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<h1>Source code for networkx.algorithms.connectivity.cuts</h1><div class="highlight"><pre>
<span class="c"># -*- coding: utf-8 -*-</span>
<span class="sd">"""</span>
<span class="sd">Flow based cut algorithms</span>
<span class="sd">"""</span>
<span class="c"># http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf</span>
<span class="c"># http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">itemgetter</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="kn">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.algorithms.connectivity.connectivity</span> <span class="kn">import</span> \
<span class="n">_aux_digraph_node_connectivity</span><span class="p">,</span> <span class="n">_aux_digraph_edge_connectivity</span><span class="p">,</span> \
<span class="n">dominating_set</span><span class="p">,</span> <span class="n">node_connectivity</span>
<span class="n">__author__</span> <span class="o">=</span> <span class="s">'</span><span class="se">\n</span><span class="s">'</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">'Jordi Torrents <jtorrents@milnou.net>'</span><span class="p">])</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span> <span class="s">'minimum_st_node_cut'</span><span class="p">,</span>
<span class="s">'minimum_node_cut'</span><span class="p">,</span>
<span class="s">'minimum_st_edge_cut'</span><span class="p">,</span>
<span class="s">'minimum_edge_cut'</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="minimum_st_edge_cut"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.cuts.minimum_st_edge_cut.html#networkx.algorithms.connectivity.cuts.minimum_st_edge_cut">[docs]</a><span class="k">def</span> <span class="nf">minimum_st_edge_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">capacity</span><span class="o">=</span><span class="s">'capacity'</span><span class="p">):</span>
<span class="sd">"""Returns the edges of the cut-set of a minimum (s, t)-cut.</span>
<span class="sd"> We use the max-flow min-cut theorem, i.e., the capacity of a minimum</span>
<span class="sd"> capacity cut is equal to the flow value of a maximum flow.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Edges of the graph are expected to have an attribute called</span>
<span class="sd"> 'capacity'. If this attribute is not present, the edge is</span>
<span class="sd"> considered to have infinite capacity.</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node for the flow.</span>
<span class="sd"> t : node</span>
<span class="sd"> Sink node for the flow.</span>
<span class="sd"> capacity: string</span>
<span class="sd"> Edges of the graph G are expected to have an attribute capacity</span>
<span class="sd"> that indicates how much flow the edge can support. If this</span>
<span class="sd"> attribute is not present, the edge is considered to have</span>
<span class="sd"> infinite capacity. Default value: 'capacity'.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cutset : set</span>
<span class="sd"> Set of edges that, if removed from the graph, will disconnect it</span>
<span class="sd"> </span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXUnbounded</span>
<span class="sd"> If the graph has a path of infinite capacity, all cuts have</span>
<span class="sd"> infinite capacity and the function raises a NetworkXError.</span>
<span class="sd"> </span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> G.add_edge('x','a', capacity = 3.0)</span>
<span class="sd"> >>> G.add_edge('x','b', capacity = 1.0)</span>
<span class="sd"> >>> G.add_edge('a','c', capacity = 3.0)</span>
<span class="sd"> >>> G.add_edge('b','c', capacity = 5.0)</span>
<span class="sd"> >>> G.add_edge('b','d', capacity = 4.0)</span>
<span class="sd"> >>> G.add_edge('d','e', capacity = 2.0)</span>
<span class="sd"> >>> G.add_edge('c','y', capacity = 2.0)</span>
<span class="sd"> >>> G.add_edge('e','y', capacity = 3.0)</span>
<span class="sd"> >>> sorted(nx.minimum_edge_cut(G, 'x', 'y'))</span>
<span class="sd"> [('c', 'y'), ('x', 'b')]</span>
<span class="sd"> >>> nx.min_cut(G, 'x', 'y')</span>
<span class="sd"> 3.0</span>
<span class="sd"> """</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">flow</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">ford_fulkerson_flow_and_auxiliary</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">capacity</span><span class="o">=</span><span class="n">capacity</span><span class="p">)</span>
<span class="n">cutset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="c"># Compute reachable nodes from source in the residual network</span>
<span class="n">reachable</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">single_source_shortest_path</span><span class="p">(</span><span class="n">H</span><span class="p">,</span><span class="n">s</span><span class="p">))</span>
<span class="c"># And unreachable nodes</span>
<span class="n">others</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="o">-</span> <span class="n">reachable</span> <span class="c"># - set([s])</span>
<span class="c"># Any edge in the original network linking these two partitions</span>
<span class="c"># is part of the edge cutset</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">nbrs</span> <span class="ow">in</span> <span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">G</span><span class="p">[</span><span class="n">n</span><span class="p">])</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">reachable</span><span class="p">):</span>
<span class="n">cutset</span><span class="o">.</span><span class="n">update</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nbrs</span> <span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">others</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cutset</span>
<span class="k">except</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">:</span>
<span class="c"># Should we raise any other exception or just let ford_fulkerson </span>
