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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/cycles.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 # Copyright (C) 2010-2019 by
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2 # Aric Hagberg <hagberg@lanl.gov>
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3 # Dan Schult <dschult@colgate.edu>
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4 # Pieter Swart <swart@lanl.gov>
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5 # All rights reserved.
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6 # BSD license.
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7 #
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8 # Authors: Jon Olav Vik <jonovik@gmail.com>
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9 # Dan Schult <dschult@colgate.edu>
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10 # Aric Hagberg <hagberg@lanl.gov>
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11 # Debsankha Manik <dmanik@nld.ds.mpg.de>
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12 """
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13 ========================
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14 Cycle finding algorithms
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15 ========================
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16 """
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17
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18 from collections import defaultdict
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19 from itertools import tee
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20
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21 import networkx as nx
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22 from networkx.utils import not_implemented_for, pairwise
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23
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24 __all__ = [
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25 'cycle_basis', 'simple_cycles',
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26 'recursive_simple_cycles', 'find_cycle',
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27 'minimum_cycle_basis',
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28 ]
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29
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30
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31 @not_implemented_for('directed')
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32 @not_implemented_for('multigraph')
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33 def cycle_basis(G, root=None):
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34 """ Returns a list of cycles which form a basis for cycles of G.
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35
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36 A basis for cycles of a network is a minimal collection of
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37 cycles such that any cycle in the network can be written
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38 as a sum of cycles in the basis. Here summation of cycles
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39 is defined as "exclusive or" of the edges. Cycle bases are
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40 useful, e.g. when deriving equations for electric circuits
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41 using Kirchhoff's Laws.
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42
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43 Parameters
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44 ----------
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45 G : NetworkX Graph
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46 root : node, optional
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47 Specify starting node for basis.
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48
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49 Returns
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50 -------
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51 A list of cycle lists. Each cycle list is a list of nodes
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52 which forms a cycle (loop) in G.
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53
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54 Examples
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55 --------
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56 >>> G = nx.Graph()
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57 >>> nx.add_cycle(G, [0, 1, 2, 3])
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58 >>> nx.add_cycle(G, [0, 3, 4, 5])
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59 >>> print(nx.cycle_basis(G, 0))
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60 [[3, 4, 5, 0], [1, 2, 3, 0]]
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61
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62 Notes
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63 -----
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64 This is adapted from algorithm CACM 491 [1]_.
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65
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66 References
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67 ----------
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68 .. [1] Paton, K. An algorithm for finding a fundamental set of
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69 cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.
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70
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71 See Also
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72 --------
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73 simple_cycles
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74 """
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75 gnodes = set(G.nodes())
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76 cycles = []
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77 while gnodes: # loop over connected components
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78 if root is None:
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79 root = gnodes.pop()
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80 stack = [root]
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81 pred = {root: root}
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82 used = {root: set()}
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83 while stack: # walk the spanning tree finding cycles
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84 z = stack.pop() # use last-in so cycles easier to find
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85 zused = used[z]
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86 for nbr in G[z]:
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87 if nbr not in used: # new node
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88 pred[nbr] = z
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89 stack.append(nbr)
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90 used[nbr] = set([z])
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91 elif nbr == z: # self loops
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92 cycles.append([z])
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93 elif nbr not in zused: # found a cycle
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94 pn = used[nbr]
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95 cycle = [nbr, z]
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96 p = pred[z]
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97 while p not in pn:
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98 cycle.append(p)
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99 p = pred[p]
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100 cycle.append(p)
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101 cycles.append(cycle)
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102 used[nbr].add(z)
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103 gnodes -= set(pred)
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104 root = None
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105 return cycles
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106
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107
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108 @not_implemented_for('undirected')
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109 def simple_cycles(G):
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110 """Find simple cycles (elementary circuits) of a directed graph.
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111
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112 A `simple cycle`, or `elementary circuit`, is a closed path where
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113 no node appears twice. Two elementary circuits are distinct if they
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114 are not cyclic permutations of each other.
