# Source code for networkx.algorithms.connectivity.cuts

# -*- coding: utf-8 -*-
"""
Flow based cut algorithms
"""
import itertools
import networkx as nx

# Define the default maximum flow function to use in all flow based
# cut algorithms.
from networkx.algorithms.flow import edmonds_karp
from networkx.algorithms.flow import build_residual_network
default_flow_func = edmonds_karp

from .utils import (build_auxiliary_node_connectivity,
build_auxiliary_edge_connectivity)

__author__ = '\n'.join(['Jordi Torrents <jtorrents@milnou.net>'])

__all__ = ['minimum_st_node_cut',
'minimum_node_cut',
'minimum_st_edge_cut',
'minimum_edge_cut']

[docs]def minimum_st_edge_cut(G, s, t, flow_func=None, auxiliary=None, residual=None): """Returns the edges of the cut-set of a minimum (s, t)-cut. This function returns the set of edges of minimum cardinality that, if removed, would destroy all paths among source and target in G. Edge weights are not considered. See :meth:minimum_cut for computing minimum cuts considering edge weights. Parameters ---------- G : NetworkX graph s : node Source node for the flow. t : node Sink node for the flow. auxiliary : NetworkX DiGraph Auxiliary digraph to compute flow based node connectivity. It has to have a graph attribute called mapping with a dictionary mapping node names in G and in the auxiliary digraph. If provided it will be reused instead of recreated. 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 :meth:node_connectivity for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. residual : NetworkX DiGraph Residual network to compute maximum flow. If provided it will be reused instead of recreated. Default value: None. Returns ------- cutset : set Set of edges that, if removed from the graph, will disconnect it. See also -------- :meth:minimum_cut :meth:minimum_node_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 Examples -------- This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the connectivity package: >>> from networkx.algorithms.connectivity import minimum_st_edge_cut We use in this example the platonic icosahedral graph, which has edge connectivity 5. >>> G = nx.icosahedral_graph() >>> len(minimum_st_edge_cut(G, 0, 6)) 5 If you need to compute local edge cuts on several pairs of nodes in the same graph, it is recommended that you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for edge connectivity, and the residual network for the underlying maximum flow computation. Example of how to compute local edge cuts among all pairs of nodes of the platonic icosahedral graph reusing the data structures. >>> import itertools >>> # You also have to explicitly import the function for >>> # building the auxiliary digraph from the connectivity package >>> from networkx.algorithms.connectivity import ( ... build_auxiliary_edge_connectivity) >>> H = build_auxiliary_edge_connectivity(G) >>> # And the function for building the residual network from the >>> # flow package >>> from networkx.algorithms.flow import build_residual_network >>> # Note that the auxiliary digraph has an edge attribute named capacity >>> R = build_residual_network(H, 'capacity') >>> result = dict.fromkeys(G, dict()) >>> # Reuse the auxiliary digraph and the residual network by passing them >>> # as parameters >>> for u, v in itertools.combinations(G, 2): ... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R)) ... result[u][v] = k >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2)) True You can also use alternative flow algorithms for computing edge cuts. For instance, 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(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path)) 5 """ if flow_func is None: flow_func = default_flow_func if auxiliary is None: H = build_auxiliary_edge_connectivity(G) else: H = auxiliary kwargs = dict(capacity='capacity', flow_func=flow_func, residual=residual) cut_value, partition = nx.minimum_cut(H, s, t, **kwargs) reachable, non_reachable = partition # Any edge in the original graph linking the two sets in the # partition is part of the edge cutset cutset = set() for u, nbrs in ((n, G[n]) for n in reachable): cutset.update((u, v) for v in nbrs if v in non_reachable) return cutset
