hSrSSKJr SSKJrJr SSKrSSKJ r /SQr \ "S5\ "S5\RS 555r S r \RS 5r\RS 5r\RS 5r\ "S5\ "S5\R"SS9SSj555r\ "S5\ "S5\R"SS9SSj555rg)z;Functions for computing and verifying matchings in a graph.)Counter) combinationsrepeatN)not_implemented_for) is_matchingis_maximal_matchingis_perfect_matchingmax_weight_matchingmin_weight_matchingmaximal_matching multigraphdirectedc[5n[5nUR5H>nUupEXB;dMXR;dMXE:wdMURU5 URU5 M@ U$)aIFind a maximal matching in the graph. A matching is a subset of edges in which no node occurs more than once. A maximal matching cannot add more edges and still be a matching. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- matching : set A maximal matching of the graph. Examples -------- >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)]) >>> sorted(nx.maximal_matching(G)) [(1, 2), (3, 5)] Notes ----- The algorithm greedily selects a maximal matching M of the graph G (i.e. no superset of M exists). It runs in $O(|E|)$ time. )setedgesaddupdateGmatchingnodesedgeuvs N/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/matching.pyr r sW<uH EE  >an LL  LL   Oc[5nUR5HDnUup4XC4U;dX!;aMX4:Xa[R"SU35eUR U5 MF U$)aConverts matching dict format to matching set format Converts a dictionary representing a matching (as returned by :func:`max_weight_matching`) to a set representing a matching (as returned by :func:`maximal_matching`). In the definition of maximal matching adopted by NetworkX, self-loops are not allowed, so the provided dictionary is expected to never have any mapping from a key to itself. However, the dictionary is expected to have mirrored key/value pairs, for example, key ``u`` with value ``v`` and key ``v`` with value ``u``. z%Selfloops cannot appear in matchings )ritemsnx NetworkXErrorr)rrrrrs rmatching_dict_to_setr!=sc EE  6U?dm  6""%J4&#QR R $ ! Lrc[U[5(a [U5n[5nUHn[ U5S:wa[ R "SU35eUupEX@;dXP;a[ R "SUS35eXE:Xa gURXE5(d gXB;dXR;a gURU5 M g)a[Return True if ``matching`` is a valid matching of ``G`` A *matching* in a graph is a set of edges in which no two distinct edges share a common endpoint. Each node is incident to at most one edge in the matching. The edges are said to be independent. Parameters ---------- G : NetworkX graph matching : dict or set A dictionary or set representing a matching. If a dictionary, it must have ``matching[u] == v`` and ``matching[v] == u`` for each edge ``(u, v)`` in the matching. If a set, it must have elements of the form ``(u, v)``, where ``(u, v)`` is an edge in the matching. Returns ------- bool Whether the given set or dictionary represents a valid matching in the graph. Raises ------ NetworkXError If the proposed matching has an edge to a node not in G. Or if the matching is not a collection of 2-tuple edges. Examples -------- >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)]) >>> nx.is_maximal_matching(G, {1: 3, 2: 4}) # using dict to represent matching True >>> nx.is_matching(G, {(1, 3), (2, 