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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/dag.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 # Copyright (C) 2006-2019 by
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3 # Aric Hagberg <hagberg@lanl.gov>
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4 # Dan Schult <dschult@colgate.edu>
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5 # Pieter Swart <swart@lanl.gov>
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6 # All rights reserved.
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7 # BSD license.
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8 #
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9 # Authors:
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10 # Aric Hagberg <aric.hagberg@gmail.com>
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11 # Dan Schult <dschult@colgate.edu>
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12 # Ben Edwards <bedwards@cs.unm.edu>
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13 # Neil Girdhar <neil.girdhar@mcgill.ca>
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14 #
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15 """Algorithms for directed acyclic graphs (DAGs).
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16
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17 Note that most of these functions are only guaranteed to work for DAGs.
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18 In general, these functions do not check for acyclic-ness, so it is up
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19 to the user to check for that.
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20 """
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21
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22 from collections import defaultdict, deque
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23 from math import gcd
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24 from functools import partial
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25 from itertools import chain
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26 from itertools import product
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27 from itertools import starmap
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28 import heapq
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29
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30 import networkx as nx
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31 from networkx.algorithms.traversal.breadth_first_search import \
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32 descendants_at_distance
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33 from networkx.generators.trees import NIL
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34 from networkx.utils import arbitrary_element
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35 from networkx.utils import consume
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36 from networkx.utils import pairwise
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37 from networkx.utils import not_implemented_for
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38
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39 __all__ = ['descendants',
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40 'ancestors',
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41 'topological_sort',
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42 'lexicographical_topological_sort',
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43 'all_topological_sorts',
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44 'is_directed_acyclic_graph',
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45 'is_aperiodic',
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46 'transitive_closure',
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47 'transitive_closure_dag',
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48 'transitive_reduction',
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49 'antichains',
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50 'dag_longest_path',
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51 'dag_longest_path_length',
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52 'dag_to_branching']
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53
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54 chaini = chain.from_iterable
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55
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56
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57 def descendants(G, source):
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58 """Returns all nodes reachable from `source` in `G`.
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59
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60 Parameters
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61 ----------
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62 G : NetworkX DiGraph
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63 A directed acyclic graph (DAG)
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64 source : node in `G`
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65
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66 Returns
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67 -------
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68 set()
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69 The descendants of `source` in `G`
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70 """
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71 if not G.has_node(source):
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72 raise nx.NetworkXError("The node %s is not in the graph." % source)
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73 des = set(n for n, d in nx.shortest_path_length(G, source=source).items())
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74 return des - {source}
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75
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76
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77 def ancestors(G, source):
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78 """Returns all nodes having a path to `source` in `G`.
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79
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80 Parameters
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81 ----------
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82 G : NetworkX DiGraph
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83 A directed acyclic graph (DAG)
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84 source : node in `G`
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85
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86 Returns
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87 -------
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88 set()
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89 The ancestors of source in G
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90 """
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91 if not G.has_node(source):
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92 raise nx.NetworkXError("The node %s is not in the graph." % source)
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93 anc = set(n for n, d in nx.shortest_path_length(G, target=source).items())
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94 return anc - {source}
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95
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96
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97 def has_cycle(G):
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98 """Decides whether the directed graph has a cycle."""
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99 try:
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100 consume(topological_sort(G))
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101 except nx.NetworkXUnfeasible:
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102 return True
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103 else:
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104 return False
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105
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106
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107 def is_directed_acyclic_graph(G):
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108 """Returns True if the graph `G` is a directed acyclic graph (DAG) or
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109 False if not.
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110
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111 Parameters
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112 ----------
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113 G : NetworkX graph
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114
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115 Returns
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116 -------
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117 bool
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118 True if `G` is a DAG, False otherwise
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119 """
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120 return G.is_directed() and not has_cycle(G)
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121
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122
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123 def topological_sort(G):
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124 """Returns a generator of nodes in topologically sorted order.
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125
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126 A topological sort is a nonunique permutation of the nodes such that an
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127 edge from u to v implies that u appears before v in the topological sort
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128 order.
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129
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130 Parameters
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131 ----------
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132 G : NetworkX digraph
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133 A directed acyclic graph (DAG)
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134
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135 Returns
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136 -------
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137 iterable
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138 An iterable of node names in topological sorted order.
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139
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140 Raises
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141 ------
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142 NetworkXError
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143 Topological sort is defined for directed graphs only. If the graph `G`
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144 is undirected, a :exc:`NetworkXError` is raised.
