Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/tree/mst.py @ 0:4f3585e2f14b draft default tip
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Algorithms for calculating min/max spanning trees/forests. |
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3 |
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4 """ |
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5 from heapq import heappop, heappush |
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6 from operator import itemgetter |
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7 from itertools import count |
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8 from math import isnan |
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9 |
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10 import networkx as nx |
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11 from networkx.utils import UnionFind, not_implemented_for |
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12 |
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13 __all__ = [ |
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14 "minimum_spanning_edges", |
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15 "maximum_spanning_edges", |
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16 "minimum_spanning_tree", |
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17 "maximum_spanning_tree", |
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18 ] |
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19 |
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20 |
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21 @not_implemented_for("multigraph") |
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22 def boruvka_mst_edges( |
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23 G, minimum=True, weight="weight", keys=False, data=True, ignore_nan=False |
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24 ): |
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25 """Iterate over edges of a Borůvka's algorithm min/max spanning tree. |
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26 |
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27 Parameters |
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28 ---------- |
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29 G : NetworkX Graph |
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30 The edges of `G` must have distinct weights, |
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31 otherwise the edges may not form a tree. |
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32 |
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33 minimum : bool (default: True) |
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34 Find the minimum (True) or maximum (False) spanning tree. |
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35 |
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36 weight : string (default: 'weight') |
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37 The name of the edge attribute holding the edge weights. |
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38 |
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39 keys : bool (default: True) |
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40 This argument is ignored since this function is not |
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41 implemented for multigraphs; it exists only for consistency |
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42 with the other minimum spanning tree functions. |
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43 |
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44 data : bool (default: True) |
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45 Flag for whether to yield edge attribute dicts. |
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46 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. |
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47 If False, yield edges `(u, v)`. |
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48 |
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49 ignore_nan : bool (default: False) |
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50 If a NaN is found as an edge weight normally an exception is raised. |
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51 If `ignore_nan is True` then that edge is ignored instead. |
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52 |
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53 """ |
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54 # Initialize a forest, assuming initially that it is the discrete |
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55 # partition of the nodes of the graph. |
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56 forest = UnionFind(G) |
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57 |
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58 def best_edge(component): |
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59 """Returns the optimum (minimum or maximum) edge on the edge |
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60 boundary of the given set of nodes. |
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61 |
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62 A return value of ``None`` indicates an empty boundary. |
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63 |
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64 """ |
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65 sign = 1 if minimum else -1 |
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66 minwt = float("inf") |
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67 boundary = None |
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68 for e in nx.edge_boundary(G, component, data=True): |
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69 wt = e[-1].get(weight, 1) * sign |
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70 if isnan(wt): |
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71 if ignore_nan: |
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72 continue |
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73 msg = f"NaN found as an edge weight. Edge {e}" |
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74 raise ValueError(msg) |
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75 if wt < minwt: |
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76 minwt = wt |
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77 boundary = e |
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78 return boundary |
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79 |
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80 # Determine the optimum edge in the edge boundary of each component |
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81 # in the forest. |
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82 best_edges = (best_edge(component) for component in forest.to_sets()) |
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83 best_edges = [edge for edge in best_edges if edge is not None] |
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84 # If each entry was ``None``, that means the graph was disconnected, |
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85 # so we are done generating the forest. |
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86 while best_edges: |
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87 # Determine the optimum edge in the edge boundary of each |
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88 # component in the forest. |
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89 # |
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90 # This must be a sequence, not an iterator. In this list, the |
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91 # same edge may appear twice, in different orientations (but |
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92 # that's okay, since a union operation will be called on the |
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93 # endpoints the first time it is seen, but not the second time). |
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94 # |
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95 # Any ``None`` indicates that the edge boundary for that |
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96 # component was empty, so that part of the forest has been |
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97 # completed. |
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98 # |
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99 # TODO This can be parallelized, both in the outer loop over |
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100 # each component in the forest and in the computation of the |
