Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/planarity.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 from collections import defaultdict |
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2 import networkx as nx |
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3 |
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4 __all__ = ["check_planarity", "PlanarEmbedding"] |
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5 |
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6 |
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7 def check_planarity(G, counterexample=False): |
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8 """Check if a graph is planar and return a counterexample or an embedding. |
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9 |
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10 A graph is planar iff it can be drawn in a plane without |
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11 any edge intersections. |
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12 |
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13 Parameters |
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14 ---------- |
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15 G : NetworkX graph |
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16 counterexample : bool |
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17 A Kuratowski subgraph (to proof non planarity) is only returned if set |
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18 to true. |
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19 |
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20 Returns |
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21 ------- |
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22 (is_planar, certificate) : (bool, NetworkX graph) tuple |
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23 is_planar is true if the graph is planar. |
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24 If the graph is planar `certificate` is a PlanarEmbedding |
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25 otherwise it is a Kuratowski subgraph. |
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26 |
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27 Notes |
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28 ----- |
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29 A (combinatorial) embedding consists of cyclic orderings of the incident |
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30 edges at each vertex. Given such an embedding there are multiple approaches |
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31 discussed in literature to drawing the graph (subject to various |
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32 constraints, e.g. integer coordinates), see e.g. [2]. |
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33 |
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34 The planarity check algorithm and extraction of the combinatorial embedding |
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35 is based on the Left-Right Planarity Test [1]. |
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36 |
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37 A counterexample is only generated if the corresponding parameter is set, |
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38 because the complexity of the counterexample generation is higher. |
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39 |
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40 References |
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41 ---------- |
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42 .. [1] Ulrik Brandes: |
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43 The Left-Right Planarity Test |
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44 2009 |
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45 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.9208 |
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46 .. [2] Takao Nishizeki, Md Saidur Rahman: |
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47 Planar graph drawing |
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48 Lecture Notes Series on Computing: Volume 12 |
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49 2004 |
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50 """ |
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51 |
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52 planarity_state = LRPlanarity(G) |
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53 embedding = planarity_state.lr_planarity() |
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54 if embedding is None: |
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55 # graph is not planar |
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56 if counterexample: |
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57 return False, get_counterexample(G) |
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58 else: |
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59 return False, None |
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60 else: |
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61 # graph is planar |
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62 return True, embedding |
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63 |
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64 |
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65 def check_planarity_recursive(G, counterexample=False): |
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66 """Recursive version of :meth:`check_planarity`.""" |
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67 planarity_state = LRPlanarity(G) |
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68 embedding = planarity_state.lr_planarity_recursive() |
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69 if embedding is None: |
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70 # graph is not planar |
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71 if counterexample: |
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72 return False, get_counterexample_recursive(G) |
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73 else: |
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74 return False, None |
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75 else: |
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76 # graph is planar |
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77 return True, embedding |
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78 |
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79 |
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80 def get_counterexample(G): |
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81 """Obtains a Kuratowski subgraph. |
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82 |
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83 Raises nx.NetworkXException if G is planar. |
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84 |
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85 The function removes edges such that the graph is still not planar. |
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86 At some point the removal of any edge would make the graph planar. |
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87 This subgraph must be a Kuratowski subgraph. |
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88 |
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89 Parameters |
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90 ---------- |
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91 G : NetworkX graph |
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92 |
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93 Returns |
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94 ------- |
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95 subgraph : NetworkX graph |
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96 A Kuratowski subgraph that proves that G is not planar. |
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97 |
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98 """ |
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99 # copy graph |
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100 G = nx.Graph(G) |
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101 |
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102 if check_planarity(G)[0]: |
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103 raise nx.NetworkXException("G is planar - no counter example.") |
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104 |
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105 # find Kuratowski subgraph |
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106 subgraph = nx.Graph() |
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107 for u in G: |
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108 nbrs = list(G[u]) |
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109 for v in nbrs: |
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110 G.remove_edge(u, v) |
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111 if check_planarity(G)[0]: |
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112 G.add_edge(u, v) |
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113 subgraph.add_edge(u, v) |
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114 |
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115 return subgraph |
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116 |
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117 |
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118 def get_counterexample_recursive(G): |
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119 """Recursive version of :meth:`get_counterexample`. |
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120 """ |
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121 |
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122 # copy graph |
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123 G = nx.Graph(G) |
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124 |
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125 if check_planarity_recursive(G)[0]: |
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126 raise nx.NetworkXException("G is planar - no counter example.") |
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127 |
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128 # find Kuratowski subgraph |
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129 subgraph = nx.Graph() |
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130 for u in G: |
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131 nbrs = list(G[u]) |
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132 for v in nbrs: |
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133 G.remove_edge(u, v) |
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134 if check_planarity_recursive(G)[0]: |
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135 G.add_edge(u, v) |
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136 subgraph.add_edge(u, v) |
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137 |
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138 return subgraph |
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139 |
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140 |
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141 class Interval: |
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142 """Represents a set of return edges. |
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143 |
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144 All return edges in an interval induce a same constraint on the contained |
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145 edges, which means that all edges must either have a left orientation or |
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146 all edges must have a right orientation. |
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147 """ |
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148 |
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149 def __init__(self, low=None, high=None): |
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150 self.low = low |
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151 self.high = high |
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152 |
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153 def empty(self): |
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154 """Check if the interval is empty""" |
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155 return self.low is None and self.high is None |
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156 |
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157 def copy(self): |
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158 """Returns a copy of this interval""" |
