hi=SrSSKJr SSKJr SSKrSSKJr SSKJ r J r J r J r J r S /rS r\R "S \"S 50S S9SSj5rg)zA Highest-label preflow-push algorithm for maximum flow problems. )deque)isliceN)arbitrary_element) CurrentEdgeGlobalRelabelThresholdLevelbuild_residual_networkdetect_unboundedness preflow_pushcX^^^^^^^^^^^^^^^TU;a#[R"S[T5S35eTU;a#[R"S[T5S35eTT:Xa[R"S5eUcSnUS:a[R"S5eUc [X5mOUm[ TTT5 TR mTR mTRmTH*nSTUS'TUR5HnSUS'M M, U4SjmT"T5mTT;aSTRS 'T$[T5m[UU4S jT55mTTT'[TTR5U5mTH-nUT;aTUOTS -TUS '[TU5TUS 'M/ UU4SjmTTR5HupyU Sn U S:dMT"TXz5 M [!ST-5V s/sH n [#5PM sn mTHbnUT:wdM UT:wdMTTUS n TUSS:aU R$R'U5 MGU R(R'U5 Md UUUU4SjmUUU4SjmUUUUUU4Sjn UUUU4SjnUUUUUUU4SjnTnUS:aTUn U R$(dUS -nOUnU n[+U R$5nU "US5nTR-5(aU"S5nUmTR/5 O>UR$(d!UR((dU"U5 US - nUmO [TU5mMUS:aMU(aTTSTRS 'T$U"S5nTR/5 UT:arTUn U R$(dUS -nOM[+U R$5nU "US5nTR-5(aU"S5nTR/5 MiUT:aMrTTSTRS 'T$s sn f)z;Implementation of the highest-label preflow-push algorithm.znode z not in graphz!source and sink are the same noderz(global_relabel_freq must be nonnegative.excessflowc>US0n[US4/5nU(agUR5up4US- nTUR5H0upVXQ;dM USUS:dMXAU'URXT45 M2 U(aMgU$)zIPerform a reverse breadth-first search from src in the residual network. rrrcapacity)rpopleftitemsappend)srcheightsquheightvattrR_preds V/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/flow/preflowpush.py reverse_bfs&preflow_push_impl..reverse_bfs5s( C8*  IA aKF!!9??,#V tJ7G(G!'AJHHa[)-a flow_valuec3<># UHoT:wdM TUv M g7f)N).0rrss r $preflow_push_impl..Ps;AFZWQZs  rr curr_edgec>TUUS==U- ss'TUUS==U-ss'TUS==U-ss'TUS==U- ss'g)z$Push flow units of flow from u to v.rrNr$)rrrR_nodesR_succs rpushpreflow_push_impl..pushZs[q ! V$q ! V$ 8$ 8$r!rc>UT:waYUT:waRTTUSnXR;a7URRU5 URRU5 gggg)zAMove a node from the inactive set to the active set of its level.rN)inactiveremoveactiveadd)rlevelr+levelsr&ts ractivate#preflow_push_impl..activaters] 6a1f71:h/0ENN"%%a(   ##6r!c>TR[TU55 [U4SjTUR555S-$)z,Relabel a node to create an admissible edge.c3T># UHupUSUS:dMTUSv M g7f)rrrNr$)r%rrr+s rr'5preflow_push_impl..relabel..~s60GA<$z"22% 8$0s((r)add_worklenminr)rr+r,grts rrelabel"preflow_push_impl..relabelzsF S^$ %ay0     r!c>>TUSnTUSnUnT URRU5 UR5upVUTUSS-:XacUSUS:aW[TUSUSUS- 5nT "XU5 T "U5 TUSS:XaT URR U5 OUR 5 MUTUS'U$![a> T "U5nU(a*UT S- :a!T URR U5 MMUnNSf=f)zDischarge a node until it becomes inactive or, during phase 1 (see below), its height reaches at least n. The node is known to have the largest height among active nodes. rr)rrrrr)r3r2getr?r1r4 move_to_next StopIteration)r is_phase1rr) next_heightrrrr+r8r6nr-rAs r discharge$preflow_push_impl..dischargesO H%AJ{+  v$$Q'mmoGAH-11d6lT*EU6U71:h/j1ADL1PQQ4  1:h'1,6N++//2 %&&(6 & 8!! %!1q56N))--a0 %  %s8CADDDc>[TUS-TS-5HnURHnTS-TUS'M URHnTS-TUS'M TTS-RRUR5 URR 5 TTS-RRUR5 URR 5 M g)zApply the gap heuristic.rrN)rr3r1updateclear)rr5rr+r6 max_heightrIs r gap_heuristic(preflow_push_impl..gap_heuristicsFFQJ Q?E\\'(1u 8$"^^'(1u 8$$ 1q5M ' ' 5 LL   1q5M " " ) )%.. 9 NN "@r!c~>U(aT OT nT "U5nU(dUT [UR55nU(a&THnXB;dM TUST :dMT S-X$'M! OUHnX$==T - ss'M UT - nX! UR5HupETUSnXV:wdMUT UR;a=T URR U5 T URR U5 O.global_relabels9a!c" )* # 8(rSr sizerrranger r3r4r1r is_reached clear_work) Gr&r7rresidualglobal_relabel_freq value_onlyrerrir5rJrPrYrrW old_levelrXr+rr,r8r@rr6rOrIr-rArs `` @@@@@@@@@@@@@rpreflow_push_implrms zs1vhm<==zs1vhm<==AvBCC"QIJJ "1 / Aq!ggG VVF VVF  8!!#AAfI$  !nG!"  AA;;;JGAJ AFFH.A BC-.'\wqzq1u 8"-fQi"8 ;%!9??$J !8 A % %QU| ,|!eg| ,F  6a1f71:h/0Eqz(#a'   #""1% $$  ''R # #FF 1*6NE<<! JI!%,,/Aq$'F~~(-#  %%i.@.@ j)#a# !V4 ; 1*F ' 8 4  E "FNN 1*6NE<<! !%,,/Aq%(F~~'.  1* $AJx0AGGL Ho-s3P'rinfT) edge_attrs returns_graphc l[XX#XEU5nSURS'[R"U5 U$)a/Find a maximum single-commodity flow using the highest-label preflow-push 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 $O(n^2 \sqrt{m})$ 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 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'. residual : NetworkX graph Residual network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None. global_relabel_freq : integer, float Relative frequency of applying the global relabeling heuristic to speed up the algorithm. If it is None, the heuristic is disabled. Default value: 1. value_only : bool If False, compute a maximum flow; otherwise, compute a maximum preflow which is enough for computing the maximum flow value. Default value: False. 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:`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 node :samp:`u` in :samp:`R`, :samp:`R.nodes[u]['excess']` represents the difference between flow into :samp:`u` and flow out of :samp:`u`. 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']`. 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 preflow_push 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 = preflow_push(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value == R.graph["flow_value"] True >>> # preflow_push also stores the maximum flow value >>> # in the excess attribute of the sink node t >>> flow_value == R.nodes["y"]["excess"] True >>> # For some problems, you might only want to compute a >>> # maximum preflow. >>> R = preflow_push(G, "x", "y", value_only=True) >>> flow_value == R.graph["flow_value"] True >>> flow_value == R.nodes["y"]["excess"] True r algorithm)rmrar[ _clear_cache)rfr&r7rrgrhrirXs rr r #s4F !XJWA)AGGKOOA Hr!)rNrF)__doc__ collectionsr itertoolsrnetworkxr[utilsrrr r r r __all__rm _dispatchablefloatr r$r!rr|sc&  J Zj%,7tLSXE ME r!