Ë ·¯wgabãó¸—dZdgZddlmZmZmZddlmZmZddl Z ddlmZGd„d«Z ed «e jd ed«ddœ¬ «dd„««Zy)z< Minimum cost flow algorithms on directed connected graphs. Únetwork_simplexé)ÚchainÚisliceÚrepeat)ÚceilÚsqrtN)Únot_implemented_forcój—eZdZ dd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Zd „Zd„Z d„Zd „Zd„Zd„Zy)Ú_DataEssentialsAndFunctionscób‡‡ —t|«|_t|j«Dcic]\}}||“Œ c}}|_|jDcgc]!}|j|j|d«‘Œ#c}|_g|_g|_|rg|_ i|_ g|_g|_|s|jd¬«}n|jdd¬«}td«Š ˆˆ fd„|D«}t|«D]é\}} |jj|j| d«|jj|j| d«|r|jj| d«||j| dd <|jj| d j‰‰ ««|jj| d j|d««Œëd|_d|_d|_d|_d|_d|_d|_d|_d|_d |_ycc}}wcc}w)NrT©Údata©rÚkeysÚinfc3ól•K—|]+}|d|dk7sŒ|dj‰‰«dk7sŒ(|–—Œ-yw)rééÿÿÿÿN©Úget)Ú.0ÚeÚcapacityrs €€úl/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/algorithms/flow/networksimplex.pyú z7_DataEssentialsAndFunctions.__init__..'s6øèø€ÒTq Q q¡T¨Q¨q©T£\°a¸±e·i±iÀÈ#Ó6NÐRSÓ6S”ÑTùsƒ4”44rérF)ÚlistÚ node_listÚ enumerateÚnode_indicesÚnodesrÚnode_demandsÚedge_sourcesÚedge_targetsÚ edge_keysÚedge_indicesÚedge_capacitiesÚedge_weightsÚedgesÚfloatÚappendÚ edge_countÚ edge_flowÚnode_potentialsÚparentÚparent_edgeÚsubtree_sizeÚ next_node_dftÚ prev_node_dftÚlast_descendent_dftÚ_spanning_tree_initialized)ÚselfÚGÚ multigraphÚdemandrÚweightÚiÚur)rrs ` @rÚ__init__z$_DataEssentialsAndFunctions.__init__sù€ô˜a›ˆŒÜ.7¸¿¹Ó.G×H¡d a¨˜Q ™TÓHˆÔà/3¯~©~ö Ø*+ˆAG‰GA‰JN‰N˜6 1Õ%ò ˆÔðˆÔØˆÔÙØˆDŒNØˆÔØ!ˆÔØˆÔáØ—G‘G GÓ&‰Eà—G‘G ¨DGÓ1ˆEäE‹lˆÜT˜EÔTˆÜ˜eÓ$ò ;‰DˆAˆqØ×Ñ×$Ñ$ T×%6Ñ%6°q¸±tÑ%<Ô=Ø×Ñ×$Ñ$ T×%6Ñ%6°q¸±tÑ%<Ô=ÙØ—‘×%Ñ% a¨¡dÔ+Ø()ˆD×Ñ˜a ˜fÑ%Ø× Ñ ×'Ñ'¨¨"©¯ © °(¸CÓ(@ÔAØ×Ñ×$Ñ$ Q r¡U§Y¡Y¨v°qÓ%9Õ:ð ;ðˆŒØˆŒØ#ˆÔØˆŒØˆÔØ ˆÔØ!ˆÔØ!ˆÔØ#'ˆÔ àð Õ'ùóQIùò s« H&Á&H,cóæ—t|j«|_tt td|j«d„|jD«««|_|jDcgc]}|dkr|n|‘Œc}|_tt td|«dg««|_ tt|j|j|z««|_tt td|«|dzg««|_tt td|«ddg««|_ ttd|««|_tt t|«|dz g««|_d|_ycc}w)Nrc3ó2K—|]}t|«–—Œyw©N©Úabs©rÚds rrzG_DataEssentialsAndFunctions.initialize_spanning_tree..Csèø€Ò.Q¸!