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145
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146 NetworkXUnfeasible
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147 If `G` is not a directed acyclic graph (DAG) no topological sort exists
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148 and a :exc:`NetworkXUnfeasible` exception is raised. This can also be
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149 raised if `G` is changed while the returned iterator is being processed
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150
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151 RuntimeError
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152 If `G` is changed while the returned iterator is being processed.
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153
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154 Examples
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155 --------
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156 To get the reverse order of the topological sort:
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157
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158 >>> DG = nx.DiGraph([(1, 2), (2, 3)])
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159 >>> list(reversed(list(nx.topological_sort(DG))))
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160 [3, 2, 1]
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161
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162 If your DiGraph naturally has the edges representing tasks/inputs
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163 and nodes representing people/processes that initiate tasks, then
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164 topological_sort is not quite what you need. You will have to change
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165 the tasks to nodes with dependence reflected by edges. The result is
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166 a kind of topological sort of the edges. This can be done
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167 with :func:`networkx.line_graph` as follows:
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168
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169 >>> list(nx.topological_sort(nx.line_graph(DG)))
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170 [(1, 2), (2, 3)]
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171
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172 Notes
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173 -----
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174 This algorithm is based on a description and proof in
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175 "Introduction to Algorithms: A Creative Approach" [1]_ .
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176
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177 See also
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178 --------
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179 is_directed_acyclic_graph, lexicographical_topological_sort
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180
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181 References
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182 ----------
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183 .. [1] Manber, U. (1989).
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184 *Introduction to Algorithms - A Creative Approach.* Addison-Wesley.
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185 """
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186 if not G.is_directed():
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187 raise nx.NetworkXError(
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188 "Topological sort not defined on undirected graphs.")
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189
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190 indegree_map = {v: d for v, d in G.in_degree() if d > 0}
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191 # These nodes have zero indegree and ready to be returned.
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192 zero_indegree = [v for v, d in G.in_degree() if d == 0]
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193
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194 while zero_indegree:
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195 node = zero_indegree.pop()
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196 if node not in G:
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197 raise RuntimeError("Graph changed during iteration")
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198 for _, child in G.edges(node):
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199 try:
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200 indegree_map[child] -= 1
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201 except KeyError:
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202 raise RuntimeError("Graph changed during iteration")
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203 if indegree_map[child] == 0:
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204 zero_indegree.append(child)
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205 del indegree_map[child]
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206
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207 yield node
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208
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209 if indegree_map:
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210 raise nx.NetworkXUnfeasible("Graph contains a cycle or graph changed "
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211 "during iteration")
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212
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213
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214 def lexicographical_topological_sort(G, key=None):
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215 """Returns a generator of nodes in lexicographically topologically sorted
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216 order.
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217
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218 A topological sort is a nonunique permutation of the nodes such that an
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219 edge from u to v implies that u appears before v in the topological sort
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220 order.
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221
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222 Parameters
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223 ----------
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224 G : NetworkX digraph
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225 A directed acyclic graph (DAG)
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226
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227 key : function, optional
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228 This function maps nodes to keys with which to resolve ambiguities in
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229 the sort order. Defaults to the identity function.
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230
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231 Returns
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232 -------
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233 iterable
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234 An iterable of node names in lexicographical topological sort order.
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235
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236 Raises
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237 ------
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238 NetworkXError
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239 Topological sort is defined for directed graphs only. If the graph `G`
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240 is undirected, a :exc:`NetworkXError` is raised.
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241
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242 NetworkXUnfeasible
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243 If `G` is not a directed acyclic graph (DAG) no topological sort exists
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244 and a :exc:`NetworkXUnfeasible` exception is raised. This can also be
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245 raised if `G` is changed while the returned iterator is being processed
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246
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247 RuntimeError
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248 If `G` is changed while the returned iterator is being processed.
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249
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250 Notes
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251 -----
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252 This algorithm is based on a description and proof in
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253 "Introduction to Algorithms: A Creative Approach" [1]_ .
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254
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255 See also
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256 --------
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257 topological_sort
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258
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259 References
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260 ----------
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261 .. [1] Manber, U. (1989).
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262 *Introduction to Algorithms - A Creative Approach.* Addison-Wesley.
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263 """
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264 if not G.is_directed():
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265 msg = "Topological sort not defined on undirected graphs."
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266 raise nx.NetworkXError(msg)
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267
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268 if key is None:
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269 def key(node):
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270 return node
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271
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272 nodeid_map = {n: i for i, n in enumerate(G)}
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273
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274 def create_tuple(node):
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275 return key(node), nodeid_map[node], node
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276
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277 indegree_map = {v: d for v, d in G.in_degree() if d > 0}
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278 # These nodes have zero indegree and ready to be returned.