<span class="c"># propagate nx.NetworkXUnbounded ?</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="s">"Infinite capacity path, no minimum cut."</span><span class="p">)</span>
</div>
<div class="viewcode-block" id="minimum_st_node_cut"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.cuts.minimum_st_node_cut.html#networkx.algorithms.connectivity.cuts.minimum_st_node_cut">[docs]</a><span class="k">def</span> <span class="nf">minimum_st_node_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns a set of nodes of minimum cardinality that disconnect source</span>
<span class="sd"> from target in G.</span>
<span class="sd"> This function returns the set of nodes of minimum cardinality that, </span>
<span class="sd"> if removed, would destroy all paths among source and target in G. </span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node.</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cutset : set</span>
<span class="sd"> Set of nodes that, if removed, would destroy all paths between </span>
<span class="sd"> source and target in G.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph has node connectivity 5 </span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> len(nx.minimum_node_cut(G, 0, 6))</span>
<span class="sd"> 5</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of minimum node cut. The algorithm </span>
<span class="sd"> is based in solving a number of max-flow problems (ie local st-node</span>
<span class="sd"> connectivity, see local_node_connectivity) to determine the capacity </span>
<span class="sd"> of the minimum cut on an auxiliary directed network that corresponds </span>
<span class="sd"> to the minimum node cut of G. It handles both directed and undirected </span>
<span class="sd"> graphs.</span>
<span class="sd"> This implementation is based on algorithm 11 in [1]_. We use the Ford </span>
<span class="sd"> and Fulkerson algorithm to compute max flow (see ford_fulkerson).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> minimum_edge_cut</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. </span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">aux_digraph</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">mapping</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">aux_digraph</span>
<span class="n">edge_cut</span> <span class="o">=</span> <span class="n">minimum_st_edge_cut</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="s">'</span><span class="si">%s</span><span class="s">B'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="s">'</span><span class="si">%s</span><span class="s">A'</span> <span class="o">%</span> <span class="n">mapping</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
<span class="c"># Each node in the original graph maps to two nodes of the auxiliary graph</span>
<span class="n">node_cut</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">H</span><span class="o">.</span><span class="n">node</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="s">'id'</span><span class="p">]</span> <span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">edge_cut</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">edge</span><span class="p">)</span>
<span class="k">return</span> <span class="n">node_cut</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">s</span><span class="p">,</span><span class="n">t</span><span class="p">])</span>
</div>
<div class="viewcode-block" id="minimum_node_cut"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.cuts.minimum_node_cut.html#networkx.algorithms.connectivity.cuts.minimum_node_cut">[docs]</a><span class="k">def</span> <span class="nf">minimum_node_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns a set of nodes of minimum cardinality that disconnects G.</span>
<span class="sd"> If source and target nodes are provided, this function returns the </span>
<span class="sd"> set of nodes of minimum cardinality that, if removed, would destroy </span>
<span class="sd"> all paths among source and target in G. If not, it returns a set </span>
<span class="sd"> of nodes of minimum cardinality that disconnects G.</span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional (default=None)</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional (default=None)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cutset : set</span>
<span class="sd"> Set of nodes that, if removed, would disconnect G. If source </span>
<span class="sd"> and target nodes are provided, the set contians the nodes that</span>
<span class="sd"> if removed, would destroy all paths between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph has node connectivity 5 </span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> len(nx.minimum_node_cut(G))</span>
<span class="sd"> 5</span>
<span class="sd"> >>> # this is the minimum over any pair of non adjacent nodes</span>
<span class="sd"> >>> from itertools import combinations</span>
<span class="sd"> >>> for u,v in combinations(G, 2):</span>
<span class="sd"> ... if v not in G[u]:</span>
<span class="sd"> ... assert(len(nx.minimum_node_cut(G,u,v)) == 5)</span>
<span class="sd"> ... </span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of minimum node cut. The algorithm </span>
<span class="sd"> is based in solving a number of max-flow problems (ie local st-node</span>
<span class="sd"> connectivity, see local_node_connectivity) to determine the capacity </span>
<span class="sd"> of the minimum cut on an auxiliary directed network that corresponds </span>
<span class="sd"> to the minimum node cut of G. It handles both directed and undirected </span>
<span class="sd"> graphs.</span>
<span class="sd"> This implementation is based on algorithm 11 in [1]_. We use the Ford </span>
<span class="sd"> and Fulkerson algorithm to compute max flow (see ford_fulkerson).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> minimum_edge_cut</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson </span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. </span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="c"># Local minimum node cut</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">t</span><span class="p">)</span>