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101 # minimum. (Same goes for the identical lines outside the loop.) |
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102 best_edges = (best_edge(component) for component in forest.to_sets()) |
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103 best_edges = [edge for edge in best_edges if edge is not None] |
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104 # Join trees in the forest using the best edges, and yield that |
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105 # edge, since it is part of the spanning tree. |
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106 # |
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107 # TODO This loop can be parallelized, to an extent (the union |
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108 # operation must be atomic). |
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109 for u, v, d in best_edges: |
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110 if forest[u] != forest[v]: |
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111 if data: |
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112 yield u, v, d |
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113 else: |
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114 yield u, v |
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115 forest.union(u, v) |
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116 |
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117 |
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118 def kruskal_mst_edges( |
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119 G, minimum, weight="weight", keys=True, data=True, ignore_nan=False |
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120 ): |
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121 """Iterate over edges of a Kruskal's algorithm min/max spanning tree. |
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122 |
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123 Parameters |
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124 ---------- |
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125 G : NetworkX Graph |
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126 The graph holding the tree of interest. |
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127 |
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128 minimum : bool (default: True) |
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129 Find the minimum (True) or maximum (False) spanning tree. |
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130 |
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131 weight : string (default: 'weight') |
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132 The name of the edge attribute holding the edge weights. |
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133 |
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134 keys : bool (default: True) |
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135 If `G` is a multigraph, `keys` controls whether edge keys ar yielded. |
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136 Otherwise `keys` is ignored. |
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137 |
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138 data : bool (default: True) |
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139 Flag for whether to yield edge attribute dicts. |
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140 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. |
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141 If False, yield edges `(u, v)`. |
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142 |
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143 ignore_nan : bool (default: False) |
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144 If a NaN is found as an edge weight normally an exception is raised. |
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145 If `ignore_nan is True` then that edge is ignored instead. |
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146 |
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147 """ |
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148 subtrees = UnionFind() |
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149 if G.is_multigraph(): |
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150 edges = G.edges(keys=True, data=True) |
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151 |
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152 def filter_nan_edges(edges=edges, weight=weight): |
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153 sign = 1 if minimum else -1 |
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154 for u, v, k, d in edges: |
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155 wt = d.get(weight, 1) * sign |
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156 if isnan(wt): |
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157 if ignore_nan: |
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158 continue |
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159 msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}" |
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160 raise ValueError(msg) |
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161 yield wt, u, v, k, d |
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162 |
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163 else: |
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164 edges = G.edges(data=True) |
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165 |
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166 def filter_nan_edges(edges=edges, weight=weight): |
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167 sign = 1 if minimum else -1 |
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168 for u, v, d in edges: |
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169 wt = d.get(weight, 1) * sign |
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170 if isnan(wt): |
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171 if ignore_nan: |
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172 continue |
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173 msg = f"NaN found as an edge weight. Edge {(u, v, d)}" |
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174 raise ValueError(msg) |
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175 yield wt, u, v, d |
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176 |
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177 edges = sorted(filter_nan_edges(), key=itemgetter(0)) |
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178 # Multigraphs need to handle edge keys in addition to edge data. |
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179 if G.is_multigraph(): |
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180 for wt, u, v, k, d in edges: |
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181 if subtrees[u] != subtrees[v]: |
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182 if keys: |
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183 if data: |
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184 yield u, v, k, d |
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185 else: |
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186 yield u, v, k |
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187 else: |
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188 if data: |
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189 yield u, v, d |
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190 else: |
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191 yield u, v |
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192 subtrees.union(u, v) |
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193 else: |
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194 for wt, u, v, d in edges: |
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195 if subtrees[u] != subtrees[v]: |
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196 if data: |
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197 yield (u, v, d) |
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198 else: |
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199 yield (u, v) |
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200 subtrees.union(u, v) |
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201 |
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202 |
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203 def prim_mst_edges(G, minimum, weight="weight", keys=True, data=True, ignore_nan=False): |
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204 """Iterate over edges of Prim's algorithm min/max spanning tree. |