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279 zero_indegree = [create_tuple(v) for v, d in G.in_degree() if d == 0]
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280 heapq.heapify(zero_indegree)
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281
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282 while zero_indegree:
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283 _, _, node = heapq.heappop(zero_indegree)
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284
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285 if node not in G:
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286 raise RuntimeError("Graph changed during iteration")
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287 for _, child in G.edges(node):
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288 try:
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289 indegree_map[child] -= 1
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290 except KeyError:
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291 raise RuntimeError("Graph changed during iteration")
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292 if indegree_map[child] == 0:
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293 heapq.heappush(zero_indegree, create_tuple(child))
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294 del indegree_map[child]
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295
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296 yield node
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297
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298 if indegree_map:
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299 msg = "Graph contains a cycle or graph changed during iteration"
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300 raise nx.NetworkXUnfeasible(msg)
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301
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302
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303 @not_implemented_for('undirected')
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304 def all_topological_sorts(G):
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305 """Returns a generator of _all_ topological sorts of the directed graph G.
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306
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307 A topological sort is a nonunique permutation of the nodes such that an
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308 edge from u to v implies that u appears before v in the topological sort
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309 order.
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310
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311 Parameters
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312 ----------
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313 G : NetworkX DiGraph
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314 A directed graph
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315
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316 Returns
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317 -------
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318 generator
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319 All topological sorts of the digraph G
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320
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321 Raises
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322 ------
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323 NetworkXNotImplemented
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324 If `G` is not directed
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325 NetworkXUnfeasible
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326 If `G` is not acyclic
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327
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328 Examples
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329 --------
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330 To enumerate all topological sorts of directed graph:
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331
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332 >>> DG = nx.DiGraph([(1, 2), (2, 3), (2, 4)])
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333 >>> list(nx.all_topological_sorts(DG))
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334 [[1, 2, 4, 3], [1, 2, 3, 4]]
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335
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336 Notes
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337 -----
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338 Implements an iterative version of the algorithm given in [1].
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339
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340 References
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341 ----------
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342 .. [1] Knuth, Donald E., Szwarcfiter, Jayme L. (1974).
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343 "A Structured Program to Generate All Topological Sorting Arrangements"
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344 Information Processing Letters, Volume 2, Issue 6, 1974, Pages 153-157,
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345 ISSN 0020-0190,
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346 https://doi.org/10.1016/0020-0190(74)90001-5.
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347 Elsevier (North-Holland), Amsterdam
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348 """
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349 if not G.is_directed():
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350 raise nx.NetworkXError(
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351 "Topological sort not defined on undirected graphs.")
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352
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353 # the names of count and D are chosen to match the global variables in [1]
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354 # number of edges originating in a vertex v
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355 count = dict(G.in_degree())
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356 # vertices with indegree 0
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357 D = deque([v for v, d in G.in_degree() if d == 0])
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358 # stack of first value chosen at a position k in the topological sort
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359 bases = []
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360 current_sort = []
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361
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362 # do-while construct
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363 while True:
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364 assert all([count[v] == 0 for v in D])
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365
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366 if len(current_sort) == len(G):
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367 yield list(current_sort)
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368
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369 # clean-up stack
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370 while len(current_sort) > 0:
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371 assert len(bases) == len(current_sort)
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372 q = current_sort.pop()
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373
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374 # "restores" all edges (q, x)
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375 # NOTE: it is important to iterate over edges instead
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376 # of successors, so count is updated correctly in multigraphs
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377 for _, j in G.out_edges(q):
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378 count[j] += 1
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379 assert count[j] >= 0
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380 # remove entries from D
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381 while len(D) > 0 and count[D[-1]] > 0:
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382 D.pop()
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383
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384 # corresponds to a circular shift of the values in D
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385 # if the first value chosen (the base) is in the first
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386 # position of D again, we are done and need to consider the
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387 # previous condition
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388 D.appendleft(q)
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389 if D[-1] == bases[-1]:
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390 # all possible values have been chosen at current position
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391 # remove corresponding marker
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392 bases.pop()
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393 else:
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394 # there are still elements that have not been fixed
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395 # at the current position in the topological sort
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396 # stop removing elements, escape inner loop
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397 break
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398
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399 else:
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400 if len(D) == 0:
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401 raise nx.NetworkXUnfeasible("Graph contains a cycle.")
