233 lines
9.2 KiB
Python
233 lines
9.2 KiB
Python
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"""
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Kanevsky all minimum node k cutsets algorithm.
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"""
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import copy
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from collections import defaultdict
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from itertools import combinations
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from operator import itemgetter
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import networkx as nx
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from networkx.algorithms.flow import (
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build_residual_network,
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edmonds_karp,
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shortest_augmenting_path,
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)
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from .utils import build_auxiliary_node_connectivity
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default_flow_func = edmonds_karp
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__all__ = ["all_node_cuts"]
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def all_node_cuts(G, k=None, flow_func=None):
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r"""Returns all minimum k cutsets of an undirected graph G.
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This implementation is based on Kanevsky's algorithm [1]_ for finding all
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minimum-size node cut-sets of an undirected graph G; ie the set (or sets)
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of nodes of cardinality equal to the node connectivity of G. Thus if
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removed, would break G into two or more connected components.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph
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k : Integer
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Node connectivity of the input graph. If k is None, then it is
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computed. Default value: None.
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flow_func : function
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Function to perform the underlying flow computations. Default value is
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:func:`~networkx.algorithms.flow.edmonds_karp`. This function performs
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better in sparse graphs with right tailed degree distributions.
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:func:`~networkx.algorithms.flow.shortest_augmenting_path` will
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perform better in denser graphs.
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Returns
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-------
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cuts : a generator of node cutsets
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Each node cutset has cardinality equal to the node connectivity of
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the input graph.
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Examples
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--------
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>>> # A two-dimensional grid graph has 4 cutsets of cardinality 2
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>>> G = nx.grid_2d_graph(5, 5)
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>>> cutsets = list(nx.all_node_cuts(G))
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>>> len(cutsets)
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4
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>>> all(2 == len(cutset) for cutset in cutsets)
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True
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>>> nx.node_connectivity(G)
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2
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Notes
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-----
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This implementation is based on the sequential algorithm for finding all
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minimum-size separating vertex sets in a graph [1]_. The main idea is to
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compute minimum cuts using local maximum flow computations among a set
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of nodes of highest degree and all other non-adjacent nodes in the Graph.
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Once we find a minimum cut, we add an edge between the high degree
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node and the target node of the local maximum flow computation to make
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sure that we will not find that minimum cut again.
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See also
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--------
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node_connectivity
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edmonds_karp
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shortest_augmenting_path
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References
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----------
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.. [1] Kanevsky, A. (1993). Finding all minimum-size separating vertex
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sets in a graph. Networks 23(6), 533--541.
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http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract
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"""
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if not nx.is_connected(G):
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raise nx.NetworkXError("Input graph is disconnected.")
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# Address some corner cases first.
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# For complete Graphs
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if nx.density(G) == 1:
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for cut_set in combinations(G, len(G) - 1):
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yield set(cut_set)
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return
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# Initialize data structures.
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# Keep track of the cuts already computed so we do not repeat them.
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seen = []
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# Even-Tarjan reduction is what we call auxiliary digraph
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# for node connectivity.
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H = build_auxiliary_node_connectivity(G)
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H_nodes = H.nodes # for speed
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mapping = H.graph["mapping"]
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# Keep a copy of original predecessors, H will be modified later.
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# Shallow copy is enough.
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original_H_pred = copy.copy(H._pred)
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R = build_residual_network(H, "capacity")
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kwargs = dict(capacity="capacity", residual=R)
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# Define default flow function
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if flow_func is None:
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flow_func = default_flow_func
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if flow_func is shortest_augmenting_path:
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kwargs["two_phase"] = True
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# Begin the actual algorithm
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# step 1: Find node connectivity k of G
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if k is None:
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k = nx.node_connectivity(G, flow_func=flow_func)
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# step 2:
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# Find k nodes with top degree, call it X:
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X = {n for n, d in sorted(G.degree(), key=itemgetter(1), reverse=True)[:k]}
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# Check if X is a k-node-cutset
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if _is_separating_set(G, X):
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seen.append(X)
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yield X
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for x in X:
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# step 3: Compute local connectivity flow of x with all other
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# non adjacent nodes in G
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non_adjacent = set(G) - X - set(G[x])
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for v in non_adjacent:
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# step 4: compute maximum flow in an Even-Tarjan reduction H of G
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# and step 5: build the associated residual network R
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R = flow_func(H, f"{mapping[x]}B", f"{mapping[v]}A", **kwargs)
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flow_value = R.graph["flow_value"]
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if flow_value == k:
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# Find the nodes incident to the flow.
