Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/approximation/treewidth.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 """Functions for computing treewidth decomposition. 
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2 
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3 Treewidth of an undirected graph is a number associated with the graph. 
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4 It can be defined as the size of the largest vertex set (bag) in a tree 
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5 decomposition of the graph minus one. 
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6 
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7 `Wikipedia: Treewidth <https://en.wikipedia.org/wiki/Treewidth>`_ 
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8 
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9 The notions of treewidth and tree decomposition have gained their 
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10 attractiveness partly because many graph and network problems that are 
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11 intractable (e.g., NPhard) on arbitrary graphs become efficiently 
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12 solvable (e.g., with a linear time algorithm) when the treewidth of the 
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13 input graphs is bounded by a constant [1]_ [2]_. 
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14 
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15 There are two different functions for computing a tree decomposition: 
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16 :func:`treewidth_min_degree` and :func:`treewidth_min_fill_in`. 
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17 
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18 .. [1] Hans L. Bodlaender and Arie M. C. A. Koster. 2010. "Treewidth 
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19 computations I.Upper bounds". Inf. Comput. 208, 3 (March 2010),259275. 
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20 http://dx.doi.org/10.1016/j.ic.2009.03.008 
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21 
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22 .. [2] Hans L. Bodlaender. "Discovering Treewidth". Institute of Information 
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23 and Computing Sciences, Utrecht University. 
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24 Technical Report UUCS2005018. 
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25 http://www.cs.uu.nl 
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26 
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27 .. [3] K. Wang, Z. Lu, and J. Hicks *Treewidth*. 
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28 http://web.eecs.utk.edu/~cphillip/cs594_spring2015_projects/treewidth.pdf 
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29 
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30 """ 
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31 
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32 import sys 
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33 
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34 import networkx as nx 
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35 from networkx.utils import not_implemented_for 
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36 from heapq import heappush, heappop, heapify 
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37 import itertools 
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38 
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39 __all__ = ["treewidth_min_degree", "treewidth_min_fill_in"] 
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40 
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41 
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42 @not_implemented_for("directed") 
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43 @not_implemented_for("multigraph") 
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44 def treewidth_min_degree(G): 
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45 """ Returns a treewidth decomposition using the Minimum Degree heuristic. 
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46 
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47 The heuristic chooses the nodes according to their degree, i.e., first 
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48 the node with the lowest degree is chosen, then the graph is updated 
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49 and the corresponding node is removed. Next, a new node with the lowest 
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50 degree is chosen, and so on. 
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51 
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52 Parameters 
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53  
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54 G : NetworkX graph 
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55 
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56 Returns 
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57  
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58 Treewidth decomposition : (int, Graph) tuple 
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59 2tuple with treewidth and the corresponding decomposed tree. 
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60 """ 
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61 deg_heuristic = MinDegreeHeuristic(G) 
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62 return treewidth_decomp(G, lambda graph: deg_heuristic.best_node(graph)) 
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63 
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64 
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65 @not_implemented_for("directed") 
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66 @not_implemented_for("multigraph") 
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67 def treewidth_min_fill_in(G): 
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68 """ Returns a treewidth decomposition using the Minimum Fillin heuristic. 
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69 
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70 The heuristic chooses a node from the graph, where the number of edges 
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71 added turning the neighbourhood of the chosen node into clique is as 
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72 small as possible. 
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73 
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74 Parameters 
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75  
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76 G : NetworkX graph 
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77 
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78 Returns 
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79  
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80 Treewidth decomposition : (int, Graph) tuple 
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81 2tuple with treewidth and the corresponding decomposed tree. 
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82 """ 
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83 return treewidth_decomp(G, min_fill_in_heuristic) 
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84 
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85 
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86 class MinDegreeHeuristic: 
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87 """ Implements the Minimum Degree heuristic. 
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88 
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89 The heuristic chooses the nodes according to their degree 
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90 (number of neighbours), i.e., first the node with the lowest degree is 
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91 chosen, then the graph is updated and the corresponding node is 
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92 removed. Next, a new node with the lowest degree is chosen, and so on. 
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93 """ 
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94 
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95 def __init__(self, graph): 
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96 self._graph = graph 
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97 
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98 # nodes that have to be updated in the heap before each iteration 
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99 self._update_nodes = [] 
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100 
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101 self._degreeq = [] # a heapq with 2tuples (degree,node) 
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102 
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103 # build heap with initial degrees 
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104 for n in graph: 
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105 self._degreeq.append((len(graph[n]), n)) 
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106 heapify(self._degreeq) 
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107 
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108 def best_node(self, graph): 
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109 # update nodes in self._update_nodes 
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110 for n in self._update_nodes: 
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111 # insert changed degrees into degreeq 
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112 heappush(self._degreeq, (len(graph[n]), n)) 
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113 
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114 # get the next valid (minimum degree) node 
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115 while self._degreeq: 
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116 (min_degree, elim_node) = heappop(self._degreeq) 
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117 if elim_node not in graph or len(graph[elim_node]) != min_degree: 
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118 # outdated entry in degreeq 
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119 continue 
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120 elif min_degree == len(graph)  1: 
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121 # fully connected: abort condition 
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122 return None 
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123 
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124 # remember to update nodes in the heap before getting the next node 
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125 self._update_nodes = graph[elim_node] 
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126 return elim_node 
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127 
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128 # the heap is empty: abort 
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129 return None 
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130 
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131 
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132 def min_fill_in_heuristic(graph): 
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133 """ Implements the Minimum Degree heuristic. 
