Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/planar_drawing.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 import networkx as nx |
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2 from collections import defaultdict |
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3 |
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4 |
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5 __all__ = ["combinatorial_embedding_to_pos"] |
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6 |
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7 |
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8 def combinatorial_embedding_to_pos(embedding, fully_triangulate=False): |
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9 """Assigns every node a (x, y) position based on the given embedding |
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10 |
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11 The algorithm iteratively inserts nodes of the input graph in a certain |
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12 order and rearranges previously inserted nodes so that the planar drawing |
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13 stays valid. This is done efficiently by only maintaining relative |
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14 positions during the node placements and calculating the absolute positions |
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15 at the end. For more information see [1]_. |
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16 |
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17 Parameters |
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18 ---------- |
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19 embedding : nx.PlanarEmbedding |
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20 This defines the order of the edges |
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21 |
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22 fully_triangulate : bool |
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23 If set to True the algorithm adds edges to a copy of the input |
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24 embedding and makes it chordal. |
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25 |
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26 Returns |
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27 ------- |
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28 pos : dict |
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29 Maps each node to a tuple that defines the (x, y) position |
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30 |
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31 References |
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32 ---------- |
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33 .. [1] M. Chrobak and T.H. Payne: |
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34 A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989 |
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35 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677 |
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36 |
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37 """ |
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38 if len(embedding.nodes()) < 4: |
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39 # Position the node in any triangle |
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40 default_positions = [(0, 0), (2, 0), (1, 1)] |
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41 pos = {} |
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42 for i, v in enumerate(embedding.nodes()): |
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43 pos[v] = default_positions[i] |
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44 return pos |
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45 |
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46 embedding, outer_face = triangulate_embedding(embedding, fully_triangulate) |
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47 |
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48 # The following dicts map a node to another node |
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49 # If a node is not in the key set it means that the node is not yet in G_k |
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50 # If a node maps to None then the corresponding subtree does not exist |
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51 left_t_child = {} |
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52 right_t_child = {} |
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53 |
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54 # The following dicts map a node to an integer |
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55 delta_x = {} |
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56 y_coordinate = {} |
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57 |
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58 node_list = get_canonical_ordering(embedding, outer_face) |
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59 |
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60 # 1. Phase: Compute relative positions |
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61 |
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62 # Initialization |
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63 v1, v2, v3 = node_list[0][0], node_list[1][0], node_list[2][0] |
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64 |
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65 delta_x[v1] = 0 |
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66 y_coordinate[v1] = 0 |
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67 right_t_child[v1] = v3 |
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68 left_t_child[v1] = None |
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69 |
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70 delta_x[v2] = 1 |
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71 y_coordinate[v2] = 0 |
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72 right_t_child[v2] = None |
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73 left_t_child[v2] = None |
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74 |
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75 delta_x[v3] = 1 |
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76 y_coordinate[v3] = 1 |
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77 right_t_child[v3] = v2 |
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78 left_t_child[v3] = None |
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79 |
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80 for k in range(3, len(node_list)): |
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81 vk, contour_neighbors = node_list[k] |
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82 wp = contour_neighbors[0] |
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83 wp1 = contour_neighbors[1] |
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84 wq = contour_neighbors[-1] |
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85 wq1 = contour_neighbors[-2] |
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86 adds_mult_tri = len(contour_neighbors) > 2 |
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87 |
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88 # Stretch gaps: |
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89 delta_x[wp1] += 1 |
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90 delta_x[wq] += 1 |
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91 |
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92 delta_x_wp_wq = sum(delta_x[x] for x in contour_neighbors[1:]) |
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93 |
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94 # Adjust offsets |
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95 delta_x[vk] = (-y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2 |
