605 lines
22 KiB
Python
605 lines
22 KiB
Python
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# -*- coding: utf-8 -*-
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"""
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Flow based cut algorithms
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"""
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import itertools
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import networkx as nx
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# Define the default maximum flow function to use in all flow based
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# cut algorithms.
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from networkx.algorithms.flow import edmonds_karp
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from networkx.algorithms.flow import build_residual_network
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default_flow_func = edmonds_karp
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from .utils import (build_auxiliary_node_connectivity,
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build_auxiliary_edge_connectivity)
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__author__ = '\n'.join(['Jordi Torrents <jtorrents@milnou.net>'])
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__all__ = ['minimum_st_node_cut',
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'minimum_node_cut',
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'minimum_st_edge_cut',
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'minimum_edge_cut']
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def minimum_st_edge_cut(G, s, t, flow_func=None, auxiliary=None,
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residual=None):
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"""Returns the edges of the cut-set of a minimum (s, t)-cut.
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This function returns the set of edges of minimum cardinality that,
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if removed, would destroy all paths among source and target in G.
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Edge weights are not considered. See :meth:`minimum_cut` for
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computing minimum cuts considering edge weights.
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Parameters
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----------
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G : NetworkX graph
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s : node
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Source node for the flow.
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t : node
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Sink node for the flow.
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auxiliary : NetworkX DiGraph
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Auxiliary digraph to compute flow based node connectivity. It has
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to have a graph attribute called mapping with a dictionary mapping
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node names in G and in the auxiliary digraph. If provided
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it will be reused instead of recreated. Default value: None.
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flow_func : function
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A function for computing the maximum flow among a pair of nodes.
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The function has to accept at least three parameters: a Digraph,
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a source node, and a target node. And return a residual network
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that follows NetworkX conventions (see :meth:`maximum_flow` for
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details). If flow_func is None, the default maximum flow function
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(:meth:`edmonds_karp`) is used. See :meth:`node_connectivity` for
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details. The choice of the default function may change from version
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to version and should not be relied on. Default value: None.
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residual : NetworkX DiGraph
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Residual network to compute maximum flow. If provided it will be
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reused instead of recreated. Default value: None.
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Returns
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-------
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cutset : set
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Set of edges that, if removed from the graph, will disconnect it.
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See also
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--------
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:meth:`minimum_cut`
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:meth:`minimum_node_cut`
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:meth:`minimum_edge_cut`
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:meth:`stoer_wagner`
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:meth:`node_connectivity`
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:meth:`edge_connectivity`
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:meth:`maximum_flow`
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:meth:`edmonds_karp`
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:meth:`preflow_push`
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:meth:`shortest_augmenting_path`
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Examples
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--------
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This function is not imported in the base NetworkX namespace, so you
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have to explicitly import it from the connectivity package:
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>>> from networkx.algorithms.connectivity import minimum_st_edge_cut
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We use in this example the platonic icosahedral graph, which has edge
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connectivity 5.
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>>> G = nx.icosahedral_graph()
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>>> len(minimum_st_edge_cut(G, 0, 6))
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5
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If you need to compute local edge cuts on several pairs of
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nodes in the same graph, it is recommended that you reuse the
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data structures that NetworkX uses in the computation: the
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auxiliary digraph for edge connectivity, and the residual
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network for the underlying maximum flow computation.
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Example of how to compute local edge cuts among all pairs of
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nodes of the platonic icosahedral graph reusing the data
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structures.
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>>> import itertools
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>>> # You also have to explicitly import the function for
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>>> # building the auxiliary digraph from the connectivity package
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>>> from networkx.algorithms.connectivity import (
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... build_auxiliary_edge_connectivity)
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>>> H = build_auxiliary_edge_connectivity(G)
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>>> # And the function for building the residual network from the
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>>> # flow package
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>>> from networkx.algorithms.flow import build_residual_network
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>>> # Note that the auxiliary digraph has an edge attribute named capacity
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>>> R = build_residual_network(H, 'capacity')
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>>> result = dict.fromkeys(G, dict())
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>>> # Reuse the auxiliary digraph and the residual network by passing them
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>>> # as parameters
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>>> for u, v in itertools.combinations(G, 2):
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... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R))
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... result[u][v] = k
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>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
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True
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You can also use alternative flow algorithms for computing edge
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cuts. For instance, in dense networks the algorithm
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:meth:`shortest_augmenting_path` will usually perform better than
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the default :meth:`edmonds_karp` which is faster for sparse
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networks with highly skewed degree distributions. Alternative flow
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functions have to be explicitly imported from the flow package.
