habSrS/rSSKJrJrJr SSKJrJr SSK r SSK J r "SS5r \ "S 5\ R"S \"S 5SS .S 9SSj55rg)z< Minimum cost flow algorithms on directed connected graphs. network_simplex)chainislicerepeat)ceilsqrtN)not_implemented_forct\rSrSrSSjrSrSrSrSrSr Sr S r S r S r S rS rSrSrSrSrg)_DataEssentialsAndFunctionsc|^^ [U5Ul[UR5VVs0sHupgXv_M snnUlURVs/sH!oqRUR US5PM# snUl/Ul/UlU(a/Ul 0Ul /Ul /Ul U(dURSS9nOURSSS9n[S5m UU 4SjU5n[U5HupiURRURU S5 URRURU S5 U(aURRU S5 X`RU SS 'URRU S R TT 55 URRU S R US55 M SUlSUlSUlSUlSUlSUlSUlSUlSUlS Ulgs snnfs snf) NrTdatarkeysinfc3x># UH/oSUS:wdMUSRTT5S:wdM+Uv M1 g7f)rNget).0ecapacityrs Y/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/flow/networksimplex.py 7_DataEssentialsAndFunctions.__init__..'s8TEqqTQqT\aeii#6NRS6SEs :: :rrF)list node_list enumerate node_indicesnodesr node_demands edge_sources edge_targets edge_keys edge_indicesedge_capacities edge_weightsedgesfloatappend edge_count edge_flownode_potentialsparent parent_edge subtree_size next_node_dft prev_node_dftlast_descendent_dft_spanning_tree_initialized) selfG multigraphdemandrweightiur+rrs ` @r__init__$_DataEssentialsAndFunctions.__init__sa.7.GH.GdaQT.GH/3~~ /=!GGAJNN61 %~  DN!GGG&EGGDG1EElTETe$DA    $ $T%6%6qt%< =    $ $T%6%6qt%< =%%ad+()  af %  ' '" (C(@ A    $ $QrUYYvq%9 :%#  !!#'  'QI s H3(H9c[UR5Ul[[ [ SUR5SUR 555UlUR Vs/sH o3S::aUOU*PM snUl[[ [ SU5S/55Ul [[URURU-55Ul [[ [ SU5US-/55Ul [[ [SU5SS/55Ul [[SU55Ul[[ [U5US- /55UlSUlgs snf)Nrc38# UHn[U5v M g7fNabsrds rrG_DataEssentialsAndFunctions.initialize_spanning_tree..Cs.Q?P!s1vv?PrrT)lenr(r.rrrr$r/r0r1ranger2r3r4r5r6r7)r8nfaux_infrGs rinitialize_spanning_tree4_DataEssentialsAndFunctions.initialize_spanning_tree@s:d//0 &DOO,.Qt?P?P.Q R 8<7H7H 7H!QHXI -7H 5A78  $//4??Q#6 7 !va|a!eW!=>! %1+Aw ' "%A,/#' %(QUG $$  +/' s(E1c~URUnURUnX4:a%URUnURUnX4:aM%X4:a%URUnURUnX4:aM%X4:XaDX:wa=URUnURUnURUnURUnOU$M)zH Find the lowest common ancestor of nodes p and q in the spanning tree. )r3r1)r8pqsize_psize_qs r find_apex%_DataEssentialsAndFunctions.find_apexVs""1%""1%/KKN**1-//KKN**1-/6 AA!..q1F AA!..q1FHcU/n/nX:waEURURU5 URUnURU5 X:waMEX44$)zH Returns the nodes and edges on the path from node p to its ancestor w. )r-r2r1)r8rQwWnWes r trace_path&_DataEssentialsAndFunctions.trace_pathlsTS f IId&&q) * AA IIaLfv rWcURX#5nURX$5upVUR5 UR5 Xa/:waURU5 URX45upxUS XW- nXh- nXV4$)z Returns the nodes and edges on the cycle containing edge i == (p, q) when the latter is added to the spanning tree. The cycle is oriented in the direction from p to q. r)rUr\reverser-) r8r=rQrRrYrZr[WnRWeRs r find_cycle&_DataEssentialsAndFunctions.find_cyclexsr NN1 &  9 IIaL??1( G  v rWc[X!5HHupEURUU:XaURU==U- ss'M1URU==U-ss'MJ g)zA Augment f units of flow along a cycle represented by Wn and We. N)zipr%r/)r8rZr[fr=rQs r augment_flow(_DataEssentialsAndFunctions.augment_flowsOKDA  #q(q!Q&!q!Q&!  