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402
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403 # choose next node
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404 q = D.pop()
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405 # "erase" all edges (q, x)
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406 # NOTE: it is important to iterate over edges instead
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407 # of successors, so count is updated correctly in multigraphs
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408 for _, j in G.out_edges(q):
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409 count[j] -= 1
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410 assert count[j] >= 0
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411 if count[j] == 0:
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412 D.append(j)
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413 current_sort.append(q)
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414
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415 # base for current position might _not_ be fixed yet
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416 if len(bases) < len(current_sort):
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417 bases.append(q)
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418
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419 if len(bases) == 0:
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420 break
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421
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422
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423 def is_aperiodic(G):
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424 """Returns True if `G` is aperiodic.
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425
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426 A directed graph is aperiodic if there is no integer k > 1 that
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427 divides the length of every cycle in the graph.
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428
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429 Parameters
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430 ----------
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431 G : NetworkX DiGraph
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432 A directed graph
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433
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434 Returns
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435 -------
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436 bool
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437 True if the graph is aperiodic False otherwise
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438
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439 Raises
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440 ------
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441 NetworkXError
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442 If `G` is not directed
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443
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444 Notes
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445 -----
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446 This uses the method outlined in [1]_, which runs in $O(m)$ time
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447 given $m$ edges in `G`. Note that a graph is not aperiodic if it is
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448 acyclic as every integer trivial divides length 0 cycles.
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449
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450 References
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451 ----------
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452 .. [1] Jarvis, J. P.; Shier, D. R. (1996),
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453 "Graph-theoretic analysis of finite Markov chains,"
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454 in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling:
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455 A Multidisciplinary Approach, CRC Press.
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456 """
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457 if not G.is_directed():
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458 raise nx.NetworkXError(
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459 "is_aperiodic not defined for undirected graphs")
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460
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461 s = arbitrary_element(G)
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462 levels = {s: 0}
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463 this_level = [s]
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464 g = 0
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465 lev = 1
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466 while this_level:
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467 next_level = []
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468 for u in this_level:
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469 for v in G[u]:
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470 if v in levels: # Non-Tree Edge
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471 g = gcd(g, levels[u] - levels[v] + 1)
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472 else: # Tree Edge
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473 next_level.append(v)
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474 levels[v] = lev
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475 this_level = next_level
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476 lev += 1
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477 if len(levels) == len(G): # All nodes in tree
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478 return g == 1
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479 else:
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480 return g == 1 and nx.is_aperiodic(G.subgraph(set(G) - set(levels)))
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481
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482
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483 @not_implemented_for('undirected')
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484 def transitive_closure(G, reflexive=False):
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485 """ Returns transitive closure of a directed graph
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486
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487 The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
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488 for all v, w in V there is an edge (v, w) in E+ if and only if there
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489 is a path from v to w in G.
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490
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491 Handling of paths from v to v has some flexibility within this definition.
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492 A reflexive transitive closure creates a self-loop for the path
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493 from v to v of length 0. The usual transitive closure creates a
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494 self-loop only if a cycle exists (a path from v to v with length > 0).
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495 We also allow an option for no self-loops.
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496
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497 Parameters
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498 ----------
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499 G : NetworkX DiGraph
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500 A directed graph
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501 reflexive : Bool or None, optional (default: False)
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502 Determines when cycles create self-loops in the Transitive Closure.
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503 If True, trivial cycles (length 0) create self-loops. The result
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504 is a reflexive tranistive closure of G.
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505 If False (the default) non-trivial cycles create self-loops.
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506 If None, self-loops are not created.
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507
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508 Returns
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509 -------
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510 NetworkX DiGraph
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511 The transitive closure of `G`
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512
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513 Raises
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514 ------
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515 NetworkXNotImplemented
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516 If `G` is not directed
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517
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518 References
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519 ----------
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520 .. [1] http://www.ics.uci.edu/~eppstein/PADS/PartialOrder.py
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521
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522 TODO this function applies to all directed graphs and is probably misplaced
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523 here in dag.py
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524 """
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525 if reflexive is None:
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diff changeset
526 TC = G.copy()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
527 for v in G:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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528 edges = ((v, u) for u in nx.dfs_preorder_nodes(G, v) if v != u)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
529 TC.add_edges_from(edges)
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diff changeset
530 return TC
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
531 if reflexive is True:
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diff changeset
532 TC = G.copy()
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parents:
diff changeset
533 for v in G:
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guerler
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diff changeset
534 edges = ((v, u) for u in nx.dfs_preorder_nodes(G, v))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
535 TC.add_edges_from(edges)
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diff changeset
536 return TC
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
537 # reflexive is False
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
538 TC = G.copy()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
539 for v in G:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
540 edges = ((v, w) for u, w in nx.edge_dfs(G, v))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
541 TC.add_edges_from(edges)
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diff changeset
542 return TC
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
543
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diff changeset
544
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diff changeset
545 @not_implemented_for('undirected')
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guerler
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diff changeset
546 def transitive_closure_dag(G, topo_order=None):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
547 """ Returns the transitive closure of a directed acyclic graph.