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159 return Interval(self.low, self.high) |
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160 |
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161 def conflicting(self, b, planarity_state): |
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162 """Returns True if interval I conflicts with edge b""" |
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163 return ( |
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164 not self.empty() |
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165 and planarity_state.lowpt[self.high] > planarity_state.lowpt[b] |
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166 ) |
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167 |
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168 |
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169 class ConflictPair: |
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170 """Represents a different constraint between two intervals. |
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171 |
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172 The edges in the left interval must have a different orientation than |
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173 the one in the right interval. |
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174 """ |
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175 |
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176 def __init__(self, left=Interval(), right=Interval()): |
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177 self.left = left |
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178 self.right = right |
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179 |
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180 def swap(self): |
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181 """Swap left and right intervals""" |
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182 temp = self.left |
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183 self.left = self.right |
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184 self.right = temp |
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185 |
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186 def lowest(self, planarity_state): |
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187 """Returns the lowest lowpoint of a conflict pair""" |
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188 if self.left.empty(): |
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189 return planarity_state.lowpt[self.right.low] |
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190 if self.right.empty(): |
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191 return planarity_state.lowpt[self.left.low] |
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192 return min( |
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193 planarity_state.lowpt[self.left.low], planarity_state.lowpt[self.right.low] |
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194 ) |
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195 |
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196 |
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197 def top_of_stack(l): |
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198 """Returns the element on top of the stack.""" |
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199 if not l: |
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200 return None |
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201 return l[-1] |
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202 |
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203 |
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204 class LRPlanarity: |
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205 """A class to maintain the state during planarity check.""" |
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206 |
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207 __slots__ = [ |
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208 "G", |
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209 "roots", |
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210 "height", |
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211 "lowpt", |
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212 "lowpt2", |
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213 "nesting_depth", |
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214 "parent_edge", |
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215 "DG", |
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216 "adjs", |
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217 "ordered_adjs", |
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218 "ref", |
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219 "side", |
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220 "S", |
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221 "stack_bottom", |
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222 "lowpt_edge", |
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223 "left_ref", |
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224 "right_ref", |
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225 "embedding", |
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226 ] |
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227 |
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228 def __init__(self, G): |
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229 # copy G without adding self-loops |
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230 self.G = nx.Graph() |
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231 self.G.add_nodes_from(G.nodes) |
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232 for e in G.edges: |
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233 if e[0] != e[1]: |
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234 self.G.add_edge(e[0], e[1]) |
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235 |
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236 self.roots = [] |
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237 |
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238 # distance from tree root |
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239 self.height = defaultdict(lambda: None) |
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240 |
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241 self.lowpt = {} # height of lowest return point of an edge |
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242 self.lowpt2 = {} # height of second lowest return point |
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243 self.nesting_depth = {} # for nesting order |
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244 |
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245 # None -> missing edge |
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246 self.parent_edge = defaultdict(lambda: None) |
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247 |
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248 # oriented DFS graph |
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249 self.DG = nx.DiGraph() |
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250 self.DG.add_nodes_from(G.nodes) |
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251 |
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252 self.adjs = {} |
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253 self.ordered_adjs = {} |
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254 |
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255 self.ref = defaultdict(lambda: None) |
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256 self.side = defaultdict(lambda: 1) |
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257 |
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258 # stack of conflict pairs |
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259 self.S = [] |
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260 self.stack_bottom = {} |
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261 self.lowpt_edge = {} |
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262 |
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263 self.left_ref = {} |
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264 self.right_ref = {} |
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265 |
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266 self.embedding = PlanarEmbedding() |
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267 |
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268 def lr_planarity(self): |
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269 """Execute the LR planarity test. |
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270 |
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271 Returns |
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272 ------- |
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273 embedding : dict |
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274 If the graph is planar an embedding is returned. Otherwise None. |
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275 """ |
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276 if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6: |
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277 # graph is not planar |
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278 return None |
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279 |
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280 # make adjacency lists for dfs |
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281 for v in self.G: |
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282 self.adjs[v] = list(self.G[v]) |
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283 |
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284 # orientation of the graph by depth first search traversal |
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285 for v in self.G: |
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286 if self.height[v] is None: |
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287 self.height[v] = 0 |
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288 self.roots.append(v) |
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289 self.dfs_orientation(v) |
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290 |
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291 # Free no longer used variables |
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292 self.G = None |
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293 self.lowpt2 = None |
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294 self.adjs = None |
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295 |
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296 # testing |
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297 for v in self.DG: # sort the adjacency lists by nesting depth |
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298 # note: this sorting leads to non linear time |
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299 self.ordered_adjs[v] = sorted( |
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300 self.DG[v], key=lambda x: self.nesting_depth[(v, x)] |
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301 ) |
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302 for v in self.roots: |
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303 if not self.dfs_testing(v): |
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304 return None |
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305 |
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306 # Free no longer used variables |
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307 self.height = None |
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308 self.lowpt = None |
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309 self.S = None |
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310 self.stack_bottom = None |
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311 self.lowpt_edge = None |
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312 |
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313 for e in self.DG.edges: |
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314 self.nesting_depth[e] = self.sign(e) * self.nesting_depth[e] |
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315 |
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316 self.embedding.add_nodes_from(self.DG.nodes) |
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317 for v in self.DG: |
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318 # sort the adjacency lists again |
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319 self.ordered_adjs[v] = sorted( |
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320 self.DG[v], key=lambda x: self.nesting_depth[(v, x)] |
