btj%dZddlmZmZddlZddlmZgdZededej dZ d Z ej d Z ej d Z ej d Zededej d ddZededej d ddZdS)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 multigraphdirectedct}t}|D]?}|\}}||vr4||vr0||kr*||||@|S)aFind 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 Z/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/matching.pyr r s{<uuH EEE 1 E>>aunna LL    LL    Oct}|D]D}|\}}||f|vs||vr||krtjd|||E|S)a?Converts 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 <s EEE  1 q6U??demm  66"#Q4#Q#QRR R $ Lrct|trt|}t}|D]}t |dkrt jd||\}}||vs||vrt jd|d||krdS|||sdS||vs||vrdS||dS)aReturn 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!!2'11 EEE   t99>>"#JD#J#JKK K1 A::!"#VT#V#V#VWW W 6655zz!Q 55 ::e55 T 4rcRt|trt|}t}t}|D]}t |dkrt jd||\}}||vs||vrt jd|d||krdS|||sdS||vs||vrdS||| || ||f|j D]\}}||f|vr||vr ||vr ||krdSdS)agReturn True if ``matching`` is a maximal matching of ``G`` A *maximal matching* in a graph is a matching in which adding any edge would cause the set to no longer be a valid matching. 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 maximal matching in the graph. Examples -------- >>> 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 rrrso>(D!!2'11 EEE EEE t99>>"#JD#J#JKK K1 A::!"#VT#V#V#VWW W 6655zz!Q 55 ::e55 T $ 1a&1 q6  ~~!5..Q!VVuu 4rct|trt|}t}|D]}t |dkrt jd||\}}||vs||vrt jd|d||krdS|||sdS||vs||vrdS||t |t |kS)aReturn True if ``matching`` is a perfect matching for ``G`` A *perfect matching* in a graph is a matching in which exactly one edge is incident upon each vertex. 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 perfect matching in the graph. Examples -------- >>> 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$r%Fr&rs rrrs @(D!!2'11 EEE   t99>>"#JD#J#JKK K1 A::!"#VT#V#V#VWW W 6655zz!Q 55 ::e55 T u::Q rweight) edge_attrscRt|jdkrt|d|S||d}dtd|Dzt j}fd|D}|||t|d|S) aCompute a minimum-weight maximum-cardinality matching of `G`. The minimum-weight maximum-cardinality matching is the matching that has the minimum weight among all maximum-cardinality matchings. Use the maximum-weight algorithm with edge weights subtracted from the maximum weight of all edges. A matching is a subset of edges in which no node occurs more than once. The weight of a matching is the sum of the weights of its edges. A maximal matching cannot add more edges and still be a matching. The cardinality of a matching is the number of matched edges. This method replaces the edge weights with 1 plus the maximum edge weight minus the original edge weight. new_weight = (max_weight + 1) - edge_weight then runs :func:`max_weight_matching` with the new weights. The max weight matching with these new weights corresponds to the min weight matching using the original weights. Adding 1 to the max edge weight keeps all edge weights positive and as integers if they started as integers. Read the documentation of `max_weight_matching` for more information. Parameters ---------- G : NetworkX graph Undirected graph weight: string, optional (default='weight') Edge data key corresponding to the edge weight. If key not found, uses 1 as weight. Returns ------- matching : set A minimal weight matching of the graph. See Also -------- max_weight_matching rT)maxcardinalityr-)datadefaultc3"K|] \}}}|V dSN).0_ws r z&min_weight_matching..7s(22wq!Q222222rc30K|]\}}}|||z fVdSr5r6)r7rrr9 max_weights rr:z&min_weight_matching..9s4 ; ;1aaJN # ; ; ; ; ; ;rr-)r)rr maxrGraphadd_weighted_edges_from)rr-G_edgesInvGrr<s @rr r s` 17||q"1T&IIIIgg61g--GS22'222222J 8::D ; ; ; ;7 ; ; ;E  v 666 tD H H HHrFc J !"#$%&'()*GddGfddt$$stSd}d}dD]c\}}}|d}||kr||kr|}|o6t t |d dd v}di(i&i'tt$$%tt$td "tt$$ itt$t|#i!ig)#fd * %&'()f d  %&'(fd}  !"%&'()*f d}  !"%&'(f d}  "(fd %&'(fd} !"#$(fd} & '  !D] }d |_  g)d d <$D].}|(vr(&%| |dd /d} )r|s) }&%|dksJ |D]m}||kr %|}%|}||kr!||fvr *||}|dkrdx||f<||f<||fvr&||d|p&|dkr.| ||}|ur| |||| ||d}n&|&|dksJd&|<||f'|<&|dkr-||*|kr||f|<-&|+||*|kr||f|<o)r||rnd}d x}x}}s#d}t#}D]U}&%|8|#*|}|dks||kr |}d}|}V"D]s}"|i&|dkrP|;*|}|r|dzdksJ|dz}n|dz }|dks||kr |}d}|}t!D]A}"|7&|dkr|dks !||kr !|}d}|}B|dkr5sJd}t#dt#}$D]a}&%|dkr#|xx|zcc<2&%|dkr#|xx|z cc<b!D]]}"|S&|dkr!|xx|z cc<4&|dkr!|xx|zcc<^|dkrn|dkr=|\}}&%|dksJdx||f<||f<)|nU|dkr=|\}}dx||f<||f<&%|dksJ)|n|dkr | |d(D]}((||ksJ|sndt!D]@}|!vr"|1&|dkr!|dkr | |dA|r | t)(S)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 ---------- .. 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