¬s°1¯vÑ.Qùó‚rrT)Úlenr&r,rrrr"r-r.r/Úranger0r1r2r3r4r5)r6ÚnÚfaux_infrDs rÚinitialize_spanning_treez4_DataEssentialsAndFunctions.initialize_spanning_tree@s7€Ü˜d×/Ñ/Ó0ˆŒÜÜ”&˜˜DŸO™OÓ,Ñ.Q¸t×?PÑ?PÔ.QÓRó ˆŒð8<×7HÑ7Hö Ø23˜˜Qš‰H X IÑ-ò ˆÔôœ5¤¨¨A£°°Ó7Ó8ˆŒÜÜ$—/‘/ 4§?¡?°QÑ#6Ó7ó ˆÔô!¤¤v¨a°£|°a¸!±e°WÓ!=Ó>ˆÔÜ!Ü”%˜˜1“+ A˜wÓ'ó ˆÔô"¤%¨¨A£,Ó/ˆÔÜ#'Ü”%˜“(˜Q ™U˜GÓ$ó$ ˆÔ ð+/ˆÕ'ùò sÁ(E.cóx—|j|}|j|} ||kr$|j|}|j|}||krŒ$||kDr$|j|}|j|}||kDrŒ$||k(rD||k7r=|j|}|j|}|j|}|j|}n|SŒœ)zX Find the lowest common ancestor of nodes p and q in the spanning tree. )r1r/)r6ÚpÚqÚsize_pÚsize_qs rÚ find_apexz%_DataEssentialsAndFunctions.find_apexVsØ€ð×"Ñ" 1Ñ%ˆØ×"Ñ" 1Ñ%ˆØØ˜6’/Ø—K‘K ‘NØ×*Ñ*¨1Ñ-ð˜6“/ð˜6’/Ø—K‘K ‘NØ×*Ñ*¨1Ñ-ð˜6“/ð˜ÒØ˜’6ØŸ™ A™AØ!×.Ñ.¨qÑ1FØŸ™ A™AØ!×.Ñ.¨qÑ1‘FàHðócó¦—|g}g}||k7rD|j|j|«|j|}|j|«||k7rŒD||fS)zX Returns the nodes and edges on the path from node p to its ancestor w. )r+r0r/)r6rLÚwÚWnÚWes rÚ trace_pathz&_DataEssentialsAndFunctions.trace_pathlsZ€ðˆSˆØ ˆØ1ŠfØI‰Id×&Ñ& qÑ)Ô*Ø—‘˜A‘ˆAØI‰IaŒLð1‹fð2ˆvˆ rQcó —|j||«}|j||«\}}|j«|j«||gk7r|j|«|j||«\}}|d=||z }||z }||fS)zÊ 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)rPrVÚreverser+) r6r;rLrMrSrTrUÚWnRÚWeRs rÚ find_cyclez&_DataEssentialsAndFunctions.find_cyclexs„€ð N‰N˜1˜aÓ ˆØ—‘ AÓ&‰ˆˆBØ ‰ ŒØ ‰ ŒØ !Š9ØI‰IaŒLØ—?‘? 1 aÓ(‰ˆˆSØˆGØ ˆc‰ ˆØ ˆc‰ ˆØ2ˆvˆ rQcó®—t||«D]F\}}|j||k(r|j|xx|z cc<Œ0|j|xx|zcc<ŒHy)zQ Augment f units of flow along a cycle represented by Wn and We. N)Úzipr#r-)r6rTrUÚfr;rLs rÚaugment_flowz(_DataEssentialsAndFunctions.augment_flow‹sX€ô˜˜B“Kò '‰DˆAˆqØ× Ñ Ñ# qÒ(Ø—‘˜qÓ! QÑ&Ô!à—‘˜qÓ! QÑ&Ô!ñ 'rQc#ópK—|–—|j|}||k7r|j|}|–—||k7rŒyyw)zD Yield the nodes in the subtree rooted at a node p. N)r4r2)r6rLÚls rÚ trace_subtreez)_DataEssentialsAndFunctions.trace_subtree•sBèø€ðŠØ×$Ñ$ QÑ'ˆØ1ŠfØ×"Ñ" 1Ñ%ˆAØŠGð1fùs‚16´6cóÊ—|j|}|j|}|j|}|j|}d|j|<d|j |<||j|<||j|<||j|<||j|<|K|j|xx|zcc<|j||k(r||j|<|j|}|ŒJyy)zT Remove an edge (s, t) where parent[t] == s from the spanning tree. N)r1r3r4r2r/r0)r6ÚsÚtÚsize_tÚprev_tÚlast_tÚnext_last_ts rÚremove_edgez'_DataEssentialsAndFunctions.remove_edgeŸsõ€ð×"Ñ" 1Ñ%ˆØ×#Ñ# AÑ&ˆØ×)Ñ)¨!