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321 ) |
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322 # initialize the embedding |
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323 previous_node = None |
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324 for w in self.ordered_adjs[v]: |
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325 self.embedding.add_half_edge_cw(v, w, previous_node) |
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326 previous_node = w |
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327 |
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328 # Free no longer used variables |
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329 self.DG = None |
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330 self.nesting_depth = None |
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331 self.ref = None |
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332 |
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333 # compute the complete embedding |
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334 for v in self.roots: |
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335 self.dfs_embedding(v) |
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336 |
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337 # Free no longer used variables |
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338 self.roots = None |
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339 self.parent_edge = None |
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340 self.ordered_adjs = None |
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341 self.left_ref = None |
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342 self.right_ref = None |
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343 self.side = None |
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344 |
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345 return self.embedding |
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346 |
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347 def lr_planarity_recursive(self): |
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348 """Recursive version of :meth:`lr_planarity`.""" |
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349 if self.G.order() > 2 and self.G.size() > 3 * self.G.order() - 6: |
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350 # graph is not planar |
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351 return None |
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352 |
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353 # orientation of the graph by depth first search traversal |
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354 for v in self.G: |
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355 if self.height[v] is None: |
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356 self.height[v] = 0 |
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357 self.roots.append(v) |
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358 self.dfs_orientation_recursive(v) |
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359 |
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360 # Free no longer used variable |
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361 self.G = None |
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362 |
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363 # testing |
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364 for v in self.DG: # sort the adjacency lists by nesting depth |
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365 # note: this sorting leads to non linear time |
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366 self.ordered_adjs[v] = sorted( |
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367 self.DG[v], key=lambda x: self.nesting_depth[(v, x)] |
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368 ) |
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369 for v in self.roots: |
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370 if not self.dfs_testing_recursive(v): |
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371 return None |
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372 |
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373 for e in self.DG.edges: |
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374 self.nesting_depth[e] = self.sign_recursive(e) * self.nesting_depth[e] |
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375 |
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376 self.embedding.add_nodes_from(self.DG.nodes) |
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377 for v in self.DG: |
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378 # sort the adjacency lists again |
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379 self.ordered_adjs[v] = sorted( |
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380 self.DG[v], key=lambda x: self.nesting_depth[(v, x)] |
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381 ) |
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382 # initialize the embedding |
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383 previous_node = None |
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384 for w in self.ordered_adjs[v]: |
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385 self.embedding.add_half_edge_cw(v, w, previous_node) |
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386 previous_node = w |
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387 |
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388 # compute the complete embedding |
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389 for v in self.roots: |
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390 self.dfs_embedding_recursive(v) |
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391 |
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392 return self.embedding |
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393 |
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394 def dfs_orientation(self, v): |
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395 """Orient the graph by DFS, compute lowpoints and nesting order. |
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396 """ |
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397 # the recursion stack |
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398 dfs_stack = [v] |
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399 # index of next edge to handle in adjacency list of each node |
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400 ind = defaultdict(lambda: 0) |
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401 # boolean to indicate whether to skip the initial work for an edge |
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402 skip_init = defaultdict(lambda: False) |
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403 |
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404 while dfs_stack: |
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405 v = dfs_stack.pop() |
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406 e = self.parent_edge[v] |
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407 |
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408 for w in self.adjs[v][ind[v] :]: |
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409 vw = (v, w) |
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410 |
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411 if not skip_init[vw]: |
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412 if (v, w) in self.DG.edges or (w, v) in self.DG.edges: |
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413 ind[v] += 1 |
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414 continue # the edge was already oriented |
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415 |
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416 self.DG.add_edge(v, w) # orient the edge |
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417 |
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418 self.lowpt[vw] = self.height[v] |
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419 self.lowpt2[vw] = self.height[v] |
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420 if self.height[w] is None: # (v, w) is a tree edge |
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421 self.parent_edge[w] = vw |
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422 self.height[w] = self.height[v] + 1 |
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423 |
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424 dfs_stack.append(v) # revisit v after finishing w |
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425 dfs_stack.append(w) # visit w next |
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426 skip_init[vw] = True # don't redo this block |
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427 break # handle next node in dfs_stack (i.e. w) |
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428 else: # (v, w) is a back edge |
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429 self.lowpt[vw] = self.height[w] |
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430 |
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431 # determine nesting graph |
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432 self.nesting_depth[vw] = 2 * self.lowpt[vw] |
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433 if self.lowpt2[vw] < self.height[v]: # chordal |
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434 self.nesting_depth[vw] += 1 |
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435 |
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436 # update lowpoints of parent edge e |
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437 if e is not None: |
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438 if self.lowpt[vw] < self.lowpt[e]: |
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439 self.lowpt2[e] = min(self.lowpt[e], self.lowpt2[vw]) |
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440 self.lowpt[e] = self.lowpt[vw] |
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441 elif self.lowpt[vw] > self.lowpt[e]: |
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442 self.lowpt2[e] = min(self.lowpt2[e], self.lowpt[vw]) |
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443 else: |
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444 self.lowpt2[e] = min(self.lowpt2[e], self.lowpt2[vw]) |
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445 |
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446 ind[v] += 1 |
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447 |
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448 def dfs_orientation_recursive(self, v): |
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449 """Recursive version of :meth:`dfs_orientation`.""" |
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450 e = self.parent_edge[v] |
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451 for w in self.G[v]: |
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452 if (v, w) in self.DG.edges or (w, v) in self.DG.edges: |
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453 continue # the edge was already oriented |
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454 vw = (v, w) |
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455 self.DG.add_edge(v, w) # orient the edge |
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456 |
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457 self.lowpt[vw] = self.height[v] |
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458 self.lowpt2[vw] = self.height[v] |
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parents:
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459 if self.height[w] is None: # (v, w) is a tree edge |
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parents:
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460 self.parent_edge[w] = vw |
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461 self.height[w] = self.height[v] + 1 |
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parents:
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462 self.dfs_orientation_recursive(w) |
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shellac
parents:
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463 else: # (v, w) is a back edge |
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parents:
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464 self.lowpt[vw] = self.height[w] |
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parents:
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changeset
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465 |
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parents:
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changeset
|
466 # determine nesting graph |