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>>> from networkx.algorithms.flow import shortest_augmenting_path
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>>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path))
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5
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"""
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if flow_func is None:
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flow_func = default_flow_func
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if auxiliary is None:
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H = build_auxiliary_edge_connectivity(G)
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else:
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H = auxiliary
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kwargs = dict(capacity='capacity', flow_func=flow_func, residual=residual)
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cut_value, partition = nx.minimum_cut(H, s, t, **kwargs)
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reachable, non_reachable = partition
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# Any edge in the original graph linking the two sets in the
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# partition is part of the edge cutset
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cutset = set()
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for u, nbrs in ((n, G[n]) for n in reachable):
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cutset.update((u, v) for v in nbrs if v in non_reachable)
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return cutset
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def minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None):
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r"""Returns a set of nodes of minimum cardinality that disconnect source
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from target in G.
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This function returns the set of nodes of minimum cardinality that,
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if removed, would destroy all paths among source and target in G.
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Parameters
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----------
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G : NetworkX graph
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s : node
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Source node.
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t : node
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Target node.
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flow_func : function
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A function for computing the maximum flow among a pair of nodes.
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The function has to accept at least three parameters: a Digraph,
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a source node, and a target node. And return a residual network
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that follows NetworkX conventions (see :meth:`maximum_flow` for
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details). If flow_func is None, the default maximum flow function
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(:meth:`edmonds_karp`) is used. See below for details. The choice
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of the default function may change from version to version and
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should not be relied on. Default value: None.
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auxiliary : NetworkX DiGraph
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Auxiliary digraph to compute flow based node connectivity. It has
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to have a graph attribute called mapping with a dictionary mapping
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node names in G and in the auxiliary digraph. If provided
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it will be reused instead of recreated. Default value: None.
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residual : NetworkX DiGraph
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Residual network to compute maximum flow. If provided it will be
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reused instead of recreated. Default value: None.
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Returns
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-------
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cutset : set
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Set of nodes that, if removed, would destroy all paths between
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source and target in G.
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Examples
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--------
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This function is not imported in the base NetworkX namespace, so you
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have to explicitly import it from the connectivity package:
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>>> from networkx.algorithms.connectivity import minimum_st_node_cut
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We use in this example the platonic icosahedral graph, which has node
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connectivity 5.
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>>> G = nx.icosahedral_graph()
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>>> len(minimum_st_node_cut(G, 0, 6))
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5
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If you need to compute local st cuts between several pairs of
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nodes in the same graph, it is recommended that you reuse the
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data structures that NetworkX uses in the computation: the
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auxiliary digraph for node connectivity and node cuts, and the
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residual network for the underlying maximum flow computation.
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Example of how to compute local st node cuts reusing the data
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structures:
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>>> # You also have to explicitly import the function for
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>>> # building the auxiliary digraph from the connectivity package
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>>> from networkx.algorithms.connectivity import (
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... build_auxiliary_node_connectivity)
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>>> H = build_auxiliary_node_connectivity(G)
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>>> # And the function for building the residual network from the
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>>> # flow package
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>>> from networkx.algorithms.flow import build_residual_network
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>>> # Note that the auxiliary digraph has an edge attribute named capacity
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>>> R = build_residual_network(H, 'capacity')
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>>> # Reuse the auxiliary digraph and the residual network by passing them
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>>> # as parameters
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>>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R))
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5
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You can also use alternative flow algorithms for computing minimum st
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node cuts. For instance, in dense networks the algorithm
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:meth:`shortest_augmenting_path` will usually perform better than
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the default :meth:`edmonds_karp` which is faster for sparse
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networks with highly skewed degree distributions. Alternative flow
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functions have to be explicitly imported from the flow package.
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>>> from networkx.algorithms.flow import shortest_augmenting_path
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>>> len(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path))
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5
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Notes
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-----
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This is a flow based implementation of minimum node cut. The algorithm
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is based in solving a number of maximum flow computations to determine
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the capacity of the minimum cut on an auxiliary directed network that
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corresponds to the minimum node cut of G. It handles both directed
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and undirected graphs. This implementation is based on algorithm 11
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in [1]_.
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See also
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--------
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:meth:`minimum_node_cut`
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:meth:`minimum_edge_cut`
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:meth:`stoer_wagner`
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:meth:`node_connectivity`
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:meth:`edge_connectivity`
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:meth:`maximum_flow`
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:meth:`edmonds_karp`
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:meth:`preflow_push`
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:meth:`shortest_augmenting_path`
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References
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----------
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.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
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http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
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"""
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if auxiliary is None:
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H = build_auxiliary_node_connectivity(G)
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else:
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H = auxiliary
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mapping = H.graph.get('mapping', None)
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if mapping is None:
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raise nx.NetworkXError('Invalid auxiliary digraph.')