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205 |
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206 Parameters |
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207 ---------- |
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208 G : NetworkX Graph |
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209 The graph holding the tree of interest. |
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210 |
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211 minimum : bool (default: True) |
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212 Find the minimum (True) or maximum (False) spanning tree. |
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213 |
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214 weight : string (default: 'weight') |
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215 The name of the edge attribute holding the edge weights. |
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216 |
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217 keys : bool (default: True) |
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218 If `G` is a multigraph, `keys` controls whether edge keys ar yielded. |
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219 Otherwise `keys` is ignored. |
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220 |
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221 data : bool (default: True) |
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222 Flag for whether to yield edge attribute dicts. |
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223 If True, yield edges `(u, v, d)`, where `d` is the attribute dict. |
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224 If False, yield edges `(u, v)`. |
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225 |
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226 ignore_nan : bool (default: False) |
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227 If a NaN is found as an edge weight normally an exception is raised. |
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228 If `ignore_nan is True` then that edge is ignored instead. |
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229 |
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230 """ |
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231 is_multigraph = G.is_multigraph() |
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232 push = heappush |
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233 pop = heappop |
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234 |
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235 nodes = set(G) |
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236 c = count() |
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237 |
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238 sign = 1 if minimum else -1 |
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239 |
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240 while nodes: |
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241 u = nodes.pop() |
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242 frontier = [] |
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243 visited = {u} |
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244 if is_multigraph: |
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245 for v, keydict in G.adj[u].items(): |
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246 for k, d in keydict.items(): |
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247 wt = d.get(weight, 1) * sign |
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248 if isnan(wt): |
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249 if ignore_nan: |
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250 continue |
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251 msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}" |
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252 raise ValueError(msg) |
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253 push(frontier, (wt, next(c), u, v, k, d)) |
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254 else: |
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255 for v, d in G.adj[u].items(): |
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256 wt = d.get(weight, 1) * sign |
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257 if isnan(wt): |
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258 if ignore_nan: |
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259 continue |
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260 msg = f"NaN found as an edge weight. Edge {(u, v, d)}" |
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261 raise ValueError(msg) |
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262 push(frontier, (wt, next(c), u, v, d)) |
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263 while frontier: |
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264 if is_multigraph: |
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265 W, _, u, v, k, d = pop(frontier) |
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266 else: |
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267 W, _, u, v, d = pop(frontier) |
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268 if v in visited or v not in nodes: |
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269 continue |
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270 # Multigraphs need to handle edge keys in addition to edge data. |
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271 if is_multigraph and keys: |
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272 if data: |
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273 yield u, v, k, d |
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274 else: |
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275 yield u, v, k |
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276 else: |
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277 if data: |
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278 yield u, v, d |
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279 else: |
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280 yield u, v |
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281 # update frontier |
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282 visited.add(v) |
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283 nodes.discard(v) |
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284 if is_multigraph: |
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285 for w, keydict in G.adj[v].items(): |
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286 if w in visited: |
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287 continue |
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288 for k2, d2 in keydict.items(): |
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289 new_weight = d2.get(weight, 1) * sign |
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290 push(frontier, (new_weight, next(c), v, w, k2, d2)) |
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291 else: |
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292 for w, d2 in G.adj[v].items(): |
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293 if w in visited: |
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294 continue |
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295 new_weight = d2.get(weight, 1) * sign |
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296 push(frontier, (new_weight, next(c), v, w, d2)) |
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297 |
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298 |
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299 ALGORITHMS = { |
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300 "boruvka": boruvka_mst_edges, |
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301 "borůvka": boruvka_mst_edges, |
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302 "kruskal": kruskal_mst_edges, |
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303 "prim": prim_mst_edges, |
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304 } |
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305 |
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306 |
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307 @not_implemented_for("directed") |
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308 def minimum_spanning_edges( |
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309 G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False |
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310 ): |
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311 """Generate edges in a minimum spanning forest of an undirected |