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96 y_coordinate[vk] = (y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2 |
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97 delta_x[wq] = delta_x_wp_wq - delta_x[vk] |
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98 if adds_mult_tri: |
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99 delta_x[wp1] -= delta_x[vk] |
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100 |
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101 # Install v_k: |
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102 right_t_child[wp] = vk |
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103 right_t_child[vk] = wq |
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104 if adds_mult_tri: |
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105 left_t_child[vk] = wp1 |
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106 right_t_child[wq1] = None |
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107 else: |
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108 left_t_child[vk] = None |
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109 |
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110 # 2. Phase: Set absolute positions |
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111 pos = dict() |
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112 pos[v1] = (0, y_coordinate[v1]) |
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113 remaining_nodes = [v1] |
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114 while remaining_nodes: |
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115 parent_node = remaining_nodes.pop() |
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116 |
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117 # Calculate position for left child |
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118 set_position( |
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119 parent_node, left_t_child, remaining_nodes, delta_x, y_coordinate, pos |
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120 ) |
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121 # Calculate position for right child |
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122 set_position( |
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123 parent_node, right_t_child, remaining_nodes, delta_x, y_coordinate, pos |
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124 ) |
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125 return pos |
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126 |
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127 |
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128 def set_position(parent, tree, remaining_nodes, delta_x, y_coordinate, pos): |
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129 """Helper method to calculate the absolute position of nodes.""" |
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130 child = tree[parent] |
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131 parent_node_x = pos[parent][0] |
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132 if child is not None: |
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133 # Calculate pos of child |
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134 child_x = parent_node_x + delta_x[child] |
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135 pos[child] = (child_x, y_coordinate[child]) |
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136 # Remember to calculate pos of its children |
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137 remaining_nodes.append(child) |
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138 |
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139 |
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140 def get_canonical_ordering(embedding, outer_face): |
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141 """Returns a canonical ordering of the nodes |
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142 |
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143 The canonical ordering of nodes (v1, ..., vn) must fulfill the following |
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144 conditions: |
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145 (See Lemma 1 in [2]_) |
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146 |
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147 - For the subgraph G_k of the input graph induced by v1, ..., vk it holds: |
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148 - 2-connected |
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149 - internally triangulated |
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150 - the edge (v1, v2) is part of the outer face |
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151 - For a node v(k+1) the following holds: |
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152 - The node v(k+1) is part of the outer face of G_k |
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153 - It has at least two neighbors in G_k |
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154 - All neighbors of v(k+1) in G_k lie consecutively on the outer face of |
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155 G_k (excluding the edge (v1, v2)). |
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156 |
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157 The algorithm used here starts with G_n (containing all nodes). It first |
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158 selects the nodes v1 and v2. And then tries to find the order of the other |
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159 nodes by checking which node can be removed in order to fulfill the |
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160 conditions mentioned above. This is done by calculating the number of |
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161 chords of nodes on the outer face. For more information see [1]_. |
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162 |
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163 Parameters |
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164 ---------- |
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165 embedding : nx.PlanarEmbedding |
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166 The embedding must be triangulated |
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167 outer_face : list |
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168 The nodes on the outer face of the graph |
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169 |
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170 Returns |
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171 ------- |
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172 ordering : list |
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173 A list of tuples `(vk, wp_wq)`. Here `vk` is the node at this position |
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174 in the canonical ordering. The element `wp_wq` is a list of nodes that |
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175 make up the outer face of G_k. |
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176 |
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177 References |
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178 ---------- |
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179 .. [1] Steven Chaplick. |
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180 Canonical Orders of Planar Graphs and (some of) Their Applications 2015 |
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181 https://wuecampus2.uni-wuerzburg.de/moodle/pluginfile.php/545727/mod_resource/content/0/vg-ss15-vl03-canonical-orders-druckversion.pdf |
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182 .. [2] M. Chrobak and T.H. Payne: |
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183 A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989 |
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184 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677 |
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185 |
