Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/approximation/treewidth.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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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., NP-hard) 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),259-275. |
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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 UU-CS-2005-018. |
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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 2-tuple 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 Fill-in 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 2-tuple 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 2-tuples (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 Fill-In |
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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 min-fill-in 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 2-tuple with treewidth and the corresponding decomposed tree. |
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190 """ |
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191 |
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192 # make dict-of-sets 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 |