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312 weighted graph. |
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313 |
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314 A minimum spanning tree is a subgraph of the graph (a tree) |
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315 with the minimum sum of edge weights. A spanning forest is a |
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316 union of the spanning trees for each connected component of the graph. |
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317 |
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318 Parameters |
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319 ---------- |
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320 G : undirected Graph |
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321 An undirected graph. If `G` is connected, then the algorithm finds a |
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322 spanning tree. Otherwise, a spanning forest is found. |
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323 |
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324 algorithm : string |
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325 The algorithm to use when finding a minimum spanning tree. Valid |
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326 choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. |
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327 |
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328 weight : string |
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329 Edge data key to use for weight (default 'weight'). |
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330 |
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331 keys : bool |
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332 Whether to yield edge key in multigraphs in addition to the edge. |
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333 If `G` is not a multigraph, this is ignored. |
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334 |
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335 data : bool, optional |
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336 If True yield the edge data along with the edge. |
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337 |
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338 ignore_nan : bool (default: False) |
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339 If a NaN is found as an edge weight normally an exception is raised. |
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340 If `ignore_nan is True` then that edge is ignored instead. |
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341 |
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342 Returns |
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343 ------- |
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344 edges : iterator |
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345 An iterator over edges in a maximum spanning tree of `G`. |
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346 Edges connecting nodes `u` and `v` are represented as tuples: |
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347 `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` |
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348 |
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349 If `G` is a multigraph, `keys` indicates whether the edge key `k` will |
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350 be reported in the third position in the edge tuple. `data` indicates |
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351 whether the edge datadict `d` will appear at the end of the edge tuple. |
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352 |
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353 If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True |
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354 or `(u, v)` if `data` is False. |
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355 |
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356 Examples |
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357 -------- |
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358 >>> from networkx.algorithms import tree |
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359 |
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360 Find minimum spanning edges by Kruskal's algorithm |
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361 |
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362 >>> G = nx.cycle_graph(4) |
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363 >>> G.add_edge(0, 3, weight=2) |
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364 >>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False) |
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365 >>> edgelist = list(mst) |
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366 >>> sorted(sorted(e) for e in edgelist) |
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367 [[0, 1], [1, 2], [2, 3]] |
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368 |
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369 Find minimum spanning edges by Prim's algorithm |
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370 |
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371 >>> G = nx.cycle_graph(4) |
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372 >>> G.add_edge(0, 3, weight=2) |
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373 >>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False) |
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374 >>> edgelist = list(mst) |
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375 >>> sorted(sorted(e) for e in edgelist) |
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376 [[0, 1], [1, 2], [2, 3]] |
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377 |
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378 Notes |
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379 ----- |
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380 For Borůvka's algorithm, each edge must have a weight attribute, and |
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381 each edge weight must be distinct. |
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382 |
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383 For the other algorithms, if the graph edges do not have a weight |
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384 attribute a default weight of 1 will be used. |
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385 |
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386 Modified code from David Eppstein, April 2006 |
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387 http://www.ics.uci.edu/~eppstein/PADS/ |
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388 |
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389 """ |
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390 try: |
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391 algo = ALGORITHMS[algorithm] |
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392 except KeyError as e: |
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393 msg = f"{algorithm} is not a valid choice for an algorithm." |
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394 raise ValueError(msg) from e |
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395 |
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396 return algo( |
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397 G, minimum=True, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan |
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398 ) |
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399 |
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400 |
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401 @not_implemented_for("directed") |
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402 def maximum_spanning_edges( |
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403 G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False |
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404 ): |
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405 """Generate edges in a maximum spanning forest of an undirected |
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406 weighted graph. |
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407 |
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408 A maximum spanning tree is a subgraph of the graph (a tree) |
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409 with the maximum possible sum of edge weights. A spanning forest is a |
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410 union of the spanning trees for each connected component of the graph. |