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186 """ |
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187 v1 = outer_face[0] |
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188 v2 = outer_face[1] |
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189 chords = defaultdict(int) # Maps nodes to the number of their chords |
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190 marked_nodes = set() |
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191 ready_to_pick = set(outer_face) |
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192 |
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193 # Initialize outer_face_ccw_nbr (do not include v1 -> v2) |
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194 outer_face_ccw_nbr = {} |
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195 prev_nbr = v2 |
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196 for idx in range(2, len(outer_face)): |
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197 outer_face_ccw_nbr[prev_nbr] = outer_face[idx] |
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198 prev_nbr = outer_face[idx] |
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199 outer_face_ccw_nbr[prev_nbr] = v1 |
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200 |
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201 # Initialize outer_face_cw_nbr (do not include v2 -> v1) |
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202 outer_face_cw_nbr = {} |
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203 prev_nbr = v1 |
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204 for idx in range(len(outer_face) - 1, 0, -1): |
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205 outer_face_cw_nbr[prev_nbr] = outer_face[idx] |
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206 prev_nbr = outer_face[idx] |
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207 |
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208 def is_outer_face_nbr(x, y): |
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209 if x not in outer_face_ccw_nbr: |
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210 return outer_face_cw_nbr[x] == y |
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211 if x not in outer_face_cw_nbr: |
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212 return outer_face_ccw_nbr[x] == y |
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213 return outer_face_ccw_nbr[x] == y or outer_face_cw_nbr[x] == y |
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214 |
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215 def is_on_outer_face(x): |
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216 return x not in marked_nodes and (x in outer_face_ccw_nbr.keys() or x == v1) |
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217 |
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218 # Initialize number of chords |
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219 for v in outer_face: |
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220 for nbr in embedding.neighbors_cw_order(v): |
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221 if is_on_outer_face(nbr) and not is_outer_face_nbr(v, nbr): |
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222 chords[v] += 1 |
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223 ready_to_pick.discard(v) |
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224 |
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225 # Initialize canonical_ordering |
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226 canonical_ordering = [None] * len(embedding.nodes()) # type: list |
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227 canonical_ordering[0] = (v1, []) |
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228 canonical_ordering[1] = (v2, []) |
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229 ready_to_pick.discard(v1) |
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230 ready_to_pick.discard(v2) |
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231 |
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232 for k in range(len(embedding.nodes()) - 1, 1, -1): |
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233 # 1. Pick v from ready_to_pick |
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234 v = ready_to_pick.pop() |
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235 marked_nodes.add(v) |
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236 |
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237 # v has exactly two neighbors on the outer face (wp and wq) |
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238 wp = None |
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239 wq = None |
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240 # Iterate over neighbors of v to find wp and wq |
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241 nbr_iterator = iter(embedding.neighbors_cw_order(v)) |
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242 while True: |
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243 nbr = next(nbr_iterator) |
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244 if nbr in marked_nodes: |
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245 # Only consider nodes that are not yet removed |
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246 continue |
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247 if is_on_outer_face(nbr): |
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248 # nbr is either wp or wq |
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249 if nbr == v1: |
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250 wp = v1 |
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251 elif nbr == v2: |
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252 wq = v2 |
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253 else: |
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254 if outer_face_cw_nbr[nbr] == v: |
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255 # nbr is wp |
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256 wp = nbr |
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257 else: |
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258 # nbr is wq |
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259 wq = nbr |
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260 if wp is not None and wq is not None: |
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261 # We don't need to iterate any further |
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262 break |
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263 |
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264 # Obtain new nodes on outer face (neighbors of v from wp to wq) |
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265 wp_wq = [wp] |
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266 nbr = wp |
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267 while nbr != wq: |
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268 # Get next next neighbor (clockwise on the outer face) |
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269 next_nbr = embedding[v][nbr]["ccw"] |
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270 wp_wq.append(next_nbr) |
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271 # Update outer face |
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272 outer_face_cw_nbr[nbr] = next_nbr |
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273 outer_face_ccw_nbr[next_nbr] = nbr |
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274 # Move to next neighbor of v |
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275 nbr = next_nbr |
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276 |
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277 if len(wp_wq) == 2: |
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278 # There was a chord between wp and wq, decrease number of chords |
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279 chords[wp] -= 1 |
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280 if chords[wp] == 0: |