4)}) # using set to represent matching True matching has non-2-tuple edge matching contains edge  with node not in GFT isinstancedictr!rlenrr has_edgerrs rrrVsR(D!!'1 EE t9>""%CD6#JK K :""%>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (3, 5)]) >>> nx.is_maximal_matching(G, {(1, 2), (3, 4)}) True r#r$r%r&FT) r(r)r!rr*rr r+rrr)rrrrrrrs rrrs>(D!!'1 EE EE t9>""%CD6#JK K :""%>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5), (4, 6)]) >>> my_match = {1: 2, 3: 5, 4: 6} >>> nx.is_perfect_matching(G, my_match) True r#r$r%r&Fr'rs rr r s@(D!!'1 EE t9>""%CD6#JK K :""%&min_weight_matching..;s2'wq!'sc38># UHupo1UTU- 4v M g7fr6r7)r8rrr: max_weights rr;r<=s ;7aJN #7sr.)r*rr maxrGraphadd_weighted_edges_from)rr.G_edgesInvGrr>s @rr r sf 177|q"1T&IIgg61g-GS2'222J 88:D ;7 ;E   6 tD HHrc ^^^^^^^^^^ ^!^"^#^$^%^&^'^(^)^*"SS5m"U4SjS5m[T5m$T$(d [5$SnSnTRSS9HVupVnURTS5nXV:waX:aUnU=(a) [ [ U55R S 5SS ;nMX 0m(0m&0m'[[T$T$55m%[[T$[S 555m"[[T$T$55m 0m[[T$[U555m#0m!0m/m)UU#U4S jm*UUUU U%U&U'U(U)4 S jmUU U%U&U'U(4Sjn UUUU U!U"U%U&U'U(U)U*4 Sjn UUUUU U!U"U%U&U'U(4 Sjn UU U"U(4SjmUUU U%U&U'U(4Sjn UU!U"U#U$U(UU4Sjn T&R5 T'R5 TR5 T!H nS Ul M TR5 /T)S S &T$H,nUT(;dM T&RT%U5bM"T"USS 5 M. SnT)(GaU(GdT)R5nT&T%US:XdeTRU5GHOnUU:XaM T%UnT%UnUU:XaMUU4T;aT*"UU5nUS::aS=TUU4'TUU4'UU4T;aT&RU5c T"USU5 MiT&RU5S:Xa'U "UU5nUTLa U "UUU5 MU "UU5 Sn OT&RU5cT&US:XdeST&U'UU4T'U'MMT&RU5S:Xa+TRU5b WT*"TU6:a UU4TU'GMGMT&RU5bGM'TRU5bWT*"TU6:dGMHUU4TU'GMR T)(a U(dGMU(aGOSnS =n=nnT(dSn[T#R55nTR!5HMnT&RT%U5bMTRU5cM.T*"TU6nUS:XdUU:dMDUnSnTUnMO T"HrnT"UbM T&RU5S:XdM"TRU5cM6T*"TU6nU(aUS-S:XdeUS-nOUS- nUS:XdUU:dMiUnSnTUnMt T!H<nT"UbM T&RU5S:XdM"US:Xd T!UU:dM3T!UnSnUnM> US:Xa.T(deSn[#S[T#R555nT$HQnT&RT%U5S:XaT#U==U-ss'M*T&RT%U5S:XdMDT#U==U- ss'MS T!HSnT"UbM T&RU5S:XaT!U==U- ss'M/T&RU5S:XdMFT!U==U-ss'MU US:XaOUS:Xa3UunnT&T%US:XdeS=TUU4'TUU4'T)R%U5 OHUS:Xa3UunnS=TUU4'TUU4'T&T%US:XdeT)R%U5 OUS:Xa U "US5 GMaT(HnT(T(UU:XaMe U(dO^[T!R'55H>nUT!;aM T"UbMT&RU5S:XdM*T!US:XdM5U "US5 M@ GMhU(aU "5 [)T(5$)aCompute a maximum-weighted matching of G. A matching is a subset of edges in which no node occurs more than once. The weight of a matching is the sum of the weights of its edges. A maximal matching cannot add more edges and still be a matching. The cardinality of a matching is the number of matched edges. Parameters ---------- G : NetworkX graph Undirected graph maxcardinality: bool, optional (default=False) If maxcardinality is True, compute the maximum-cardinality matching with maximum weight among all maximum-cardinality matchings. weight: string, optional (default='weight') Edge data key corresponding to the edge weight. If key not found, uses 1 as weight. Returns ------- matching : set A maximal matching of the graph. Examples -------- >>> G = nx.Graph() >>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)] >>> G.add_weighted_edges_from(edges) >>> sorted(nx.max_weight_matching(G)) [(2, 4), (5, 3)] Notes ----- If G has edges