-1]!r} has negative capacity")
###########################################################################
# Initialization
###########################################################################
# Add a dummy node -1 and connect all existing nodes to it with infinite-
# capacity dummy edges. Node -1 will serve as the root of the
# spanning tree of the network simplex method. The new edges will used to
# trivially satisfy the node demands and create an initial strongly
# feasible spanning tree.
for i, d in enumerate(DEAF.node_demands):
# Must be greater-than here. Zero-demand nodes must have
# edges pointing towards the root to ensure strong feasibility.
if d > 0:
DEAF.edge_sources.append(-1)
DEAF.edge_targets.append(i)
else:
DEAF.edge_sources.append(i)
DEAF.edge_targets.append(-1)
faux_inf = (
3
* max(
chain(
[
sum(c for c in DEAF.edge_capacities if c < inf),
sum(abs(w) for w in DEAF.edge_weights),
],
(abs(d) for d in DEAF.node_demands),
)
)
or 1
)
n = len(DEAF.node_list) # number of nodes
DEAF.edge_weights.extend(repeat(faux_inf, n))
DEAF.edge_capacities.extend(repeat(faux_inf, n))
# Construct the initial spanning tree.
DEAF.initialize_spanning_tree(n, faux_inf)
###########################################################################
# Pivot loop
###########################################################################
for i, p, q in DEAF.find_entering_edges():
Wn, We = DEAF.find_cycle(i, p, q)
j, s, t = DEAF.find_leaving_edge(Wn, We)
DEAF.augment_flow(Wn, We, DEAF.residual_capacity(j, s))
# Do nothing more if the entering edge is the same as the leaving edge.
if i != j:
if DEAF.parent[t] != s:
# Ensure that s is the parent of t.
s, t = t, s
if We.index(i) > We.index(j):
# Ensure that q is in the subtree rooted at t.
p, q = q, p
DEAF.remove_edge(s, t)
DEAF.make_root(q)
DEAF.add_edge(i, p, q)
DEAF.update_potentials(i, p, q)
###########################################################################
# Infeasibility and unboundedness detection
###########################################################################
if any(DEAF.edge_flow[i] != 0 for i in range(-n, 0)):
raise nx.NetworkXUnfeasible("no flow satisfies all node demands")
if any(DEAF.edge_flow[i] * 2 >= faux_inf for i in range(DEAF.edge_count)) or any(
e[-1].get(capacity, inf) == inf and e[-1].get(weight, 0) < 0
for e in nx.selfloop_edges(G, data=True)
):
raise nx.NetworkXUnbounded("negative cycle with infinite capacity found")
###########################################################################
# Flow cost calculation and flow dict construction
###########################################################################
del DEAF.edge_flow[DEAF.edge_count :]
flow_cost = sum(w * x for w, x in zip(DEAF.edge_weights, DEAF.edge_flow))
flow_dict = {n: {} for n in DEAF.node_list}
def add_entry(e):
"""Add a flow dict entry."""
d = flow_dict[e[0]]
for k in e[1:-2]:
try:
d = d[k]
except KeyError:
t = {}
d[k] = t
d = t
d[e[-2]] = e[-1]
DEAF.edge_sources = (
DEAF.node_list[s] for s in DEAF.edge_sources
) # Use original nodes.
DEAF.edge_targets = (
DEAF.node_list[t] for t in DEAF.edge_targets
) # Use original nodes.
if not multigraph:
for e in zip(
DEAF.edge_sources,
DEAF.edge_targets,
DEAF.edge_flow,
):
add_entry(e)
edges = G.edges(data=True)
else:
for e in zip(
DEAF.edge_sources,
DEAF.edge_targets,
DEAF.edge_keys,
DEAF.edge_flow,
):
add_entry(e)
edges = G.edges(data=True, keys=True)
for e in edges:
if e[0] != e[1]:
if e[-1].get(capacity, inf) == 0:
add_entry(e[:-1] + (0,))
else:
w = e[-1].get(weight, 0)
if w >= 0:
add_entry(e[:-1] + (0,))
else:
c = e[-1][capacity]
flow_cost += w * c
add_entry(e[:-1] + (c,))
return flow_cost, flow_dict
def bellman_ford(G,
inicio,
peso,
padre=None,
caminos=None,
dist=None,
meta=None,
heuristic=True):
for s in inicio:
if s not in G:
raise nx.NodeNotFound(f"Source {s} not in G")
if padre is None:
padre = {v: [] for v in inicio}
if dist is None:
dist = {v: 0 for v in inicio}
# Heuristic Storage setup. Note: use None because nodes cannot be None
nonexistent_edge = (None, None)
pred_edge = {v: None for v in inicio}
recent_update = {v: nonexistent_edge for v in inicio}
G_succ = G.succ if G.is_directed() else G.adj
inf = float("inf")
n = len(G)
count = {}
q = deque(inicio)
in_q = set(inicio)
while q:
u = q.popleft()
in_q.remove(u)
if all(pred_u not in in_q for pred_u in padre[u]):
dist_u = dist[u]
for v, e in G_succ[u].items():
dist_v = dist_u + peso(u, v, e)
if dist_v < dist.get(v, inf):
if heuristic:
if v in recent_update[u]:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
if v in pred_edge and pred_edge[v] == u:
recent_update[v] = recent_update[u]
else:
recent_update[v] = (u, v)
if v not in in_q:
q.append(v)
in_q.add(v)
count_v = count.get(v, 0) + 1
if count_v == n:
raise nx.NetworkXUnbounded(
"Negative cost cycle detected.")
