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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/euler.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 #
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3 # Copyright (C) 2010 by
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4 # Aric Hagberg <hagberg@lanl.gov>
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5 # Dan Schult <dschult@colgate.edu>
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6 # Pieter Swart <swart@lanl.gov>
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7 # All rights reserved.
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8 # BSD license.
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9 #
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10 # Authors:
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11 # Nima Mohammadi <nima.irt@gmail.com>
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12 # Aric Hagberg <hagberg@lanl.gov>
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13 # Mike Trenfield <william.trenfield@utsouthwestern.edu>
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14 """
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15 Eulerian circuits and graphs.
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16 """
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17 from itertools import combinations
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18
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19 import networkx as nx
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20 from ..utils import arbitrary_element, not_implemented_for
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21
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22 __all__ = ['is_eulerian', 'eulerian_circuit', 'eulerize',
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23 'is_semieulerian', 'has_eulerian_path', 'eulerian_path',
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24 ]
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25
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26
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27 def is_eulerian(G):
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28 """Returns True if and only if `G` is Eulerian.
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29
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30 A graph is *Eulerian* if it has an Eulerian circuit. An *Eulerian
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31 circuit* is a closed walk that includes each edge of a graph exactly
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32 once.
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33
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34 Parameters
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35 ----------
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36 G : NetworkX graph
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37 A graph, either directed or undirected.
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38
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39 Examples
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40 --------
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41 >>> nx.is_eulerian(nx.DiGraph({0: [3], 1: [2], 2: [3], 3: [0, 1]}))
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42 True
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43 >>> nx.is_eulerian(nx.complete_graph(5))
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44 True
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45 >>> nx.is_eulerian(nx.petersen_graph())
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46 False
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47
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48 Notes
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49 -----
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50 If the graph is not connected (or not strongly connected, for
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51 directed graphs), this function returns False.
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52
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53 """
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54 if G.is_directed():
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55 # Every node must have equal in degree and out degree and the
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56 # graph must be strongly connected
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57 return (all(G.in_degree(n) == G.out_degree(n) for n in G) and
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58 nx.is_strongly_connected(G))
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59 # An undirected Eulerian graph has no vertices of odd degree and
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60 # must be connected.
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61 return all(d % 2 == 0 for v, d in G.degree()) and nx.is_connected(G)
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62
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63
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64 def is_semieulerian(G):
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65 """Return True iff `G` is semi-Eulerian.
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66
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67 G is semi-Eulerian if it has an Eulerian path but no Eulerian circuit.
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68 """
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69 return has_eulerian_path(G) and not is_eulerian(G)
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70
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71
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72 def _find_path_start(G):
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73 """Return a suitable starting vertex for an Eulerian path.
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74
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75 If no path exists, return None.
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76 """
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77 if not has_eulerian_path(G):
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78 return None
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79
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80 if is_eulerian(G):
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81 return arbitrary_element(G)
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82
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83 if G.is_directed():
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84 v1, v2 = [v for v in G if G.in_degree(v) != G.out_degree(v)]
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85 # Determines which is the 'start' node (as opposed to the 'end')
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86 if G.out_degree(v1) > G.in_degree(v1):
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87 return v1
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88 else:
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89 return v2
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90
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91 else:
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92 # In an undirected graph randomly choose one of the possibilities
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93 start = [v for v in G if G.degree(v) % 2 != 0][0]
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94 return start
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95
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96
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97 def _simplegraph_eulerian_circuit(G, source):
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98 if G.is_directed():
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99 degree = G.out_degree
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100 edges = G.out_edges
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101 else:
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102 degree = G.degree
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103 edges = G.edges
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104 vertex_stack = [source]
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105 last_vertex = None
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106 while vertex_stack:
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107 current_vertex = vertex_stack[-1]
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108 if degree(current_vertex) == 0:
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109 if last_vertex is not None:
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110 yield (last_vertex, current_vertex)
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111 last_vertex = current_vertex
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112 vertex_stack.pop()
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113 else:
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114 _, next_vertex = arbitrary_element(edges(current_vertex))
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115 vertex_stack.append(next_vertex)
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116 G.remove_edge(current_vertex, next_vertex)
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117
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118
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119 def _multigraph_eulerian_circuit(G, source):
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120 if G.is_directed():
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121 degree = G.out_degree
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122 edges = G.out_edges
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123 else:
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124 degree = G.degree
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125 edges = G.edges
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126 vertex_stack = [(source, None)]
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127 last_vertex = None
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128 last_key = None
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129 while vertex_stack:
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130 current_vertex, current_key = vertex_stack[-1]
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131 if degree(current_vertex) == 0:
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132 if last_vertex is not None:
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133 yield (last_vertex, current_vertex, last_key)
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134 last_vertex, last_key = current_vertex, current_key
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135 vertex_stack.pop()
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136 else:
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137 triple = arbitrary_element(edges(current_vertex, keys=True))
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138 _, next_vertex, next_key = triple
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139 vertex_stack.append((next_vertex, next_key))
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140 G.remove_edge(current_vertex, next_vertex, next_key)
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141
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142
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143 def eulerian_circuit(G, source=None, keys=False):
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144 """Returns an iterator over the edges of an Eulerian circuit in `G`.
