D^(habdZdgZddlmZmZmZddlmZmZddl Z ddl m Z GddZ e d e jd ed dd  ddZy)z< Minimum cost flow algorithms on directed connected graphs. network_simplex)chainislicerepeat)ceilsqrtN)not_implemented_forcjeZdZ ddZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZdZy)_DataEssentialsAndFunctionscb t||_t|jDcic]\}}|| c}}|_|jDcgc]!}|j|j |d#c}|_g|_g|_|rg|_ i|_ g|_ g|_ |s|jd}n|jdd}td  fd|D}t|D]\}} |jj|j| d|jj|j| d|r|jj| d||j| dd <|jj| d j  |jj| d j |dd|_d|_d|_d|_d|_d|_d|_d|_d|_d |_ycc}}wcc}w) NrTdatarkeysinfc3lK|]+}|d|dk7s|djdk7s(|-yw)rNget).0ecapacityrs e/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/networkx/algorithms/flow/networksimplex.py z7_DataEssentialsAndFunctions.__init__..'s6TqQqTQqT\aeii#6NRS6STs 444rrF)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__z$_DataEssentialsAndFunctions.__init__sa.7.GHdaQTH/3~~ *+AGGAJNN61 %  DN!GGG&EGGDG1EElTETe$ ;DAq    $ $T%6%6qt%< =    $ $T%6%6qt%< =%%ad+()D  af %  ' '" (C(@ A    $ $QrUYYvq%9 : ;#  !!#'  'QI s H&&H,ct|j|_tt t d|jd|j D|_|j Dcgc] }|dkr|n| c}|_tt t d|dg|_ tt|j|j|z|_ tt t d||dzg|_ tt td|ddg|_ ttd||_tt t||dz g|_d|_ycc}w)Nrc32K|]}t|ywNabsrds rrzG_DataEssentialsAndFunctions.initialize_spanning_tree..Cs.Q!s1v.QrrT)lenr&r,rrrr"r-r.r/ranger0r1r2r3r4r5)r6nfaux_infrDs rinitialize_spanning_treez4_DataEssentialsAndFunctions.initialize_spanning_tree@s7d//0 &DOO,.Qt?P?P.Q R 8<7H7H 23QHXI - 5A78  $//4??Q#6 7 !va|a!eW!=>! %1+Aw ' "%A,/#' %(QUG $$  +/' s(E.cx|j|}|j|} ||kr$|j|}|j|}||kr$||kDr$|j|}|j|}||kDr$||k(rD||k7r=|j|}|j|}|j|}|j|}n|S)zX Find the lowest common ancestor of nodes p and q in the spanning tree. )r1r/)r6pqsize_psize_qs r find_apexz%_DataEssentialsAndFunctions.find_apexVs""1%""1%6/KKN**1-6/6/KKN**1-6/6 AA!..q1F AA!..q1FHc|g}g}||k7rD|j|j||j|}|j|||k7rD||fS)zX Returns the nodes and edges on the path from node p to its ancestor w. )r+r0r/)r6rLwWnWes r trace_pathz&_DataEssentialsAndFunctions.trace_pathlsZS 1f IId&&q) * AA IIaL1f2v rQc |j||}|j||\}}|j|j||gk7r|j||j||\}}|d=||z }||z }||fS)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)rPrVreverser+) r6r;rLrMrSrTrUWnRWeRs r find_cyclez&_DataEssentialsAndFunctions.find_cyclexs NN1a A&B  !9 IIaL??1a(S G c  c 2v rQct||D]F\}}|j||k(r|j|xx|z cc<0|j|xx|zcc<Hy)zQ Augment f units of flow along a cycle represented by Wn and We. N)zipr#r-)r6rTrUfr;rLs r augment_flowz(_DataEssentialsAndFunctions.augment_flowsXBK 'DAq  #q(q!Q&!q!Q&!  'rQc#pK||j|}||k7r|j|}|||k7ryyw)zD Yield the nodes in the subtree rooted at a node p. N)r4r2)r6rLls r trace_subtreez)_DataEssentialsAndFunctions.trace_subtreesB  $ $Q '1f""1%AG1fs166c|j|}|j|}|j|}|j|}d|j|<d|j |<||j|<||j|<||j|<||j|<|K|j|xx|zcc<|j||k(r||j|<|j|}|Jyy)zT Remove an edge (s, t) where parent[t] == s from the spanning tree. N)r1r3r4r2r/r0)r6stsize_tprev_tlast_t next_last_ts r remove_edgez'_DataEssentialsAndFunctions.remove_edges""1%##A&))!,((0  A"%06"*0;'%&6" &1m   a F * ''*f4.4((+ AA mrQcrg}|#|j||j|}|#|jt|t |ddD]e\}}|j |}|j |}|j|}|j |}|j|}||j|<d|j|<|j||j|<d|j|<||j |z |j |<||j |<||j|<||j|<||j|<||j|<||k(r||j |<|}||j|<||j|<||j|<||j|<||j |<hy)zC Make a node q the root of its containing subtree. Nr) r+r/rXr]rr1r4r3r2r0) r6rM ancestorsrLrNlast_pprev_qlast_q next_last_qs r make_rootz%_DataEssentialsAndFunctions.make_roots m   Q  AAm  6)Q#=> 1DAq&&q)F--a0F''*F--a0F,,V4KDKKN!DKKN"&"2"21"5D  Q "&D  Q #)D,=,=a,@#@D  a #)D  a )4D  v &.4D  { +)*D  v &$*D  q !.4((+%+D  q !)