Ñ,ˆØ×(Ñ(¨Ñ0ˆàˆ‰A‰Ø"ˆ×Ñ˜Ñà%0ˆ×Ñ˜6Ñ"Ø*0ˆ×Ñ˜;Ñ'Ø%&ˆ×Ñ˜6Ñ"Ø &ˆ×Ñ˜1ÑðˆmØ×Ñ˜aÓ FÑ*Ó Ø×'Ñ'¨Ñ*¨fÒ4Ø.4×(Ñ(¨Ñ+Ø—‘˜A‘ˆAð ‹mrQcór—g}|#|j|«|j|}|Œ#|j«t|t |dd««D]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|<Œhy)zC Make a node q the root of its containing subtree. Nr) r+r/rXr]rr1r4r3r2r0) r6rMÚ ancestorsrLrNÚlast_pÚprev_qÚlast_qÚnext_last_qs rÚ make_rootz%_DataEssentialsAndFunctions.make_root·sÕ€ðˆ ØˆmØ×Ñ˜QÔØ—‘˜A‘ˆAð‰mð ×ÑÔÜ˜ ¤6¨)°Q¸Ó#=Ó>ó 1‰DˆAˆqØ×&Ñ& qÑ)ˆFØ×-Ñ-¨aÑ0ˆFØ×'Ñ'¨Ñ*ˆFØ×-Ñ-¨aÑ0ˆFØ×,Ñ,¨VÑ4ˆKàˆDK‰K˜‰NØ!ˆDK‰K˜‰NØ"&×"2Ñ"2°1Ñ"5ˆD×Ñ˜QÑØ"&ˆD×Ñ˜QÑØ#)¨D×,=Ñ,=¸aÑ,@Ñ#@ˆD×Ñ˜aÑ Ø#)ˆD×Ñ˜aÑ à)4ˆD×Ñ˜vÑ&Ø.4ˆD×Ñ˜{Ñ+Ø)*ˆD×Ñ˜vÑ&Ø$*ˆD×Ñ˜qÑ!Ø˜ÒØ.4×(Ñ(¨Ñ+Øð%+ˆD×Ñ˜qÑ!Ø)*ˆD×Ñ˜vÑ&Ø)*ˆD×Ñ˜vÑ&Ø$*ˆD×Ñ˜qÑ!Ø*0ˆD×$Ñ$ QÓ'ñ7 1rQcóÊ—|j|}|j|}|j|}|j|}||j|<||j|<||j|<||j |<||j |<||j|<|K|j|xx|z cc<|j||k(r||j|<|j|}|ŒJyy)z[ Add an edge (p, q) to the spanning tree where q is the root of a subtree. N)r4r2r1r/r0r3)r6r;rLrMrmÚnext_last_prOros rÚadd_edgez$_DataEssentialsAndFunctions.add_edgeÝsõ€ð×)Ñ)¨!Ñ,ˆØ×(Ñ(¨Ñ0ˆØ×"Ñ" 1Ñ%ˆØ×)Ñ)¨!Ñ,ˆàˆ‰A‰Øˆ×Ñ˜Ñà%&ˆ×Ñ˜6Ñ"Ø &ˆ×Ñ˜1ÑØ*0ˆ×Ñ˜;Ñ'Ø%0ˆ×Ñ˜6Ñ"ðˆmØ×Ñ˜aÓ FÑ*Ó Ø×'Ñ'¨Ñ*¨fÒ4Ø.4×(Ñ(¨Ñ+Ø—‘˜A‘ˆAð ‹mrQcó@—||j|k(r0|j||j|z |j|z }n/|j||j|z|j|z }|j|«D]}|j|xx|z cc<Œy)z† Update the potentials of the nodes in the subtree rooted at a node q connected to its parent p by an edge i. N)r$r.r(rb)r6r;rLrMrDs rÚupdate_potentialsz-_DataEssentialsAndFunctions.update_potentialsõs©€ð ×!Ñ! !Ñ$Ò$Ø×$Ñ$ QÑ'¨$×*;Ñ*;¸AÑ*>Ñ>À×AUÑAUÐVWÑAXÑX‰Aà×$Ñ$ QÑ'¨$×*;Ñ*;¸AÑ*>Ñ>À×AUÑAUÐVWÑAXÑXˆAØ×#Ñ# AÓ&ò )ˆAØ× Ñ Ó# qÑ(Ô#ñ )rQcóÂ—|j||j|j|z |j|j|z}|j|dk(r|S|S)z&Returns the reduced cost of an edge i.r)r(r.r#r$r-)r6r;Úcs rÚreduced_costz(_DataEssentialsAndFunctions.reduced_costsr€ð ×Ñ˜aÑ Ø×"Ñ" 4×#4Ñ#4°QÑ#7Ñ8ñ 9à×"Ñ" 4×#4Ñ#4°QÑ#7Ñ8ñ 9ð ð —N‘N 1Ñ%¨Ò*ˆqÐ2°°Ð2rQc#óK—|jdk(ryttt|j«««}|j|zdz |z}d}d}||krí||z}||jkr t ||«}n8||jz}tt ||j«t |««}|}t ||j¬«}|j|«}|dk\r|dz }nX|j|dk(r|j|} |j|} n|j|} |j|} || | f–—d}||krŒìyyw)z-Yield entering edges until none can be found.rNr©Úkey)r,ÚintrrrGrÚminryr-r#r$)r6ÚBÚMÚmr^rar)r;rxrLrMs rÚfind_entering_edgesz/_DataEssentialsAndFunctions.find_entering_edges sDèø€à?