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145
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146 An *Eulerian circuit* is a closed walk that includes each edge of a
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147 graph exactly once.
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148
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149 Parameters
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150 ----------
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151 G : NetworkX graph
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152 A graph, either directed or undirected.
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153
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154 source : node, optional
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155 Starting node for circuit.
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156
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157 keys : bool
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158 If False, edges generated by this function will be of the form
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159 ``(u, v)``. Otherwise, edges will be of the form ``(u, v, k)``.
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160 This option is ignored unless `G` is a multigraph.
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161
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162 Returns
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163 -------
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164 edges : iterator
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165 An iterator over edges in the Eulerian circuit.
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166
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167 Raises
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168 ------
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169 NetworkXError
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170 If the graph is not Eulerian.
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171
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172 See Also
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173 --------
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174 is_eulerian
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175
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176 Notes
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177 -----
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178 This is a linear time implementation of an algorithm adapted from [1]_.
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179
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180 For general information about Euler tours, see [2]_.
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181
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182 References
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183 ----------
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184 .. [1] J. Edmonds, E. L. Johnson.
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185 Matching, Euler tours and the Chinese postman.
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186 Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
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187 .. [2] https://en.wikipedia.org/wiki/Eulerian_path
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188
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189 Examples
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190 --------
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191 To get an Eulerian circuit in an undirected graph::
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192
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193 >>> G = nx.complete_graph(3)
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194 >>> list(nx.eulerian_circuit(G))
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195 [(0, 2), (2, 1), (1, 0)]
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196 >>> list(nx.eulerian_circuit(G, source=1))
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197 [(1, 2), (2, 0), (0, 1)]
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198
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199 To get the sequence of vertices in an Eulerian circuit::
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200
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201 >>> [u for u, v in nx.eulerian_circuit(G)]
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202 [0, 2, 1]
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203
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204 """
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205 if not is_eulerian(G):
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206 raise nx.NetworkXError("G is not Eulerian.")
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207 if G.is_directed():
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208 G = G.reverse()
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209 else:
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210 G = G.copy()
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211 if source is None:
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212 source = arbitrary_element(G)
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213 if G.is_multigraph():
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214 for u, v, k in _multigraph_eulerian_circuit(G, source):
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215 if keys:
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216 yield u, v, k
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217 else:
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218 yield u, v
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219 else:
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220 for u, v in _simplegraph_eulerian_circuit(G, source):
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221 yield u, v
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222
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223
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224 def has_eulerian_path(G):
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225 """Return True iff `G` has an Eulerian path.
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226
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227 An Eulerian path is a path in a graph which uses each edge of a graph
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228 exactly once.
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229
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230 A directed graph has an Eulerian path iff:
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231 - at most one vertex has out_degree - in_degree = 1,
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232 - at most one vertex has in_degree - out_degree = 1,
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233 - every other vertex has equal in_degree and out_degree,
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234 - and all of its vertices with nonzero degree belong to a
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235 - single connected component of the underlying undirected graph.
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236
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237 An undirected graph has an Eulerian path iff:
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238 - exactly zero or two vertices have odd degree,
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239 - and all of its vertices with nonzero degree belong to a
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240 - single connected component.
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241
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242 Parameters
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243 ----------
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244 G : NetworkX Graph
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245 The graph to find an euler path in.
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246
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247 Returns
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248 -------
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249 Bool : True if G has an eulerian path.
