ó |£*^c@sƒdZddlZddlZddlmZddddgZedƒZd „Z d „Z d „Z d „Z d „Z e ZdS(sProvides functions for computing a maximum cardinality matching in a bipartite graph. If you don't care about the particular implementation of the maximum matching algorithm, simply use the :func:`maximum_matching`. If you do care, you can import one of the named maximum matching algorithms directly. For example, to find a maximum matching in the complete bipartite graph with two vertices on the left and three vertices on the right: >>> import networkx as nx >>> G = nx.complete_bipartite_graph(2, 3) >>> left, right = nx.bipartite.sets(G) >>> list(left) [0, 1] >>> list(right) [2, 3, 4] >>> nx.bipartite.maximum_matching(G) {0: 2, 1: 3, 2: 0, 3: 1} The dictionary returned by :func:`maximum_matching` includes a mapping for vertices in both the left and right vertex sets. iÿÿÿÿN(tsetstmaximum_matchingthopcroft_karp_matchingteppstein_matchingtto_vertex_covertinfcs‡‡‡‡‡‡fd†}‡‡‡‡‡fd†‰tˆƒ\‰}d„ˆDƒ‰d„|Dƒ‰i‰tjƒ‰d}xJ|ƒrÏx:ˆD]2}ˆ|d kr–ˆ|ƒrÈ|d7}qÈq–q–Wq†Wd„ˆjƒDƒ‰d„ˆjƒDƒ‰ttjˆjƒˆjƒƒƒS( s‰Returns the maximum cardinality matching of the bipartite graph `G`. Parameters ---------- G : NetworkX graph Undirected bipartite graph Returns ------- matches : dictionary The matching is returned as a dictionary, `matches`, such that ``matches[v] == w`` if node ``v`` is matched to node ``w``. Unmatched nodes do not occur as a key in mate. Notes ----- This function is implemented with the `Hopcroft--Karp matching algorithm `_ for bipartite graphs. See Also -------- eppstein_matching References ---------- .. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for Maximum Matchings in Bipartite Graphs" In: **SIAM Journal of Computing** 2.4 (1973), pp. 225--231. . csàxBˆD]:}ˆ|dkr7dˆ|<ˆj|ƒqtˆ|€s cSsi|]}d|“qS(N(R(RR ((sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pys s iicSs+i|]!\}}|dk r||“qS(N(R(RtkR ((sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pys s cSs+i|]!\}}|dk r||“qS(N(R(RRR ((sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pys s N(tbipartite_setst collectionstdequeRtitemstdictt itertoolstchain(R Rtrighttnum_matched_pairsR ((R RR RRRRsz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyR7s *     csîi‰x=|D]5}x,||D] }|ˆkr|ˆ|`_. See Also -------- hopcroft_karp_matching csi|]}ˆ|“qS(((RR (t unmatched(sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pys Ës csw|ˆkrsˆj|ƒ}xU|D]J}|ˆkr"ˆj|ƒ}|ˆks[ˆ|ƒrl|ˆ| s cs„|ˆkrtS|dkr tS|r,ˆnˆ}xKˆ|D]?}||f|ksg||f|kr=ˆ||d| ƒSq=WtS(sÖReturns ``True`` if and only if `u` is connected to one of the targets by an alternating path. `u` is a vertex in the graph `G`. `depth` specifies the maximum recursion depth of the depth-first search. If `along_matched` is ``True``, this step of the depth-first search will continue only through edges in the given matching. Otherwise, it will continue only through edges *not* in the given matching. ii(RR(R tdeptht along_matchedt valid_edgesR (R t_alternating_dfst matched_edgesttargetstunmatched_edges(sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyR1#s  $R/(RtsettedgesRtlenR(R R R%R3((R R1R2R3R4sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyt!_is_connected_by_alternating_path s (cs‡‡‡fd†ˆDƒS(s Returns the set of vertices that are connected to one of the target vertices by an alternating path in `G`. An *alternating path* is a path in which every other edge is in the specified maximum matching (and the remaining edges in the path are not in the matching). An alternating path may have matched edges in the even positions or in the odd positions, as long as the edges alternate between 'matched' and 'unmatched'. `G` is an undirected bipartite NetworkX graph. `matching` is a dictionary representing a maximum matching in `G`, as returned by, for example, :func:`maximum_matching`. `targets` is a set of vertices. cs+h|]!}tˆ|ˆˆƒr|’qS((R8(RR (R R%R3(sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pys bs ((R R%R3((R R%R3sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyt_connected_by_alternating_pathsOscCsTt|ƒ\}}t|ƒt|ƒ}||@}t|||ƒ}||||@BS(sTReturns the minimum vertex cover corresponding to the given maximum matching of the bipartite graph `G`. Parameters ---------- G : NetworkX graph Undirected bipartite graph matching : dictionary A dictionary whose keys are vertices in `G` and whose values are the distinct neighbors comprising the maximum matching for `G`, as returned by, for example, :func:`maximum_matching`. The dictionary *must* represent the maximum matching. Returns ------- vertex_cover : :class:`set` The minimum vertex cover in `G`. Notes ----- This function is implemented using the procedure guaranteed by `Konig's theorem `_, which proves an equivalence between a maximum matching and a minimum vertex cover in bipartite graphs. Since a minimum vertex cover is the complement of a maximum independent set for any graph, one can compute the maximum independent set of a bipartite graph this way: >>> import networkx as nx >>> G = nx.complete_bipartite_graph(2, 3) >>> matching = nx.bipartite.maximum_matching(G) >>> vertex_cover = nx.bipartite.to_vertex_cover(G, matching) >>> independent_set = set(G) - vertex_cover >>> print(list(independent_set)) [2, 3, 4] (RR5R9(R R%R#tRtunmatched_verticestUtZ((sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyRfs 1 (t__doc__RRtnetworkx.algorithms.bipartiteRRt__all__tfloatRRRR8R9RR(((sz/home/gitlab-runner/builds/8480fa44/0/bergerc/fluidmanager-web/art-framework/bin/networkx/algorithms/bipartite/matching.pyt+s      d n F  ?