Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/euler.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Eulerian circuits and graphs. |
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3 """ |
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4 from itertools import combinations |
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5 |
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6 import networkx as nx |
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7 from ..utils import arbitrary_element, not_implemented_for |
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8 |
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9 __all__ = [ |
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10 "is_eulerian", |
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11 "eulerian_circuit", |
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12 "eulerize", |
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13 "is_semieulerian", |
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14 "has_eulerian_path", |
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15 "eulerian_path", |
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16 ] |
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17 |
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18 |
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19 def is_eulerian(G): |
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20 """Returns True if and only if `G` is Eulerian. |
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21 |
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22 A graph is *Eulerian* if it has an Eulerian circuit. An *Eulerian |
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23 circuit* is a closed walk that includes each edge of a graph exactly |
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24 once. |
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25 |
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26 Parameters |
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27 ---------- |
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28 G : NetworkX graph |
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29 A graph, either directed or undirected. |
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30 |
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31 Examples |
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32 -------- |
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33 >>> nx.is_eulerian(nx.DiGraph({0: [3], 1: [2], 2: [3], 3: [0, 1]})) |
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34 True |
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35 >>> nx.is_eulerian(nx.complete_graph(5)) |
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36 True |
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37 >>> nx.is_eulerian(nx.petersen_graph()) |
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38 False |
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39 |
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40 Notes |
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41 ----- |
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42 If the graph is not connected (or not strongly connected, for |
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43 directed graphs), this function returns False. |
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44 |
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45 """ |
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46 if G.is_directed(): |
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47 # Every node must have equal in degree and out degree and the |
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48 # graph must be strongly connected |
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49 return all( |
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50 G.in_degree(n) == G.out_degree(n) for n in G |
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51 ) and nx.is_strongly_connected(G) |
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52 # An undirected Eulerian graph has no vertices of odd degree and |
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53 # must be connected. |
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54 return all(d % 2 == 0 for v, d in G.degree()) and nx.is_connected(G) |
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55 |
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56 |
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57 def is_semieulerian(G): |
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58 """Return True iff `G` is semi-Eulerian. |
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59 |
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60 G is semi-Eulerian if it has an Eulerian path but no Eulerian circuit. |
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61 """ |
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62 return has_eulerian_path(G) and not is_eulerian(G) |
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63 |
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64 |
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65 def _find_path_start(G): |
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66 """Return a suitable starting vertex for an Eulerian path. |
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67 |
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68 If no path exists, return None. |
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69 """ |
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70 if not has_eulerian_path(G): |
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71 return None |
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72 |
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73 if is_eulerian(G): |
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74 return arbitrary_element(G) |
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75 |
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76 if G.is_directed(): |
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77 v1, v2 = [v for v in G if G.in_degree(v) != G.out_degree(v)] |
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78 # Determines which is the 'start' node (as opposed to the 'end') |
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79 if G.out_degree(v1) > G.in_degree(v1): |
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80 return v1 |
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81 else: |
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82 return v2 |
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83 |
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84 else: |
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85 # In an undirected graph randomly choose one of the possibilities |
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86 start = [v for v in G if G.degree(v) % 2 != 0][0] |
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87 return start |
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88 |
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89 |
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90 def _simplegraph_eulerian_circuit(G, source): |
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91 if G.is_directed(): |
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92 degree = G.out_degree |
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93 edges = G.out_edges |
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94 else: |
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95 degree = G.degree |
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96 edges = G.edges |
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97 vertex_stack = [source] |
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98 last_vertex = None |
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99 while vertex_stack: |
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100 current_vertex = vertex_stack[-1] |
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101 if degree(current_vertex) == 0: |
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102 if last_vertex is not None: |
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103 yield (last_vertex, current_vertex) |
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104 last_vertex = current_vertex |
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105 vertex_stack.pop() |
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106 else: |
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107 _, next_vertex = arbitrary_element(edges(current_vertex)) |
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108 vertex_stack.append(next_vertex) |
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109 G.remove_edge(current_vertex, next_vertex) |
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110 |
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111 |
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112 def _multigraph_eulerian_circuit(G, source): |
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113 if G.is_directed(): |
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114 degree = G.out_degree |
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115 edges = G.out_edges |
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116 else: |
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117 degree = G.degree |
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118 edges = G.edges |
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119 vertex_stack = [(source, None)] |
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120 last_vertex = None |
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121 last_key = None |
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122 while vertex_stack: |
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123 current_vertex, current_key = vertex_stack[-1] |
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124 if degree(current_vertex) == 0: |
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125 if last_vertex is not None: |
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126 yield (last_vertex, current_vertex, last_key) |
