U 9v f|3@sdZddlZddlZddlmZddlmZmZddddd d gZ Gd d d ej Z ed ed ddZ ej fddZddZddZddZddZddZdej fddZed dd ZdS)z Algorithms for chordal graphs. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). https://en.wikipedia.org/wiki/Chordal_graph N)connected_components)arbitrary_elementnot_implemented_for is_chordalfind_induced_nodeschordal_graph_cliqueschordal_graph_treewidthNetworkXTreewidthBoundExceededcomplete_to_chordal_graphc@seZdZdZdS)r zVException raised when a treewidth bound has been provided and it has been exceededN)__name__ __module__ __qualname____doc__rr?/tmp/pip-unpacked-wheel-_lngutwb/networkx/algorithms/chordal.pyr sZdirectedZ multigraphcCstt|dkS)uChecks whether G is a chordal graph. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). Parameters ---------- G : graph A NetworkX graph. Returns ------- chordal : bool True if G is a chordal graph and False otherwise. Raises ------ NetworkXNotImplemented The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... ] >>> G = nx.Graph(e) >>> nx.is_chordal(G) True Notes ----- The routine tries to go through every node following maximum cardinality search. It returns False when it finds that the separator for any node is not a clique. Based on the algorithms in [1]_. References ---------- .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984), pp. 566–579. r)len_find_chordality_breakerGrrrrs6c Cst|stdt|}|||t}t|||}|r~|\}}} |||D]} | |krV||| qVt|||}q:|r||||D]*}t |t||@dkr||qq|S)aReturns the set of induced nodes in the path from s to t. Parameters ---------- G : graph A chordal NetworkX graph s : node Source node to look for induced nodes t : node Destination node to look for induced nodes treewidth_bound: float Maximum treewidth acceptable for the graph H. The search for induced nodes will end as soon as the treewidth_bound is exceeded. Returns ------- induced_nodes : Set of nodes The set of induced nodes in the path from s to t in G Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a :exc:`NetworkXError` is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> G = nx.Graph() >>> G = nx.generators.classic.path_graph(10) >>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2) >>> sorted(induced_nodes) [1, 2, 3, 4, 5, 6, 7, 8, 9] Notes ----- G must be a chordal graph and (s,t) an edge that is not in G. If a treewidth_bound is provided, the search for induced nodes will end as soon as the treewidth_bound is exceeded. The algorithm is inspired by Algorithm 4 in [1]_. A formal definition of induced node can also be found on that reference. References ---------- .. [1] Learning Bounded Treewidth Bayesian Networks. Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008. http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf Input graph is not chordal.) rnx NetworkXErrorZGraphZadd_edgesetrupdateaddr) rsttreewidth_boundHZ induced_nodesZtripletuvwnrrrrVs(5         c#sfddtDD]}|dkrNt|dkr>tdt|Vqt|}t|}| ||h}|h}|rt |||}| || |t| ||@}| |}t|r| |||kst|V|}qxtdqxt|VqdS)aUReturns all maximal cliques of a chordal graph. The algorithm breaks the graph in connected components and performs a maximum cardinality search in each component to get the cliques. Parameters ---------- G : graph A NetworkX graph Yields ------ frozenset of nodes Maximal cliques, each of which is a frozenset of nodes in `G`. The order of cliques is arbitrary. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> cliques = [c for c in chordal_graph_cliques(G)] >>> cliques[0] frozenset({1, 2, 3}) c3s|]}|VqdSN)subgraphcopy).0crrr sz(chordal_graph_cliques..rrN)rnumber_of_nodesrnumber_of_selfloopsr frozensetnodesrrremove_max_cardinality_noderZ neighborsr%_is_complete_graph)rC unnumberedr!numberedclique_wanna_beZnew_clique_wanna_besgrrrrs.-           cCs<t|stdd}t|D]}t|t|}q |dS)aReturns the treewidth of the chordal graph G. Parameters ---------- G : graph A NetworkX graph Returns ------- treewidth : int The size of the largest clique in the graph minus one. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> nx.chordal_graph_treewidth(G) 3 References ---------- .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth rr*)rrrrmaxr)rZ max_cliqueZcliquerrrrs , cCsLt|dkrtd|}|dkr,dS|}||dd}||kS)z&Returns True if G is a complete graph.rz'Self loop found in _is_complete_graph()rTr*)rr,rr+Znumber_of_edges)rr#eZ max_edgesrrrr1 s r1cCsHt|}|D]6}|tt|||g}|r ||fSq dS)z5Given a non-complete graph G, returns a missing edge.N)rlistkeyspop)rr.r missingrrr_find_missing_edge,s r>cs<d}|D].}tfdd||D}||kr|}|}q|S)z`Returns a the node in choices that has more connections in G to nodes in wanna_connect. r7csg|]}|kr|qSrr)r'y wanna_connectrr ;sz)_max_cardinality_node..)r)rchoicesrAZ max_numberxnumberZmax_cardinality_noderr@rr05sr0c Cst|dkrtdt|}|dkr0t|}|||h}d}|rt|||}||||t|||@}||}t |rt |t |}||krt d|qDt |\} } | || fSqDdS)a'Given a graph G, starts a max cardinality search (starting from s if s is given and from an arbitrary node otherwise) trying to find a non-chordal cycle. If it does find one, it returns (u,v,w) where u,v,w are the three nodes that together with s are involved in the cycle. rrNr7ztreewidth_bound exceeded: r)rr,rrrr/r0rr%r1r8rr r>) rrrr3r4Zcurrent_treewidthr!r5r6r r"rrrrBs.        rc s0|}dd|D}t|r(||fSt}dd|Dt|}tt|ddD]}t|fddd}| ||||<g}|D]l}| ||r| |q|fd d |D} t | | ||g||r| ||||fq|D]} | d 7<qq`||||fS) aReturn a copy of G completed to a chordal graph Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is called chordal if for each cycle with length bigger than 3, there exist two non-adjacent nodes connected by an edge (called a chord). Parameters ---------- G : NetworkX graph Undirected graph Returns ------- H : NetworkX graph The chordal enhancement of G alpha : Dictionary The elimination ordering of nodes of G Notes ----- There are different approaches to calculate the chordal enhancement of a graph. The algorithm used here is called MCS-M and gives at least minimal (local) triangulation of graph. Note that this triangulation is not necessarily a global minimum. https://en.wikipedia.org/wiki/Chordal_graph References ---------- .. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004) Maximum Cardinality Search for Computing Minimal Triangulations of Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3. Examples -------- >>> from networkx.algorithms.chordal import complete_to_chordal_graph >>> G = nx.wheel_graph(10) >>> H, alpha = complete_to_chordal_graph(G) cSsi|] }|dqSrrr'noderrr sz-complete_to_chordal_graph..cSsi|] }|dqSrFrrGrrrrIsrr7cs|Sr$r)rH)weightrrz+complete_to_chordal_graph..)keycsg|]}|kr|qSrrrGrJZy_weightrrrBs z-complete_to_chordal_graph..r*)r&rrrr.r:rangerr8r/Zhas_edgeappendZhas_pathr%rZadd_edges_from) rralphaZchordsZunnumbered_nodesizZ update_nodesr?Z lower_nodesrHrrNrr gs4)        )rsysZnetworkxrZnetworkx.algorithms.componentsrZnetworkx.utilsrr__all__ZNetworkXExceptionr rmaxsizerrrr1r>r0rr rrrrs0   7MH5   %