Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/chordal.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 """ |
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2 Algorithms for chordal graphs. |
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3 |
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4 A graph is chordal if every cycle of length at least 4 has a chord |
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5 (an edge joining two nodes not adjacent in the cycle). |
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6 https://en.wikipedia.org/wiki/Chordal_graph |
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7 """ |
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8 import sys |
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9 import warnings |
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10 |
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11 import networkx as nx |
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12 from networkx.algorithms.components import connected_components |
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13 from networkx.utils import arbitrary_element, not_implemented_for |
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14 |
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15 |
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16 __all__ = [ |
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17 "is_chordal", |
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18 "find_induced_nodes", |
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19 "chordal_graph_cliques", |
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20 "chordal_graph_treewidth", |
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21 "NetworkXTreewidthBoundExceeded", |
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22 "complete_to_chordal_graph", |
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23 ] |
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24 |
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25 |
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26 class NetworkXTreewidthBoundExceeded(nx.NetworkXException): |
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27 """Exception raised when a treewidth bound has been provided and it has |
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28 been exceeded""" |
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29 |
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30 |
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31 def is_chordal(G): |
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32 """Checks whether G is a chordal graph. |
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33 |
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34 A graph is chordal if every cycle of length at least 4 has a chord |
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35 (an edge joining two nodes not adjacent in the cycle). |
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36 |
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37 Parameters |
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38 ---------- |
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39 G : graph |
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40 A NetworkX graph. |
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41 |
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42 Returns |
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43 ------- |
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44 chordal : bool |
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45 True if G is a chordal graph and False otherwise. |
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46 |
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47 Raises |
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48 ------ |
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49 NetworkXError |
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50 The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. |
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51 If the input graph is an instance of one of these classes, a |
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52 :exc:`NetworkXError` is raised. |
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53 |
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54 Examples |
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55 -------- |
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56 >>> e = [ |
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57 ... (1, 2), |
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58 ... (1, 3), |
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59 ... (2, 3), |
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60 ... (2, 4), |
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61 ... (3, 4), |
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62 ... (3, 5), |
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63 ... (3, 6), |
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64 ... (4, 5), |
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65 ... (4, 6), |
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66 ... (5, 6), |
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67 ... ] |
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68 >>> G = nx.Graph(e) |
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69 >>> nx.is_chordal(G) |
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70 True |
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71 |
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72 Notes |
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73 ----- |
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74 The routine tries to go through every node following maximum cardinality |
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75 search. It returns False when it finds that the separator for any node |
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76 is not a clique. Based on the algorithms in [1]_. |
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77 |
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78 References |
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79 ---------- |
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80 .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms |
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81 to test chordality of graphs, test acyclicity of hypergraphs, and |
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82 selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984), |
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83 pp. 566–579. |
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84 """ |
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85 if G.is_directed(): |
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86 raise nx.NetworkXError("Directed graphs not supported") |
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87 if G.is_multigraph(): |
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88 raise nx.NetworkXError("Multiply connected graphs not supported.") |
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89 if len(_find_chordality_breaker(G)) == 0: |
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90 return True |
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91 else: |
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92 return False |
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93 |
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94 |
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95 def find_induced_nodes(G, s, t, treewidth_bound=sys.maxsize): |
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96 """Returns the set of induced nodes in the path from s to t. |
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97 |
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98 Parameters |
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99 ---------- |
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100 G : graph |
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101 A chordal NetworkX graph |
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102 s : node |
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103 Source node to look for induced nodes |
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104 t : node |
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105 Destination node to look for induced nodes |
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106 treewith_bound: float |
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107 Maximum treewidth acceptable for the graph H. The search |
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108 for induced nodes will end as soon as the treewidth_bound is exceeded. |
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109 |
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110 Returns |
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111 ------- |
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112 Induced_nodes : Set of nodes |
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113 The set of induced nodes in the path from s to t in G |
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114 |
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115 Raises |
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116 ------ |
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117 NetworkXError |
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118 The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. |
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119 If the input graph is an instance of one of these classes, a |
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120 :exc:`NetworkXError` is raised. |
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121 The algorithm can only be applied to chordal graphs. If the input |
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122 graph is found to be non-chordal, a :exc:`NetworkXError` is raised. |
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123 |
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124 Examples |
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125 -------- |