rWc#r# Uv URUnX:waURUnUv X:waMgg7f)z4 Yield the nodes in the subtree rooted at a node p. N)r6r4)r8rQls r trace_subtree)_DataEssentialsAndFunctions.trace_subtrees=  $ $Q 'f""1%AGfs177cURUnURUnURUnURUnSURU'SUR U'X`RU'X@RU'X RU'XPRU'UbMURU==U-ss'URUU:XaX@RU'URUnUbMLgg)zD Remove an edge (s, t) where parent[t] == s from the spanning tree. N)r3r5r6r4r1r2)r8stsize_tprev_tlast_t next_last_ts r remove_edge'_DataEssentialsAndFunctions.remove_edges""1%##A&))!,((0  A"%06"*0;'%&6" &1m   a F * ''*f4.4((+ AA mrWc`/nUb%URU5 URUnUbM%UR5 [U[ USS55GHXup1UR UnUR UnURUnUR UnURUnXRU'SURU'URUURU'SURU'X@R U- UR U'X@R U'XRU'X`RU'XRU'XpRU'XW:XaX`R U'UnXpRU'X0RU'XRU'XPRU'XPR U'GM[ g)z3 Make a node q the root of its containing subtree. Nr) r-r1r_rerr3r6r5r4r2) r8rR ancestorsrQrSlast_pprev_qlast_q next_last_qs r make_root%_DataEssentialsAndFunctions.make_roots m   Q  AAm  6)Q#=>DA&&q)F--a0F''*F--a0F,,V4KKKN!DKKN"&"2"21"5D  Q "&D  Q #),=,=a,@#@D  a #)  a )4  v &.4  { +)*  v &$*  q !.4((+%+  q !)*  v &)*  v &$*  q !*0 $ $Q '7?rWcURUnURUnURUnURUnX RU'XRU'X0RU'X@R U'XpR U'XPRU'UbMURU==U- ss'URUU:XaXpRU'URUnUbMLgg)zK Add an edge (p, q) to the spanning tree where q is the root of a subtree. N)r6r4r3r1r2r5)r8r=rQrRrx next_last_prTrzs radd_edge$_DataEssentialsAndFunctions.add_edges))!,((0 ""1%))!, A%&6" &1*0;'%06"m   a F * ''*f4.4((+ AA mrWcDX0RU:Xa0URUURU- URU- nO/URUURU-URU- nURU5HnURU==U- ss'M g)zn Update the potentials of the nodes in the subtree rooted at a node q connected to its parent p by an edge i. N)r&r0r*rk)r8r=rQrRrGs rupdate_potentials-_DataEssentialsAndFunctions.update_potentialss !!!$ $$$Q'$*;*;A*>>AUAUVWAXXA$$Q'$*;*;A*>>AUAUVWAXXA##A&A   #q ( #'rWcURUURURU- URURU-nURUS:XaU$U*$)z&Returns the reduced cost of an edge i.r)r*r0r%r&r/)r8r=cs r reduced_cost(_DataEssentialsAndFunctions.reduced_costsr   a ""4#4#4Q#78 9""4#4#4Q#78 9 NN1%*q22rWc## URS:Xag[[[UR555nURU-S- U-nSnSnX2:aXA-nXPR::a [ XE5nO6XPR-n[ [ X@R5[ U55nUn[ X`RS9nURU5nUS:aUS- nOXURUS:XaURUn URUn OURUn URUn XyU 4v SnX2:aMgg7f)z-Yield entering edges until none can be found.rNrkey) r.intrrrKrminrr/r%r&) r8BMmrfrjr+r=rrQrRs rfind_entering_edges/_DataEssentialsAndFunctions.find_entering_edges s1 ??a   T$//*+ , __q 1 $ *  eAOO#a __$eA7qBAE001A!!!$AAvQ>>!$)))!,A))!,A))!,A))!,AAg 1es D;E?EcURUU:XaURUURU- $URU$)zVReturns the residual capacity of an edge i in the direction away from its endpoint p. )r%r)r/)r8r=rQs rresidual_capacity-_DataEssentialsAndFunctions.residual_capacity4sM   #q(   #dnnQ&7 7 " rWc^[[[U5[U55U4SjS9up4TRUU:XaTRUOTRUnX4U4$)z=Returns the leaving edge in a cycle represented by Wn and We.c">TR"U6$rC)r)i_pr8s r?