h DSrSSKrSSKrSSKJrJrJr SSKrSSK J r SS/r \ "S5\ "S5\R"S S 9S 555r \ "S5\ "S5\R"S S 9S 555r"S S5rSr\R"S S 9\4Sj5rg)a,Functions for computing treewidth decomposition. Treewidth of an undirected graph is a number associated with the graph. It can be defined as the size of the largest vertex set (bag) in a tree decomposition of the graph minus one. `Wikipedia: Treewidth `_ The notions of treewidth and tree decomposition have gained their attractiveness partly because many graph and network problems that are intractable (e.g., NP-hard) on arbitrary graphs become efficiently solvable (e.g., with a linear time algorithm) when the treewidth of the input graphs is bounded by a constant [1]_ [2]_. There are two different functions for computing a tree decomposition: :func:`treewidth_min_degree` and :func:`treewidth_min_fill_in`. .. [1] Hans L. Bodlaender and Arie M. C. A. Koster. 2010. "Treewidth computations I.Upper bounds". Inf. Comput. 208, 3 (March 2010),259-275. http://dx.doi.org/10.1016/j.ic.2009.03.008 .. [2] Hans L. Bodlaender. "Discovering Treewidth". Institute of Information and Computing Sciences, Utrecht University. Technical Report UU-CS-2005-018. http://www.cs.uu.nl .. [3] K. Wang, Z. Lu, and J. Hicks *Treewidth*. https://web.archive.org/web/20210507025929/http://web.eecs.utk.edu/~cphill25/cs594_spring2015_projects/treewidth.pdf N)heapifyheappopheappush)not_implemented_fortreewidth_min_degreetreewidth_min_fill_indirected multigraphT) returns_graphc:^[U5m[UU4Sj5$)aReturns a treewidth decomposition using the Minimum Degree heuristic. The heuristic chooses the nodes according to their degree, i.e., first the node with the lowest degree is chosen, then the graph is updated and the corresponding node is removed. Next, a new node with the lowest degree is chosen, and so on. Parameters ---------- G : NetworkX graph Returns ------- Treewidth decomposition : (int, Graph) tuple 2-tuple with treewidth and the corresponding decomposed tree. c&>TRU5$N) best_node)graph deg_heuristics ]/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/approximation/treewidth.py&treewidth_min_degree..?s]-D-DU-K)MinDegreeHeuristictreewidth_decomp)Grs @rrr*s('q)M AK LLrc"[U[5$)aReturns a treewidth decomposition using the Minimum Fill-in heuristic. The heuristic chooses a node from the graph, where the number of edges added turning the neighborhood of the chosen node into clique is as small as possible. Parameters ---------- G : NetworkX graph Returns ------- Treewidth decomposition : (int, Graph) tuple 2-tuple with treewidth and the corresponding decomposed tree. )rmin_fill_in_heuristic)rs rrrBs& A4 55rc$\rSrSrSrSrSrSrg)rXa1Implements the Minimum Degree heuristic. The heuristic chooses the nodes according to their degree (number of neighbors), i.e., first the node with the lowest degree is chosen, then the graph is updated and the corresponding node is removed. Next, a new node with the lowest degree is chosen, and so on. cXl/Ul/Ul[R"5UlUH?nURR [ X5[UR5U45 MA [UR5 gr) _graph _update_nodes_degreeq itertoolscountappendlennextr)selfrns r__init__MinDegreeHeuristic.__init__asf   __& A MM #eh-djj1A1!E F rcVURH:n[UR[X5[ UR 5U45 M< UR(aN[ UR5up4nXQ;d[X5U:waMAU[U5S- :XagXUlU$g)N)rrr r$r%r"r)r&rr' min_degree_ elim_nodes rrMinDegreeHeuristic.best_nodeos##A T]]S]D4Da$H I$ mm)0)? &ZI%U-=)>*)Ls5zA~-"'!1D  r)r rrr"N)__name__ __module__ __qualname____firstlineno____doc__r(r__static_attributes__rrrrXs rrcZ^[T5S:XagSn[Rn[TU4SjS9n[TUS5nU[T5S- :XagUHOnSnTUnUH%nU[UTU- 5S- - nUSU-:dM% O US-nXb:dMAUS:XaUs $UnUnMQ U$)abImplements the Minimum Degree heuristic. Returns the node from the graph, where the number of edges added when turning the neighborhood of the chosen node into clique is as small as possible. This algorithm chooses the nodes using the Minimum Fill-In heuristic. The running time of the algorithm is :math:`O(V^3)` and it uses additional constant memory.rNc >[TU5$r)r$)xrs rr'min_fill_in_heuristic..s #eAh-r)keyr+)r$sysmaxsizesorted) rmin_fill_in_node min_fill_innodes_by_degreer,node num_fill_innbrsnbrs ` rrrs 5zQ++KU(?@OU?1-./JSZ!^# T{C 3teCj01A5 5Ka+o-  q  $a %K# ! $ rcUVs0sHo"[X5U1- _M nn/nU"U5nUbX5n[R"US5H"upxXU;dMX7RU5 M$ UR XV45 X5HnX7R U5 M X5 U"U5nUbM[ R"5n [UR55n U RU 5 [U 5S- n U(aUR5upSn U RH nXn::dM Un O U cU n URU 5 [U5n[U [U5S- 5n U RX5 U(aMX4$s snf)aReturns a treewidth decomposition using the passed heuristic. Parameters ---------- G : NetworkX graph heuristic : heuristic function Returns ------- Treewidth decomposition : (int, Graph) tuple 2-tuple with treewidth and the corresponding decomposed tree. Nr<r+)setr! permutationsaddr#removenxGraph frozensetkeysadd_noder$popnodesmaxadd_edge)r heuristicr'r node_stackr.rEuvdecomp first_bag treewidth curr_nodeold_bagbagnew_bags rrrs *+ +AAD QC E +J% I  **43DAa  Q4 9+,!A HOOI &"  e$   $XXZF%**,'I OOII"I &NN,<res> ,,. !#: ;Z \"%M&#!M*Z \"%6&#!6&,,^*Z%"7G&Gr