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126 >>> G = nx.Graph() |
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127 >>> G = nx.generators.classic.path_graph(10) |
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128 >>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2) |
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129 >>> sorted(induced_nodes) |
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130 [1, 2, 3, 4, 5, 6, 7, 8, 9] |
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131 |
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132 Notes |
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133 ----- |
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134 G must be a chordal graph and (s,t) an edge that is not in G. |
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135 |
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136 If a treewidth_bound is provided, the search for induced nodes will end |
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137 as soon as the treewidth_bound is exceeded. |
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138 |
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139 The algorithm is inspired by Algorithm 4 in [1]_. |
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140 A formal definition of induced node can also be found on that reference. |
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141 |
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142 References |
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143 ---------- |
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144 .. [1] Learning Bounded Treewidth Bayesian Networks. |
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145 Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008. |
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146 http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf |
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147 """ |
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148 if not is_chordal(G): |
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149 raise nx.NetworkXError("Input graph is not chordal.") |
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150 |
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151 H = nx.Graph(G) |
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152 H.add_edge(s, t) |
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153 Induced_nodes = set() |
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154 triplet = _find_chordality_breaker(H, s, treewidth_bound) |
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155 while triplet: |
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156 (u, v, w) = triplet |
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157 Induced_nodes.update(triplet) |
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158 for n in triplet: |
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159 if n != s: |
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160 H.add_edge(s, n) |
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161 triplet = _find_chordality_breaker(H, s, treewidth_bound) |
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162 if Induced_nodes: |
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163 # Add t and the second node in the induced path from s to t. |
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164 Induced_nodes.add(t) |
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165 for u in G[s]: |
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166 if len(Induced_nodes & set(G[u])) == 2: |
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167 Induced_nodes.add(u) |
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168 break |
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169 return Induced_nodes |
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170 |
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171 |
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172 def chordal_graph_cliques(G): |
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173 """Returns the set of maximal cliques of a chordal graph. |
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174 |
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175 The algorithm breaks the graph in connected components and performs a |
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176 maximum cardinality search in each component to get the cliques. |
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177 |
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178 Parameters |
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179 ---------- |
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180 G : graph |
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181 A NetworkX graph |
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182 |
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183 Returns |
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184 ------- |
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185 cliques : A set containing the maximal cliques in G. |
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186 |
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187 Raises |
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188 ------ |
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189 NetworkXError |
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190 The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. |
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191 If the input graph is an instance of one of these classes, a |
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192 :exc:`NetworkXError` is raised. |
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193 The algorithm can only be applied to chordal graphs. If the input |
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194 graph is found to be non-chordal, a :exc:`NetworkXError` is raised. |
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195 |
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196 Examples |
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197 -------- |
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198 >>> e = [ |
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199 ... (1, 2), |
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200 ... (1, 3), |
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201 ... (2, 3), |
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202 ... (2, 4), |
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203 ... (3, 4), |
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204 ... (3, 5), |
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205 ... (3, 6), |
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206 ... (4, 5), |
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207 ... (4, 6), |
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208 ... (5, 6), |
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209 ... (7, 8), |
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210 ... ] |
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211 >>> G = nx.Graph(e) |
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212 >>> G.add_node(9) |
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213 >>> setlist = nx.chordal_graph_cliques(G) |
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214 """ |
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215 msg = "This will return a generator in 3.0." |
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216 warnings.warn(msg, DeprecationWarning) |
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217 return {c for c in _chordal_graph_cliques(G)} |
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218 |
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219 |
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220 def chordal_graph_treewidth(G): |
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221 """Returns the treewidth of the chordal graph G. |
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222 |
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223 Parameters |
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224 ---------- |
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225 G : graph |
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226 A NetworkX graph |
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227 |
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228 Returns |
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229 ------- |
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230 treewidth : int |
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231 The size of the largest clique in the graph minus one. |
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232 |
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233 Raises |
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234 ------ |
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235 NetworkXError |
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236 The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. |
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237 If the input graph is an instance of one of these classes, a |
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238 :exc:`NetworkXError` is raised. |
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239 The algorithm can only be applied to chordal graphs. If the input |
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240 graph is found to be non-chordal, a :exc:`NetworkXError` is raised. |
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241 |
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242 Examples |
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243 -------- |
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244 >>> e = [ |
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245 ... (1, 2), |
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246 ... (1, 3), |
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247 ... (2, 3), |
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248 ... (2, 4), |
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249 ... (3, 4), |