<span class="k">return</span> <span class="n">minimum_st_node_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="c"># Global minimum node cut</span>
<span class="c"># Analog to the algoritm 11 for global node connectivity in [1]</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'Input graph is not connected'</span><span class="p">)</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span>
<span class="k">def</span> <span class="nf">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
<span class="k">return</span> <span class="n">itertools</span><span class="o">.</span><span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span><span class="p">([</span><span class="n">G</span><span class="o">.</span><span class="n">predecessors_iter</span><span class="p">(</span><span class="n">v</span><span class="p">),</span>
<span class="n">G</span><span class="o">.</span><span class="n">successors_iter</span><span class="p">(</span><span class="n">v</span><span class="p">)])</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'Input graph is not connected'</span><span class="p">)</span>
<span class="n">iter_func</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">combinations</span>
<span class="n">neighbors</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">neighbors_iter</span>
<span class="c"># Choose a node with minimum degree</span>
<span class="n">deg</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
<span class="n">min_deg</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">deg</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">v</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span><span class="n">d</span> <span class="ow">in</span> <span class="n">deg</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="n">min_deg</span><span class="p">)</span>
<span class="c"># Initial node cutset is all neighbors of the node with minimum degree</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="c"># Reuse the auxiliary digraph</span>
<span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">_aux_digraph_node_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="c"># compute st node cuts between v and all its non-neighbors nodes in G</span>
<span class="c"># and store the minimum</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">v</span><span class="p">]):</span>
<span class="n">this_cut</span> <span class="o">=</span> <span class="n">minimum_st_node_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">min_cut</span><span class="p">)</span> <span class="o">>=</span> <span class="nb">len</span><span class="p">(</span><span class="n">this_cut</span><span class="p">):</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">this_cut</span>
<span class="c"># Same for non adjacent pairs of neighbors of v</span>
<span class="k">for</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="ow">in</span> <span class="n">iter_func</span><span class="p">(</span><span class="n">neighbors</span><span class="p">(</span><span class="n">v</span><span class="p">),</span><span class="mi">2</span><span class="p">):</span>
<span class="k">if</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">x</span><span class="p">]:</span> <span class="k">continue</span>
<span class="n">this_cut</span> <span class="o">=</span> <span class="n">minimum_st_node_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">aux_digraph</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">mapping</span><span class="o">=</span><span class="n">mapping</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">min_cut</span><span class="p">)</span> <span class="o">>=</span> <span class="nb">len</span><span class="p">(</span><span class="n">this_cut</span><span class="p">):</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">this_cut</span>
<span class="k">return</span> <span class="n">min_cut</span>
</div>
<div class="viewcode-block" id="minimum_edge_cut"><a class="viewcode-back" href="../../../../reference/generated/networkx.algorithms.connectivity.cuts.minimum_edge_cut.html#networkx.algorithms.connectivity.cuts.minimum_edge_cut">[docs]</a><span class="k">def</span> <span class="nf">minimum_edge_cut</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">t</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">r"""Returns a set of edges of minimum cardinality that disconnects G.</span>
<span class="sd"> If source and target nodes are provided, this function returns the </span>
<span class="sd"> set of edges of minimum cardinality that, if removed, would break </span>
<span class="sd"> all paths among source and target in G. If not, it returns a set of </span>
<span class="sd"> edges of minimum cardinality that disconnects G.</span>
<span class="sd"> </span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> s : node</span>
<span class="sd"> Source node. Optional (default=None)</span>
<span class="sd"> t : node</span>
<span class="sd"> Target node. Optional (default=None)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cutset : set</span>
<span class="sd"> Set of edges that, if removed, would disconnect G. If source </span>
<span class="sd"> and target nodes are provided, the set contians the edges that</span>
<span class="sd"> if removed, would destroy all paths between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> # Platonic icosahedral graph has edge connectivity 5</span>
<span class="sd"> >>> G = nx.icosahedral_graph()</span>
<span class="sd"> >>> len(nx.minimum_edge_cut(G))</span>
<span class="sd"> 5</span>
<span class="sd"> >>> # this is the minimum over any pair of nodes</span>
<span class="sd"> >>> from itertools import combinations</span>
<span class="sd"> >>> for u,v in combinations(G, 2):</span>
<span class="sd"> ... assert(len(nx.minimum_edge_cut(G,u,v)) == 5)</span>
<span class="sd"> ... </span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a flow based implementation of minimum edge cut. For</span>