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115
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116 This is a nonrecursive, iterator/generator version of Johnson's
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117 algorithm [1]_. There may be better algorithms for some cases [2]_ [3]_.
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118
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119 Parameters
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120 ----------
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121 G : NetworkX DiGraph
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122 A directed graph
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123
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124 Returns
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125 -------
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126 cycle_generator: generator
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127 A generator that produces elementary cycles of the graph.
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128 Each cycle is represented by a list of nodes along the cycle.
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129
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130 Examples
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131 --------
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132 >>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
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133 >>> G = nx.DiGraph(edges)
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134 >>> len(list(nx.simple_cycles(G)))
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135 5
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136
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137 To filter the cycles so that they don't include certain nodes or edges,
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138 copy your graph and eliminate those nodes or edges before calling
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139
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140 >>> copyG = G.copy()
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141 >>> copyG.remove_nodes_from([1])
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142 >>> copyG.remove_edges_from([(0, 1)])
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143 >>> len(list(nx.simple_cycles(copyG)))
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144 3
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145
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146
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147 Notes
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148 -----
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149 The implementation follows pp. 79-80 in [1]_.
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150
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151 The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
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152 elementary circuits.
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153
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154 References
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155 ----------
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156 .. [1] Finding all the elementary circuits of a directed graph.
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157 D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
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158 https://doi.org/10.1137/0204007
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159 .. [2] Enumerating the cycles of a digraph: a new preprocessing strategy.
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160 G. Loizou and P. Thanish, Information Sciences, v. 27, 163-182, 1982.
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161 .. [3] A search strategy for the elementary cycles of a directed graph.
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162 J.L. Szwarcfiter and P.E. Lauer, BIT NUMERICAL MATHEMATICS,
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163 v. 16, no. 2, 192-204, 1976.
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164
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165 See Also
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166 --------
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167 cycle_basis
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168 """
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169 def _unblock(thisnode, blocked, B):
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170 stack = set([thisnode])
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171 while stack:
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172 node = stack.pop()
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173 if node in blocked:
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174 blocked.remove(node)
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175 stack.update(B[node])
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176 B[node].clear()
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177
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178 # Johnson's algorithm requires some ordering of the nodes.
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179 # We assign the arbitrary ordering given by the strongly connected comps
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180 # There is no need to track the ordering as each node removed as processed.
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181 # Also we save the actual graph so we can mutate it. We only take the
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182 # edges because we do not want to copy edge and node attributes here.
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183 subG = type(G)(G.edges())
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184 sccs = [scc for scc in nx.strongly_connected_components(subG)
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185 if len(scc) > 1]
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186
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187 # Johnson's algorithm exclude self cycle edges like (v, v)
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188 # To be backward compatible, we record those cycles in advance
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189 # and then remove from subG
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190 for v in subG:
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191 if subG.has_edge(v, v):
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192 yield [v]
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193 subG.remove_edge(v, v)
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194
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195 while sccs:
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196 scc = sccs.pop()
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197 sccG = subG.subgraph(scc)
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198 # order of scc determines ordering of nodes
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199 startnode = scc.pop()
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200 # Processing node runs "circuit" routine from recursive version
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201 path = [startnode]
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202 blocked = set() # vertex: blocked from search?