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548
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549 This function is faster than the function `transitive_closure`, but fails
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550 if the graph has a cycle.
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551
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552 The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
553 for all v, w in V there is an edge (v, w) in E+ if and only if there
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
554 is a non-null path from v to w in G.
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guerler
parents:
diff changeset
555
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diff changeset
556 Parameters
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parents:
diff changeset
557 ----------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
558 G : NetworkX DiGraph
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
559 A directed acyclic graph (DAG)
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diff changeset
560
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
561 topo_order: list or tuple, optional
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diff changeset
562 A topological order for G (if None, the function will compute one)
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guerler
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563
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564 Returns
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
565 -------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
566 NetworkX DiGraph
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
567 The transitive closure of `G`
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
568
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
569 Raises
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
570 ------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
571 NetworkXNotImplemented
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
572 If `G` is not directed
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parents:
diff changeset
573 NetworkXUnfeasible
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guerler
parents:
diff changeset
574 If `G` has a cycle
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diff changeset
575
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
576 Notes
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parents:
diff changeset
577 -----
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
578 This algorithm is probably simple enough to be well-known but I didn't find
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
579 a mention in the literature.
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diff changeset
580 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
581 if topo_order is None:
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guerler
parents:
diff changeset
582 topo_order = list(topological_sort(G))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
583
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diff changeset
584 TC = G.copy()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
585
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diff changeset
586 # idea: traverse vertices following a reverse topological order, connecting
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
587 # each vertex to its descendants at distance 2 as we go
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diff changeset
588 for v in reversed(topo_order):
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diff changeset
589 TC.add_edges_from((v, u) for u in descendants_at_distance(TC, v, 2))
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guerler
parents:
diff changeset
590
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parents:
diff changeset
591 return TC
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guerler
parents:
diff changeset
592
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
593
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diff changeset
594 @not_implemented_for('undirected')
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parents:
diff changeset
595 def transitive_reduction(G):
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parents:
diff changeset
596 """ Returns transitive reduction of a directed graph
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diff changeset
597
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
598 The transitive reduction of G = (V,E) is a graph G- = (V,E-) such that
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
599 for all v,w in V there is an edge (v,w) in E- if and only if (v,w) is
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
600 in E and there is no path from v to w in G with length greater than 1.
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guerler
parents:
diff changeset
601
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diff changeset
602 Parameters
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
603 ----------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
604 G : NetworkX DiGraph
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
605 A directed acyclic graph (DAG)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
606
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
607 Returns
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
608 -------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
609 NetworkX DiGraph
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
610 The transitive reduction of `G`
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
611
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
612 Raises
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
613 ------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
614 NetworkXError
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
615 If `G` is not a directed acyclic graph (DAG) transitive reduction is
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
616 not uniquely defined and a :exc:`NetworkXError` exception is raised.
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guerler
parents:
diff changeset
617
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
618 References
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
619 ----------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
620 https://en.wikipedia.org/wiki/Transitive_reduction
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diff changeset
621
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
622 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
623 if not is_directed_acyclic_graph(G):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
624 msg = "Directed Acyclic Graph required for transitive_reduction"
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guerler
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diff changeset
625 raise nx.NetworkXError(msg)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
626 TR = nx.DiGraph()
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diff changeset
627 TR.add_nodes_from(G.nodes())
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diff changeset
628 descendants = {}
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
629 # count before removing set stored in descendants
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
630 check_count = dict(G.in_degree)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
631 for u in G:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
632 u_nbrs = set(G[u])
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
633 for v in G[u]:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
634 if v in u_nbrs:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
635 if v not in descendants:
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guerler
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diff changeset
636 descendants[v] = {y for x, y in nx.dfs_edges(G, v)}
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
637 u_nbrs -= descendants[v]
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diff changeset
638 check_count[v] -= 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
639 if check_count[v] == 0:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
640 del descendants[v]
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guerler
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diff changeset
641 TR.add_edges_from((u, v) for v in u_nbrs)
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diff changeset
642 return TR
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guerler
parents:
diff changeset
643
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
644
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diff changeset
645 @not_implemented_for('undirected')
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
646 def antichains(G, topo_order=None):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
647 """Generates antichains from a directed acyclic graph (DAG).
