hpSSKJr SSKrSSKJrJr SS/r\"S5\R"SSS 9SS j55r S r S r S r \ \ S.r \"S5\R"SSS9SSj55rg))chainN)not_implemented_forpairwisemetric_closure steiner_treedirectedweightT) edge_attrs returns_graphc [R"5n[U5n[R"XS9n[ U5unupgU[U5- (aSn[R "U5eUR U5 UHn URXYXiXyS9 M UH7unupgUR U5 UHn URXYXiXyS9 M M9 U$)a'Return the metric closure of a graph. The metric closure of a graph *G* is the complete graph in which each edge is weighted by the shortest path distance between the nodes in *G* . Parameters ---------- G : NetworkX graph Returns ------- NetworkX graph Metric closure of the graph `G`. r z:G is not a connected graph. metric_closure is not defined.)distancepath)nxGraphsetall_pairs_dijkstranext NetworkXErrorremoveadd_edge) Gr MGnodesall_paths_iterurrmsgvs _/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/approximation/steinertree.pyrr s$  A VF**1^^# UH"ummTT[TTTUUUU4SjS94v M$ g7f)c >TTTUT$Nkrrrr s r2_mehlhorn_steiner_tree...Qad1gaj.@r keyNmin.0rrrr s @@r )_mehlhorn_steiner_tree..Ps/ OVtq!Q3qtAw$@A Bw*/)rmulti_source_dijkstra_pathnodeslenredgesgethas_edgerr0minimum_spanning_edges shortest_pathrlist is_multigraph edge_subgraphcopy_remove_nonterminal_leaves)rterminal_nodesr pathsd_1sr G_1_primerr#susv weight_here new_weightG_2G_3drn1n2G_3_mstG_4s` ` r_mehlhorn_steiner_treerS2s  ) )!^ G OV  //' " ' ' )CsN3 99;r c^^[TTS9nURU5n[R"USSS9n[R "SU55nTR 5(a UU4SjU5nTRU5n[R"USSS9nTRU5R5n [X5 U R5$) Nr rT)r r#c3D# UHupn[US5v M g7f)rN)r)r2rrrNs rr3$_kou_steiner_tree..as 'U9a6(;(;9s c 3b>^^# UH"ummTT[TTTUUUU4SjS94v M$ g7f)c >TTTUT$r&r'r(s rr*-_kou_steiner_tree...dr,r r-Nr/r1s @@rr3rVcs3 %13qtAw$@A B%r5r F) rsubgraphrr<r from_iterabler?r@rArBr9) rrCr rH mst_edges mst_all_edgesG_ST_ST_Hs ` ` r_kou_steiner_treerbXsq(A >"A))!JTJI'''U9'UUM % //- (C # #Cu EC //#  # # %Cs3 99;r c ^[U5nTVs1sH&n[[TU5U1- 5S:XdM$UiM( nnXB- nU(av[R"U4SjU56nXeU--nTRU5 UVs1sH&n[[TU5U1- 5S:XdM$UiM( nnXB- nU(aMuggs snfs snf)Nr"c3@># UHn[TU5v M g7fr&)r)r2nrs rr3-_remove_nonterminal_leaves..|s&M:LQs1Q4yy:Ls)rr8unionremove_nodes_from)r terminals terminal_setreleavesnonterminal_leavescandidate_leavess` rrBrBssy>L 8Ac#ad)qc/2a7aF 8. 99&M:L&MN== ./-K-S1Y!_1E1J!-K#2  9Ls#CC#C+C)koumehlhorn)preserve_all_attrsr c^^UcSn[UnU"TUT5nTR5(a UU4SjU5nTR U5nU$![an[US35UeSnAff=f)u1 Return an approximation to the minimum Steiner tree of a graph. The minimum Steiner tree of `G` w.r.t a set of `terminal_nodes` (also *S*) is a tree within `G` that spans those nodes and has minimum size (sum of edge weights) among all such trees. The approximation algorithm is specified with the `method` keyword argument. All three available algorithms produce a tree whose weight is within a ``(2 - (2 / l))`` factor of the weight of the optimal Steiner tree, where ``l`` is the minimum number of leaf nodes across all possible Steiner trees. * ``"kou"`` [2]_ (runtime $O(|S| |V|^2)$) computes the minimum spanning tree of the subgraph of the metric closure of *G* induced by the terminal nodes, where the metric closure of *G* is the complete graph in which each edge is weighted by the shortest path distance between the nodes in *G*. * ``"mehlhorn"`` [3]_ (runtime $O(|E|+|V|\log|V|)$) modifies Kou et al.'s algorithm, beginning by finding the closest terminal node for each non-terminal. This data is used to create a complete graph containing only the terminal nodes, in which edge is weighted with the shortest path distance between them. The algorithm then proceeds in the same way as Kou et al.. Parameters ---------- G : NetworkX graph terminal_nodes : list A list of terminal nodes for which minimum steiner tree is to be found. weight : string (default = 'weight') Use the edge attribute specified by this string as the edge weight. Any edge attribute not present defaults to 1. method : string, optional (default = 'mehlhorn') The algorithm to use to approximate the Steiner tree. Supported options: 'kou', 'mehlhorn'. Other inputs produce a ValueError. Returns ------- NetworkX graph Approximation to the minimum steiner tree of `G` induced by `terminal_nodes` . Raises ------ NetworkXNotImplemented If `G` is directed. ValueError If the specified `method` is not supported. Notes ----- For multigraphs, the edge between two nodes with minimum weight is the edge put into the Steiner tree. References ---------- .. [1] Steiner_tree_problem on Wikipedia. https://en.wikipedia.org/wiki/Steiner_tree_problem .. [2] Kou, L., G. Markowsky, and L. Berman. 1981. ‘A Fast Algorithm for Steiner Trees’. Acta Informatica 15 (2): 141–45. https://doi.org/10.1007/BF00288961. .. [3] Mehlhorn, Kurt. 1988. ‘A Faster Approximation Algorithm for the Steiner Problem in Graphs’. Information Processing Letters 27 (3): 125–28. https://doi.org/10.1016/0020-0190(88)90066-X. Nroz( is not a valid choice for an algorithm.c 3b>^^# UH"ummTT[TTTUUUU4SjS94v M$ g7f)c >TTTUT$r&r'r(s rr*(steiner_tree...r,r r-Nr/r1s @@rr3steiner_tree..s/ OTtq!Q3qtAw$@A Bur5) ALGORITHMSKeyError ValueErrorr?r@)rrCr methodalgoer9Ts` ` rrrsZ~U&! NF +E OT  A H UF8#KLMSTTUs A A/A**A/r )r N) itertoolsrnetworkxrnetworkx.utilsrr__all__ _dispatchablerrSrbrBrvrr'r rrs8 ^ ,Z XT:$ ;!$ N#L63& & Z T>Z ?!Z r