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250 ... (3, 5), |
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251 ... (3, 6), |
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252 ... (4, 5), |
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253 ... (4, 6), |
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254 ... (5, 6), |
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255 ... (7, 8), |
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256 ... ] |
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257 >>> G = nx.Graph(e) |
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258 >>> G.add_node(9) |
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259 >>> nx.chordal_graph_treewidth(G) |
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260 3 |
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261 |
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262 References |
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263 ---------- |
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264 .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth |
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265 """ |
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266 if not is_chordal(G): |
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267 raise nx.NetworkXError("Input graph is not chordal.") |
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268 |
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269 max_clique = -1 |
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270 for clique in nx.chordal_graph_cliques(G): |
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271 max_clique = max(max_clique, len(clique)) |
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272 return max_clique - 1 |
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273 |
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274 |
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275 def _is_complete_graph(G): |
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276 """Returns True if G is a complete graph.""" |
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277 if nx.number_of_selfloops(G) > 0: |
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278 raise nx.NetworkXError("Self loop found in _is_complete_graph()") |
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279 n = G.number_of_nodes() |
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280 if n < 2: |
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281 return True |
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282 e = G.number_of_edges() |
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283 max_edges = (n * (n - 1)) / 2 |
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284 return e == max_edges |
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285 |
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286 |
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287 def _find_missing_edge(G): |
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288 """ Given a non-complete graph G, returns a missing edge.""" |
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289 nodes = set(G) |
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290 for u in G: |
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291 missing = nodes - set(list(G[u].keys()) + [u]) |
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292 if missing: |
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293 return (u, missing.pop()) |
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294 |
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295 |
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296 def _max_cardinality_node(G, choices, wanna_connect): |
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297 """Returns a the node in choices that has more connections in G |
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298 to nodes in wanna_connect. |
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299 """ |
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300 max_number = -1 |
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301 for x in choices: |
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302 number = len([y for y in G[x] if y in wanna_connect]) |
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303 if number > max_number: |
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304 max_number = number |
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305 max_cardinality_node = x |
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306 return max_cardinality_node |
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307 |
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308 |
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309 def _find_chordality_breaker(G, s=None, treewidth_bound=sys.maxsize): |
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310 """ Given a graph G, starts a max cardinality search |
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311 (starting from s if s is given and from an arbitrary node otherwise) |
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312 trying to find a non-chordal cycle. |
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313 |
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314 If it does find one, it returns (u,v,w) where u,v,w are the three |
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315 nodes that together with s are involved in the cycle. |
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316 """ |
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317 |
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318 unnumbered = set(G) |
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319 if s is None: |
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320 s = arbitrary_element(G) |
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321 unnumbered.remove(s) |
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322 numbered = {s} |
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323 current_treewidth = -1 |
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324 while unnumbered: # and current_treewidth <= treewidth_bound: |
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325 v = _max_cardinality_node(G, unnumbered, numbered) |
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326 unnumbered.remove(v) |
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327 numbered.add(v) |
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328 clique_wanna_be = set(G[v]) & numbered |
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329 sg = G.subgraph(clique_wanna_be) |
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330 if _is_complete_graph(sg): |
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331 # The graph seems to be chordal by now. We update the treewidth |
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332 current_treewidth = max(current_treewidth, len(clique_wanna_be)) |
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333 if current_treewidth > treewidth_bound: |
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334 raise nx.NetworkXTreewidthBoundExceeded( |
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335 f"treewidth_bound exceeded: {current_treewidth}" |
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336 ) |
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337 else: |
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338 # sg is not a clique, |
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339 # look for an edge that is not included in sg |
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340 (u, w) = _find_missing_edge(sg) |
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341 return (u, v, w) |
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342 return () |
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343 |
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344 |
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345 def _chordal_graph_cliques(G): |
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346 """Returns all maximal cliques of a chordal graph. |
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347 |
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348 The algorithm breaks the graph in connected components and performs a |
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349 maximum cardinality search in each component to get the cliques. |
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350 |
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351 Parameters |
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352 ---------- |
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353 G : graph |
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354 A NetworkX graph |
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355 |
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356 Returns |
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357 ------- |
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358 iterator |
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359 An iterator over maximal cliques, each of which is a frozenset of |
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360 nodes in `G`. The order of cliques is arbitrary. |
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361 |
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362 Raises |
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363 ------ |
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364 NetworkXError |
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365 The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. |
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366 If the input graph is an instance of one of these classes, a |
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367 :exc:`NetworkXError` is raised. |
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368 The algorithm can only be applied to chordal graphs. If the input |
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369 graph is found to be non-chordal, a :exc:`NetworkXError` is raised. |