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guerler
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diff changeset
648
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guerler
parents:
diff changeset
649 An antichain is a subset of a partially ordered set such that any
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
650 two elements in the subset are incomparable.
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diff changeset
651
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diff changeset
652 Parameters
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diff changeset
653 ----------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
654 G : NetworkX DiGraph
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
655 A directed acyclic graph (DAG)
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diff changeset
656
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657 topo_order: list or tuple, optional
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diff changeset
658 A topological order for G (if None, the function will compute one)
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diff changeset
659
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660 Returns
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661 -------
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diff changeset
662 generator object
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663
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diff changeset
664 Raises
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665 ------
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diff changeset
666 NetworkXNotImplemented
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diff changeset
667 If `G` is not directed
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668
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diff changeset
669 NetworkXUnfeasible
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diff changeset
670 If `G` contains a cycle
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diff changeset
671
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diff changeset
672 Notes
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diff changeset
673 -----
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diff changeset
674 This function was originally developed by Peter Jipsen and Franco Saliola
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parents:
diff changeset
675 for the SAGE project. It's included in NetworkX with permission from the
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diff changeset
676 authors. Original SAGE code at:
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diff changeset
677
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diff changeset
678 https://github.com/sagemath/sage/blob/master/src/sage/combinat/posets/hasse_diagram.py
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679
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680 References
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681 ----------
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682 .. [1] Free Lattices, by R. Freese, J. Jezek and J. B. Nation,
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diff changeset
683 AMS, Vol 42, 1995, p. 226.
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diff changeset
684 """
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685 if topo_order is None:
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diff changeset
686 topo_order = list(nx.topological_sort(G))
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diff changeset
687
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diff changeset
688 TC = nx.transitive_closure_dag(G, topo_order)
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diff changeset
689 antichains_stacks = [([], list(reversed(topo_order)))]
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690
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691 while antichains_stacks:
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diff changeset
692 (antichain, stack) = antichains_stacks.pop()
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diff changeset
693 # Invariant:
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diff changeset
694 # - the elements of antichain are independent
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diff changeset
695 # - the elements of stack are independent from those of antichain
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diff changeset
696 yield antichain
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697 while stack:
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698 x = stack.pop()
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parents:
diff changeset
699 new_antichain = antichain + [x]
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diff changeset
700 new_stack = [
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parents:
diff changeset
701 t for t in stack if not ((t in TC[x]) or (x in TC[t]))]
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parents:
diff changeset
702 antichains_stacks.append((new_antichain, new_stack))
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diff changeset
703
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704
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diff changeset
705 @not_implemented_for('undirected')
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diff changeset
706 def dag_longest_path(G, weight='weight', default_weight=1, topo_order=None):
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diff changeset
707 """Returns the longest path in a directed acyclic graph (DAG).
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diff changeset
708
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diff changeset
709 If `G` has edges with `weight` attribute the edge data are used as
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diff changeset
710 weight values.
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diff changeset
711
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diff changeset
712 Parameters
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diff changeset
713 ----------
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diff changeset
714 G : NetworkX DiGraph
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diff changeset
715 A directed acyclic graph (DAG)
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diff changeset
716
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diff changeset
717 weight : str, optional
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diff changeset
718 Edge data key to use for weight
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diff changeset
719
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diff changeset
720 default_weight : int, optional
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diff changeset
721 The weight of edges that do not have a weight attribute
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diff changeset
722
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diff changeset
723 topo_order: list or tuple, optional
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diff changeset
724 A topological order for G (if None, the function will compute one)
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diff changeset
725
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726 Returns
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diff changeset
727 -------
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diff changeset
728 list
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diff changeset
729 Longest path
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diff changeset
730
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diff changeset
731 Raises
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diff changeset
732 ------
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diff changeset
733 NetworkXNotImplemented
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diff changeset
734 If `G` is not directed
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diff changeset
735
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diff changeset
736 See also
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diff changeset
737 --------
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diff changeset
738 dag_longest_path_length
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diff changeset
739
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740 """
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diff changeset
741 if not G:
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diff changeset
742 return []
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diff changeset
743
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744 if topo_order is None:
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diff changeset
745 topo_order = nx.topological_sort(G)
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diff changeset
746
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diff changeset
747 dist = {} # stores {v : (length, u)}
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parents:
diff changeset
748 for v in topo_order:
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parents:
diff changeset
749 us = [(dist[u][0] + data.get(weight, default_weight), u)
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guerler
parents:
diff changeset
750 for u, data in G.pred[v].items()]
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diff changeset
751
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diff changeset
752 # Use the best predecessor if there is one and its distance is
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parents:
diff changeset
753 # non-negative, otherwise terminate.