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parents:
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467 self.nesting_depth[vw] = 2 * self.lowpt[vw] |
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468 if self.lowpt2[vw] < self.height[v]: # chordal |
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parents:
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469 self.nesting_depth[vw] += 1 |
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parents:
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470 |
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parents:
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changeset
|
471 # update lowpoints of parent edge e |
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parents:
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472 if e is not None: |
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shellac
parents:
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changeset
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473 if self.lowpt[vw] < self.lowpt[e]: |
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parents:
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changeset
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474 self.lowpt2[e] = min(self.lowpt[e], self.lowpt2[vw]) |
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shellac
parents:
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changeset
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475 self.lowpt[e] = self.lowpt[vw] |
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parents:
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476 elif self.lowpt[vw] > self.lowpt[e]: |
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parents:
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changeset
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477 self.lowpt2[e] = min(self.lowpt2[e], self.lowpt[vw]) |
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parents:
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478 else: |
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parents:
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479 self.lowpt2[e] = min(self.lowpt2[e], self.lowpt2[vw]) |
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parents:
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480 |
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parents:
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changeset
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481 def dfs_testing(self, v): |
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482 """Test for LR partition.""" |
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parents:
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483 # the recursion stack |
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484 dfs_stack = [v] |
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parents:
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changeset
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485 # index of next edge to handle in adjacency list of each node |
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486 ind = defaultdict(lambda: 0) |
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parents:
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changeset
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487 # boolean to indicate whether to skip the initial work for an edge |
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488 skip_init = defaultdict(lambda: False) |
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parents:
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489 |
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parents:
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490 while dfs_stack: |
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491 v = dfs_stack.pop() |
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492 e = self.parent_edge[v] |
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changeset
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493 # to indicate whether to skip the final block after the for loop |
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494 skip_final = False |
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parents:
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changeset
|
495 |
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shellac
parents:
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changeset
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496 for w in self.ordered_adjs[v][ind[v] :]: |
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497 ei = (v, w) |
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parents:
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498 |
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shellac
parents:
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changeset
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499 if not skip_init[ei]: |
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parents:
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500 self.stack_bottom[ei] = top_of_stack(self.S) |
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shellac
parents:
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501 |
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shellac
parents:
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changeset
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502 if ei == self.parent_edge[w]: # tree edge |
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parents:
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503 dfs_stack.append(v) # revisit v after finishing w |
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504 dfs_stack.append(w) # visit w next |
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505 skip_init[ei] = True # don't redo this block |
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506 skip_final = True # skip final work after breaking |
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507 break # handle next node in dfs_stack (i.e. w) |
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parents:
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508 else: # back edge |
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shellac
parents:
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509 self.lowpt_edge[ei] = ei |
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parents:
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510 self.S.append(ConflictPair(right=Interval(ei, ei))) |
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shellac
parents:
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changeset
|
511 |
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shellac
parents:
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changeset
|
512 # integrate new return edges |
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shellac
parents:
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changeset
|
513 if self.lowpt[ei] < self.height[v]: |
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shellac
parents:
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changeset
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514 if w == self.ordered_adjs[v][0]: # e_i has return edge |
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shellac
parents:
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changeset
|
515 self.lowpt_edge[e] = self.lowpt_edge[ei] |
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shellac
parents:
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516 else: # add constraints of e_i |
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shellac
parents:
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changeset
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517 if not self.add_constraints(ei, e): |
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shellac
parents:
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changeset
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518 # graph is not planar |
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519 return False |
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shellac
parents:
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changeset
|
520 |
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shellac
parents:
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changeset
|
521 ind[v] += 1 |
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shellac
parents:
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changeset
|
522 |
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shellac
parents:
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changeset
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523 if not skip_final: |
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shellac
parents:
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changeset
|
524 # remove back edges returning to parent |
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shellac
parents:
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changeset
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525 if e is not None: # v isn't root |
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shellac
parents:
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changeset
|
526 self.remove_back_edges(e) |
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shellac
parents:
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changeset
|
527 |
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shellac
parents:
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changeset
|
528 return True |
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shellac
parents:
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changeset
|
529 |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
530 def dfs_testing_recursive(self, v): |
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shellac
parents:
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changeset
|
531 """Recursive version of :meth:`dfs_testing`.""" |
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shellac
parents:
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changeset
|
532 e = self.parent_edge[v] |
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shellac
parents:
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changeset
|
533 for w in self.ordered_adjs[v]: |
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shellac
parents:
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changeset
|
534 ei = (v, w) |
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shellac
parents:
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changeset
|
535 self.stack_bottom[ei] = top_of_stack(self.S) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
536 if ei == self.parent_edge[w]: # tree edge |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
537 if not self.dfs_testing_recursive(w): |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
538 return False |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
539 else: # back edge |
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shellac
parents:
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changeset
|
540 self.lowpt_edge[ei] = ei |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
541 self.S.append(ConflictPair(right=Interval(ei, ei))) |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
542 |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
543 # integrate new return edges |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
544 if self.lowpt[ei] < self.height[v]: |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
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545 if w == self.ordered_adjs[v][0]: # e_i has return edge |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
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546 self.lowpt_edge[e] = self.lowpt_edge[ei] |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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547 else: # add constraints of e_i |
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shellac
parents:
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changeset
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548 if not self.add_constraints(ei, e): |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
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549 # graph is not planar |
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shellac
parents:
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changeset
|
550 return False |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
551 |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
552 # remove back edges returning to parent |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
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553 if e is not None: # v isn't root |
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shellac