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if G.has_edge(s, t) or G.has_edge(t, s):
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return []
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kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H)
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# The edge cut in the auxiliary digraph corresponds to the node cut in the
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# original graph.
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edge_cut = minimum_st_edge_cut(H, '%sB' % mapping[s], '%sA' % mapping[t],
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**kwargs)
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# Each node in the original graph maps to two nodes of the auxiliary graph
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node_cut = set(H.nodes[node]['id'] for edge in edge_cut for node in edge)
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return node_cut - set([s, t])
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def minimum_node_cut(G, s=None, t=None, flow_func=None):
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r"""Returns a set of nodes of minimum cardinality that disconnects G.
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If source and target nodes are provided, this function returns the
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set of nodes of minimum cardinality that, if removed, would destroy
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all paths among source and target in G. If not, it returns a set
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of nodes of minimum cardinality that disconnects G.
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Parameters
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----------
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G : NetworkX graph
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s : node
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Source node. Optional. Default value: None.
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t : node
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Target node. Optional. Default value: None.
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flow_func : function
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A function for computing the maximum flow among a pair of nodes.
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The function has to accept at least three parameters: a Digraph,
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a source node, and a target node. And return a residual network
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that follows NetworkX conventions (see :meth:`maximum_flow` for
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details). If flow_func is None, the default maximum flow function
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(:meth:`edmonds_karp`) is used. See below for details. The
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choice of the default function may change from version
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to version and should not be relied on. Default value: None.
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Returns
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-------
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cutset : set
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Set of nodes that, if removed, would disconnect G. If source
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and target nodes are provided, the set contains the nodes that
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if removed, would destroy all paths between source and target.
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Examples
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--------
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>>> # Platonic icosahedral graph has node connectivity 5
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>>> G = nx.icosahedral_graph()
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>>> node_cut = nx.minimum_node_cut(G)
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>>> len(node_cut)
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5
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You can use alternative flow algorithms for the underlying maximum
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flow computation. In dense networks the algorithm
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:meth:`shortest_augmenting_path` will usually perform better
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than the default :meth:`edmonds_karp`, which is faster for
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sparse networks with highly skewed degree distributions. Alternative
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flow functions have to be explicitly imported from the flow package.
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>>> from networkx.algorithms.flow import shortest_augmenting_path
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>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
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True
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If you specify a pair of nodes (source and target) as parameters,
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this function returns a local st node cut.
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>>> len(nx.minimum_node_cut(G, 3, 7))
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5
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If you need to perform several local st cuts among different
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pairs of nodes on the same graph, it is recommended that you reuse
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the data structures used in the maximum flow computations. See
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:meth:`minimum_st_node_cut` for details.
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Notes
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-----
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This is a flow based implementation of minimum node cut. The algorithm
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||
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is based in solving a number of maximum flow computations to determine
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the capacity of the minimum cut on an auxiliary directed network that
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corresponds to the minimum node cut of G. It handles both directed
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|
and undirected graphs. This implementation is based on algorithm 11
|
||
|
|
in [1]_.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
:meth:`minimum_st_node_cut`
|
||
|
|
:meth:`minimum_cut`
|
||
|
|
:meth:`minimum_edge_cut`
|
||
|
|
:meth:`stoer_wagner`
|
||
|
|
:meth:`node_connectivity`
|
||
|
|
:meth:`edge_connectivity`
|
||
|
|
:meth:`maximum_flow`
|
||
|
|
:meth:`edmonds_karp`
|
||
|
|
:meth:`preflow_push`
|
||
|
|
:meth:`shortest_augmenting_path`
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
|
||
|
|
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
if (s is not None and t is None) or (s is None and t is not None):
|
||
|
|
raise nx.NetworkXError('Both source and target must be specified.')
|
||
|
|
|
||
|
|
# Local minimum node cut.
|
||
|
|
if s is not None and t is not None:
|
||
|
|
if s not in G:
|
||
|
|
raise nx.NetworkXError('node %s not in graph' % s)
|
||
|
|
if t not in G:
|
||
|
|
raise nx.NetworkXError('node %s not in graph' % t)
|
||
|
|
return minimum_st_node_cut(G, s, t, flow_func=flow_func)
