h99SrSSKrSSKJrJrJr SSKJr \r SSK J r SSK JrJr SS /r\R""SS S .S S \"S500S9SSj5r\R""SS S .S SS00S 1S9SSj5rSrg)z(Flow based node and edge disjoint paths.N) edmonds_karp preflow_pushshortest_augmenting_path)NetworkXNoPath) filterfalse)!build_auxiliary_edge_connectivity!build_auxiliary_node_connectivityedge_disjoint_pathsnode_disjoint_paths)Gz auxiliary? auxiliarycapacityinf)graphspreserve_edge_attrsc## X;a[R"SUS35eX ;a[R"SUS35eUc[nUc [U5nOUn[ UR U5UR U55nU(d[eUcUnO [ XH5nSUUSS.n U[LaU S U[LaSU S'U"XqU40U D6n U RS S :Xa[eU RSS 9V V V s/sH"upn U SU S :XdMU S S :dMX4PM$ nn n n UVVs0sHoHnU0_M M nnnUH upS UU U 'M S n[UU5Hzn UU:a gU/nX:XaURU 5 Uv M*U n X:wa/URU 5 UU R5un nX:waM/URU5 Uv US - nM| gs sn n n fs snnf![a Mf=f7f)aReturns the edges disjoint paths between source and target. Edge disjoint paths are paths that do not share any edge. The number of edge disjoint paths between source and target is equal to their edge connectivity. Parameters ---------- G : NetworkX graph s : node Source node for the flow. t : node Sink node for the flow. 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. The choice of the default function may change from version to version and should not be relied on. Default value: None. cutoff : integer or None (default: None) Maximum number of paths to yield. If specified, the maximum flow algorithm will terminate when the flow value reaches or exceeds the cutoff. This only works for flows that support the cutoff parameter (most do) and is ignored otherwise. auxiliary : NetworkX DiGraph Auxiliary digraph to compute flow based edge 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 ------- paths : generator A generator of edge independent paths. Raises ------ NetworkXNoPath If there is no path between source and target. NetworkXError If source or target are not in the graph G. See also -------- :meth:`node_disjoint_paths` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` Examples -------- We use in this example the platonic icosahedral graph, which has node edge connectivity 5, thus there are 5 edge disjoint paths between any pair of nodes. >>> G = nx.icosahedral_graph() >>> len(list(nx.edge_disjoint_paths(G, 0, 6))) 5 If you need to compute edge disjoint paths 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 edge disjoint paths 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 = {n: {} for n in G} >>> # Reuse the auxiliary digraph and the residual network by passing them >>> # as arguments >>> for u, v in itertools.combinations(G, 2): ... k = len(list(nx.edge_disjoint_paths(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 disjoint paths. 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(list(nx.edge_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path))) 5 Notes ----- This is a flow based implementation of edge disjoint paths. We compute the maximum flow between source and target on an auxiliary directed network. The saturated edges in the residual network after running the maximum flow algorithm correspond to edge disjoint paths between source and target in the original network. This function handles both directed and undirected graphs, and can use all flow algorithms from NetworkX flow package. node not in graphNrT)rresidualcutoff value_onlyr two_phase flow_valuer)dataflowr)nx NetworkXErrordefault_flow_funcr min out_degree in_degreerrrgraphedgeslistappendpopitemKeyError)rst flow_funcrrrHpossiblekwargsRuvdcutsetedgen flow_dict paths_foundpath_s a/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/connectivity/disjoint_paths.pyr r sIJ zqc788zqc788%  -a 0 1<<?AKKN3H  ~V&  F L 8 ,,"{!$V$Aww|! wwDw))GA! Z=AfI % *+F)a- ) $*8644aB46I8 ! QK )A,  &  s 6 KKNJ  f KKN  |++-1f KKNJ 1 K) 9*  sUC8H:G'G'G'# H,G.A(H+G4H,H4 H>HHHid)rpreserve_node_attrspreserve_graph_attrsc#J^ # X;a[R"SUS35eX ;a[R"SUS35eUc [U5m OUm T RR SS5nUc[R"S5e[ T R XqS35T RXrS355nU(d[eUcUnO [ XH5nUUT US.n [T XqS3XrS340U D6n U H#n [[U 4S jU 555v M% g7f) aComputes node disjoint paths between source and target. Node disjoint paths are paths that only share their first and last nodes. The number of node independent paths between two nodes is equal to their local node connectivity. 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. cutoff : integer or None (default: None) Maximum number of paths to yield. If specified, the maximum flow algorithm will terminate when the flow value reaches or exceeds the cutoff. This only works for flows that support the cutoff parameter (most do) and is ignored otherwise. 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 ------- paths : generator Generator of node disjoint paths. Raises ------ NetworkXNoPath If there is no path between source and target. NetworkXError If source or target are not in the graph G. Examples -------- We use in this example the platonic icosahedral graph, which has node connectivity 5, thus there are 5 node disjoint paths between any pair of non neighbor nodes. >>> G = nx.icosahedral_graph() >>> len(list(nx.node_disjoint_paths(G, 0, 6))) 5 If you need to compute node disjoint paths 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 node disjoint paths 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 arguments >>> len(list(nx.node_disjoint_paths(G, 0, 6, auxiliary=H, residual=R))) 5 You can also use alternative flow algorithms for computing node disjoint paths. 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(list(nx.node_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path))) 5 Notes ----- This is a flow based implementation of node disjoint paths. We compute the maximum flow between source and target on an auxiliary directed network. The saturated edges in the residual network after running the maximum flow algorithm correspond to node disjoint paths between source and target in the original network. This function handles both directed and undirected graphs, and can use all flow algorithms from NetworkX flow package. See also -------- :meth:`edge_disjoint_paths` :meth:`node_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` rrNmappingzInvalid auxiliary digraph.BA)r,rrrc3H># UHnTRUSv M g7f)r<N)nodes).0noder-s r; &node_disjoint_paths..s#IDDAGGDM$$7Ds") rrr r$getr!r"r#rr r&_unique_everseen) rr*r+r,rrrr@r.r/ paths_edgesr9r-s @r;r r s3~ zqc788zqc788 -a 0 ggkk)T*G;<<1<<7:,a 011;;'*Q?O3PQH  ~V& F&aGJr`s. . 1X "7 8! $$z5<&@A DHL L^! $$tTl3% DH^K  ^KBr_