*D  v &)*D  v &$*D  q !*0D $ $Q '7 1rQc|j|}|j|}|j|}|j|}||j|<||j|<||j|<||j |<||j |<||j|<|K|j|xx|z cc<|j||k(r||j|<|j|}|Jyy)z[ Add an edge (p, q) to the spanning tree where q is the root of a subtree. N)r4r2r1r/r0r3)r6r;rLrMrm next_last_prOros radd_edgez$_DataEssentialsAndFunctions.add_edges))!,((0 ""1%))!, A%&6" &1*0;'%06"m   a F * ''*f4.4((+ AA mrQc@||j|k(r0|j||j|z |j|z }n/|j||j|z|j|z }|j|D]}|j|xx|z cc<y)z 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$r.r(rb)r6r;rLrMrDs rupdate_potentialsz-_DataEssentialsAndFunctions.update_potentialss !!!$ $$$Q'$*;*;A*>>AUAUVWAXXA$$Q'$*;*;A*>>AUAUVWAXXA##A& )A   #q ( # )rQc|j||j|j|z |j|j|z}|j|dk(r|S| S)z&Returns the reduced cost of an edge i.r)r(r.r#r$r-)r6r;cs r reduced_costz(_DataEssentialsAndFunctions.reduced_costsr   a ""4#4#4Q#78 9""4#4#4Q#78 9 NN1%*q22rQc#K|jdk(ryttt|j}|j|zdz |z}d}d}||kr||z}||jkr t ||}n8||jz}t t ||jt |}|}t ||j}|j|}|dk\r|dz }nX|j|dk(r|j|} |j|} n|j|} |j|} || | fd}||kryyw)z-Yield entering edges until none can be found.rNrkey) r,intrrrGrminryr-r#r$) r6BMmr^rar)r;rxrLrMs rfind_entering_edgesz/_DataEssentialsAndFunctions.find_entering_edges sD ??a   T$//*+ , __q 1 $ *  !eAADOO#a T__$eAt7qBAEt001A!!!$AAvQ>>!$)))!,A))!,A))!,A))!,AAg 1!es EEEc|j||k(r|j||j|z S|j|S)zfReturns the residual capacity of an edge i in the direction away from its endpoint p. )r#r'r-)r6r;rLs rresidual_capacityz-_DataEssentialsAndFunctions.residual_capacity4sM   #q(   #dnnQ&7 7 " rQcttt|t|fd\}}j||k(rj|nj|}|||fS)z=Returns the leaving edge in a cycle represented by Wn and We.c"j|Sr@)r)i_pr6s rz?_DataEssentialsAndFunctions.find_leaving_edge..Bs2D22C8rQr{)r~r]reversedr#r$)r6rTrUjrdres` rfind_leaving_edgez-_DataEssentialsAndFunctions.find_leaving_edge>sc  hrl +8 1%)$5$5a$8A$=D  a 4CTCTUVCW!QwrQNr9rr:)__name__ __module__ __qualname__r=rJrPrVr[r_rbrjrqrtrvryrrrrQrr r sSJR/ b/,, &'0$1L0 )3&T rQr undirectedr9r)rr:) node_attrs edge_attrsc t|dk(rtjd|j}t |||t dt jjD],\}}t|k(stjd|dt jjD],\}}t|k(stjd|d|stj|d } ntj|d d } | D]?}t|d jdk(s%tjd|d d dtjdk7rtjdt jj D]#\}} | dks tjd|d|stj|d } ntj|d d } | D]6}|d jdkstjd|d d dt#jD]w\} }|dkDr7j$j'd j(j'| Bj$j'| j(j'd ydt+t-tfdj DtdjDgdjDzxsdtj} jj/t1| j j/t1| j3| j5D]\} } }j7| | |\}}j9||\}}}j;||j=||| |k7s\j>||k7r||}}|jA| |jA|kDr|| }} jC||jE|jG| | |jI| | |tKfdtM| dDrtjdtKfdtMjNDs+tKfdtj|d DrtjPdjRjNd =tdt jjRD}jD cic]} | ic} fd}fdj$D_fdj(D_|sKt j$j(jRD] }|| |jUd } nVt j$j(jVjRD] }|| |jUd d } | D]}}|d|dk7r(|d jdk(s'||d d dz6|d jd}|dk\r||d d dz_|d } ||| zz }||d d | fz|fScc} w)adFind 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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