+ j`"Rt^RIHt^RIt^RIHt^RIHtR.t ]P!R] !R4/RRR 7R R l4t R t R#) z. Dinitz' algorithm for maximum flow problems. )dequeN)build_residual_network)pairwisedinitzcapacityinfT) edge_attrs returns_graphpreserve_edge_attrscj\WW#WF4pRVPR&\P!V4V#)aFind a maximum single-commodity flow using Dinitz' algorithm. This function returns the residual network resulting after computing the maximum flow. See below for details about the conventions NetworkX uses for defining residual networks. This algorithm has a running time of $O(n^2 m)$ for $n$ nodes and $m$ edges [1]_. 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 or function (default= 'capacity') If this is a string, then edge capacity will be accessed via the edge attribute with this key (that is, the capacity of the edge joining `u` to `v` will be ``G.edges[u, v][capacity]``). If no such edge attribute exists, the capacity of the edge is assumed to be infinite. If this is a function, the capacity of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge. residual : NetworkX graph Residual network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None. value_only : bool If True compute only the value of the maximum flow. This parameter will be ignored by this algorithm because it is not applicable. cutoff : integer, float If specified, the algorithm will terminate when the flow value reaches or exceeds the cutoff. In this case, it may be unable to immediately determine a minimum cut. Default value: None. Returns ------- R : NetworkX DiGraph Residual network after computing the maximum flow. 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. See also -------- :meth:`maximum_flow` :meth:`minimum_cut` :meth:`preflow_push` :meth:`shortest_augmenting_path` Notes ----- The residual network :samp:`R` from an input graph :samp:`G` has the same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists in :samp:`G`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists in :samp:`G` or zero otherwise. If the capacity is infinite, :samp:`R[u][v]['capacity']` will have a high arbitrary finite value that does not affect the solution of the problem. This value is stored in :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. The flow value, defined as the total flow into :samp:`t`, the sink, is stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum :samp:`s`-:samp:`t` cut. Examples -------- >>> from networkx.algorithms.flow import dinitz The functions that implement flow algorithms and output a residual network, such as this one, are not imported to the base NetworkX namespace, so you have to explicitly import them from the flow package. >>> 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) >>> R = dinitz(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value 3.0 >>> flow_value == R.graph["flow_value"] True References ---------- .. [1] Dinitz' Algorithm: The Original Version and Even's Version. 2006. Yefim Dinitz. In Theoretical Computer Science. Lecture Notes in Computer Science. Volume 3895. pp 218-240. https://doi.org/10.1007/11685654_10 r algorithm) dinitz_implgraphnx _clear_cache)Gstrresidual value_onlycutoffRs&&&&&&& d/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/flow/dinitz_alg.pyrrs2H A!x8A#AGGKOOA HcaaaaaSV9d$\P!R\S4 R24hSV9d$\P!R\S4 R24hSS8Xd\P!R4hVf \W4pMTpVF%pWg,P 4Fp^VR&K K' VP R,oVfSpVP oVPoVVV3Rlp VVVV3Rlp ^p W8dFV !4p SV 9dM6V !V 4p V ^,S8d\P!R4hW, p KKWP R&V#) znode z not in graphz!source and sink are the same nodeflowrc</pS^/p\S^3.4pV'dS V9dV#VP4wr4SV,P4FwrVVR,VR,, ^8gK#WP9d0W,V^,8XdW,PV4KVKX\V.4W&V^,W&VPWT^,34K KV#)rr)rpopleftitemsappend) parents vertex_distqueueudistvattrR_succrrs rbreath_first_search(dinitz_impl..breath_first_searchs!f 1vhG|mmoGA!!9??, #d6l2Q6|&>TAX5#J--a06&+A3Z )-  a]3-rc<^pS pV.p\W,4^8d"W,^,pVPV4MBVP4\V4^8XdV#VR,pW,P4VS8XEd8Sp\ V4FBwr$\ VSV,V,R,SV,V,R,, 4pKD \ \ V44Fwr$SV,V,R;;,V, uu&SV,V,R;;,V,uu&SV,V,R,SV,V,R,, ^8XgKW,P4VR,V8wgKVP4K" W, pVR,pTpEK)z-Build a path using DFS starting from the sinkrr)lenr poprrminreversed) r! total_flowr$pathr&rINFR_predrrs & rdepth_first_search'dinitz_impl..depth_first_searchsb s7:"JqM A t9>&%H ""$Av$TNDAtVAYq\*%=q ! V@T%TUD+$Xd^4DA1IaL(D0(1IaL(D0(ay|J/&)A,v2FF!K **,"2h!m HHJ5" HArz-Infinite capacity path, flow unbounded above. flow_value) r NetworkXErrorstrrvaluesrsuccpredNetworkXUnbounded)rrrrrrrr$er)r5r7r! this_flowr3r4r(s&ff&&& @@@rr r s>zs1vhm<==zs1vhm<==AvBCC "1 / AAfI ''%.C ~ VVF VVF(""HJ  %' G  &w/ q=3 &&'VW W &GGL Hr)rNFN)__doc__ collectionsrnetworkxrnetworkx.algorithms.flow.utilsrnetworkx.utilsr__all__ _dispatchablefloatrr rrrIsXA# *E%L)SWD D N_ r