<span class="sd"> undirected graphs the algorithm works by finding a 'small' dominating</span>
<span class="sd"> set of nodes of G (see algorithm 7 in [1]_) and computing the maximum</span>
<span class="sd"> flow between an arbitrary node in the dominating set and the rest of</span>
<span class="sd"> nodes in it. This is an implementation of algorithm 6 in [1]_.</span>
<span class="sd"> For directed graphs, the algorithm does n calls to the max flow function.</span>
<span class="sd"> This is an implementation of algorithm 8 in [1]_. We use the Ford and</span>
<span class="sd"> Fulkerson algorithm to compute max flow (see ford_fulkerson).</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> node_connectivity</span>
<span class="sd"> edge_connectivity</span>
<span class="sd"> minimum_node_cut</span>
<span class="sd"> max_flow</span>
<span class="sd"> ford_fulkerson</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.</span>
<span class="sd"> http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf</span>
<span class="sd"> """</span>
<span class="c"># reuse auxiliary digraph</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">_aux_digraph_edge_connectivity</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="c"># Local minimum edge cut if s and t are not None</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'node </span><span class="si">%s</span><span class="s"> not in graph'</span> <span class="o">%</span> <span class="n">t</span><span class="p">)</span>
<span class="k">return</span> <span class="n">minimum_st_edge_cut</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
<span class="c"># Global minimum edge cut</span>
<span class="c"># Analog to the algoritm for global edge connectivity</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="c"># Based on algorithm 8 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weakly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'Input graph is not connected'</span><span class="p">)</span>
<span class="c"># Initial cutset is all edges of a node with minimum degree</span>
<span class="n">deg</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
<span class="n">min_deg</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">deg</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">node</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span><span class="n">d</span> <span class="ow">in</span> <span class="n">deg</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span><span class="o">==</span><span class="n">min_deg</span><span class="p">)</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<span class="n">nodes</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">()</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodes</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">this_cut</span> <span class="o">=</span> <span class="n">minimum_st_edge_cut</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">this_cut</span><span class="p">)</span> <span class="o"><=</span> <span class="nb">len</span><span class="p">(</span><span class="n">min_cut</span><span class="p">):</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">this_cut</span>
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span> <span class="c"># Last node!</span>
<span class="n">this_cut</span> <span class="o">=</span> <span class="n">minimum_st_edge_cut</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">this_cut</span><span class="p">)</span> <span class="o"><=</span> <span class="nb">len</span><span class="p">(</span><span class="n">min_cut</span><span class="p">):</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">this_cut</span>
<span class="k">return</span> <span class="n">min_cut</span>
<span class="k">else</span><span class="p">:</span> <span class="c"># undirected</span>
<span class="c"># Based on algorithm 6 in [1]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s">'Input graph is not connected'</span><span class="p">)</span>
<span class="c"># Initial cutset is all edges of a node with minimum degree</span>
<span class="n">deg</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
<span class="n">min_deg</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">deg</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="n">node</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span><span class="n">d</span> <span class="ow">in</span> <span class="n">deg</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span><span class="o">==</span><span class="n">min_deg</span><span class="p">)</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<span class="c"># A dominating set is \lambda-covering</span>
<span class="c"># We need a dominating set with at least two nodes</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">dominating_set</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">start_with</span><span class="o">=</span><span class="n">node</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">D</span><span class="p">:</span> <span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="c"># in complete graphs the dominating set will always be of one node</span>
<span class="c"># thus we return min_cut, which now contains the edges of a node</span>
<span class="c"># with minimum degree</span>
<span class="k">return</span> <span class="n">min_cut</span>
<span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">D</span><span class="p">:</span>
<span class="n">this_cut</span> <span class="o">=</span> <span class="n">minimum_st_edge_cut</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">this_cut</span><span class="p">)</span> <span class="o"><=</span> <span class="nb">len</span><span class="p">(</span><span class="n">min_cut</span><span class="p">):</span>
<span class="n">min_cut</span> <span class="o">=</span> <span class="n">this_cut</span>
<span class="k">return</span> <span class="n">min_cut</span></div>
</pre></div>
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