|
||
|
|
|
||
|
|
# Global minimum node cut.
|
||
|
|
# Analog to the algorithm 11 for global node connectivity in [1].
|
||
|
|
if G.is_directed():
|
||
|
|
if not nx.is_weakly_connected(G):
|
||
|
|
raise nx.NetworkXError('Input graph is not connected')
|
||
|
|
iter_func = itertools.permutations
|
||
|
|
|
||
|
|
def neighbors(v):
|
||
|
|
return itertools.chain.from_iterable([G.predecessors(v),
|
||
|
|
G.successors(v)])
|
||
|
|
else:
|
||
|
|
if not nx.is_connected(G):
|
||
|
|
raise nx.NetworkXError('Input graph is not connected')
|
||
|
|
iter_func = itertools.combinations
|
||
|
|
neighbors = G.neighbors
|
||
|
|
|
||
|
|
# Reuse the auxiliary digraph and the residual network.
|
||
|
|
H = build_auxiliary_node_connectivity(G)
|
||
|
|
R = build_residual_network(H, 'capacity')
|
||
|
|
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
|
||
|
|
|
||
|
|
# Choose a node with minimum degree.
|
||
|
|
v = min(G, key=G.degree)
|
||
|
|
# Initial node cutset is all neighbors of the node with minimum degree.
|
||
|
|
min_cut = set(G[v])
|
||
|
|
# Compute st node cuts between v and all its non-neighbors nodes in G.
|
||
|
|
for w in set(G) - set(neighbors(v)) - set([v]):
|
||
|
|
this_cut = minimum_st_node_cut(G, v, w, **kwargs)
|
||
|
|
if len(min_cut) >= len(this_cut):
|
||
|
|
min_cut = this_cut
|
||
|
|
# Also for non adjacent pairs of neighbors of v.
|
||
|
|
for x, y in iter_func(neighbors(v), 2):
|
||
|
|
if y in G[x]:
|
||
|
|
continue
|
||
|
|
this_cut = minimum_st_node_cut(G, x, y, **kwargs)
|
||
|
|
if len(min_cut) >= len(this_cut):
|
||
|
|
min_cut = this_cut
|
||
|
|
|
||
|
|
return min_cut
|
||
|
|
|
||
|
|
|
||
|
|
def minimum_edge_cut(G, s=None, t=None, flow_func=None):
|
||
|
|
r"""Returns a set of edges of minimum cardinality that disconnects G.
|
||
|
|
|
||
|
|
If source and target nodes are provided, this function returns the
|
||
|
|
set of edges of minimum cardinality that, if removed, would break
|
||
|
|
all paths among source and target in G. If not, it returns a set of
|
||
|
|
edges of minimum cardinality that disconnects G.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
|
||
|
|
s : node
|
||
|
|
Source node. Optional. Default value: None.
|
||
|
|
|
||
|
|
t : node
|
||
|
|
Target node. Optional. Default value: None.
|
||
|
|
|
||
|
|
flow_func : function
|
||
|
|
A function for computing the maximum flow among a pair of nodes.
|
||
|
|
The function has to accept at least three parameters: a Digraph,
|
||
|
|
a source node, and a target node. And return a residual network
|
||
|
|
that follows NetworkX conventions (see :meth:`maximum_flow` for
|
||
|
|
details). If flow_func is None, the default maximum flow function
|
||
|
|
(:meth:`edmonds_karp`) is used. See below for details. The
|
||
|
|
choice of the default function may change from version
|
||
|
|
to version and should not be relied on. Default value: None.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
cutset : set
|
||
|
|
Set of edges that, if removed, would disconnect G. If source
|
||
|
|
and target nodes are provided, the set contains the edges that
|
||
|
|
if removed, would destroy all paths between source and target.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> # Platonic icosahedral graph has edge connectivity 5
|
||
|
|
>>> G = nx.icosahedral_graph()
|
||
|
|
>>> len(nx.minimum_edge_cut(G))
|
||
|
|
5
|
||
|
|
|
||
|
|
You can use alternative flow algorithms for the underlying
|
||
|
|
maximum flow computation. In dense networks the algorithm
|
||
|
|
:meth:`shortest_augmenting_path` will usually perform better
|
||
|
|
than the default :meth:`edmonds_karp`, which is faster for
|
||
|
|
sparse networks with highly skewed degree distributions.
|
||
|
|
Alternative flow functions have to be explicitly imported
|
||
|
|
from the flow package.
|
||
|
|
|
||
|
|
>>> from networkx.algorithms.flow import shortest_augmenting_path
|
||
|
|
>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
|
||
|
|
5
|
||
|
|
|
||
|
|
If you specify a pair of nodes (source and target) as parameters,
|
||
|
|
this function returns the value of local edge connectivity.