parents:
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changeset
|
554 self.remove_back_edges(e) |
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shellac
parents:
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changeset
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555 return True |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
556 |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
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557 def add_constraints(self, ei, e): |
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shellac
parents:
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558 P = ConflictPair() |
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shellac
parents:
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changeset
|
559 # merge return edges of e_i into P.right |
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shellac
parents:
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changeset
|
560 while True: |
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shellac
parents:
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561 Q = self.S.pop() |
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shellac
parents:
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changeset
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562 if not Q.left.empty(): |
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shellac
parents:
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changeset
|
563 Q.swap() |
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shellac
parents:
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changeset
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564 if not Q.left.empty(): # not planar |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
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565 return False |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
566 if self.lowpt[Q.right.low] > self.lowpt[e]: |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
567 # merge intervals |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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changeset
|
568 if P.right.empty(): # topmost interval |
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569 P.right = Q.right.copy() |
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parents:
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|
570 else: |
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parents:
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571 self.ref[P.right.low] = Q.right.high |
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572 P.right.low = Q.right.low |
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573 else: # align |
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574 self.ref[Q.right.low] = self.lowpt_edge[e] |
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parents:
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575 if top_of_stack(self.S) == self.stack_bottom[ei]: |
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parents:
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576 break |
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parents:
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changeset
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577 # merge conflicting return edges of e_1,...,e_i-1 into P.L |
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578 while top_of_stack(self.S).left.conflicting(ei, self) or top_of_stack( |
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parents:
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579 self.S |
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parents:
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580 ).right.conflicting(ei, self): |
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581 Q = self.S.pop() |
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582 if Q.right.conflicting(ei, self): |
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parents:
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583 Q.swap() |
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584 if Q.right.conflicting(ei, self): # not planar |
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585 return False |
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parents:
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586 # merge interval below lowpt(e_i) into P.R |
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587 self.ref[P.right.low] = Q.right.high |
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588 if Q.right.low is not None: |
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589 P.right.low = Q.right.low |
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590 |
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591 if P.left.empty(): # topmost interval |
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592 P.left = Q.left.copy() |
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593 else: |
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594 self.ref[P.left.low] = Q.left.high |
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595 P.left.low = Q.left.low |
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596 |
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597 if not (P.left.empty() and P.right.empty()): |
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598 self.S.append(P) |
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599 return True |
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600 |
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601 def remove_back_edges(self, e): |
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602 u = e[0] |
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603 # trim back edges ending at parent u |
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604 # drop entire conflict pairs |
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605 while self.S and top_of_stack(self.S).lowest(self) == self.height[u]: |
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606 P = self.S.pop() |
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607 if P.left.low is not None: |
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608 self.side[P.left.low] = -1 |
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609 |
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610 if self.S: # one more conflict pair to consider |
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611 P = self.S.pop() |
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612 # trim left interval |
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613 while P.left.high is not None and P.left.high[1] == u: |
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614 P.left.high = self.ref[P.left.high] |
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615 if P.left.high is None and P.left.low is not None: |
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616 # just emptied |
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617 self.ref[P.left.low] = P.right.low |
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618 self.side[P.left.low] = -1 |
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619 P.left.low = None |
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620 # trim right interval |
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621 while P.right.high is not None and P.right.high[1] == u: |
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622 P.right.high = self.ref[P.right.high] |
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623 if P.right.high is None and P.right.low is not None: |
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parents:
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624 # just emptied |
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625 self.ref[P.right.low] = P.left.low |
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parents:
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626 self.side[P.right.low] = -1 |
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627 P.right.low = None |
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628 self.S.append(P) |
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629 |
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630 # side of e is side of a highest return edge |
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631 if self.lowpt[e] < self.height[u]: # e has return edge |
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parents:
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632 hl = top_of_stack(self.S).left.high |
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parents:
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633 hr = top_of_stack(self.S).right.high |
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parents:
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634 |
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parents:
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635 if hl is not None and (hr is None or self.lowpt[hl] > self.lowpt[hr]): |
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parents:
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|
636 self.ref[e] = hl |
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637 else: |
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parents:
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638 self.ref[e] = hr |
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parents:
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|
639 |
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parents:
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640 def dfs_embedding(self, v): |
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641 """Completes the embedding.""" |
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parents:
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642 # the recursion stack |
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643 dfs_stack = [v] |
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parents:
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644 # index of next edge to handle in adjacency list of each node |
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645 ind = defaultdict(lambda: 0) |
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646 |
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parents:
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647 while dfs_stack: |
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648 v = dfs_stack.pop() |
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649 |
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shellac
parents:
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650 for w in self.ordered_adjs[v][ind[v] :]: |
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parents:
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651 ind[v] += 1 |
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652 ei = (v, w) |
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parents:
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653 |
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parents:
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654 if ei == self.parent_edge[w]: # tree edge |
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655 self.embedding.add_half_edge_first(w, v) |
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656 self.left_ref[v] = w |
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657 self.right_ref[v] = w |
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658 |
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shellac
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659 dfs_stack.append(v) # revisit v after finishing w |
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660 dfs_stack.append(w) # visit w next |
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661 break # handle next node in dfs_stack (i.e. w) |
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662 else: # back edge |
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663 if self.side[ei] == 1: |
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parents:
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664 self.embedding.add_half_edge_cw(w, v, self.right_ref[w]) |
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665 else: |
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shellac
parents:
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666 self.embedding.add_half_edge_ccw(w, v, self.left_ref[w]) |
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667 self.left_ref[w] = v |
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parents:
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668 |
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parents:
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669 def dfs_embedding_recursive(self, v): |
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shellac