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diff changeset
754 maxu = max(us, key=lambda x: x[0]) if us else (0, v)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
755 dist[v] = maxu if maxu[0] >= 0 else (0, v)
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diff changeset
756
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757 u = None
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diff changeset
758 v = max(dist, key=lambda x: dist[x][0])
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diff changeset
759 path = []
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diff changeset
760 while u != v:
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diff changeset
761 path.append(v)
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diff changeset
762 u = v
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diff changeset
763 v = dist[v][1]
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diff changeset
764
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diff changeset
765 path.reverse()
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diff changeset
766 return path
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diff changeset
767
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diff changeset
768
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diff changeset
769 @not_implemented_for('undirected')
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diff changeset
770 def dag_longest_path_length(G, weight='weight', default_weight=1):
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parents:
diff changeset
771 """Returns the longest path length in a DAG
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diff changeset
772
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773 Parameters
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diff changeset
774 ----------
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
775 G : NetworkX DiGraph
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diff changeset
776 A directed acyclic graph (DAG)
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diff changeset
777
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778 weight : string, optional
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diff changeset
779 Edge data key to use for weight
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diff changeset
780
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781 default_weight : int, optional
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diff changeset
782 The weight of edges that do not have a weight attribute
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783
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784 Returns
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785 -------
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786 int
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787 Longest path length
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diff changeset
788
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parents:
diff changeset
789 Raises
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diff changeset
790 ------
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parents:
diff changeset
791 NetworkXNotImplemented
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diff changeset
792 If `G` is not directed
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diff changeset
793
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diff changeset
794 See also
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diff changeset
795 --------
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diff changeset
796 dag_longest_path
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parents:
diff changeset
797 """
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diff changeset
798 path = nx.dag_longest_path(G, weight, default_weight)
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parents:
diff changeset
799 path_length = 0
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diff changeset
800 for (u, v) in pairwise(path):
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guerler
parents:
diff changeset
801 path_length += G[u][v].get(weight, default_weight)
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parents:
diff changeset
802
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diff changeset
803 return path_length
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parents:
diff changeset
804
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diff changeset
805
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diff changeset
806 def root_to_leaf_paths(G):
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parents:
diff changeset
807 """Yields root-to-leaf paths in a directed acyclic graph.
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parents:
diff changeset
808
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diff changeset
809 `G` must be a directed acyclic graph. If not, the behavior of this
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parents:
diff changeset
810 function is undefined. A "root" in this graph is a node of in-degree
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parents:
diff changeset
811 zero and a "leaf" a node of out-degree zero.
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parents:
diff changeset
812
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diff changeset
813 When invoked, this function iterates over each path from any root to
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parents:
diff changeset
814 any leaf. A path is a list of nodes.
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parents:
diff changeset
815
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816 """
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817 roots = (v for v, d in G.in_degree() if d == 0)
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diff changeset
818 leaves = (v for v, d in G.out_degree() if d == 0)
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parents:
diff changeset
819 all_paths = partial(nx.all_simple_paths, G)
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parents:
diff changeset
820 # TODO In Python 3, this would be better as `yield from ...`.
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parents:
diff changeset
821 return chaini(starmap(all_paths, product(roots, leaves)))
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diff changeset
822
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823
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diff changeset
824 @not_implemented_for('multigraph')
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parents:
diff changeset
825 @not_implemented_for('undirected')
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parents:
diff changeset
826 def dag_to_branching(G):
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parents:
diff changeset
827 """Returns a branching representing all (overlapping) paths from
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parents:
diff changeset
828 root nodes to leaf nodes in the given directed acyclic graph.
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parents:
diff changeset
829
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diff changeset
830 As described in :mod:`networkx.algorithms.tree.recognition`, a
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diff changeset
831 *branching* is a directed forest in which each node has at most one
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diff changeset
832 parent. In other words, a branching is a disjoint union of
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diff changeset
833 *arborescences*. For this function, each node of in-degree zero in
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parents:
diff changeset
834 `G` becomes a root of one of the arborescences, and there will be
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parents:
diff changeset
835 one leaf node for each distinct path from that root to a leaf node
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diff changeset
836 in `G`.