with weight attributes the edge data are used as weight values else the weights are assumed to be 1. This function takes time O(number_of_nodes ** 3). If all edge weights are integers, the algorithm uses only integer computations. If floating point weights are used, the algorithm could return a slightly suboptimal matching due to numeric precision errors. This method is based on the "blossom" method for finding augmenting paths and the "primal-dual" method for finding a matching of maximum weight, both methods invented by Jack Edmonds [1]_. Bipartite graphs can also be matched using the functions present in :mod:`networkx.algorithms.bipartite.matching`. References ---------- .. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs", Zvi Galil, ACM Computing Surveys, 1986. c\rSrSrSrSrg)#max_weight_matching..NoNodeiz-Dummy value which is different from any node.r7N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r7rrNoNoderGs;rrNc.>\rSrSrSr/SQrU4SjrSrg)$max_weight_matching..Blossomiz7Representation of a non-trivial blossom or sub-blossom.)childsr mybestedgesc3># /URQnU(aKUR5n[UT5(aURUR5 OUv U(aMJgg7fr6)rQpopr(extend)selfstacktBlossoms rleaves+max_weight_matching..Blossom.leavessJ"dkkNEIIKa))LL*G %s AA%#A%r7N)rHrIrJrKrL __slots__rZrM)rYsrrYrPsE6   rrYrTr3r2')intlongNcR>TUTU-STUURTS5-- $)Nr#r2)get)rr:rdualvarr.s rslack"max_weight_matching..slacks3qzGAJ&QqT!W[[-C)CCCrcz> T UnT RU5cT RU5beU=T U'T U'Ub X 4=T U'T U'O S=T U'T U'S=TU'TU'US:XaC[UT5(a T RUR55 gT R U5 gUS:XaTUnT"T USU5 gg)Nr2r#)rbr(rUrZappend)r:rXrbbaserY assignLabelbestedge blossombase inblossomlabel labeledgematequeues rrj(max_weight_matching..assignLabels aLyy|# ! (<<<a58 =+,& 0IaL9Q<*. .IaL9Q<$(( hqk 6!W%% QXXZ( Q !Vq>D T At , rcT>/nTnUTLaTUnTUS-(aTUnOwTUS:XdeURU5 STU'T UcTUT ;deTnO4T UST TU:XdeT USnTUnTUS:XdeT USnUTLaXpUTLaMUHnSTU'M U$)Nr2rr#)rg) rr:pathrirhrNrlrmrnrorps r scanBlossom(max_weight_matching..scanBlossomsvo! AQx!|"1~8q= = KKNE!H|#"1~T111 |A${1~*>>>>aLOaLQx1}$}aLO1/vo2AE!H rc> TUnTUnTUnT"5nUTU'STU'UTU'/=UlnX4/=UlnXC:wadUTU'URU5 URTU5 TUS:XdTUS:XaTUSTTU:XdeTUSnTUnXC:waMdURU5 UR5 UR5 XS:waoUTU'URU5 URTUSTUS45 TUS:XdTUS:XaTUSTTU:XdeTUSnTUnXS:waMoTUS:XdeSTU'TUTU'STU'UR 5H%nTTUS:XaTRU5 UTU'M' 0n UHn[ UT5(aeUR bUR n SUlOmUR 5VVs/sH&nTRU5H o!U:wdM X4PM M( n nnO)TRU5Vs/sH o$U:wdM XB4PM n nU HQn U upTU U:XaXpTU nX:wdMTRU5S:XdM5X;dT"X5T"X6:dMMXU'MS STU'GM [U R55UlSnSTU'UR Hn T"U 6nUbUW:dMU nUnM UTU'gs snnfs snf)Nr#r2r) rQrrgreverserZr(rR neighborsrblistvalues)rirr:bbbvbwrhrvedgs bestedgetonblistkijbj mybestedgekslack mybestslackrYrrkrl blossomdual blossomparentrmrnrorprqrds r addBlossom'max_weight_matching..addBlossomCsn t_ q\ q\ I A a b4&!$h !M"  KKO KK " &9>b Q9R=#3tKO7L#L " a A1Bh B  h !M"  KKO KK2q)9R=+;< =9>b Q9R=#3tKO7L#L " a A1BhRyA~~a } !  AAYq\"a' QIaL  B"g&&>>-^^F%)BN )+ (31Q[[^TUv^ F,-;;r?F?aAg'2'?FQ<1$qq\G " *.5; AW3W%&rN HRL78Z..01   AAYF!Vk%9 $  ! 