|
||
|
|
|
||
|
|
>>> nx.edge_connectivity(G, 3, 7)
|
||
|
|
5
|
||
|
|
|
||
|
|
If you need to perform several local computations among different
|
||
|
|
pairs of nodes on the same graph, it is recommended that you reuse
|
||
|
|
the data structures used in the maximum flow computations. See
|
||
|
|
:meth:`local_edge_connectivity` for details.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is a flow based implementation of minimum edge cut. For
|
||
|
|
undirected graphs the algorithm works by finding a 'small' dominating
|
||
|
|
set of nodes of G (see algorithm 7 in [1]_) and computing the maximum
|
||
|
|
flow between an arbitrary node in the dominating set and the rest of
|
||
|
|
nodes in it. This is an implementation of algorithm 6 in [1]_. For
|
||
|
|
directed graphs, the algorithm does n calls to the max flow function.
|
||
|
|
The function raises an error if the directed graph is not weakly
|
||
|
|
connected and returns an empty set if it is weakly connected.
|
||
|
|
It is an implementation of algorithm 8 in [1]_.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
:meth:`minimum_st_edge_cut`
|
||
|
|
:meth:`minimum_node_cut`
|
||
|
|
:meth:`stoer_wagner`
|
||
|
|
:meth:`node_connectivity`
|
||
|
|
:meth:`edge_connectivity`
|
||
|
|
:meth:`maximum_flow`
|
||
|
|
:meth:`edmonds_karp`
|
||
|
|
:meth:`preflow_push`
|
||
|
|
:meth:`shortest_augmenting_path`
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
|
||
|
|
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
if (s is not None and t is None) or (s is None and t is not None):
|
||
|
|
raise nx.NetworkXError('Both source and target must be specified.')
|
||
|
|
|
||
|
|
# reuse auxiliary digraph and residual network
|
||
|
|
H = build_auxiliary_edge_connectivity(G)
|
||
|
|
R = build_residual_network(H, 'capacity')
|
||
|
|
kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)
|
||
|
|
|
||
|
|
# Local minimum edge cut if s and t are not None
|
||
|
|
if s is not None and t is not None:
|
||
|
|
if s not in G:
|
||
|
|
raise nx.NetworkXError('node %s not in graph' % s)
|
||
|
|
if t not in G:
|
||
|
|
raise nx.NetworkXError('node %s not in graph' % t)
|
||
|
|
return minimum_st_edge_cut(H, s, t, **kwargs)
|
||
|
|
|
||
|
|
# Global minimum edge cut
|
||
|
|
# Analog to the algorithm for global edge connectivity
|
||
|
|
if G.is_directed():
|
||
|
|
# Based on algorithm 8 in [1]
|
||
|
|
if not nx.is_weakly_connected(G):
|
||
|
|
raise nx.NetworkXError('Input graph is not connected')
|
||
|
|
|
||
|
|
# Initial cutset is all edges of a node with minimum degree
|
||
|
|
node = min(G, key=G.degree)
|
||
|
|
min_cut = set(G.edges(node))
|
||
|
|
nodes = list(G)
|
||
|
|
n = len(nodes)
|
||
|
|
for i in range(n):
|
||
|
|
try:
|
||
|
|
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1], **kwargs)
|
||
|
|
if len(this_cut) <= len(min_cut):
|
||
|
|
min_cut = this_cut
|
||
|
|
except IndexError: # Last node!
|
||
|
|
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)
|
||
|
|
if len(this_cut) <= len(min_cut):
|
||
|
|
min_cut = this_cut
|
||
|
|
|
||
|
|
return min_cut
|
||
|
|
|
||
|
|
else: # undirected
|
||
|
|
# Based on algorithm 6 in [1]
|
||
|
|
if not nx.is_connected(G):
|
||
|
|
raise nx.NetworkXError('Input graph is not connected')
|
||
|
|
|
||
|
|
# Initial cutset is all edges of a node with minimum degree
|
||
|
|
node = min(G, key=G.degree)
|
||
|
|
min_cut = set(G.edges(node))
|
||
|
|
# A dominating set is \lambda-covering
|
||
|
|
# We need a dominating set with at least two nodes
|
||
|
|
for node in G:
|
||
|
|
D = nx.dominating_set(G, start_with=node)
|
||
|
|
v = D.pop()
|
||
|
|
if D:
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
# in complete graphs the dominating set will always be of one node
|
||
|
|
# thus we return min_cut, which now contains the edges of a node
|
||
|
|
# with minimum degree
|
||
|
|
return min_cut
|
||
|
|
for w in D:
|
||
|
|
this_cut = minimum_st_edge_cut(H, v, w, **kwargs)
|
||
|
|
if len(this_cut) <= len(min_cut):
|
||
|
|
min_cut = this_cut
|
||
|
|
|
||
|
|
return min_cut
|