parents:
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670 """Recursive version of :meth:`dfs_embedding`.""" |
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shellac
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671 for w in self.ordered_adjs[v]: |
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shellac
parents:
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672 ei = (v, w) |
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shellac
parents:
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673 if ei == self.parent_edge[w]: # tree edge |
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shellac
parents:
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674 self.embedding.add_half_edge_first(w, v) |
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parents:
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675 self.left_ref[v] = w |
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676 self.right_ref[v] = w |
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677 self.dfs_embedding_recursive(w) |
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678 else: # back edge |
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679 if self.side[ei] == 1: |
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680 # place v directly after right_ref[w] in embed. list of w |
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681 self.embedding.add_half_edge_cw(w, v, self.right_ref[w]) |
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682 else: |
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683 # place v directly before left_ref[w] in embed. list of w |
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684 self.embedding.add_half_edge_ccw(w, v, self.left_ref[w]) |
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685 self.left_ref[w] = v |
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686 |
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687 def sign(self, e): |
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688 """Resolve the relative side of an edge to the absolute side.""" |
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689 # the recursion stack |
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690 dfs_stack = [e] |
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691 # dict to remember reference edges |
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692 old_ref = defaultdict(lambda: None) |
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693 |
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694 while dfs_stack: |
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695 e = dfs_stack.pop() |
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696 |
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697 if self.ref[e] is not None: |
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698 dfs_stack.append(e) # revisit e after finishing self.ref[e] |
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699 dfs_stack.append(self.ref[e]) # visit self.ref[e] next |
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700 old_ref[e] = self.ref[e] # remember value of self.ref[e] |
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701 self.ref[e] = None |
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702 else: |
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703 self.side[e] *= self.side[old_ref[e]] |
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704 |
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705 return self.side[e] |
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706 |
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707 def sign_recursive(self, e): |
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708 """Recursive version of :meth:`sign`.""" |
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709 if self.ref[e] is not None: |
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710 self.side[e] = self.side[e] * self.sign_recursive(self.ref[e]) |
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711 self.ref[e] = None |
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712 return self.side[e] |
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713 |
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714 |
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715 class PlanarEmbedding(nx.DiGraph): |
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716 """Represents a planar graph with its planar embedding. |
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717 |
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718 The planar embedding is given by a `combinatorial embedding |
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719 <https://en.wikipedia.org/wiki/Graph_embedding#Combinatorial_embedding>`_. |
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720 |
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721 **Neighbor ordering:** |
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722 |
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723 In comparison to a usual graph structure, the embedding also stores the |
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724 order of all neighbors for every vertex. |
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725 The order of the neighbors can be given in clockwise (cw) direction or |
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726 counterclockwise (ccw) direction. This order is stored as edge attributes |
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727 in the underlying directed graph. For the edge (u, v) the edge attribute |
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728 'cw' is set to the neighbor of u that follows immediately after v in |
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729 clockwise direction. |
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730 |
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731 In order for a PlanarEmbedding to be valid it must fulfill multiple |
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732 conditions. It is possible to check if these conditions are fulfilled with |
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733 the method :meth:`check_structure`. |
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734 The conditions are: |
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735 |
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736 * Edges must go in both directions (because the edge attributes differ) |
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737 * Every edge must have a 'cw' and 'ccw' attribute which corresponds to a |
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738 correct planar embedding. |
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739 * A node with non zero degree must have a node attribute 'first_nbr'. |
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740 |
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741 As long as a PlanarEmbedding is invalid only the following methods should |
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742 be called: |
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743 |
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744 * :meth:`add_half_edge_ccw` |
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745 * :meth:`add_half_edge_cw` |
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746 * :meth:`connect_components` |
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747 * :meth:`add_half_edge_first` |
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748 |
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749 Even though the graph is a subclass of nx.DiGraph, it can still be used |
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750 for algorithms that require undirected graphs, because the method |
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751 :meth:`is_directed` is overridden. This is possible, because a valid |
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752 PlanarGraph must have edges in both directions. |
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753 |
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754 **Half edges:** |
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755 |
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756 In methods like `add_half_edge_ccw` the term "half-edge" is used, which is |
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757 a term that is used in `doubly connected edge lists |
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758 <https://en.wikipedia.org/wiki/Doubly_connected_edge_list>`_. It is used |
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759 to emphasize that the edge is only in one direction and there exists |
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760 another half-edge in the opposite direction. |
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761 While conventional edges always have two faces (including outer face) next |
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762 to them, it is possible to assign each half-edge *exactly one* face. |
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763 For a half-edge (u, v) that is orientated such that u is below v then the |
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764 face that belongs to (u, v) is to the right of this half-edge. |
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765 |
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766 Examples |
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767 -------- |
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768 |
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769 Create an embedding of a star graph (compare `nx.star_graph(3)`): |
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770 |
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771 >>> G = nx.PlanarEmbedding() |
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772 >>> G.add_half_edge_cw(0, 1, None) |
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773 >>> G.add_half_edge_cw(0, 2, 1) |
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774 >>> G.add_half_edge_cw(0, 3, 2) |
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775 >>> G.add_half_edge_cw(1, 0, None) |
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776 >>> G.add_half_edge_cw(2, 0, None) |
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777 >>> G.add_half_edge_cw(3, 0, None) |
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778 |
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779 Alternatively the same embedding can also be defined in counterclockwise |
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780 orientation. The following results in exactly the same PlanarEmbedding: |
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781 |
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782 >>> G = nx.PlanarEmbedding() |
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783 >>> G.add_half_edge_ccw(0, 1, None) |
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784 >>> G.add_half_edge_ccw(0, 3, 1) |
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785 >>> G.add_half_edge_ccw(0, 2, 3) |
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786 >>> G.add_half_edge_ccw(1, 0, None) |
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787 >>> G.add_half_edge_ccw(2, 0, None) |
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788 >>> G.add_half_edge_ccw(3, 0, None) |
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789 |
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790 After creating a graph, it is possible to validate that the PlanarEmbedding |
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791 object is correct: |
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792 |
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793 >>> G.check_structure() |
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794 |
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795 """ |
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796 |
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797 def get_data(self): |
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798 """Converts the adjacency structure into a better readable structure. |
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799 |
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800 Returns |
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801 ------- |
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802 embedding : dict |
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803 A dict mapping all nodes to a list of neighbors sorted in |
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804 clockwise order. |
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805 |
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806 See Also |
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807 -------- |
