"""
Algorithms for finding k-edge-connected components and subgraphs.
A k-edge-connected component (k-edge-cc) is a maximal set of nodes in G, such
that all pairs of node have an edge-connectivity of at least k.
A k-edge-connected subgraph (k-edge-subgraph) is a maximal set of nodes in G,
such that the subgraph of G defined by the nodes has an edge-connectivity at
least k.
"""
import itertools as it
from functools import partial
import networkx as nx
from networkx.utils import arbitrary_element, not_implemented_for
__all__ = [
"k_edge_components",
"k_edge_subgraphs",
"bridge_components",
"EdgeComponentAuxGraph",
]
[docs]
@not_implemented_for("multigraph")
@nx._dispatchable
def k_edge_components(G, k):
"""Generates nodes in each maximal k-edge-connected component in G.
Parameters
----------
G : NetworkX graph
k : Integer
Desired edge connectivity
Returns
-------
k_edge_components : a generator of k-edge-ccs. Each set of returned nodes
will have k-edge-connectivity in the graph G.
See Also
--------
:func:`local_edge_connectivity`
:func:`k_edge_subgraphs` : similar to this function, but the subgraph
defined by the nodes must also have k-edge-connectivity.
:func:`k_components` : similar to this function, but uses node-connectivity
instead of edge-connectivity
Raises
------
NetworkXNotImplemented
If the input graph is a multigraph.
ValueError:
If k is less than 1
Notes
-----
Attempts to use the most efficient implementation available based on k.
If k=1, this is simply connected components for directed graphs and
connected components for undirected graphs.
If k=2 on an efficient bridge connected component algorithm from _[1] is
run based on the chain decomposition.
Otherwise, the algorithm from _[2] is used.
Examples
--------
>>> import itertools as it
>>> from networkx.utils import pairwise
>>> paths = [
... (1, 2, 4, 3, 1, 4),
... (5, 6, 7, 8, 5, 7, 8, 6),
... ]
>>> G = nx.Graph()
>>> G.add_nodes_from(it.chain(*paths))
>>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
>>> # note this returns {1, 4} unlike k_edge_subgraphs
>>> sorted(map(sorted, nx.k_edge_components(G, k=3)))
[[1, 4], [2], [3], [5, 6, 7, 8]]
References
----------
.. [1] https://en.wikipedia.org/wiki/Bridge_%28graph_theory%29
.. [2] Wang, Tianhao, et al. (2015) A simple algorithm for finding all
k-edge-connected components.
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264
"""
# Compute k-edge-ccs using the most efficient algorithms available.
if k < 1:
raise ValueError("k cannot be less than 1")
if G.is_directed():
if k == 1:
return nx.strongly_connected_components(G)
else:
# TODO: investigate https://arxiv.org/abs/1412.6466 for k=2
aux_graph = EdgeComponentAuxGraph.construct(G)
return aux_graph.k_edge_components(k)
else:
if k == 1:
return nx.connected_components(G)
elif k == 2:
return bridge_components(G)
else:
aux_graph = EdgeComponentAuxGraph.construct(G)
return aux_graph.k_edge_components(k)
[docs]
@not_implemented_for("multigraph")
@nx._dispatchable
def k_edge_subgraphs(G, k):
"""Generates nodes in each maximal k-edge-connected subgraph in G.
Parameters
----------
G : NetworkX graph
k : Integer
Desired edge connectivity
Returns
-------
k_edge_subgraphs : a generator of k-edge-subgraphs
Each k-edge-subgraph is a maximal set of nodes that defines a subgraph
of G that is k-edge-connected.
See Also
--------
:func:`edge_connectivity`
:func:`k_edge_components` : similar to this function, but nodes only
need to have k-edge-connectivity within the graph G and the subgraphs
might not be k-edge-connected.
Raises
------
NetworkXNotImplemented
If the input graph is a multigraph.
ValueError:
If k is less than 1
Notes
-----
Attempts to use the most efficient implementation available based on k.
If k=1, or k=2 and the graph is undirected, then this simply calls
`k_edge_components`. Otherwise the algorithm from _[1] is used.
