ó ú£Õ\c@svdZdZddlZddddgZdd „Zdd „Zdd „Zdd „Zdd „Z dd„Z dS(sB Maximum flow (and minimum cut) algorithms on capacitated graphs. s'Loïc Séguin-C. iÿÿÿÿNtford_fulkersontford_fulkerson_flowtmax_flowtmin_cuttcapacitycCsƒtjƒ}|j|ƒi}tj|ƒrªxH|jdtƒD]b}||dkr~|d|dkr£|j|Œq£qA|j|Œd||d|df|||||||Wnëxè|jƒD]Ú\}}|j||ƒr||||krçt||||||||ƒ|||>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge('x','a', capacity=3.0) >>> G.add_edge('x','b', capacity=1.0) >>> G.add_edge('a','c', capacity=3.0) >>> G.add_edge('b','c', capacity=5.0) >>> G.add_edge('b','d', capacity=4.0) >>> G.add_edge('d','e', capacity=2.0) >>> G.add_edge('c','y', capacity=2.0) >>> G.add_edge('e','y', capacity=3.0) >>> flow, F = nx.ford_fulkerson(G, 'x', 'y') >>> flow 3.0 s0MultiGraph and MultiDiGraph not supported (yet).snode %s not in graphRiiÿÿÿÿis-Infinite capacity path, flow unbounded above.(t is_multigraphRt NetworkXErrortstrRR tbidirectional_shortest_pathtNetworkXNoPathtlisttziptmint ValueErrortNetworkXUnboundedt remove_edgeRR R(R tsttRRRt flow_valuet path_nodest path_edgesRRt path_capacityt edge_attrt flow_dict((sX/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/flow/maxflow.pyRVsL@      3    " cCst|||d|ƒdS(s“Return a maximum flow for a single-commodity flow problem. Parameters ---------- G : NetworkX graph Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity. s : node Source node for the flow. t : node Sink node for the flow. capacity: string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. Returns ------- flow_dict : dictionary Dictionary of dictionaries keyed by nodes such that flow_dict[u][v] is the flow edge (u, v). Raises ------ NetworkXError The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded If the graph has a path of infinite capacity, the value of a feasible flow on the graph is unbounded above and the function raises a NetworkXUnbounded. Examples -------- >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge('x','a', capacity=3.0) >>> G.add_edge('x','b', capacity=1.0) >>> G.add_edge('a','c', capacity=3.0) >>> G.add_edge('b','c', capacity=5.0) >>> G.add_edge('b','d', capacity=4.0) >>> G.add_edge('d','e', capacity=2.0) >>> G.add_edge('c','y', capacity=2.0) >>> G.add_edge('e','y', capacity=3.0) >>> F = nx.ford_fulkerson_flow(G, 'x', 'y') >>> for u, v in G.edges_iter(): ... print('(%s, %s) %.2f' % (u, v, F[u][v])) ... (a, c) 2.00 (c, y) 2.00 (b, c) 0.00 (b, d) 1.00 (e, y) 1.00 (d, e) 1.00 (x, a) 2.00 (x, b) 1.00 Ri(R(R R(R)R((sX/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/flow/maxflow.pyRÐsAcCst|||d|ƒdS(s˜Find the value of a maximum single-commodity flow. Parameters ---------- G : NetworkX graph Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity. s : node Source node for the flow. t : node Sink node for the flow. capacity: string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. Returns ------- flow_value : integer, float Value of the maximum flow, i.e., net outflow from the source. Raises ------ NetworkXError The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded If the graph has a path of infinite capacity, the value of a feasible flow on the graph is unbounded above and the function raises a NetworkXUnbounded. Examples -------- >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge('x','a', capacity=3.0) >>> G.add_edge('x','b', capacity=1.0) >>> G.add_edge('a','c', capacity=3.0) >>> G.add_edge('b','c', capacity=5.0) >>> G.add_edge('b','d', capacity=4.0) >>> G.add_edge('d','e', capacity=2.0) >>> G.add_edge('c','y', capacity=2.0) >>> G.add_edge('e','y', capacity=3.0) >>> flow = nx.max_flow(G, 'x', 'y') >>> flow 3.0 Ri(R(R R(R)R((sX/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/flow/maxflow.pyRs7cCsHyt|||d|ƒdSWn#tjk rCtjdƒ‚nXdS(sCompute the value of a minimum (s, t)-cut. Use the max-flow min-cut theorem, i.e., the capacity of a minimum capacity cut is equal to the flow value of a maximum flow. Parameters ---------- G : NetworkX graph Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity. s : node Source node for the flow. t : node Sink node for the flow. capacity: string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. Returns ------- cutValue : integer, float Value of the minimum cut. Raises ------ NetworkXUnbounded If the graph has a path of infinite capacity, all cuts have infinite capacity and the function raises a NetworkXError. Examples -------- >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge('x','a', capacity = 3.0) >>> G.add_edge('x','b', capacity = 1.0) >>> G.add_edge('a','c', capacity = 3.0) >>> G.add_edge('b','c', capacity = 5.0) >>> G.add_edge('b','d', capacity = 4.0) >>> G.add_edge('d','e', capacity = 2.0) >>> G.add_edge('c','y', capacity = 2.0) >>> G.add_edge('e','y', capacity = 3.0) >>> nx.min_cut(G, 'x', 'y') 3.0 Ris'Infinite capacity path, no minimum cut.N(RRR&(R R(R)R((sX/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/flow/maxflow.pyRNs 4( t__doc__t __author__tnetworkxRt__all__RRRRRR(((sX/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/flow/maxflow.pyts    $ z D :