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808 set_data |
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809 |
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810 """ |
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811 embedding = dict() |
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812 for v in self: |
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813 embedding[v] = list(self.neighbors_cw_order(v)) |
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814 return embedding |
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815 |
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816 def set_data(self, data): |
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817 """Inserts edges according to given sorted neighbor list. |
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818 |
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819 The input format is the same as the output format of get_data(). |
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820 |
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821 Parameters |
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822 ---------- |
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823 data : dict |
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824 A dict mapping all nodes to a list of neighbors sorted in |
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825 clockwise order. |
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826 |
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827 See Also |
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828 -------- |
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829 get_data |
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830 |
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831 """ |
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832 for v in data: |
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833 for w in reversed(data[v]): |
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834 self.add_half_edge_first(v, w) |
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835 |
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836 def neighbors_cw_order(self, v): |
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837 """Generator for the neighbors of v in clockwise order. |
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838 |
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839 Parameters |
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840 ---------- |
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841 v : node |
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842 |
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843 Yields |
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844 ------ |
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845 node |
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846 |
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847 """ |
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848 if len(self[v]) == 0: |
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849 # v has no neighbors |
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850 return |
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851 start_node = self.nodes[v]["first_nbr"] |
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852 yield start_node |
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853 current_node = self[v][start_node]["cw"] |
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854 while start_node != current_node: |
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855 yield current_node |
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856 current_node = self[v][current_node]["cw"] |
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857 |
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858 def check_structure(self): |
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859 """Runs without exceptions if this object is valid. |
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860 |
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861 Checks that the following properties are fulfilled: |
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862 |
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863 * Edges go in both directions (because the edge attributes differ). |
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864 * Every edge has a 'cw' and 'ccw' attribute which corresponds to a |
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865 correct planar embedding. |
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866 * A node with a degree larger than 0 has a node attribute 'first_nbr'. |
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867 |
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868 Running this method verifies that the underlying Graph must be planar. |
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869 |
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870 Raises |
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871 ------ |
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872 NetworkXException |
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873 This exception is raised with a short explanation if the |
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874 PlanarEmbedding is invalid. |
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875 """ |
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876 # Check fundamental structure |
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877 for v in self: |
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878 try: |
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879 sorted_nbrs = set(self.neighbors_cw_order(v)) |
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880 except KeyError as e: |
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881 msg = f"Bad embedding. Missing orientation for a neighbor of {v}" |
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882 raise nx.NetworkXException(msg) from e |
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883 |
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884 unsorted_nbrs = set(self[v]) |
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885 if sorted_nbrs != unsorted_nbrs: |
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886 msg = "Bad embedding. Edge orientations not set correctly." |
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887 raise nx.NetworkXException(msg) |
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888 for w in self[v]: |
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889 # Check if opposite half-edge exists |
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890 if not self.has_edge(w, v): |
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891 msg = "Bad embedding. Opposite half-edge is missing." |
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892 raise nx.NetworkXException(msg) |
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893 |
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894 # Check planarity |
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895 counted_half_edges = set() |
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896 for component in nx.connected_components(self): |
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897 if len(component) == 1: |
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898 # Don't need to check single node component |
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899 continue |
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900 num_nodes = len(component) |
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901 num_half_edges = 0 |
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902 num_faces = 0 |
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903 for v in component: |
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904 for w in self.neighbors_cw_order(v): |
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905 num_half_edges += 1 |
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906 if (v, w) not in counted_half_edges: |
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907 # We encountered a new face |
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908 num_faces += 1 |
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909 # Mark all half-edges belonging to this face |
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910 self.traverse_face(v, w, counted_half_edges) |
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911 num_edges = num_half_edges // 2 # num_half_edges is even |
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912 if num_nodes - num_edges + num_faces != 2: |
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913 # The result does not match Euler's formula |
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914 msg = "Bad embedding. The graph does not match Euler's formula" |
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915 raise nx.NetworkXException(msg) |
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916 |
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917 def add_half_edge_ccw(self, start_node, end_node, reference_neighbor): |
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918 """Adds a half-edge from start_node to end_node. |
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919 |
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920 The half-edge is added counter clockwise next to the existing half-edge |
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921 (start_node, reference_neighbor). |
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922 |
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923 Parameters |
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924 ---------- |
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925 start_node : node |
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926 Start node of inserted edge. |
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927 end_node : node |
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928 End node of inserted edge. |
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929 reference_neighbor: node |
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930 End node of reference edge. |
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931 |
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932 Raises |
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933 ------ |
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934 NetworkXException |
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935 If the reference_neighbor does not exist. |
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936 |
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937 See Also |
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938 -------- |
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939 add_half_edge_cw |
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940 connect_components |
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941 add_half_edge_first |
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942 |
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943 """ |
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944 if reference_neighbor is None: |
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945 # The start node has no neighbors |
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946 self.add_edge(start_node, end_node) # Add edge to graph |
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947 self[start_node][end_node]["cw"] = end_node |
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948 self[start_node][end_node]["ccw"] = end_node |
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949 self.nodes[start_node]["first_nbr"] = end_node |
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950 else: |
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951 ccw_reference = self[start_node][reference_neighbor]["ccw"] |
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952 self.add_half_edge_cw(start_node, end_node, ccw_reference) |
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953 |
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954 if reference_neighbor == self.nodes[start_node].get("first_nbr", None): |
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955 # Update first neighbor |
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956 self.nodes[start_node]["first_nbr"] = end_node |
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957 |
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958 def add_half_edge_cw(self, start_node, end_node, reference_neighbor): |
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959 """Adds a half-edge from start_node to end_node. |
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960 |
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961 The half-edge is added clockwise next to the existing half-edge |