count[v] = count_v
dist[v] = dist_v
padre[v] = [u]
pred_edge[v] = u
elif dist.get(v) is not None and dist_v == dist.get(v):
padre[v].append(u)
if caminos is not None:
inicios = set(inicio)
dsts = [meta] if meta is not None else padre
for dst in dsts:
gen = reconstruir_caminos_por_padres(inicios, dst, padre)
caminos[dst] = next(gen)
return dist
def bellman_ford(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.bellman_ford(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
dist = {source: 0}
pred = {source: None}
if len(G) == 1:
return pred, dist
return _bellman_ford_relaxation(G, pred, dist, [source], weight)
def dinitz_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
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
# Use an arbitrary high value as infinite. It is computed
# when building the residual network. Useful when checking
# for infinite capacity paths.
INF = R.graph['inf']
if cutoff is None:
cutoff = INF
R_succ = R.succ
def breath_first_search(G, R, s, t):
rank = {}
rank[s] = 0
queue = deque([s])
while queue:
if t in rank:
break
u = queue.popleft()
for v in R_succ[u]:
attr = R_succ[u][v]
if v not in rank and attr['capacity'] - attr['flow'] > 0:
rank[v] = rank[u] + 1
queue.append(v)
return rank
def depth_first_search(G, R, u, t, flow, rank):
if u == t:
return flow
for v in (n for n in R_succ[u] if n in rank and rank[n] == rank[u] + 1):
attr = R_succ[u][v]
if attr['capacity'] > attr['flow']:
min_flow = min(flow, attr['capacity'] - attr['flow'])
this_flow = depth_first_search(G, R, v, t, min_flow, rank)
if this_flow > 0:
R_succ[u][v]['flow'] += this_flow
R_succ[v][u]['flow'] -= this_flow
return this_flow
return 0
flow_value = 0
while flow_value < cutoff:
rank = breath_first_search(G, R, s, t)
if t not in rank:
break
while flow_value < cutoff:
blocking_flow = depth_first_search(G, R, s, t, INF, rank)
if blocking_flow * 2 > INF:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
elif blocking_flow == 0:
break
flow_value += blocking_flow
R.graph['flow_value'] = flow_value
return R
def goldberg_radzik(G, source, weight='weight'):
"""Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of O(mn) where n is the number of
nodes and m is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NetworkXUnbounded
If the (di)graph contains a negative cost (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative cost (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cost cycle.
Examples
--------
>>> import networkx as nx
>>> G = nx.path_graph(5, create_using = nx.DiGraph())
>>> pred, dist = nx.goldberg_radzik(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> from nose.tools import assert_raises
>>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
>>> G[1][2]['weight'] = -7
>>> assert_raises(nx.NetworkXUnbounded, nx.goldberg_radzik, G, 0)
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative cost (di)cycle, it
will not be detected.
"""
if source not in G:
raise KeyError("Node %s is not found in the graph" % source)
for u, v, attr in G.selfloop_edges(data=True):
if attr.get(weight, 1) < 0:
raise nx.NetworkXUnbounded("Negative cost cycle detected.")
if len(G) == 1:
return {source: None}, {source: 0}
if G.is_multigraph():
def get_weight(edge_dict):
return min(attr.get(weight, 1) for attr in edge_dict.values())
else:
def get_weight(edge_dict):
return edge_dict.get(weight, 1)
if G.is_directed():
G_succ = G.succ
else:
G_succ = G.adj
inf = float('inf')
d = dict((u, inf) for u in G)
d[source] = 0
pred = {source: None}
def topo_sort(relabeled):
"""Topologically sort nodes relabeled in the previous round and detect
negative cycles.