‰?˜aÒØô ””T˜$Ÿ/™/Ó*Ó+Ó,ˆØ _‰_˜qÑ 1Ñ $¨Ñ*ˆØ ˆà ˆØ!ŠeàA‘ˆAØD—O‘OÒ#Ü˜a ›‘àT—_‘_Ñ$Üœe A t§¡Ó7¼¸q»ÓBØˆAäE˜t×0Ñ0Ô1ˆAØ×!Ñ! !Ó$ˆAØAŠvàQ‘‘ð—>‘> !Ñ$¨Ò)Ø×)Ñ)¨!Ñ,AØ×)Ñ)¨!Ñ,‘Aà×)Ñ)¨!Ñ,AØ×)Ñ)¨!Ñ,AØ˜˜Ag’ Øð1!eùs‚EEÅEcó‚—|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 rÚresidual_capacityz-_DataEssentialsAndFunctions.residual_capacity4sM€ð× Ñ Ñ# qÒ(ð × Ñ Ñ# d§n¡n°QÑ&7Ñ7ð ð—‘ Ñ"ð rQcóÌ‡—ttt|«t|««ˆfd„¬«\}}‰j||k(r‰j|n‰j|}|||fS)z=Returns the leaving edge in a cycle represented by Wn and We.có"•—‰j|ŽSr@)r„)Úi_pr6s €rúz?_DataEssentialsAndFunctions.find_leaving_edge..Bsø€Ð2˜D×2Ñ2°CÐ8€rQr{)r~r]Úreversedr#r$)r6rTrUÚjrdres` rÚfind_leaving_edgez-_DataEssentialsAndFunctions.find_leaving_edge>scø€äÜ”˜“œh r›lÓ+Û8ô ‰ˆˆ1ð%)×$5Ñ$5°aÑ$8¸AÒ$=ˆD×Ñ˜aÒ À4×CTÑCTÐUVÑCWˆØ!QˆwˆrQN©r9rr:)Ú__name__Ú __module__Ú__qualname__r=rJrPrVr[r_rbrjrqrtrvryr‚r„r‹©rQrrrsS„àJRó/ òb/ò,ò, òò&'òòò0$1òLò0 )ò3ò&òT órQrÚ undirectedr9r)rr:)Ú node_attrsÚ edge_attrsc ó‡‡‡‡‡‡—t|«dk(rtjd«‚|j«}t |||‰‰¬«Štd«Št ‰j‰j«D],\}}t|«‰k(sŒtjd|›d«‚t ‰j‰j«D],\}}t|«‰k(sŒtjd|›d«‚|stj|d ¬ «} ntj|d d ¬«} | D]?}t|dj‰d««‰k(sŒ%tjd|d d›d«‚t‰j«dk7rtjd«‚t ‰j‰j «D]#\}} | dksŒtjd|›d«‚|stj|d ¬ «} ntj|d d ¬«} | D]6}|dj‰‰«dksŒtjd|d d›d«‚t#‰j«D]w\}}|dkDr7‰j$j'd«‰j(j'|«ŒB‰j$j'|«‰j(j'd«Œydt+t-tˆfd„‰j D««td„‰jD««gd„‰jD«««zxsdŠt‰j«}‰jj/t1‰|««‰j j/t1‰|««‰j3|‰«‰j5«D]á\}} }‰j7|| |«\}}‰j9||«\}}}‰j;||‰j=||««||k7sŒ\‰j>||k7r||}}|jA|«|jA|«kDr|| }} ‰jC||«‰jE|«‰jG|| |«‰jI|| |«ŒãtKˆfd„tM|d«D««rtjd«‚tKˆˆfd„tM‰jN«D««s+tKˆˆˆfd„tj|d ¬ «D««rtjPd«‚‰jR‰jNd …=td„t ‰j‰jR«D««}‰jDcic]}|i“Œc}Šˆfd„}ˆfd„‰j$D«‰_ˆfd„‰j(D«‰_|sKt ‰j$‰j(‰jR«D] }||«Œ|jUd ¬ «} nVt ‰j$‰j(‰jV‰jR«D] }||«Œ|jUd d ¬«} | D]}}|d|dk7r(|dj‰‰«dk(sŒ'||d ddz«Œ6|dj‰d«}|dk\r||d ddz«Œ_|d‰} ||| zz }||d d| fz«Œ|‰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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