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203 closed = set() # nodes involved in a cycle
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204 blocked.add(startnode)
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205 B = defaultdict(set) # graph portions that yield no elementary circuit
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206 stack = [(startnode, list(sccG[startnode]))] # sccG gives comp nbrs
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207 while stack:
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208 thisnode, nbrs = stack[-1]
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209 if nbrs:
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210 nextnode = nbrs.pop()
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211 if nextnode == startnode:
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212 yield path[:]
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213 closed.update(path)
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214 # print "Found a cycle", path, closed
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215 elif nextnode not in blocked:
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216 path.append(nextnode)
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217 stack.append((nextnode, list(sccG[nextnode])))
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218 closed.discard(nextnode)
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219 blocked.add(nextnode)
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220 continue
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221 # done with nextnode... look for more neighbors
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222 if not nbrs: # no more nbrs
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223 if thisnode in closed:
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224 _unblock(thisnode, blocked, B)
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225 else:
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226 for nbr in sccG[thisnode]:
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227 if thisnode not in B[nbr]:
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228 B[nbr].add(thisnode)
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229 stack.pop()
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230 # assert path[-1] == thisnode
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231 path.pop()
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232 # done processing this node
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233 H = subG.subgraph(scc) # make smaller to avoid work in SCC routine
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234 sccs.extend(scc for scc in nx.strongly_connected_components(H)
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235 if len(scc) > 1)
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236
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237
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238 @not_implemented_for('undirected')
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239 def recursive_simple_cycles(G):
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240 """Find simple cycles (elementary circuits) of a directed graph.
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241
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242 A `simple cycle`, or `elementary circuit`, is a closed path where
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243 no node appears twice. Two elementary circuits are distinct if they
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244 are not cyclic permutations of each other.
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245
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246 This version uses a recursive algorithm to build a list of cycles.
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247 You should probably use the iterator version called simple_cycles().
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248 Warning: This recursive version uses lots of RAM!
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249
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250 Parameters
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251 ----------
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252 G : NetworkX DiGraph
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253 A directed graph
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254
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255 Returns
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256 -------
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257 A list of cycles, where each cycle is represented by a list of nodes
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258 along the cycle.
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259
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260 Example:
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261
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262 >>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
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263 >>> G = nx.DiGraph(edges)
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264 >>> nx.recursive_simple_cycles(G)
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265 [[0], [2], [0, 1, 2], [0, 2], [1, 2]]
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266
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267 See Also
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268 --------
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269 cycle_basis (for undirected graphs)
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270
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271 Notes
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272 -----
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273 The implementation follows pp. 79-80 in [1]_.
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274
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275 The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
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276 elementary circuits.
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277
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278 References
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279 ----------
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280 .. [1] Finding all the elementary circuits of a directed graph.
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281 D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
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282 https://doi.org/10.1137/0204007
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283
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284 See Also
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285 --------
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286 simple_cycles, cycle_basis
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287 """
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288 # Jon Olav Vik, 2010-08-09
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289 def _unblock(thisnode):
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290 """Recursively unblock and remove nodes from B[thisnode]."""
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291 if blocked[thisnode]:
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292 blocked[thisnode] = False
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293 while B[thisnode]:
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294 _unblock(B[thisnode].pop())
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295
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296 def circuit(thisnode, startnode, component):
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297 closed = False # set to True if elementary path is closed
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298 path.append(thisnode)
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299 blocked[thisnode] = True
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300 for nextnode in component[thisnode]: # direct successors of thisnode
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301 if nextnode == startnode:
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302 result.append(path[:])
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303 closed = True
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304 elif not blocked[nextnode]:
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305 if circuit(nextnode, startnode, component):
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306 closed = True
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307 if closed:
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308 _unblock(thisnode)
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309 else:
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310 for nextnode in component[thisnode]:
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311 if thisnode not in B[nextnode]: # TODO: use set for speedup?
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312 B[nextnode].append(thisnode)
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313 path.pop() # remove thisnode from path
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314 return closed
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315
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316 path = [] # stack of nodes in current path
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317 blocked = defaultdict(bool) # vertex: blocked from search?
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318 B = defaultdict(list) # graph portions that yield no elementary circuit
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319 result = [] # list to accumulate the circuits found
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320
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321 # Johnson's algorithm exclude self cycle edges like (v, v)
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322 # To be backward compatible, we record those cycles in advance
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323 # and then remove from subG
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324 for v in G:
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325 if G.has_edge(v, v):
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326 result.append([v])
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327 G.remove_edge(v, v)
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328
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329 # Johnson's algorithm requires some ordering of the nodes.