7Gs+K8 K8/ K><K>c > UUUU U U U U UUU4 SjnU"X5/nU(a?USnUHnURU"XQ55 O UR5 U(aM>gg)Nc3> # URHRnSTU'[UT 5(a4U(aTUS:XaUv M/UR5HnUTU'M MMUTU'MT U(GdTRU5S:XGaTTUSnURR U5nUS-(aU[ UR5-nSnOSnTUup7US:waUS:XaUR UupOUR US- upSTU'STU 'T"USU5 S=T X4'T X4'XV- nUS:XaUR Uup7OUR US- upsS=T X74'T Xs4'XV- nUS:waMURUn S=TU'TU 'X74=TU'TU 'STU 'XV- nURUU:waURUn TRU 5S:XaXV- nM=[U T 5(a0U R5HnTRU5(dM O OU nTRU5(a6TUS:XdeTUU :XdeSTU'STTTU 'T"USTUS5 XV- nURUU:waMTRUS5 TRUS5 TRUS5 TU TU TU g7f)Nrr#r2T)rQr(rZrbindexr*rrT)rhendstagesr entrychildrjstepr:pqrrrY allowedgerjrkrlrrrmrnrorps r_recurse.expandBlossom.._recursesXX#' a a))KNa$7!"A+,IaL",$%IaL%))A,!"3 'y|A7 HHNN:.q5QXX&AEE |1fz wwqz1 wwq1u~#E!H#E!H1a(<@@Iqf% 1&(9JAz wwqz1 wwq1u~<@@Iqf% 1&(9JA%1f*XXa['((a5901v5 ! y}#  hhqkZ/!Byy})  !"g..!#A$yy|| %"-yy||$Qx1},}(|r111#'a7;d;r?34#Aq)A,q/:JA1hhqkZ/4 IIa  MM!T " LLD !a AAsE-K2B"KA*KAKrrgrT)rhrrrWtoprrYrrjrkrlrrrmrnrorps r expandBlossom*max_weight_matching..expandBlossoms[ [ [ D!&')C Xa23 erc>UUUU 4SjnU"X5/nU(a<USnUHnURU"U65 O UR5 U(aM;gg)Nc3># UnT UU:waT UnT UU:waM[UT5(aX!4v URRU5=p4US-(aU[UR5-nSnOSnUS:waXE- nURUnUS:XaURUupgOURUS- upv[UT5(aX&4v XE- nURUn[UT5(aX'4v UT U'UT U'US:waMURUSURSU-UlURUSURSU-UlT URST U'T UU:Xdeg7f)Nr2rr)r(rQrr*r) rhrrXrrrr:xrYrlrrps rr=max_weight_matching..augmentBlossom.._recursesA"a'!!$ "a'!W%%f HHNN1% %A1uS]"q& HHQKA:771:DAq771q5>DAa))&L HHQKa))&LQQ#q&&xx|ahhrl2AHggabkAGGBQK/AG(!5KNq>Q& &&sE7C-E7A)E7rr) rhrrrWrargsrYrlrrps raugmentBlossom+max_weight_matching..augmentBlossomsW ) ') '`! )C Xt_- erch>X4X44Hup#T UnT US:XdeT Uc T UT ;dT UST T U:Xde[UT5(aT"XB5 UT U'T UcM_T USnT UnT US:XdeT Uup#T UU:Xde[UT5(aT"Xc5 UT U'M g)Nr2rr#)r()rr:rrbsrXbtrYrrlrmrnrorps raugmentMatching,max_weight_matching..augmentMatchingWsVaV$DAq\RyA~%~!" -+b/2MbM!$[_(==b'**"2)QR=(bM!$q\RyA~%~ }"2!+++b'**"2)Q3 %rcr>T(a%[S[TR55*5nOSn[TR55U-S:de[T 5S:Xd[T R55S:deT R SS9GH$upnUR TS5nX:XaM TUTU-SU-- nU/nU/nTUSb"UR TUS5 TUSbM"TUSb"UR TUS5 TUSbM"UR5 UR5 [Xg5HupX:wa OUST U-- nM US:deTR U5U:XdTR U5U:XdGMTUU:Xa TUU:XdeUS:XaGM%e THn U T;aM TU U-S:XaMe T H[n T U S:dM[U R5S-S:XdeU RSSS2HupTUU:Xa TUU:XaMe M] g)NrTr]r2r#r) r@minr}r*rrbrgrzzip) vdualoffsetrrdwtr iblossoms jblossomsbirrrhrrrrcgnodesrpr1r.s r verifyOptimum*max_weight_matching..verifyOptimumxs` a#gnn&6"7!78KK7>>#${2a777;1$K,>,>,@(AQ(FFFwwDw)GA!vq!Bv WQZ'!b&0AII " .:  y}!=> " .: " .:  y}!=> " .:        i38QR((46M6xx{a488A;!#3Aw!|Q1 44Av v)*,AI'!*{":a"? ??A1~!177|a'1,,,GGADqDMDA7a1>'(E!H,-q6IaL22!+$<<+3vxPR|@T7T,-q6HRL8U1-$<<?2fuhqk?R6R+,a&HQKi(  x I/3 3E 3I " GNN,-WWY99Yq\*2x||A7Rx{+A B!e) !$% $,QK #!!$, ! ) Q3"HQK0F! & q000"aK"SL B!e) !$% $,QK #$!!!$, ! )"bKNU,B'NE !I#$L!B&%~ As7>>#34599Yq\*a/AJ%'JYYy|,1AJ%'J ! #+yy|q(#A%/1*#A%/!A~a"AYq\*a///8<< 1a&!Iq!f$5 Qa"A8<< 1a&!Iq!f$5Yq\*a/// QalE2QZAQ=A% %% k&&()A #Q'EIIaLA,=+a.TUBUa& *c p  %%rr?)Fr.)rL collectionsr itertoolsrrnetworkxrnetworkx.utilsr__all__ _dispatchabler r!rrr r r r7rrrsA*. \"Z $!#$N299x::z0 0 f\"Z X&7I'!#7It\"Z X&{ &'!#{ &r