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411 |
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412 Parameters |
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413 ---------- |
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414 G : undirected Graph |
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415 An undirected graph. If `G` is connected, then the algorithm finds a |
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416 spanning tree. Otherwise, a spanning forest is found. |
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417 |
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418 algorithm : string |
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419 The algorithm to use when finding a maximum spanning tree. Valid |
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420 choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'. |
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421 |
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422 weight : string |
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423 Edge data key to use for weight (default 'weight'). |
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424 |
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425 keys : bool |
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426 Whether to yield edge key in multigraphs in addition to the edge. |
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427 If `G` is not a multigraph, this is ignored. |
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428 |
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429 data : bool, optional |
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430 If True yield the edge data along with the edge. |
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431 |
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432 ignore_nan : bool (default: False) |
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433 If a NaN is found as an edge weight normally an exception is raised. |
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434 If `ignore_nan is True` then that edge is ignored instead. |
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435 |
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436 Returns |
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437 ------- |
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438 edges : iterator |
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439 An iterator over edges in a maximum spanning tree of `G`. |
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440 Edges connecting nodes `u` and `v` are represented as tuples: |
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441 `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)` |
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442 |
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443 If `G` is a multigraph, `keys` indicates whether the edge key `k` will |
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444 be reported in the third position in the edge tuple. `data` indicates |
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445 whether the edge datadict `d` will appear at the end of the edge tuple. |
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446 |
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447 If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True |
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448 or `(u, v)` if `data` is False. |
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449 |
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450 Examples |
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451 -------- |
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452 >>> from networkx.algorithms import tree |
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453 |
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454 Find maximum spanning edges by Kruskal's algorithm |
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455 |
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456 >>> G = nx.cycle_graph(4) |
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457 >>> G.add_edge(0, 3, weight=2) |
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458 >>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False) |
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459 >>> edgelist = list(mst) |
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460 >>> sorted(sorted(e) for e in edgelist) |
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461 [[0, 1], [0, 3], [1, 2]] |
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462 |
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463 Find maximum spanning edges by Prim's algorithm |
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464 |
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465 >>> G = nx.cycle_graph(4) |
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466 >>> G.add_edge(0, 3, weight=2) # assign weight 2 to edge 0-3 |
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467 >>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False) |
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468 >>> edgelist = list(mst) |
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469 >>> sorted(sorted(e) for e in edgelist) |
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470 [[0, 1], [0, 3], [2, 3]] |
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471 |
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472 Notes |
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473 ----- |
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474 For Borůvka's algorithm, each edge must have a weight attribute, and |
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475 each edge weight must be distinct. |
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476 |
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477 For the other algorithms, if the graph edges do not have a weight |
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478 attribute a default weight of 1 will be used. |
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479 |
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480 Modified code from David Eppstein, April 2006 |
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481 http://www.ics.uci.edu/~eppstein/PADS/ |
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482 """ |
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483 try: |
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484 algo = ALGORITHMS[algorithm] |
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485 except KeyError as e: |
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486 msg = f"{algorithm} is not a valid choice for an algorithm." |
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487 raise ValueError(msg) from e |
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488 |
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489 return algo( |
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490 G, minimum=False, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan |
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491 ) |
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492 |
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493 |
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494 def minimum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): |
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495 """Returns a minimum spanning tree or forest on an undirected graph `G`. |
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496 |
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497 Parameters |
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498 ---------- |
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499 G : undirected graph |
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500 An undirected graph. If `G` is connected, then the algorithm finds a |
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501 spanning tree. Otherwise, a spanning forest is found. |
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502 |
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503 weight : str |
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504 Data key to use for edge weights. |
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505 |
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506 algorithm : string |
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507 The algorithm to use when finding a minimum spanning tree. Valid |
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508 choices are 'kruskal', 'prim', or 'boruvka'. The default is |
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509 'kruskal'. |