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330 # They might not be sortable so we assign an arbitrary ordering.
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331 ordering = dict(zip(G, range(len(G))))
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332 for s in ordering:
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333 # Build the subgraph induced by s and following nodes in the ordering
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334 subgraph = G.subgraph(node for node in G
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335 if ordering[node] >= ordering[s])
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336 # Find the strongly connected component in the subgraph
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337 # that contains the least node according to the ordering
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338 strongcomp = nx.strongly_connected_components(subgraph)
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339 mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
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340 component = G.subgraph(mincomp)
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341 if len(component) > 1:
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342 # smallest node in the component according to the ordering
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343 startnode = min(component, key=ordering.__getitem__)
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344 for node in component:
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345 blocked[node] = False
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346 B[node][:] = []
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347 dummy = circuit(startnode, startnode, component)
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348 return result
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349
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350
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351 def find_cycle(G, source=None, orientation=None):
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352 """Returns a cycle found via depth-first traversal.
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353
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354 The cycle is a list of edges indicating the cyclic path.
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355 Orientation of directed edges is controlled by `orientation`.
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356
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357 Parameters
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358 ----------
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359 G : graph
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360 A directed/undirected graph/multigraph.
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361
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362 source : node, list of nodes
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363 The node from which the traversal begins. If None, then a source
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364 is chosen arbitrarily and repeatedly until all edges from each node in
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365 the graph are searched.
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366
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367 orientation : None | 'original' | 'reverse' | 'ignore' (default: None)
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368 For directed graphs and directed multigraphs, edge traversals need not
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369 respect the original orientation of the edges.
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370 When set to 'reverse' every edge is traversed in the reverse direction.
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371 When set to 'ignore', every edge is treated as undirected.
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372 When set to 'original', every edge is treated as directed.
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373 In all three cases, the yielded edge tuples add a last entry to
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374 indicate the direction in which that edge was traversed.
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375 If orientation is None, the yielded edge has no direction indicated.
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376 The direction is respected, but not reported.
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377
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378 Returns
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379 -------
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380 edges : directed edges
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381 A list of directed edges indicating the path taken for the loop.
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382 If no cycle is found, then an exception is raised.
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383 For graphs, an edge is of the form `(u, v)` where `u` and `v`
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384 are the tail and head of the edge as determined by the traversal.
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385 For multigraphs, an edge is of the form `(u, v, key)`, where `key` is
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386 the key of the edge. When the graph is directed, then `u` and `v`
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387 are always in the order of the actual directed edge.
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388 If orientation is not None then the edge tuple is extended to include
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389 the direction of traversal ('forward' or 'reverse') on that edge.
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390
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391 Raises
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392 ------
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393 NetworkXNoCycle
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394 If no cycle was found.
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395
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396 Examples
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397 --------
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398 In this example, we construct a DAG and find, in the first call, that there
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399 are no directed cycles, and so an exception is raised. In the second call,
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400 we ignore edge orientations and find that there is an undirected cycle.
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401 Note that the second call finds a directed cycle while effectively
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402 traversing an undirected graph, and so, we found an "undirected cycle".
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403 This means that this DAG structure does not form a directed tree (which
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404 is also known as a polytree).
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405
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406 >>> import networkx as nx
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407 >>> G = nx.DiGraph([(0, 1), (0, 2), (1, 2)])
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408 >>> try:
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409 ... nx.find_cycle(G, orientation='original')
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410 ... except:
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411 ... pass
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412 ...