Examples
--------
>>> import itertools as it
>>> from networkx.utils import pairwise
>>> paths = [
... (1, 2, 4, 3, 1, 4),
... (5, 6, 7, 8, 5, 7, 8, 6),
... ]
>>> G = nx.Graph()
>>> G.add_nodes_from(it.chain(*paths))
>>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
>>> # note this does not return {1, 4} unlike k_edge_components
>>> sorted(map(sorted, nx.k_edge_subgraphs(G, k=3)))
[[1], [2], [3], [4], [5, 6, 7, 8]]
References
----------
.. [1] Zhou, Liu, et al. (2012) Finding maximal k-edge-connected subgraphs
from a large graph. ACM International Conference on Extending Database
Technology 2012 480-–491.
https://openproceedings.org/2012/conf/edbt/ZhouLYLCL12.pdf
"""
if k < 1:
raise ValueError("k cannot be less than 1")
if G.is_directed():
if k <= 1:
# For directed graphs ,
# When k == 1, k-edge-ccs and k-edge-subgraphs are the same
return k_edge_components(G, k)
else:
return _k_edge_subgraphs_nodes(G, k)
else:
if k <= 2:
# For undirected graphs,
# when k <= 2, k-edge-ccs and k-edge-subgraphs are the same
return k_edge_components(G, k)
else:
return _k_edge_subgraphs_nodes(G, k)
def _k_edge_subgraphs_nodes(G, k):
"""Helper to get the nodes from the subgraphs.
This allows k_edge_subgraphs to return a generator.
"""
for C in general_k_edge_subgraphs(G, k):
yield set(C.nodes())
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def bridge_components(G):
"""Finds all bridge-connected components G.
Parameters
----------
G : NetworkX undirected graph
Returns
-------
bridge_components : a generator of 2-edge-connected components
See Also
--------
:func:`k_edge_subgraphs` : this function is a special case for an
undirected graph where k=2.
:func:`biconnected_components` : similar to this function, but is defined
using 2-node-connectivity instead of 2-edge-connectivity.
Raises
------
NetworkXNotImplemented
If the input graph is directed or a multigraph.
Notes
-----
Bridge-connected components are also known as 2-edge-connected components.
Examples
--------
>>> # The barbell graph with parameter zero has a single bridge
>>> G = nx.barbell_graph(5, 0)
>>> from networkx.algorithms.connectivity.edge_kcomponents import bridge_components
>>> sorted(map(sorted, bridge_components(G)))
[[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]
"""
H = G.copy()
H.remove_edges_from(nx.bridges(G))
yield from nx.connected_components(H)
[docs]
class EdgeComponentAuxGraph:
r"""A simple algorithm to find all k-edge-connected components in a graph.
Constructing the auxiliary graph (which may take some time) allows for the
k-edge-ccs to be found in linear time for arbitrary k.
Notes
-----
This implementation is based on [1]_. The idea is to construct an auxiliary
graph from which the k-edge-ccs can be extracted in linear time. The
auxiliary graph is constructed in $O(|V|\cdot F)$ operations, where F is the
complexity of max flow. Querying the components takes an additional $O(|V|)$
operations. This algorithm can be slow for large graphs, but it handles an
arbitrary k and works for both directed and undirected inputs.
The undirected case for k=1 is exactly connected components.
The undirected case for k=2 is exactly bridge connected components.
The directed case for k=1 is exactly strongly connected components.
References
----------
.. [1] Wang, Tianhao, et al. (2015) A simple algorithm for finding all
k-edge-connected components.
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264
Examples
--------
>>> import itertools as it
>>> from networkx.utils import pairwise
>>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph
>>> # Build an interesting graph with multiple levels of k-edge-ccs
>>> paths = [
... (1, 2, 3, 4, 1, 3, 4, 2), # a 3-edge-cc (a 4 clique)
... (5, 6, 7, 5), # a 2-edge-cc (a 3 clique)
... (1, 5), # combine first two ccs into a 1-edge-cc
... (0,), # add an additional disconnected 1-edge-cc
... ]
>>> G = nx.Graph()
>>> G.add_nodes_from(it.chain(*paths))
>>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
>>> # Constructing the AuxGraph takes about O(n ** 4)
>>> aux_graph = EdgeComponentAuxGraph.construct(G)
>>> # Once constructed, querying takes O(n)
>>> sorted(map(sorted, aux_graph.k_edge_components(k=1)))
[[0], [1, 2, 3, 4, 5, 6, 7]]
>>> sorted(map(sorted, aux_graph.k_edge_components(k=2)))
[[0], [1, 2, 3, 4], [5, 6, 7]]
>>> sorted(map(sorted, aux_graph.k_edge_components(k=3)))
[[0], [1, 2, 3, 4], [5], [6], [7]]
>>> sorted(map(sorted, aux_graph.k_edge_components(k=4)))
[[0], [1], [2], [3], [4], [5], [6], [7]]
The auxiliary graph is primarily used for k-edge-ccs but it
can also speed up the queries of k-edge-subgraphs by refining the
search space.