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250
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251 See Also
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252 --------
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253 is_eulerian
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254 eulerian_path
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255 """
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256 if G.is_directed():
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257 ins = G.in_degree
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258 outs = G.out_degree
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259 semibalanced_ins = sum(ins(v) - outs(v) == 1 for v in G)
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260 semibalanced_outs = sum(outs(v) - ins(v) == 1 for v in G)
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261 return (semibalanced_ins <= 1 and
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262 semibalanced_outs <= 1 and
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263 sum(G.in_degree(v) != G.out_degree(v) for v in G) <= 2 and
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264 nx.is_weakly_connected(G))
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265 else:
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266 return (sum(d % 2 == 1 for v, d in G.degree()) in (0, 2)
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267 and nx.is_connected(G))
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268
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269
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270 def eulerian_path(G, source=None, keys=False):
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271 """Return an iterator over the edges of an Eulerian path in `G`.
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272
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273 Parameters
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274 ----------
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275 G : NetworkX Graph
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276 The graph in which to look for an eulerian path.
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277 source : node or None (default: None)
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278 The node at which to start the search. None means search over all
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279 starting nodes.
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280 keys : Bool (default: False)
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281 Indicates whether to yield edge 3-tuples (u, v, edge_key).
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282 The default yields edge 2-tuples
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283
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284 Yields
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285 ------
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286 Edge tuples along the eulerian path.
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287
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288 Warning: If `source` provided is not the start node of an Euler path
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289 will raise error even if an Euler Path exists.
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290 """
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291 if not has_eulerian_path(G):
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292 raise nx.NetworkXError("Graph has no Eulerian paths.")
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293 if G.is_directed():
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294 G = G.reverse()
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295 else:
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296 G = G.copy()
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297 if source is None:
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298 source = _find_path_start(G)
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299 if G.is_multigraph():
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300 for u, v, k in _multigraph_eulerian_circuit(G, source):
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301 if keys:
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302 yield u, v, k
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303 else:
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304 yield u, v
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305 else:
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306 for u, v in _simplegraph_eulerian_circuit(G, source):
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307 yield u, v
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308
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309
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310 @not_implemented_for('directed')
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311 def eulerize(G):
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312 """Transforms a graph into an Eulerian graph
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313
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314 Parameters
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315 ----------
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316 G : NetworkX graph
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317 An undirected graph
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318
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319 Returns
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320 -------
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321 G : NetworkX multigraph
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322
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323 Raises
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324 ------
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325 NetworkXError
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326 If the graph is not connected.
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327
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328 See Also
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329 --------
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330 is_eulerian
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331 eulerian_circuit
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332
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333 References
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334 ----------
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335 .. [1] J. Edmonds, E. L. Johnson.
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336 Matching, Euler tours and the Chinese postman.
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337 Mathematical programming, Volume 5, Issue 1 (1973), 111-114.
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338 [2] https://en.wikipedia.org/wiki/Eulerian_path
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339 .. [3] http://web.math.princeton.edu/math_alive/5/Notes1.pdf
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340
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341 Examples
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342 --------
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343 >>> G = nx.complete_graph(10)
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344 >>> H = nx.eulerize(G)
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345 >>> nx.is_eulerian(H)
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346 True
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347
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348 """
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349 if G.order() == 0:
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350 raise nx.NetworkXPointlessConcept("Cannot Eulerize null graph")
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351 if not nx.is_connected(G):
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352 raise nx.NetworkXError("G is not connected")
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353 odd_degree_nodes = [n for n, d in G.degree() if d % 2 == 1]
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354 G = nx.MultiGraph(G)
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355 if len(odd_degree_nodes) == 0:
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356 return G
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357
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358 # get all shortest paths between vertices of odd degree
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359 odd_deg_pairs_paths = [(m,
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360 {n: nx.shortest_path(G, source=m, target=n)}
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361 )
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362 for m, n in combinations(odd_degree_nodes, 2)]
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363
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364 # use inverse path lengths as edge-weights in a new graph
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365 # store the paths in the graph for easy indexing later
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366 Gp = nx.Graph()
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367 for n, Ps in odd_deg_pairs_paths:
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368 for m, P in Ps.items():
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369 if n != m:
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370 Gp.add_edge(m, n, weight=1/len(P), path=P)
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371
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372 # find the minimum weight matching of edges in the weighted graph
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373 best_matching = nx.Graph(list(nx.max_weight_matching(Gp)))
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374
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375 # duplicate each edge along each path in the set of paths in Gp
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376 for m, n in best_matching.edges():
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377 path = Gp[m][n]["path"]
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378 G.add_edges_from(nx.utils.pairwise(path))
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379 return G