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134 
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135 Returns the node from the graph, where the number of edges added when 
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136 turning the neighbourhood of the chosen node into clique is as small as 
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137 possible. This algorithm chooses the nodes using the Minimum FillIn 
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138 heuristic. The running time of the algorithm is :math:`O(V^3)` and it uses 
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139 additional constant memory.""" 
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140 
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141 if len(graph) == 0: 
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142 return None 
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143 
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144 min_fill_in_node = None 
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145 
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146 min_fill_in = sys.maxsize 
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147 
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148 # create sorted list of (degree, node) 
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149 degree_list = [(len(graph[node]), node) for node in graph] 
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150 degree_list.sort() 
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151 
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152 # abort condition 
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153 min_degree = degree_list[0][0] 
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154 if min_degree == len(graph)  1: 
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155 return None 
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156 
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157 for (_, node) in degree_list: 
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158 num_fill_in = 0 
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159 nbrs = graph[node] 
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160 for nbr in nbrs: 
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161 # count how many nodes in nbrs current nbr is not connected to 
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162 # subtract 1 for the node itself 
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163 num_fill_in += len(nbrs  graph[nbr])  1 
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164 if num_fill_in >= 2 * min_fill_in: 
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165 break 
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166 
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167 num_fill_in /= 2 # divide by 2 because of double counting 
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168 
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169 if num_fill_in < min_fill_in: # update minfillin node 
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170 if num_fill_in == 0: 
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171 return node 
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172 min_fill_in = num_fill_in 
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173 min_fill_in_node = node 
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174 
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175 return min_fill_in_node 
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176 
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177 
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178 def treewidth_decomp(G, heuristic=min_fill_in_heuristic): 
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179 """ Returns a treewidth decomposition using the passed heuristic. 
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180 
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181 Parameters 
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182  
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183 G : NetworkX graph 
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184 heuristic : heuristic function 
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185 
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186 Returns 
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187  
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188 Treewidth decomposition : (int, Graph) tuple 
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189 2tuple with treewidth and the corresponding decomposed tree. 
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190 """ 
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191 
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192 # make dictofsets structure 
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193 graph = {n: set(G[n])  {n} for n in G} 
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194 
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195 # stack containing nodes and neighbors in the order from the heuristic 
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196 node_stack = [] 
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197 
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198 # get first node from heuristic 
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199 elim_node = heuristic(graph) 
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200 while elim_node is not None: 
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201 # connect all neighbours with each other 
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202 nbrs = graph[elim_node] 
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203 for u, v in itertools.permutations(nbrs, 2): 
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204 if v not in graph[u]: 
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205 graph[u].add(v) 
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206 
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207 # push node and its current neighbors on stack 
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208 node_stack.append((elim_node, nbrs)) 
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209 
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210 # remove node from graph 
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211 for u in graph[elim_node]: 
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212 graph[u].remove(elim_node) 
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213 
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214 del graph[elim_node] 
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215 elim_node = heuristic(graph) 
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216 
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217 # the abort condition is met; put all remaining nodes into one bag 
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218 decomp = nx.Graph() 
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219 first_bag = frozenset(graph.keys()) 
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220 decomp.add_node(first_bag) 
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221 
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222 treewidth = len(first_bag)  1 
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223 
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224 while node_stack: 
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225 # get node and its neighbors from the stack 
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226 (curr_node, nbrs) = node_stack.pop() 
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227 
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228 # find a bag all neighbors are in 
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229 old_bag = None 
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230 for bag in decomp.nodes: 
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231 if nbrs <= bag: 
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232 old_bag = bag 
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233 break 
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234 
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235 if old_bag is None: 
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236 # no old_bag was found: just connect to the first_bag 
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237 old_bag = first_bag 
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238 
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239 # create new node for decomposition 
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240 nbrs.add(curr_node) 
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241 new_bag = frozenset(nbrs) 
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242 
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243 # update treewidth 
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244 treewidth = max(treewidth, len(new_bag)  1) 
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245 
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246 # add edge to decomposition (implicitly also adds the new node) 
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247 decomp.add_edge(old_bag, new_bag) 
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248 
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249 return treewidth, decomp 