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962 (start_node, reference_neighbor). |
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963 |
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964 Parameters |
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965 ---------- |
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966 start_node : node |
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967 Start node of inserted edge. |
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968 end_node : node |
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969 End node of inserted edge. |
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970 reference_neighbor: node |
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971 End node of reference edge. |
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972 |
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973 Raises |
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974 ------ |
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975 NetworkXException |
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976 If the reference_neighbor does not exist. |
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977 |
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978 See Also |
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979 -------- |
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980 add_half_edge_ccw |
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981 connect_components |
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982 add_half_edge_first |
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983 """ |
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984 self.add_edge(start_node, end_node) # Add edge to graph |
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985 |
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986 if reference_neighbor is None: |
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987 # The start node has no neighbors |
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988 self[start_node][end_node]["cw"] = end_node |
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989 self[start_node][end_node]["ccw"] = end_node |
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990 self.nodes[start_node]["first_nbr"] = end_node |
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991 return |
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992 |
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993 if reference_neighbor not in self[start_node]: |
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994 raise nx.NetworkXException( |
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995 "Cannot add edge. Reference neighbor does not exist" |
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996 ) |
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997 |
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998 # Get half-edge at the other side |
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999 cw_reference = self[start_node][reference_neighbor]["cw"] |
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1000 # Alter half-edge data structures |
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1001 self[start_node][reference_neighbor]["cw"] = end_node |
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1002 self[start_node][end_node]["cw"] = cw_reference |
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1003 self[start_node][cw_reference]["ccw"] = end_node |
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1004 self[start_node][end_node]["ccw"] = reference_neighbor |
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1005 |
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1006 def connect_components(self, v, w): |
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1007 """Adds half-edges for (v, w) and (w, v) at some position. |
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1008 |
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1009 This method should only be called if v and w are in different |
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1010 components, or it might break the embedding. |
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1011 This especially means that if `connect_components(v, w)` |
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1012 is called it is not allowed to call `connect_components(w, v)` |
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1013 afterwards. The neighbor orientations in both directions are |
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1014 all set correctly after the first call. |
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1015 |
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1016 Parameters |
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1017 ---------- |
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1018 v : node |
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1019 w : node |
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1020 |
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1021 See Also |
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1022 -------- |
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1023 add_half_edge_ccw |
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1024 add_half_edge_cw |
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1025 add_half_edge_first |
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1026 """ |
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1027 self.add_half_edge_first(v, w) |
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1028 self.add_half_edge_first(w, v) |
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1029 |
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1030 def add_half_edge_first(self, start_node, end_node): |
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1031 """The added half-edge is inserted at the first position in the order. |
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1032 |
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1033 Parameters |
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1034 ---------- |
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1035 start_node : node |
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1036 end_node : node |
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1037 |
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1038 See Also |
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1039 -------- |
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1040 add_half_edge_ccw |
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1041 add_half_edge_cw |
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1042 connect_components |
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1043 """ |
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1044 if start_node in self and "first_nbr" in self.nodes[start_node]: |
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1045 reference = self.nodes[start_node]["first_nbr"] |
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1046 else: |
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1047 reference = None |
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1048 self.add_half_edge_ccw(start_node, end_node, reference) |
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1049 |
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1050 def next_face_half_edge(self, v, w): |
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1051 """Returns the following half-edge left of a face. |
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1052 |
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1053 Parameters |
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1054 ---------- |
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1055 v : node |
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1056 w : node |
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1057 |
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1058 Returns |
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1059 ------- |
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1060 half-edge : tuple |
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1061 """ |
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1062 new_node = self[w][v]["ccw"] |
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1063 return w, new_node |
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1064 |
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1065 def traverse_face(self, v, w, mark_half_edges=None): |
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1066 """Returns nodes on the face that belong to the half-edge (v, w). |
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1067 |
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1068 The face that is traversed lies to the right of the half-edge (in an |
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1069 orientation where v is below w). |
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1070 |
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1071 Optionally it is possible to pass a set to which all encountered half |
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1072 edges are added. Before calling this method, this set must not include |
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1073 any half-edges that belong to the face. |
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1074 |
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1075 Parameters |
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1076 ---------- |
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1077 v : node |
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1078 Start node of half-edge. |
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1079 w : node |
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1080 End node of half-edge. |
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1081 mark_half_edges: set, optional |
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1082 Set to which all encountered half-edges are added. |
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1083 |
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1084 Returns |
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1085 ------- |
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1086 face : list |
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1087 A list of nodes that lie on this face. |
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1088 """ |
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1089 if mark_half_edges is None: |
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1090 mark_half_edges = set() |
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1091 |
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1092 face_nodes = [v] |
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1093 mark_half_edges.add((v, w)) |
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1094 prev_node = v |
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1095 cur_node = w |
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1096 # Last half-edge is (incoming_node, v) |
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1097 incoming_node = self[v][w]["cw"] |
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1098 |
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1099 while cur_node != v or prev_node != incoming_node: |
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1100 face_nodes.append(cur_node) |
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1101 prev_node, cur_node = self.next_face_half_edge(prev_node, cur_node) |
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1102 if (prev_node, cur_node) in mark_half_edges: |
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1103 raise nx.NetworkXException("Bad planar embedding. Impossible face.") |
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1104 mark_half_edges.add((prev_node, cur_node)) |
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1105 |
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1106 return face_nodes |
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1107 |
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1108 def is_directed(self): |
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1109 """A valid PlanarEmbedding is undirected. |
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1110 |
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1111 All reverse edges are contained, i.e. for every existing |
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1112 half-edge (v, w) the half-edge in the opposite direction (w, v) is also |
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1113 contained. |
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1114 """ |
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1115 return False |