"""
# List of nodes to scan in this round. Denoted by A in Goldberg and
# Radzik's paper.
to_scan = []
# In the DFS in the loop below, neg_count records for each node the
# number of edges of negative reduced costs on the path from a DFS root
# to the node in the DFS forest. The reduced cost of an edge (u, v) is
# defined as d[u] + weight[u][v] - d[v].
#
# neg_count also doubles as the DFS visit marker array.
neg_count = {}
for u in relabeled:
# Skip visited nodes.
if u in neg_count:
continue
d_u = d[u]
# Skip nodes without out-edges of negative reduced costs.
if all(d_u + get_weight(e) >= d[v] for v, e in G_succ[u].items()):
continue
# Nonrecursive DFS that inserts nodes reachable from u via edges of
# nonpositive reduced costs into to_scan in (reverse) topological
# order.
stack = [(u, iter(G_succ[u].items()))]
in_stack = set([u])
neg_count[u] = 0
while stack:
u, it = stack[-1]
try:
v, e = next(it)
except StopIteration:
to_scan.append(u)
stack.pop()
in_stack.remove(u)
continue
t = d[u] + get_weight(e)
d_v = d[v]
if t <= d_v:
is_neg = t < d_v
d[v] = t
pred[v] = u
if v not in neg_count:
neg_count[v] = neg_count[u] + int(is_neg)
stack.append((v, iter(G_succ[v].items())))
in_stack.add(v)
elif (v in in_stack
and neg_count[u] + int(is_neg) > neg_count[v]):
# (u, v) is a back edge, and the cycle formed by the
# path v to u and (u, v) contains at least one edge of
# negative reduced cost. The cycle must be of negative
# cost.
raise nx.NetworkXUnbounded(
'Negative cost cycle detected.')
to_scan.reverse()
return to_scan
def relax(to_scan):
"""Relax out-edges of relabeled nodes.
"""
relabeled = set()
# Scan nodes in to_scan in topological order and relax incident
# out-edges. Add the relabled nodes to labeled.
for u in to_scan:
d_u = d[u]
for v, e in G_succ[u].items():
w_e = get_weight(e)
if d_u + w_e < d[v]:
d[v] = d_u + w_e
pred[v] = u
relabeled.add(v)
return relabeled
# Set of nodes relabled in the last round of scan operations. Denoted by B
# in Goldberg and Radzik's paper.
relabeled = set([source])
while relabeled:
to_scan = topo_sort(relabeled)
relabeled = relax(to_scan)
d = dict((u, d[u]) for u in pred)
return pred, d
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
if not is_lowest_hierarchy:
logger.debug("Trying to resolve the routing restrictions")
path_set = set()
it = iter(path)
u = next(it)
for v in it:
if u == "super_ingress" or v == "super_egress":
u = v
continue
# translate the tuple to a path identifier via the path map
path_id = next((path_key for path_key, p in path_map.items()
if p == (u, v) or p == (v, u)), None)
logger.debug("Path {} resolves to path id {}".format((u, v),
path_id))
if path_id == None:
continue
if path_map[path_id] == (u, v):
path_set.add(path_id)
else:
logger.debug(
"Conflict! edge {} was added as {}. Trying to deduce the path heuristically."
.format((u, v), (v, u)))
path_id = utils.get_backward_edge_id(
forward_edge_id=path_id)
logger.debug("Resolved {} to {}".format((v, u), path_id))
path_set.add(path_id)
u = v
logger.debug(
"Edmonds karp found a new path: {} with flow: {}".format(
path, flow))
for routing_restriction in routing_restrictions:
intersection = set(routing_restriction.paths) & path_set
if len(intersection) > 2:
logger.warn(
"Panic! The path {0} contains paths {1}, which are part of the same routing restriction {2}."
"Hence, they use the same bottleneck. The flow will be reduced now."
.format(path_set, intersection,
routing_restriction.identifier))
raise RuntimeError()
for path_id in path_set:
if path_id in routing_restriction.paths:
intersection = set(routing_restriction.paths) & set(
path_map.keys())
logger.debug(
"Path {} was part of the routing restriction {}.".
format(path_id, routing_restriction.identifier))
logger.debug(
"Thus, the flow of paths {} will be reduced by {}".
format(intersection, flow))
for p_prime in intersection:
if path_id != p_prime:
(src, dst) = path_map[p_prime]
logger.debug(
"Edge {} will be charged with {} cap".
format((src, dst), flow))
R_succ[src][dst]['capacity'] = max(
R_succ[src][dst]['capacity'] - flow, 0)
logger.debug(
"New flow of edge {} is {}".format(
(src, dst),
R_succ[src][dst]['capacity']))
return flow