_DataEssentialsAndFunctions.find_leaving_edge..BsD22C8rWr)rrereversedr%r&)r8rZr[jrnros` rfind_leaving_edge-_DataEssentialsAndFunctions.find_leaving_edge>s_  hrl +8 %)$5$5a$8A$=D  a 4CTCTUVCWQwrW)r7r)r.r/r(r'r%r&r*r6r4r$r"r r0r1r2r5r3Nr;rr<)__name__ __module__ __qualname____firstlineno__r?rNrUr\rbrgrkrtr|rrrrrr__static_attributes__rWrr r sSJR/ b/,, &'0$1L0 )3&T rWr undirectedr;r)rr<) node_attrs edge_attrsc ^^^^^^[U5S:Xa[R"S5eUR5n[ XUTTS9m[ S5m[ TRTR5H/upV[U5T:XdM[R"SU<S35e [ TRTR5H/upx[U5T:XdM[R"SU<S35e U(d[R"US S 9n O[R"US S S 9n U HCn[US RTS55T:XdM'[R"SUS S <S35e [TR5S:wa[R"S5e[ TRTR 5H&upzU S:dM [R"SU<S35e U(d[R"US S 9n O[R"US S S 9n U H:nUS RTT5S:dM[R"SUS S <S35e [#TR5HyupUS:a8TR$R'S 5 TR(R'U 5 MCTR$R'U 5 TR(R'S 5 M{ S[+[-[U4SjTR 55[STR55/STR555-=(d Sm[TR5n TRR/[1TU 55 TR R/[1TU 55 TR3U T5 TR55HupnTR7XU5unnTR9UU5unnnTR;UUTR=UU55 U U:wdM\TR>UU:waUUnnURAU 5URAU5:aXpTRCUU5 TREU5 TRGXU5 TRIXU5 M [KU4Sj[MU *S555(a[R"S5e[KUU4Sj[MTRN555(d/[KUUU4Sj[R"US S 955(a[RP"S5eTRRTRNS 2 [S[ TRTRR555nTRV s0sHo0_M sn mU4SjnU4SjTR$5TlU4SjTR(5TlU(dJ[ TR$TR(TRR5H nU"U5 M URUS S 9n OU[ TR$TR(TRVTRR5H nU"U5 M URUS S S 9n U HnUSUS:wa+US RTT5S:XaU"US S S-5 M8M:US RTS5nUS:aU"US S S-5 MeUS Tn UX-- nU"US S U 4-5 M UT4$s sn f)a8Find a minimum cost flow satisfying all demands in digraph G. This is a primal network simplex algorithm that uses the leaving arc rule to prevent cycling. G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive some amount of flow. A negative demand means that the node wants to send flow, a positive demand means that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is equal to the demand of that node. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. demand : string Nodes of the graph G are expected to have an attribute demand that indicates how much flow a node wants to send (negative demand) or receive (positive demand). Note that the sum of the demands should be 0 otherwise the problem in not feasible. If this attribute is not present, a node is considered to have 0 demand. Default value: 'demand'. 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'. weight : string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. Returns ------- flowCost : integer, float Cost of a minimum cost flow satisfying all demands. flowDict : dictionary Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u, v). Raises ------ NetworkXError This exception is raised if the input graph is not directed or not connected. NetworkXUnfeasible This exception is raised in the following situations: * The sum of the demands is not zero. Then, there is no flow satisfying all demands. * There is no flow satisfying all demand. NetworkXUnbounded This exception is raised if the digraph G has a cycle of negative cost and infinite capacity. Then, the cost of a flow satisfying all demands is unbounded below. Notes ----- This algorithm is not guaranteed to work if edge weights or demands are floating point numbers (overflows and