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127 last_vertex, last_key = current_vertex, current_key |
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128 vertex_stack.pop() |
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129 else: |
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130 triple = arbitrary_element(edges(current_vertex, keys=True)) |
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131 _, next_vertex, next_key = triple |
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132 vertex_stack.append((next_vertex, next_key)) |
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133 G.remove_edge(current_vertex, next_vertex, next_key) |
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134 |
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135 |
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136 def eulerian_circuit(G, source=None, keys=False): |
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137 """Returns an iterator over the edges of an Eulerian circuit in `G`. |
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138 |
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139 An *Eulerian circuit* is a closed walk that includes each edge of a |
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140 graph exactly once. |
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141 |
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142 Parameters |
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143 ---------- |
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144 G : NetworkX graph |
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145 A graph, either directed or undirected. |
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146 |
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147 source : node, optional |
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148 Starting node for circuit. |
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149 |
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150 keys : bool |
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151 If False, edges generated by this function will be of the form |
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152 ``(u, v)``. Otherwise, edges will be of the form ``(u, v, k)``. |
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153 This option is ignored unless `G` is a multigraph. |
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154 |
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155 Returns |
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156 ------- |
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157 edges : iterator |
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158 An iterator over edges in the Eulerian circuit. |
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159 |
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160 Raises |
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161 ------ |
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162 NetworkXError |
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163 If the graph is not Eulerian. |
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164 |
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165 See Also |
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166 -------- |
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167 is_eulerian |
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168 |
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169 Notes |
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170 ----- |
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171 This is a linear time implementation of an algorithm adapted from [1]_. |
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172 |
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173 For general information about Euler tours, see [2]_. |
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174 |
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175 References |
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176 ---------- |
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177 .. [1] J. Edmonds, E. L. Johnson. |
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178 Matching, Euler tours and the Chinese postman. |
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179 Mathematical programming, Volume 5, Issue 1 (1973), 111-114. |
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180 .. [2] https://en.wikipedia.org/wiki/Eulerian_path |
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181 |
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182 Examples |
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183 -------- |
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184 To get an Eulerian circuit in an undirected graph:: |
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185 |
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186 >>> G = nx.complete_graph(3) |
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187 >>> list(nx.eulerian_circuit(G)) |
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188 [(0, 2), (2, 1), (1, 0)] |
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189 >>> list(nx.eulerian_circuit(G, source=1)) |
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190 [(1, 2), (2, 0), (0, 1)] |
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191 |
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192 To get the sequence of vertices in an Eulerian circuit:: |
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193 |
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194 >>> [u for u, v in nx.eulerian_circuit(G)] |
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195 [0, 2, 1] |
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196 |
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197 """ |
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198 if not is_eulerian(G): |
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199 raise nx.NetworkXError("G is not Eulerian.") |
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200 if G.is_directed(): |
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201 G = G.reverse() |
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202 else: |
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203 G = G.copy() |
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204 if source is None: |
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205 source = arbitrary_element(G) |
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206 if G.is_multigraph(): |
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207 for u, v, k in _multigraph_eulerian_circuit(G, source): |
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208 if keys: |
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209 yield u, v, k |
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210 else: |
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211 yield u, v |
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212 else: |
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213 yield from _simplegraph_eulerian_circuit(G, source) |
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214 |
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215 |
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216 def has_eulerian_path(G): |
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217 """Return True iff `G` has an Eulerian path. |
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218 |
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219 An Eulerian path is a path in a graph which uses each edge of a graph |
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220 exactly once. |
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221 |
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222 A directed graph has an Eulerian path iff: |
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223 - at most one vertex has out_degree - in_degree = 1, |
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224 - at most one vertex has in_degree - out_degree = 1, |
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225 - every other vertex has equal in_degree and out_degree, |
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226 - and all of its vertices with nonzero degree belong to a |
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227 - single connected component of the underlying undirected graph. |
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228 |
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229 An undirected graph has an Eulerian path iff: |
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230 - exactly zero or two vertices have odd degree, |
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231 - and all of its vertices with nonzero degree belong to a |
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232 - single connected component. |
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233 |
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234 Parameters |
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235 ---------- |
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236 G : NetworkX Graph |
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237 The graph to find an euler path in. |
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238 |
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239 Returns |
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240 ------- |
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241 Bool : True if G has an eulerian path. |
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242 |
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243 See Also |
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244 -------- |
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245 is_eulerian |
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246 eulerian_path |
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247 """ |
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248 if G.is_directed(): |
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249 ins = G.in_degree |