>>> import itertools as it
>>> from networkx.utils import pairwise
>>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph
>>> paths = [
... (1, 2, 4, 3, 1, 4),
... ]
>>> G = nx.Graph()
>>> G.add_nodes_from(it.chain(*paths))
>>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
>>> aux_graph = EdgeComponentAuxGraph.construct(G)
>>> sorted(map(sorted, aux_graph.k_edge_subgraphs(k=3)))
[[1], [2], [3], [4]]
>>> sorted(map(sorted, aux_graph.k_edge_components(k=3)))
[[1, 4], [2], [3]]
"""
# @not_implemented_for('multigraph') # TODO: fix decor for classmethods
[docs]
@classmethod
def construct(EdgeComponentAuxGraph, G):
"""Builds an auxiliary graph encoding edge-connectivity between nodes.
Notes
-----
Given G=(V, E), initialize an empty auxiliary graph A.
Choose an arbitrary source node s. Initialize a set N of available
nodes (that can be used as the sink). The algorithm picks an
arbitrary node t from N - {s}, and then computes the minimum st-cut
(S, T) with value w. If G is directed the minimum of the st-cut or
the ts-cut is used instead. Then, the edge (s, t) is added to the
auxiliary graph with weight w. The algorithm is called recursively
first using S as the available nodes and s as the source, and then
using T and t. Recursion stops when the source is the only available
node.
Parameters
----------
G : NetworkX graph
"""
# workaround for classmethod decorator
not_implemented_for("multigraph")(lambda G: G)(G)
def _recursive_build(H, A, source, avail):
# Terminate once the flow has been compute to every node.
if {source} == avail:
return
# pick an arbitrary node as the sink
sink = arbitrary_element(avail - {source})
# find the minimum cut and its weight
value, (S, T) = nx.minimum_cut(H, source, sink)
if H.is_directed():
# check if the reverse direction has a smaller cut
value_, (T_, S_) = nx.minimum_cut(H, sink, source)
if value_ < value:
value, S, T = value_, S_, T_
# add edge with weight of cut to the aux graph
A.add_edge(source, sink, weight=value)
# recursively call until all but one node is used
_recursive_build(H, A, source, avail.intersection(S))
_recursive_build(H, A, sink, avail.intersection(T))
# Copy input to ensure all edges have unit capacity
H = G.__class__()
H.add_nodes_from(G.nodes())
H.add_edges_from(G.edges(), capacity=1)
# A is the auxiliary graph to be constructed
# It is a weighted undirected tree
A = nx.Graph()
# Pick an arbitrary node as the source
if H.number_of_nodes() > 0:
source = arbitrary_element(H.nodes())
# Initialize a set of elements that can be chosen as the sink
avail = set(H.nodes())
# This constructs A
_recursive_build(H, A, source, avail)
# This class is a container the holds the auxiliary graph A and
# provides access the k_edge_components function.
self = EdgeComponentAuxGraph()
self.A = A
self.H = H
return self
[docs]
def k_edge_components(self, k):
"""Queries the auxiliary graph for k-edge-connected components.
Parameters
----------
k : Integer
Desired edge connectivity
Returns
-------
k_edge_components : a generator of k-edge-ccs
Notes
-----
Given the auxiliary graph, the k-edge-connected components can be
determined in linear time by removing all edges with weights less than
k from the auxiliary graph. The resulting connected components are the
k-edge-ccs in the original graph.
"""
if k < 1:
raise ValueError("k cannot be less than 1")
A = self.A
# "traverse the auxiliary graph A and delete all edges with weights less
# than k"
aux_weights = nx.get_edge_attributes(A, "weight")
# Create a relevant graph with the auxiliary edges with weights >= k
R = nx.Graph()
R.add_nodes_from(A.nodes())
R.add_edges_from(e for e, w in aux_weights.items() if w >= k)
# Return the nodes that are k-edge-connected in the original graph
yield from nx.connected_components(R)
[docs]
def k_edge_subgraphs(self, k):
"""Queries the auxiliary graph for k-edge-connected subgraphs.
Parameters
----------
k : Integer
Desired edge connectivity
Returns
-------
k_edge_subgraphs : a generator of k-edge-subgraphs
Notes
-----
Refines the k-edge-ccs into k-edge-subgraphs. The running time is more
than $O(|V|)$.
For single values of k it is faster to use `nx.k_edge_subgraphs`.
But for multiple values of k, it can be faster to build AuxGraph and
then use this method.
