9i%SrSSKJrJr SSKrSSKJr /SQr\"S5\"S5\RS555r S r \RS 5r \RS 5r \RS 5r\"S5\"S5\R"S S9SSj555r\"S5\"S5\R"S S9SSj555rg)z;Functions for computing and verifying matchings in a graph.) 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 \/var/www/html/land-doc-ocr/venv/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 rlenrrhas_edgerrs rrrUsR(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$r%FT) r'r(r rr)rrr*rrr)rrrrrrrs rrrs>(D!!'1 EE EE t9>""%CD6#JK K :""%>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5), (4, 6)]) >>> my_match = {1: 2, 3: 5, 4: 6} >>> nx.is_perfect_matching(G, my_match) True r"r#r$r%Fr&rs rrrs@(D!!'1 EE t9>""%CD6#JK K :""%&min_weight_matching..7s2'wq!'sc38># UHupo1UTU- 4v M g7fr5r6)r7rrr9 max_weights rr:r;9s ;7aJN #7sr-)r)rr maxrGraphadd_weighted_edges_from)rr-G_edgesInvGrr=s @rr r s` 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.r6N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r6rrNoNoderFs;rrMc.>\rSrSrSr/SQrU4SjrSrg)$max_weight_matching..Blossomiz7Representation of a non-trivial blossom or sub-blossom.)childsr mybestedgesc3># /URQnU(aKUR5n[UT5(aURUR5 OUv U(aMJgg7fr5)rPpopr'extend)selfstacktBlossoms rleaves+max_weight_matching..Blossom.leavessJ"dkkNEIIKa))LL*G %s AA%#A%r6N)rGrHrIrJrK __slots__rYrL)rXsrrXrOsE6   rrXrTr2r1')intlongNcR>TUTU-STUURTS5-- $)Nr"r1)get)rr9rdualvarr-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)Nr1r")rar'rTrYappend)r9rWrbbaserX assignLabelbestedge blossombase inblossomlabel labeledgematequeues rri(max_weight_matching..assignLabels aLyy|# ! 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