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250 outs = G.out_degree |
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251 semibalanced_ins = sum(ins(v) - outs(v) == 1 for v in G) |
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252 semibalanced_outs = sum(outs(v) - ins(v) == 1 for v in G) |
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253 return ( |
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254 semibalanced_ins <= 1 |
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255 and semibalanced_outs <= 1 |
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256 and sum(G.in_degree(v) != G.out_degree(v) for v in G) <= 2 |
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257 and nx.is_weakly_connected(G) |
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258 ) |
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259 else: |
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260 return sum(d % 2 == 1 for v, d in G.degree()) in (0, 2) and nx.is_connected(G) |
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261 |
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262 |
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263 def eulerian_path(G, source=None, keys=False): |
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264 """Return an iterator over the edges of an Eulerian path in `G`. |
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265 |
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266 Parameters |
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267 ---------- |
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268 G : NetworkX Graph |
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269 The graph in which to look for an eulerian path. |
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270 source : node or None (default: None) |
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271 The node at which to start the search. None means search over all |
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272 starting nodes. |
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273 keys : Bool (default: False) |
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274 Indicates whether to yield edge 3-tuples (u, v, edge_key). |
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275 The default yields edge 2-tuples |
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276 |
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277 Yields |
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278 ------ |
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279 Edge tuples along the eulerian path. |
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280 |
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281 Warning: If `source` provided is not the start node of an Euler path |
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282 will raise error even if an Euler Path exists. |
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283 """ |
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284 if not has_eulerian_path(G): |
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285 raise nx.NetworkXError("Graph has no Eulerian paths.") |
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286 if G.is_directed(): |
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287 G = G.reverse() |
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288 else: |
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289 G = G.copy() |
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290 if source is None: |
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291 source = _find_path_start(G) |
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292 if G.is_multigraph(): |
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293 for u, v, k in _multigraph_eulerian_circuit(G, source): |
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294 if keys: |
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295 yield u, v, k |
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296 else: |
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297 yield u, v |
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298 else: |
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299 yield from _simplegraph_eulerian_circuit(G, source) |
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300 |
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301 |
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302 @not_implemented_for("directed") |
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303 def eulerize(G): |
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304 """Transforms a graph into an Eulerian graph |
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305 |
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306 Parameters |
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307 ---------- |
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308 G : NetworkX graph |
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309 An undirected graph |
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310 |
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311 Returns |
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312 ------- |
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313 G : NetworkX multigraph |
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314 |
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315 Raises |
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316 ------ |
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317 NetworkXError |
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318 If the graph is not connected. |
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319 |
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320 See Also |
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321 -------- |
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322 is_eulerian |
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323 eulerian_circuit |
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324 |
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325 References |
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326 ---------- |
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327 .. [1] J. Edmonds, E. L. Johnson. |
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328 Matching, Euler tours and the Chinese postman. |
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329 Mathematical programming, Volume 5, Issue 1 (1973), 111-114. |
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330 [2] https://en.wikipedia.org/wiki/Eulerian_path |
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331 .. [3] http://web.math.princeton.edu/math_alive/5/Notes1.pdf |
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332 |
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333 Examples |
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334 -------- |
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335 >>> G = nx.complete_graph(10) |
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336 >>> H = nx.eulerize(G) |
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337 >>> nx.is_eulerian(H) |
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338 True |
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339 |
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340 """ |
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341 if G.order() == 0: |
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342 raise nx.NetworkXPointlessConcept("Cannot Eulerize null graph") |
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343 if not nx.is_connected(G): |
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344 raise nx.NetworkXError("G is not connected") |
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345 odd_degree_nodes = [n for n, d in G.degree() if d % 2 == 1] |
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346 G = nx.MultiGraph(G) |
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347 if len(odd_degree_nodes) == 0: |
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348 return G |
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349 |
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350 # get all shortest paths between vertices of odd degree |
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351 odd_deg_pairs_paths = [ |
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352 (m, {n: nx.shortest_path(G, source=m, target=n)}) |
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353 for m, n in combinations(odd_degree_nodes, 2) |
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354 ] |
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355 |
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356 # use inverse path lengths as edge-weights in a new graph |
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357 # store the paths in the graph for easy indexing later |
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358 Gp = nx.Graph() |
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359 for n, Ps in odd_deg_pairs_paths: |
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360 for m, P in Ps.items(): |
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361 if n != m: |
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362 Gp.add_edge(m, n, weight=1 / len(P), path=P) |
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363 |
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364 # find the minimum weight matching of edges in the weighted graph |
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365 best_matching = nx.Graph(list(nx.max_weight_matching(Gp))) |
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366 |
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367 # duplicate each edge along each path in the set of paths in Gp |
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368 for m, n in best_matching.edges(): |
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369 path = Gp[m][n]["path"] |
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370 G.add_edges_from(nx.utils.pairwise(path)) |
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371 return G |