"""
if k < 1:
raise ValueError("k cannot be less than 1")
H = self.H
A = self.A
# "traverse the auxiliary graph A and delete all edges with weights less
# than k"
aux_weights = nx.get_edge_attributes(A, "weight")
# Create a relevant graph with the auxiliary edges with weights >= k
R = nx.Graph()
R.add_nodes_from(A.nodes())
R.add_edges_from(e for e, w in aux_weights.items() if w >= k)
# Return the components whose subgraphs are k-edge-connected
for cc in nx.connected_components(R):
if len(cc) < k:
# Early return optimization
for node in cc:
yield {node}
else:
# Call subgraph solution to refine the results
C = H.subgraph(cc)
yield from k_edge_subgraphs(C, k)
def _low_degree_nodes(G, k, nbunch=None):
"""Helper for finding nodes with degree less than k."""
# Nodes with degree less than k cannot be k-edge-connected.
if G.is_directed():
# Consider both in and out degree in the directed case
seen = set()
for node, degree in G.out_degree(nbunch):
if degree < k:
seen.add(node)
yield node
for node, degree in G.in_degree(nbunch):
if node not in seen and degree < k:
seen.add(node)
yield node
else:
# Only the degree matters in the undirected case
for node, degree in G.degree(nbunch):
if degree < k:
yield node
def _high_degree_components(G, k):
"""Helper for filtering components that can't be k-edge-connected.
Removes and generates each node with degree less than k. Then generates
remaining components where all nodes have degree at least k.
"""
# Iteratively remove parts of the graph that are not k-edge-connected
H = G.copy()
singletons = set(_low_degree_nodes(H, k))
while singletons:
# Only search neighbors of removed nodes
nbunch = set(it.chain.from_iterable(map(H.neighbors, singletons)))
nbunch.difference_update(singletons)
H.remove_nodes_from(singletons)
for node in singletons:
yield {node}
singletons = set(_low_degree_nodes(H, k, nbunch))
# Note: remaining connected components may not be k-edge-connected
if G.is_directed():
yield from nx.strongly_connected_components(H)
else:
yield from nx.connected_components(H)
@nx._dispatchable(returns_graph=True)
def general_k_edge_subgraphs(G, k):
"""General algorithm to find all maximal k-edge-connected subgraphs in `G`.
Parameters
----------
G : nx.Graph
Graph in which all maximal k-edge-connected subgraphs will be found.
k : int
Yields
------
k_edge_subgraphs : Graph instances that are k-edge-subgraphs
Each k-edge-subgraph contains a maximal set of nodes that defines a
subgraph of `G` that is k-edge-connected.
Notes
-----
Implementation of the basic algorithm from [1]_. The basic idea is to find
a global minimum cut of the graph. If the cut value is at least k, then the
graph is a k-edge-connected subgraph and can be added to the results.
Otherwise, the cut is used to split the graph in two and the procedure is
applied recursively. If the graph is just a single node, then it is also
added to the results. At the end, each result is either guaranteed to be
a single node or a subgraph of G that is k-edge-connected.
This implementation contains optimizations for reducing the number of calls
to max-flow, but there are other optimizations in [1]_ that could be
implemented.
References
----------
.. [1] Zhou, Liu, et al. (2012) Finding maximal k-edge-connected subgraphs
from a large graph. ACM International Conference on Extending Database
Technology 2012 480-–491.
https://openproceedings.org/2012/conf/edbt/ZhouLYLCL12.pdf
Examples
--------
>>> from networkx.utils import pairwise
>>> paths = [
... (11, 12, 13, 14, 11, 13, 14, 12), # a 4-clique
... (21, 22, 23, 24, 21, 23, 24, 22), # another 4-clique
... # connect the cliques with high degree but low connectivity
... (50, 13),
... (12, 50, 22),
... (13, 102, 23),
... (14, 101, 24),
... ]
>>> G = nx.Graph(it.chain(*[pairwise(path) for path in paths]))
>>> sorted(len(k_sg) for k_sg in k_edge_subgraphs(G, k=3))
[1, 1, 1, 4, 4]
"""
if k < 1:
raise ValueError("k cannot be less than 1")
# Node pruning optimization (incorporates early return)
# find_ccs is either connected_components/strongly_connected_components
find_ccs = partial(_high_degree_components, k=k)
# Quick return optimization
if G.number_of_nodes() < k:
for node in G.nodes():
yield G.subgraph([node]).copy()
return
# Intermediate results
R0 = {G.subgraph(cc).copy() for cc in find_ccs(G)}
# Subdivide CCs in the intermediate results until they are k-conn
while R0:
G1 = R0.pop()
if G1.number_of_nodes() == 1:
yield G1
else:
# Find a global minimum cut
cut_edges = nx.minimum_edge_cut(G1)
cut_value = len(cut_edges)
if cut_value < k:
# G1 is not k-edge-connected, so subdivide it
G1.remove_edges_from(cut_edges)
for cc in find_ccs(G1):
R0.add(G1.subgraph(cc).copy())
else:
# Otherwise we found a k-edge-connected subgraph
yield G1