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413 >>> list(nx.find_cycle(G, orientation='ignore'))
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414 [(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]
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415
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416 """
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417 if not G.is_directed() or orientation in (None, 'original'):
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418 def tailhead(edge):
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419 return edge[:2]
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420 elif orientation == 'reverse':
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421 def tailhead(edge):
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422 return edge[1], edge[0]
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423 elif orientation == 'ignore':
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424 def tailhead(edge):
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425 if edge[-1] == 'reverse':
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426 return edge[1], edge[0]
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427 return edge[:2]
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428
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429 explored = set()
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430 cycle = []
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431 final_node = None
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432 for start_node in G.nbunch_iter(source):
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433 if start_node in explored:
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434 # No loop is possible.
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435 continue
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436
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437 edges = []
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438 # All nodes seen in this iteration of edge_dfs
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439 seen = {start_node}
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440 # Nodes in active path.
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441 active_nodes = {start_node}
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442 previous_head = None
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443
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444 for edge in nx.edge_dfs(G, start_node, orientation):
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445 # Determine if this edge is a continuation of the active path.
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446 tail, head = tailhead(edge)
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447 if head in explored:
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448 # Then we've already explored it. No loop is possible.
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449 continue
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450 if previous_head is not None and tail != previous_head:
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451 # This edge results from backtracking.
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452 # Pop until we get a node whose head equals the current tail.
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453 # So for example, we might have:
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454 # (0, 1), (1, 2), (2, 3), (1, 4)
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455 # which must become:
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456 # (0, 1), (1, 4)
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457 while True:
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458 try:
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459 popped_edge = edges.pop()
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460 except IndexError:
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461 edges = []
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462 active_nodes = {tail}
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463 break
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464 else:
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465 popped_head = tailhead(popped_edge)[1]
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466 active_nodes.remove(popped_head)
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467
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468 if edges:
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469 last_head = tailhead(edges[-1])[1]
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470 if tail == last_head:
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471 break
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472 edges.append(edge)
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473
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474 if head in active_nodes:
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475 # We have a loop!
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476 cycle.extend(edges)
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477 final_node = head
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478 break
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479 else:
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480 seen.add(head)
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481 active_nodes.add(head)
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482 previous_head = head
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483
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484 if cycle:
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485 break
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486 else:
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487 explored.update(seen)
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488
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489 else:
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490 assert(len(cycle) == 0)
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491 raise nx.exception.NetworkXNoCycle('No cycle found.')
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492
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493 # We now have a list of edges which ends on a cycle.
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494 # So we need to remove from the beginning edges that are not relevant.
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495
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496 for i, edge in enumerate(cycle):
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497 tail, head = tailhead(edge)
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498 if tail == final_node:
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499 break
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500
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501 return cycle[i:]
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502
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503
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504 @not_implemented_for('directed')
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505 @not_implemented_for('multigraph')
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506 def minimum_cycle_basis(G, weight=None):
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507 """ Returns a minimum weight cycle basis for G
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508
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509 Minimum weight means a cycle basis for which the total weight
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510 (length for unweighted graphs) of all the cycles is minimum.
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511
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512 Parameters
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513 ----------
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514 G : NetworkX Graph
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515 weight: string
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516 name of the edge attribute to use for edge weights
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517
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518 Returns
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519 -------
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520 A list of cycle lists. Each cycle list is a list of nodes
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521 which forms a cycle (loop) in G. Note that the nodes are not
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522 necessarily returned in a order by which they appear in the cycle
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523
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524 Examples
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525 --------
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526 >>> G=nx.Graph()
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527 >>> nx.add_cycle(G, [0,1,2,3])
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528 >>> nx.add_cycle(G, [0,3,4,5])
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529 >>> print([sorted(c) for c in nx.minimum_cycle_basis(G)])
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530 [[0, 1, 2, 3], [0, 3, 4, 5]]
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531
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532 References:
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533 [1] Kavitha, Telikepalli, et al. "An O(m^2n) Algorithm for
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534 Minimum Cycle Basis of Graphs."
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535 http://link.springer.com/article/10.1007/s00453-007-9064-z
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536 [2] de Pina, J. 1995. Applications of shortest path methods.