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E1 = flowed_edges = [
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(u, w) for (u, w, d) in R.edges(data=True) if d["flow"] != 0
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]
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VE1 = incident_nodes = {n for edge in E1 for n in edge}
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# Remove saturated edges form the residual network.
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# Note that reversed edges are introduced with capacity 0
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# in the residual graph and they need to be removed too.
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saturated_edges = [
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(u, w, d)
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for (u, w, d) in R.edges(data=True)
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if d["capacity"] == d["flow"] or d["capacity"] == 0
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]
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R.remove_edges_from(saturated_edges)
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R_closure = nx.transitive_closure(R)
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# step 6: shrink the strongly connected components of
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# residual flow network R and call it L.
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L = nx.condensation(R)
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cmap = L.graph["mapping"]
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inv_cmap = defaultdict(list)
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for n, scc in cmap.items():
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inv_cmap[scc].append(n)
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# Find the incident nodes in the condensed graph.
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VE1 = {cmap[n] for n in VE1}
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# step 7: Compute all antichains of L;
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# they map to closed sets in H.
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# Any edge in H that links a closed set is part of a cutset.
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for antichain in nx.antichains(L):
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# Only antichains that are subsets of incident nodes counts.
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# Lemma 8 in reference.
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if not set(antichain).issubset(VE1):
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continue
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# Nodes in an antichain of the condensation graph of
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# the residual network map to a closed set of nodes that
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# define a node partition of the auxiliary digraph H
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# through taking all of antichain's predecessors in the
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# transitive closure.
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S = set()
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for scc in antichain:
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S.update(inv_cmap[scc])
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S_ancestors = set()
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for n in S:
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S_ancestors.update(R_closure._pred[n])
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S.update(S_ancestors)
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if f"{mapping[x]}B" not in S or f"{mapping[v]}A" in S:
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continue
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# Find the cutset that links the node partition (S,~S) in H
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cutset = set()
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for u in S:
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cutset.update((u, w) for w in original_H_pred[u] if w not in S)
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# The edges in H that form the cutset are internal edges
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# (ie edges that represent a node of the original graph G)
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if any([H_nodes[u]["id"] != H_nodes[w]["id"] for u, w in cutset]):
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continue
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node_cut = {H_nodes[u]["id"] for u, _ in cutset}
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if len(node_cut) == k:
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# The cut is invalid if it includes internal edges of
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# end nodes. The other half of Lemma 8 in ref.
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if x in node_cut or v in node_cut:
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continue
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if node_cut not in seen:
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yield node_cut
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seen.append(node_cut)
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# Add an edge (x, v) to make sure that we do not
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# find this cutset again. This is equivalent
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# of adding the edge in the input graph
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# G.add_edge(x, v) and then regenerate H and R:
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# Add edges to the auxiliary digraph.
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# See build_residual_network for convention we used
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# in residual graphs.
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H.add_edge(f"{mapping[x]}B", f"{mapping[v]}A", capacity=1)
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H.add_edge(f"{mapping[v]}B", f"{mapping[x]}A", capacity=1)
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# Add edges to the residual network.
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R.add_edge(f"{mapping[x]}B", f"{mapping[v]}A", capacity=1)
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R.add_edge(f"{mapping[v]}A", f"{mapping[x]}B", capacity=0)
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R.add_edge(f"{mapping[v]}B", f"{mapping[x]}A", capacity=1)
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R.add_edge(f"{mapping[x]}A", f"{mapping[v]}B", capacity=0)
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# Add again the saturated edges to reuse the residual network
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R.add_edges_from(saturated_edges)
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def _is_separating_set(G, cut):
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"""Assumes that the input graph is connected"""
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if len(cut) == len(G) - 1:
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return True
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H = nx.restricted_view(G, cut, [])
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if nx.is_connected(H):
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return False
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return True
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