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281 ready_to_pick.add(wp) |
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282 chords[wq] -= 1 |
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283 if chords[wq] == 0: |
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284 ready_to_pick.add(wq) |
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285 else: |
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286 # Update all chords involving w_(p+1) to w_(q-1) |
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287 new_face_nodes = set(wp_wq[1:-1]) |
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288 for w in new_face_nodes: |
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289 # If we do not find a chord for w later we can pick it next |
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290 ready_to_pick.add(w) |
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291 for nbr in embedding.neighbors_cw_order(w): |
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292 if is_on_outer_face(nbr) and not is_outer_face_nbr(w, nbr): |
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293 # There is a chord involving w |
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294 chords[w] += 1 |
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295 ready_to_pick.discard(w) |
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296 if nbr not in new_face_nodes: |
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297 # Also increase chord for the neighbor |
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298 # We only iterator over new_face_nodes |
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299 chords[nbr] += 1 |
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300 ready_to_pick.discard(nbr) |
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301 # Set the canonical ordering node and the list of contour neighbors |
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302 canonical_ordering[k] = (v, wp_wq) |
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303 |
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304 return canonical_ordering |
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305 |
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306 |
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307 def triangulate_face(embedding, v1, v2): |
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308 """Triangulates the face given by half edge (v, w) |
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309 |
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310 Parameters |
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311 ---------- |
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312 embedding : nx.PlanarEmbedding |
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313 v1 : node |
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314 The half-edge (v1, v2) belongs to the face that gets triangulated |
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315 v2 : node |
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316 """ |
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317 _, v3 = embedding.next_face_half_edge(v1, v2) |
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318 _, v4 = embedding.next_face_half_edge(v2, v3) |
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319 if v1 == v2 or v1 == v3: |
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320 # The component has less than 3 nodes |
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321 return |
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322 while v1 != v4: |
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323 # Add edge if not already present on other side |
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324 if embedding.has_edge(v1, v3): |
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325 # Cannot triangulate at this position |
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326 v1, v2, v3 = v2, v3, v4 |
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327 else: |
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328 # Add edge for triangulation |
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329 embedding.add_half_edge_cw(v1, v3, v2) |
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330 embedding.add_half_edge_ccw(v3, v1, v2) |
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331 v1, v2, v3 = v1, v3, v4 |
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332 # Get next node |
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333 _, v4 = embedding.next_face_half_edge(v2, v3) |
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334 |
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335 |
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336 def triangulate_embedding(embedding, fully_triangulate=True): |
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337 """Triangulates the embedding. |
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338 |
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339 Traverses faces of the embedding and adds edges to a copy of the |
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340 embedding to triangulate it. |
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341 The method also ensures that the resulting graph is 2-connected by adding |
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342 edges if the same vertex is contained twice on a path around a face. |
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343 |
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344 Parameters |
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345 ---------- |
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346 embedding : nx.PlanarEmbedding |
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347 The input graph must contain at least 3 nodes. |
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348 |
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349 fully_triangulate : bool |
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350 If set to False the face with the most nodes is chooses as outer face. |
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351 This outer face does not get triangulated. |
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352 |
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353 Returns |
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354 ------- |
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355 (embedding, outer_face) : (nx.PlanarEmbedding, list) tuple |
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356 The element `embedding` is a new embedding containing all edges from |
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357 the input embedding and the additional edges to triangulate the graph. |
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358 The element `outer_face` is a list of nodes that lie on the outer face. |
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359 If the graph is fully triangulated these are three arbitrary connected |
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360 nodes. |
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361 |
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362 """ |
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363 if len(embedding.nodes) <= 1: |
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364 return embedding, list(embedding.nodes) |
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365 embedding = nx.PlanarEmbedding(embedding) |
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366 |
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367 # Get a list with a node for each connected component |
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368 component_nodes = [next(iter(x)) for x in nx.connected_components(embedding)] |
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369 |
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370 # 1. Make graph a single component (add edge between components) |