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537 Ph.D. thesis, University of Amsterdam, Netherlands
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538
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539 See Also
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540 --------
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541 simple_cycles, cycle_basis
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542 """
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543 # We first split the graph in commected subgraphs
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544 return sum((_min_cycle_basis(G.subgraph(c), weight) for c in
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545 nx.connected_components(G)), [])
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546
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547
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548 def _min_cycle_basis(comp, weight):
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549 cb = []
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550 # We extract the edges not in a spanning tree. We do not really need a
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551 # *minimum* spanning tree. That is why we call the next function with
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552 # weight=None. Depending on implementation, it may be faster as well
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553 spanning_tree_edges = list(nx.minimum_spanning_edges(comp, weight=None,
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554 data=False))
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555 edges_excl = [frozenset(e) for e in comp.edges()
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556 if e not in spanning_tree_edges]
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557 N = len(edges_excl)
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558
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559 # We maintain a set of vectors orthogonal to sofar found cycles
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560 set_orth = [set([edge]) for edge in edges_excl]
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561 for k in range(N):
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562 # kth cycle is "parallel" to kth vector in set_orth
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563 new_cycle = _min_cycle(comp, set_orth[k], weight=weight)
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564 cb.append(list(set().union(*new_cycle)))
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565 # now update set_orth so that k+1,k+2... th elements are
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566 # orthogonal to the newly found cycle, as per [p. 336, 1]
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567 base = set_orth[k]
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568 set_orth[k + 1:] = [orth ^ base if len(orth & new_cycle) % 2 else orth
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569 for orth in set_orth[k + 1:]]
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570 return cb
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571
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572
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573 def _min_cycle(G, orth, weight=None):
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574 """
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575 Computes the minimum weight cycle in G,
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576 orthogonal to the vector orth as per [p. 338, 1]
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577 """
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578 T = nx.Graph()
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579
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580 nodes_idx = {node: idx for idx, node in enumerate(G.nodes())}
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581 idx_nodes = {idx: node for node, idx in nodes_idx.items()}
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582
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583 nnodes = len(nodes_idx)
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584
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585 # Add 2 copies of each edge in G to T. If edge is in orth, add cross edge;
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586 # otherwise in-plane edge
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587 for u, v, data in G.edges(data=True):
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588 uidx, vidx = nodes_idx[u], nodes_idx[v]
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589 edge_w = data.get(weight, 1)
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590 if frozenset((u, v)) in orth:
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591 T.add_edges_from(
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592 [(uidx, nnodes + vidx), (nnodes + uidx, vidx)], weight=edge_w)
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593 else:
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594 T.add_edges_from(
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595 [(uidx, vidx), (nnodes + uidx, nnodes + vidx)], weight=edge_w)
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596
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597 all_shortest_pathlens = dict(nx.shortest_path_length(T, weight=weight))
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598 cross_paths_w_lens = {n: all_shortest_pathlens[n][nnodes + n]
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599 for n in range(nnodes)}
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600
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601 # Now compute shortest paths in T, which translates to cyles in G
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602 start = min(cross_paths_w_lens, key=cross_paths_w_lens.get)
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603 end = nnodes + start
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604 min_path = nx.shortest_path(T, source=start, target=end, weight='weight')
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605
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606 # Now we obtain the actual path, re-map nodes in T to those in G
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607 min_path_nodes = [node if node < nnodes else node - nnodes
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608 for node in min_path]
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609 # Now remove the edges that occur two times
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610 mcycle_pruned = _path_to_cycle(min_path_nodes)
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611
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612 return {frozenset((idx_nodes[u], idx_nodes[v])) for u, v in mcycle_pruned}
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613
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614
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615 def _path_to_cycle(path):
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616 """
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617 Removes the edges from path that occur even number of times.
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618 Returns a set of edges
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619 """
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620 edges = set()
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621 for edge in pairwise(path):
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622 # Toggle whether to keep the current edge.
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623 edges ^= {edge}
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624 return edges