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510 |
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511 ignore_nan : bool (default: False) |
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512 If a NaN is found as an edge weight normally an exception is raised. |
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513 If `ignore_nan is True` then that edge is ignored instead. |
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514 |
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515 Returns |
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516 ------- |
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517 G : NetworkX Graph |
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518 A minimum spanning tree or forest. |
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519 |
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520 Examples |
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521 -------- |
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522 >>> G = nx.cycle_graph(4) |
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523 >>> G.add_edge(0, 3, weight=2) |
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524 >>> T = nx.minimum_spanning_tree(G) |
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525 >>> sorted(T.edges(data=True)) |
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526 [(0, 1, {}), (1, 2, {}), (2, 3, {})] |
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527 |
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528 |
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529 Notes |
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530 ----- |
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531 For Borůvka's algorithm, each edge must have a weight attribute, and |
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532 each edge weight must be distinct. |
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533 |
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534 For the other algorithms, if the graph edges do not have a weight |
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535 attribute a default weight of 1 will be used. |
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536 |
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537 There may be more than one tree with the same minimum or maximum weight. |
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538 See :mod:`networkx.tree.recognition` for more detailed definitions. |
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539 |
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540 Isolated nodes with self-loops are in the tree as edgeless isolated nodes. |
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541 |
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542 """ |
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543 edges = minimum_spanning_edges( |
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544 G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan |
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545 ) |
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546 T = G.__class__() # Same graph class as G |
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547 T.graph.update(G.graph) |
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548 T.add_nodes_from(G.nodes.items()) |
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549 T.add_edges_from(edges) |
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550 return T |
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551 |
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552 |
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553 def maximum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False): |
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554 """Returns a maximum spanning tree or forest on an undirected graph `G`. |
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555 |
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556 Parameters |
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557 ---------- |
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558 G : undirected graph |
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559 An undirected graph. If `G` is connected, then the algorithm finds a |
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560 spanning tree. Otherwise, a spanning forest is found. |
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561 |
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562 weight : str |
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563 Data key to use for edge weights. |
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564 |
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565 algorithm : string |
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566 The algorithm to use when finding a maximum spanning tree. Valid |
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567 choices are 'kruskal', 'prim', or 'boruvka'. The default is |
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568 'kruskal'. |
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569 |
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570 ignore_nan : bool (default: False) |
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571 If a NaN is found as an edge weight normally an exception is raised. |
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572 If `ignore_nan is True` then that edge is ignored instead. |
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573 |
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574 |
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575 Returns |
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576 ------- |
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577 G : NetworkX Graph |
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578 A maximum spanning tree or forest. |
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579 |
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580 |
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581 Examples |
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582 -------- |
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583 >>> G = nx.cycle_graph(4) |
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584 >>> G.add_edge(0, 3, weight=2) |
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585 >>> T = nx.maximum_spanning_tree(G) |
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586 >>> sorted(T.edges(data=True)) |
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587 [(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})] |
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588 |
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589 |
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590 Notes |
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591 ----- |
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592 For Borůvka's algorithm, each edge must have a weight attribute, and |
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593 each edge weight must be distinct. |
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594 |
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595 For the other algorithms, if the graph edges do not have a weight |
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596 attribute a default weight of 1 will be used. |
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597 |
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598 There may be more than one tree with the same minimum or maximum weight. |
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599 See :mod:`networkx.tree.recognition` for more detailed definitions. |
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600 |
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601 Isolated nodes with self-loops are in the tree as edgeless isolated nodes. |
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602 |
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603 """ |
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604 edges = maximum_spanning_edges( |
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605 G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan |
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606 ) |
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607 edges = list(edges) |
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608 T = G.__class__() # Same graph class as G |
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609 T.graph.update(G.graph) |
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610 T.add_nodes_from(G.nodes.items()) |
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611 T.add_edges_from(edges) |
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612 return T |