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370 |
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371 Examples |
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372 -------- |
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373 >>> e = [ |
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374 ... (1, 2), |
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375 ... (1, 3), |
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376 ... (2, 3), |
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377 ... (2, 4), |
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378 ... (3, 4), |
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379 ... (3, 5), |
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380 ... (3, 6), |
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381 ... (4, 5), |
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382 ... (4, 6), |
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383 ... (5, 6), |
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384 ... (7, 8), |
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385 ... ] |
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386 >>> G = nx.Graph(e) |
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387 >>> G.add_node(9) |
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388 >>> cliques = [c for c in _chordal_graph_cliques(G)] |
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389 >>> cliques[0] |
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390 frozenset({1, 2, 3}) |
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391 """ |
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392 if not is_chordal(G): |
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393 raise nx.NetworkXError("Input graph is not chordal.") |
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394 |
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395 for C in (G.subgraph(c).copy() for c in connected_components(G)): |
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396 if C.number_of_nodes() == 1: |
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397 yield frozenset(C.nodes()) |
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398 else: |
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399 unnumbered = set(C.nodes()) |
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400 v = arbitrary_element(C) |
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401 unnumbered.remove(v) |
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402 numbered = {v} |
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403 clique_wanna_be = {v} |
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404 while unnumbered: |
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405 v = _max_cardinality_node(C, unnumbered, numbered) |
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406 unnumbered.remove(v) |
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407 numbered.add(v) |
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408 new_clique_wanna_be = set(C.neighbors(v)) & numbered |
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409 sg = C.subgraph(clique_wanna_be) |
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410 if _is_complete_graph(sg): |
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411 new_clique_wanna_be.add(v) |
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412 if not new_clique_wanna_be >= clique_wanna_be: |
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413 yield frozenset(clique_wanna_be) |
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414 clique_wanna_be = new_clique_wanna_be |
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415 else: |
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416 raise nx.NetworkXError("Input graph is not chordal.") |
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417 yield frozenset(clique_wanna_be) |
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418 |
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419 |
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420 @not_implemented_for("directed") |
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421 def complete_to_chordal_graph(G): |
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422 """Return a copy of G completed to a chordal graph |
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423 |
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424 Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is |
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425 called chordal if for each cycle with length bigger than 3, there exist |
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426 two non-adjacent nodes connected by an edge (called a chord). |
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427 |
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428 Parameters |
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429 ---------- |
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430 G : NetworkX graph |
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431 Undirected graph |
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432 |
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433 Returns |
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434 ------- |
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435 H : NetworkX graph |
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436 The chordal enhancement of G |
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437 alpha : Dictionary |
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438 The elimination ordering of nodes of G |
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439 |
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440 Notes |
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441 ------ |
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442 There are different approaches to calculate the chordal |
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443 enhancement of a graph. The algorithm used here is called |
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444 MCS-M and gives at least minimal (local) triangulation of graph. Note |
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445 that this triangulation is not necessarily a global minimum. |
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446 |
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447 https://en.wikipedia.org/wiki/Chordal_graph |
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448 |
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449 References |
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450 ---------- |
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451 .. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004) |
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452 Maximum Cardinality Search for Computing Minimal Triangulations of |
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453 Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3. |
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454 |
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455 Examples |
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456 -------- |
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457 >>> from networkx.algorithms.chordal import complete_to_chordal_graph |
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458 >>> G = nx.wheel_graph(10) |
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459 >>> H, alpha = complete_to_chordal_graph(G) |
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460 """ |
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461 H = G.copy() |
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462 alpha = {node: 0 for node in H} |
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463 if nx.is_chordal(H): |
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464 return H, alpha |
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465 chords = set() |
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466 weight = {node: 0 for node in H.nodes()} |
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467 unnumbered_nodes = list(H.nodes()) |
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468 for i in range(len(H.nodes()), 0, -1): |
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469 # get the node in unnumbered_nodes with the maximum weight |
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470 z = max(unnumbered_nodes, key=lambda node: weight[node]) |
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471 unnumbered_nodes.remove(z) |
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472 alpha[z] = i |
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473 update_nodes = [] |
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474 for y in unnumbered_nodes: |
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475 if G.has_edge(y, z): |
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476 update_nodes.append(y) |
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477 else: |
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478 # y_weight will be bigger than node weights between y and z |
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479 y_weight = weight[y] |
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480 lower_nodes = [ |
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481 node for node in unnumbered_nodes if weight[node] < y_weight |
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482 ] |
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483 if nx.has_path(H.subgraph(lower_nodes + [z, y]), y, z): |
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484 update_nodes.append(y) |
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485 chords.add((z, y)) |
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486 # during calculation of paths the weights should not be updated |
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487 for node in update_nodes: |
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488 weight[node] += 1 |
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489 H.add_edges_from(chords) |
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490 return H, alpha |