roundoff errors can cause problems). As a workaround you can use integer numbers by multiplying the relevant edge attributes by a convenient constant factor (eg 100). See also -------- cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost Examples -------- A simple example of a min cost flow problem. >>> G = nx.DiGraph() >>> G.add_node("a", demand=-5) >>> G.add_node("d", demand=5) >>> G.add_edge("a", "b", weight=3, capacity=4) >>> G.add_edge("a", "c", weight=6, capacity=10) >>> G.add_edge("b", "d", weight=1, capacity=9) >>> G.add_edge("c", "d", weight=2, capacity=5) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost 24 >>> flowDict {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} The mincost flow algorithm can also be used to solve shortest path problems. To find the shortest path between two nodes u and v, give all edges an infinite capacity, give node u a demand of -1 and node v a demand a 1. Then run the network simplex. The value of a min cost flow will be the distance between u and v and edges carrying positive flow will indicate the path. >>> G = nx.DiGraph() >>> G.add_weighted_edges_from( ... [ ... ("s", "u", 10), ... ("s", "x", 5), ... ("u", "v", 1), ... ("u", "x", 2), ... ("v", "y", 1), ... ("x", "u", 3), ... ("x", "v", 5), ... ("x", "y", 2), ... ("y", "s", 7), ... ("y", "v", 6), ... ] ... ) >>> G.add_node("s", demand=-1) >>> G.add_node("v", demand=1) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight") True >>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0]) [('s', 'x'), ('u', 'v'), ('x', 'u')] >>> nx.shortest_path(G, "s", "v", weight="weight") ['s', 'x', 'u', 'v'] It is possible to change the name of the attributes used for the algorithm. >>> G = nx.DiGraph() >>> G.add_node("p", spam=-4) >>> G.add_node("q", spam=2) >>> G.add_node("a", spam=-2) >>> G.add_node("d", spam=-1) >>> G.add_node("t", spam=2) >>> G.add_node("w", spam=3) >>> G.add_edge("p", "q", cost=7, vacancies=5) >>> G.add_edge("p", "a", cost=1, vacancies=4) >>> G.add_edge("q", "d", cost=2, vacancies=3) >>> G.add_edge("t", "q", cost=1, vacancies=2) >>> G.add_edge("a", "t", cost=2, vacancies=4) >>> G.add_edge("d", "w", cost=3, vacancies=4) >>> G.add_edge("t", "w", cost=4, vacancies=1) >>> flowCost, flowDict = nx.network_simplex( ... G, demand="spam", capacity="vacancies", weight="cost" ... ) >>> flowCost 37 >>> flowDict {'p': {'q': 2, 'a': 2}, 'q': {'d': 1}, 'a': {'t': 4}, 'd': {'w': 2}, 't': {'q': 1, 'w': 1}, 'w': {}} References ---------- .. [1] Z. Kiraly, P. Kovacs. Efficient implementation of minimum-cost flow algorithms. Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012. .. [2] R. Barr, F. Glover, D. Klingman. Enhancement of spanning tree labeling procedures for network optimization. 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