h8SrS/rSSKJr SSKJr SSKrSSKJ r J r J r Sr \ "S 5S 5r S r\R"S \"S 5SS.S9S SS\ 4Sj5rg)z/ Capacity scaling minimum cost flow algorithm. capacity_scaling)chain)logN) BinaryHeaparbitrary_elementnot_implemented_forc[R"5nURU5 URSn[ S5nUHinXR 5HPupVUnUR 5HupUSU:XdM[ XvS5nM Xs:wdM@URXEUS9 MR Mk [R"U5(a[R"S5eg)z)Detect infinite-capacity negative cycles.infcapacityweight)r UNegative cost cycle of infinite capacity found. Min cost flow may be unbounded below.N) nxDiGraphadd_nodes_fromgraphfloatitemsminadd_edgenegative_edge_cycleNetworkXUnbounded) RGr f_infuvewks Z/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/flow/capacityscaling.py_detect_unboundednessr"s AQ ''%.C %LE DJJLDAA Z=C'A{+A"z 1 *! a  "" 4  ! undirectedc ^^^^ ^ [UU4SjT55S:wa[R"S5e[R"5m T R UU4SjT55 [ S5m [R "TSS9HJupEnURUS5S:dMURTT 5T :XdM6[R"S5e TR5(aKTRSSS 9VVVVs/sH)upEpvXE:wdM URTT 5S:dM%XEXv4PM+ nnnnnOHTRSS9VVVs/sH*upEnXE:wdM URTT 5S:dM%XESU4PM, nnnn[[U 4S jT 55S [UU 4S jU55-5=(d S m UH^upEpv[URTT 5T 5n URUS5n T RXEUS4XSS9 T RXTUS4SU *SS9 M` T T RS'[T 5 T $s snnnnfs snnnf)z4Build a residual network and initialize a zero flow.c3b># UH$nTRURTS5v M& g7frNnodesget.0rrdemands r! *_build_residual_network..,s' 0a1771:>>&! $ $as,/rzSum of the demands should be 0.c3l># UH)oTRURTS5*SS.4v M+ g7f)r)excess potentialNr(r+s r!r.r/0s2MN vq11 BCQs14r Tdatar)r4keysc3Z># UH n[TRUS5v M" g7f)r1N)absr)r,rrs r!r.r/Qs&51aAGGAJx())1s(+c3Z># UH upp4TU;dMUTT:wdMUTv M" g7f)N)r,rrr rr r s r!r.r/Ss:"+JA!q=%&x[C%7( "+s ++ +)keyr r flowF)sumrNetworkXUnfeasible MultiDiGraphrrselfloop_edgesr*r is_multigraphedgesmaxrrrr") rr-r r rrrr edge_listrrrr s ``` @@r!_build_residual_networkrH)sI 0a 00A5##$EFF AMN ,C$$QT2a 55 a AEE(C$8C$?&&8 3  gg4dg; ; av %%#.2 Q1L;  777- -av %%#.2 Q1aL-   515 5 "+    a h$c * EE&!  1aY1 E 1aZ!QBQ G AGGEN! HU  s$? H< H< 'H<  II0 Ic 6[S5n0nUR5(aUHn0XV'XR5HdupxUR5V V s0sH<upU Xg:wd+U RX$5S::dU RUS5S:aSOX_M> sn n XVU'Mf XR5H:upxXuU;dMXVUR SUR555 M< M U$UHnXR5VV s0sH<upzUXg:wd+U RX$5S::dU RUS5S:aSOX_M> sn nXV'XVR SXR555 M U$s sn n fs sn nf)z0Build a flow dictionary from a residual network.r rc3P# UHupUSS:dMUSUS4v M g7fr>rNr;)r,r rs r!r.#_build_flow_dict..s0+6@daAfIPQM)1qy)js&&c3x# UH0upUR5HnUSS:dMXS4v M M2 g7frK)values)r,resrs r!r.rLs? )EAAV9q=fI$)s#::)rrCrr*update) rrr r r flow_dictrrrOr rs r!_build_flow_dictrRls ,CIAIL!#  #!+FaeeH&:a&?155QRCSWXCX[ )!+ # Q&! $LO**+68hhj+&H !ADJJL)DA !