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Shortest augmenting path algorithm for maximum flow problems. )dequeN)edmonds_karp_core) CurrentEdgebuild_residual_networkshortest_augmenting_pathcaaaaW9d$\P!R\V4 R24hW 9d$\P!R\V4 R24hW8Xd\P!R4hVf \W4pMTpVPoVP pVP oVF&p SV ,P4Fp ^V R&K K( V^/p \V^3.4p V 'drV P4wrV ^, p W,P4F:wrW9gK VR,VR,8gK$WV&V PW34K< KyW9d^VPR&V#\V4oVP4^, pVF9p W9d W,MSSV ,R&\SV ,4SV ,R&K; ^.^S,^, ,pVF&p VSV ,R,;;,^, uu&K( VPR ,oVV3R lpVVV3R lpVf \!R 4p^pV.pTp V'gSM)\#\%VR ,^SR,,44pSV,R,V8pV'gSV ,R,p SV ,R,pVP'4wrV SV,R,^,8Xd*VR,VR,8dVPV4Tp MVP)4KkW8XgKVV!V4, pVV8dVVPR&V#V.pTp KV\/WqW&V, 4, pVVPR&V# \*dTT ;;,^,uu&TT ,^8XdTTPR&Tu#T!T 4p Y8Xd(T T8d!T'gTTPR&Tu#R pKTT ;;,^, uu&T ST ,R&Y8wdTP-4TR,p EKEKi;i)z9Implementation of the shortest augmenting path algorithm.znode z not in graphz!source and sink are the same nodeflowcapacity flow_valueheight curr_edgeinfc<Sp\V4p\V4pVF5pSV,V,p\WR,VR,, 4pTpK7 V^,S8d\P!R4h\V4p\V4pVFKpSV,V,R;;,V, uu&SV,V,R;;,V,uu&TpKM V#)z&Augment flow along a path from s to t.r rz-Infinite capacity path, flow unbounded above.)iternextminnxNetworkXUnbounded)pathrituvattrR_succr s& p/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/flow/shortestaugmentingpath.pyaugment.shortest_augmenting_path_impl..augmentFs $Z HA!9Q&&'VW W $Z HA 1IaL D ( 1IaL D ( A c<S^, pSV,P4F6wr#VR,VR,8gK\VSV,R,4pK8 V^,#)z,Relabel a node to create an admissible edge.rr r )itemsr)rr rrR_nodesrns& rrelabel.shortest_augmenting_path_impl..relabel[sUQay(GAF|d:..VWQZ%9:)zrg?TgUUUUUU?)r NetworkXErrorstrrnodespredsuccvaluesrpopleftrappendgraphlensizerfloatintrget move_to_next StopIterationpopr)Gstr residual two_phasecutoffRR_predreheightsqr rrmcountsrr"r rddoner r rr r!s&&&&&&& @@@@rshortest_augmenting_path_implrEszs1vhm<==zs1vhm<==vBCC "1 / ggG VVF VVF!!#AAfI$ !fG 1vhA IIK ! y(GADL4 3C$C# !%)  !"  AA 1 A-.\wzq 8"-fQi"8 ; SAEAI F wqz(#$)$ ''%.C*~uJ 3D AC3A'N0B$C DA 1:h 1 $DH%AJ{+ mmoGAH-11d6lT*EU6U A &&(8 6 '$- 'JV#(2 %3DA#A!j-@AAJ&AGGL HS! v!#&>Q&-7AGGL)H 6fk$1; -  $v!#'- 8$6HHJRA + s+N?QQ-QQAQ Qr r T) edge_attrs returns_graphpreserve_edge_attrsc l\WW#WFV4pRVPR&\P!V4V#)aFind a maximum single-commodity flow using the shortest augmenting path 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. 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. two_phase : bool If True, a two-phase variant is used. The two-phase variant improves the running time on unit-capacity networks from $O(nm)$ to $O(\min(n^{2/3}, m^{1/2}) m)$. Default value: False. 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:`edmonds_karp` :meth:`preflow_push` 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 shortest_augmenting_path 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 = shortest_augmenting_path(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value 3.0 >>> flow_value == R.graph["flow_value"] True r algorithm)rEr-r _clear_cache) r6r7r8r r9 value_onlyr:r;r<s &&&&&&&& rrrs4X &aAfUA5AGGKOOA Hr)r NFFN)__doc__ collectionsrnetworkxr edmondskarprutilsrr__all__rE _dispatchabler0rrrrUsY*6 % &T nE%L)SWL L r