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837
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838 Each node `v` in `G` with *k* parents becomes *k* distinct nodes in
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diff changeset
839 the returned branching, one for each parent, and the sub-DAG rooted
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diff changeset
840 at `v` is duplicated for each copy. The algorithm then recurses on
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parents:
diff changeset
841 the children of each copy of `v`.
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842
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843 Parameters
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844 ----------
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845 G : NetworkX graph
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diff changeset
846 A directed acyclic graph.
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847
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848 Returns
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diff changeset
849 -------
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diff changeset
850 DiGraph
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diff changeset
851 The branching in which there is a bijection between root-to-leaf
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diff changeset
852 paths in `G` (in which multiple paths may share the same leaf)
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parents:
diff changeset
853 and root-to-leaf paths in the branching (in which there is a
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diff changeset
854 unique path from a root to a leaf).
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diff changeset
855
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diff changeset
856 Each node has an attribute 'source' whose value is the original
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diff changeset
857 node to which this node corresponds. No other graph, node, or
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parents:
diff changeset
858 edge attributes are copied into this new graph.
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diff changeset
859
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diff changeset
860 Raises
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diff changeset
861 ------
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diff changeset
862 NetworkXNotImplemented
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diff changeset
863 If `G` is not directed, or if `G` is a multigraph.
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parents:
diff changeset
864
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diff changeset
865 HasACycle
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diff changeset
866 If `G` is not acyclic.
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parents:
diff changeset
867
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diff changeset
868 Examples
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diff changeset
869 --------
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diff changeset
870 To examine which nodes in the returned branching were produced by
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parents:
diff changeset
871 which original node in the directed acyclic graph, we can collect
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diff changeset
872 the mapping from source node to new nodes into a dictionary. For
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diff changeset
873 example, consider the directed diamond graph::
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diff changeset
874
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diff changeset
875 >>> from collections import defaultdict
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parents:
diff changeset
876 >>> from operator import itemgetter
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diff changeset
877 >>>
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parents:
diff changeset
878 >>> G = nx.DiGraph(nx.utils.pairwise('abd'))
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parents:
diff changeset
879 >>> G.add_edges_from(nx.utils.pairwise('acd'))
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parents:
diff changeset
880 >>> B = nx.dag_to_branching(G)
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parents:
diff changeset
881 >>>
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parents:
diff changeset
882 >>> sources = defaultdict(set)
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parents:
diff changeset
883 >>> for v, source in B.nodes(data='source'):
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parents:
diff changeset
884 ... sources[source].add(v)
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parents:
diff changeset
885 >>> len(sources['a'])
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parents:
diff changeset
886 1
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parents:
diff changeset
887 >>> len(sources['d'])
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diff changeset
888 2
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diff changeset
889
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diff changeset
890 To copy node attributes from the original graph to the new graph,
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diff changeset
891 you can use a dictionary like the one constructed in the above
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diff changeset
892 example::
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diff changeset
893
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diff changeset
894 >>> for source, nodes in sources.items():
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parents:
diff changeset
895 ... for v in nodes:
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diff changeset
896 ... B.nodes[v].update(G.nodes[source])
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parents:
diff changeset
897
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diff changeset
898 Notes
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diff changeset
899 -----
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diff changeset
900 This function is not idempotent in the sense that the node labels in
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diff changeset
901 the returned branching may be uniquely generated each time the
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parents:
diff changeset
902 function is invoked. In fact, the node labels may not be integers;
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parents:
diff changeset
903 in order to relabel the nodes to be more readable, you can use the
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guerler
parents:
diff changeset
904 :func:`networkx.convert_node_labels_to_integers` function.
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parents:
diff changeset
905
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diff changeset
906 The current implementation of this function uses
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diff changeset
907 :func:`networkx.prefix_tree`, so it is subject to the limitations of
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diff changeset
908 that function.
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diff changeset
909
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diff changeset
910 """
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diff changeset
911 if has_cycle(G):
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diff changeset
912 msg = 'dag_to_branching is only defined for acyclic graphs'
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diff changeset
913 raise nx.HasACycle(msg)
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diff changeset
914 paths = root_to_leaf_paths(G)
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diff changeset
915 B, root = nx.prefix_tree(paths)
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parents:
diff changeset
916 # Remove the synthetic `root` and `NIL` nodes in the prefix tree.
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guerler
parents:
diff changeset
917 B.remove_node(root)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
918 B.remove_node(NIL)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
919 return B