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371 for i in range(len(component_nodes) - 1): |
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372 v1 = component_nodes[i] |
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373 v2 = component_nodes[i + 1] |
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374 embedding.connect_components(v1, v2) |
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375 |
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376 # 2. Calculate faces, ensure 2-connectedness and determine outer face |
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377 outer_face = [] # A face with the most number of nodes |
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378 face_list = [] |
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379 edges_visited = set() # Used to keep track of already visited faces |
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380 for v in embedding.nodes(): |
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381 for w in embedding.neighbors_cw_order(v): |
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382 new_face = make_bi_connected(embedding, v, w, edges_visited) |
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383 if new_face: |
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384 # Found a new face |
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385 face_list.append(new_face) |
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386 if len(new_face) > len(outer_face): |
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387 # The face is a candidate to be the outer face |
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388 outer_face = new_face |
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389 |
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390 # 3. Triangulate (internal) faces |
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391 for face in face_list: |
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392 if face is not outer_face or fully_triangulate: |
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393 # Triangulate this face |
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394 triangulate_face(embedding, face[0], face[1]) |
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395 |
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396 if fully_triangulate: |
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397 v1 = outer_face[0] |
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398 v2 = outer_face[1] |
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399 v3 = embedding[v2][v1]["ccw"] |
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400 outer_face = [v1, v2, v3] |
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401 |
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402 return embedding, outer_face |
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403 |
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404 |
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405 def make_bi_connected(embedding, starting_node, outgoing_node, edges_counted): |
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406 """Triangulate a face and make it 2-connected |
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407 |
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408 This method also adds all edges on the face to `edges_counted`. |
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409 |
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410 Parameters |
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411 ---------- |
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412 embedding: nx.PlanarEmbedding |
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413 The embedding that defines the faces |
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414 starting_node : node |
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415 A node on the face |
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416 outgoing_node : node |
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417 A node such that the half edge (starting_node, outgoing_node) belongs |
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418 to the face |
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419 edges_counted: set |
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420 Set of all half-edges that belong to a face that have been visited |
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421 |
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422 Returns |
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423 ------- |
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424 face_nodes: list |
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425 A list of all nodes at the border of this face |
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426 """ |
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427 |
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428 # Check if the face has already been calculated |
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429 if (starting_node, outgoing_node) in edges_counted: |
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430 # This face was already counted |
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431 return [] |
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432 edges_counted.add((starting_node, outgoing_node)) |
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433 |
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434 # Add all edges to edges_counted which have this face to their left |
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435 v1 = starting_node |
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436 v2 = outgoing_node |
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437 face_list = [starting_node] # List of nodes around the face |
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438 face_set = set(face_list) # Set for faster queries |
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439 _, v3 = embedding.next_face_half_edge(v1, v2) |
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440 |
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441 # Move the nodes v1, v2, v3 around the face: |
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442 while v2 != starting_node or v3 != outgoing_node: |
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443 if v1 == v2: |
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444 raise nx.NetworkXException("Invalid half-edge") |
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445 # cycle is not completed yet |
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446 if v2 in face_set: |
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447 # v2 encountered twice: Add edge to ensure 2-connectedness |
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448 embedding.add_half_edge_cw(v1, v3, v2) |
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449 embedding.add_half_edge_ccw(v3, v1, v2) |
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450 edges_counted.add((v2, v3)) |
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451 edges_counted.add((v3, v1)) |
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452 v2 = v1 |
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453 else: |
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454 face_set.add(v2) |
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455 face_list.append(v2) |
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456 |
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457 # set next edge |
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458 v1 = v2 |
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459 v2, v3 = embedding.next_face_half_edge(v2, v3) |
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460 |
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461 # remember that this edge has been counted |
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462 edges_counted.add((v1, v2)) |
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463 |
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464 return face_list |