%%"6!";quuVQ?OST?T% )IL L   TZZ\   C #$sAF AFr-r )r r ) node_attrs edge_attrsr r c  ^^^&^'[XTT5m&[S5m'[UU'U4Sj[R"USS955n[ [ T'*/ST&RSS9555nUT'*:XaU[UT&TT54$T&RnT&RnS[[US55-n U S:Ga;T&Hn XzSn XR5HupU R5HupUS US - nUS U - X|S-S :dM&US US - nUU :dM9US ==U- ss'XU US US(+4S ==U-ss'XzS ==U-ss'X|S ==U- ss'M M M [5n[5nURnUR nURnUR nT&H+n XzS nUU :a U"U 5 MUU *::dM#U"U 5 M- U(GaU(Ga[#U5nSn0nUS0nU"5nUR$nUR&nU"US 5 U(aUR)5un nUUU 'U U;aU nOXzSn XR5H|upU U;aM T'n U R5H)upUS US - U :dMUS n!U!U :dM#U!n Un"Un#M+ U T':XaMTUU -U - X|S-n$U"U U$5(dMtU W"W#4UU 'M~ U(aMUbW U:waAU n UU upnUS ==U - ss'XU US US(+4S ==U -ss'U U:waMAUUS ==U -ss'UUS ==U - ss'UUS U :aU"U5 UUS U *:aU"U5 UUn%UR5Hun nXzS==UU%- -ss'M OU"U5 U(a U(aGMU S-n U S:aGM;[+U&4SjT&55(a[R,"S5eT&HMn XR5H4upU R/5HnUS nUS :dMUUUS -- nM M6 MO U[UT&TT54$)aPFind a minimum cost flow satisfying all demands in digraph G. This is a capacity scaling successive shortest augmenting path algorithm. 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 or MultiDiGraph 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'. heap : class Type of heap to be used in the algorithm. It should be a subclass of :class:`MinHeap` or implement a compatible interface. If a stock heap implementation is to be used, :class:`BinaryHeap` is recommended over :class:`PairingHeap` for Python implementations without optimized attribute accesses (e.g., CPython) despite a slower asymptotic running time. For Python implementations with optimized attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better performance. Default value: :class:`BinaryHeap`. Returns ------- flowCost : integer Cost of a minimum cost flow satisfying all demands. flowDict : dictionary If G is a digraph, a dict-of-dicts keyed by nodes such that flowDict[u][v] is the flow on edge (u, v). If G is a MultiDiGraph, a dict-of-dicts-of-dicts keyed by nodes so that flowDict[u][v][key] is the flow on edge (u, v, key). Raises ------ NetworkXError This exception is raised if the input graph is not directed, 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 does not work if edge weights are floating-point numbers. See also -------- :meth:`network_simplex` 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.capacity_scaling(G) >>> flowCost 24 >>> flowDict {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}} 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.capacity_scaling( ... 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': {}} r c3># UHAupnURTT5S::dURTS5S:aSO UTUT-v MC g7fr')r*)r,rrrr r r s r!r.#capacity_scaling..!s\7GA! 553 1 $fa(8A(= x[1V9 $ %7sA A Tr3c30# UH upo3Sv M g7f)r Nr;)r,rrrs r!r.rW)sN;Ma m;Msr9r<r2r r>r rr1Nc3N># UHnTRUSS:gv M g7f)r1rN)r)r8s r!r.rWs# 0a1771:h 1 $as"%zNo flow satisfying all demands.)rHrr?rrBrErrDrRr)succintrrsetaddremoverinsertr*popanyr@rN)(rr-r r heap flow_costwmaxR_nodesR_succdeltarp_urrOr rr>STS_addS_removeT_addT_remover1stdpredhh_inserth_getd_uwminrkminemind_vd_trr s( `` @@r!rrsJ 8V >>r#)__doc____all__ itertoolsrmathrnetworkxrutilsrrr r"rHrR _dispatchablerrr;r#r!rs  GG 4\"? #? D)Xu$KH:|?|?r#