[docs]def minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None): r"""Returns a set of nodes of minimum cardinality that disconnect source from target in G. This function returns the set of nodes of minimum cardinality that, if removed, would destroy all paths among source and target in G. Parameters ---------- G : NetworkX graph s : node Source node. t : node Target node. 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. auxiliary : NetworkX DiGraph Auxiliary digraph to compute flow based node connectivity. It has to have a graph attribute called mapping with a dictionary mapping node names in G and in the auxiliary digraph. If provided it will be reused instead of recreated. Default value: None. residual : NetworkX DiGraph Residual network to compute maximum flow. If provided it will be reused instead of recreated. Default value: None. Returns ------- cutset : set Set of nodes that, if removed, would destroy all paths between source and target in G. Examples -------- This function is not imported in the base NetworkX namespace, so you have to explicitly import it from the connectivity package: >>> from networkx.algorithms.connectivity import minimum_st_node_cut We use in this example the platonic icosahedral graph, which has node connectivity 5. >>> G = nx.icosahedral_graph() >>> len(minimum_st_node_cut(G, 0, 6)) 5 If you need to compute local st cuts between several pairs of nodes in the same graph, it is recommended that you reuse the data structures that NetworkX uses in the computation: the auxiliary digraph for node connectivity and node cuts, and the residual network for the underlying maximum flow computation. Example of how to compute local st node cuts reusing the data structures: >>> # You also have to explicitly import the function for >>> # building the auxiliary digraph from the connectivity package >>> from networkx.algorithms.connectivity import ( ... build_auxiliary_node_connectivity) >>> H = build_auxiliary_node_connectivity(G) >>> # And the function for building the residual network from the >>> # flow package >>> from networkx.algorithms.flow import build_residual_network >>> # Note that the auxiliary digraph has an edge attribute named capacity >>> R = build_residual_network(H, 'capacity') >>> # Reuse the auxiliary digraph and the residual network by passing them >>> # as parameters >>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R)) 5 You can also use alternative flow algorithms for computing minimum st node cuts. For instance, 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(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path)) 5 Notes ----- This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of maximum flow computations to determine the capacity of the minimum cut on an auxiliary directed network that corresponds to the minimum node cut of G. It handles both directed and undirected graphs. This implementation is based on algorithm 11 in [1]_. See also -------- :meth:minimum_node_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 auxiliary is None: H = build_auxiliary_node_connectivity(G) else: H = auxiliary mapping = H.graph.get('mapping', None) if mapping is None: raise nx.NetworkXError('Invalid auxiliary digraph.') if G.has_edge(s, t) or G.has_edge(t, s): return [] kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H) # The edge cut in the auxiliary digraph corresponds to the node cut in the # original graph. edge_cut = minimum_st_edge_cut(H, '%sB' % mapping[s], '%sA' % mapping[t], **kwargs) # Each node in the original graph maps to two nodes of the auxiliary graph node_cut = set(H.nodes[node]['id'] for edge in edge_cut for node in edge) return node_cut - set([s, t])
[docs]def minimum_node_cut(G, s=None, t=None, flow_func=None): r"""Returns a set of nodes of minimum cardinality that disconnects G. If source and target nodes are provided, this function returns the set of nodes of minimum cardinality that, if removed, would destroy all paths among source and target in G. If not, it returns a set of nodes 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 nodes that, if removed, would disconnect G. If source and target nodes are provided, the set contains the nodes that if removed, would destroy all paths between source and target. Examples -------- >>> # Platonic icosahedral graph has node connectivity 5 >>> G = nx.icosahedral_graph() >>> node_cut = nx.minimum_node_cut(G) >>> len(node_cut) 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 >>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path) True If you specify a pair of nodes (source and target) as parameters, this function returns a local st node cut. >>> len(nx.minimum_node_cut(G, 3, 7)) 5 If you need to perform several local st cuts 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:minimum_st_node_cut for details. Notes ----- This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of maximum flow computations to determine the capacity of the minimum cut on an auxiliary directed network that corresponds to the minimum node cut of G. It handles both directed 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
[docs]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