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

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # Copyright (C) 2018
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2 # Robert Gmyr <robert@gmyr.net>
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3 # All rights reserved.
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4 # BSD license.
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5 """Functions for computing sparsifiers of graphs."""
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6 import math
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7 import networkx as nx
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8 from networkx.utils import not_implemented_for, py_random_state
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9
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10 __all__ = ['spanner']
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11
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12
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13 @py_random_state(3)
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14 @not_implemented_for('directed')
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15 @not_implemented_for('multigraph')
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16 def spanner(G, stretch, weight=None, seed=None):
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17 """Returns a spanner of the given graph with the given stretch.
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18
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19 A spanner of a graph G = (V, E) with stretch t is a subgraph
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20 H = (V, E_S) such that E_S is a subset of E and the distance between
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21 any pair of nodes in H is at most t times the distance between the
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22 nodes in G.
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23
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24 Parameters
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25 ----------
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26 G : NetworkX graph
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27 An undirected simple graph.
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28
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29 stretch : float
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30 The stretch of the spanner.
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31
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32 weight : object
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33 The edge attribute to use as distance.
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34
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35 seed : integer, random_state, or None (default)
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36 Indicator of random number generation state.
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37 See :ref:`Randomness<randomness>`.
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38
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39 Returns
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40 -------
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41 NetworkX graph
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42 A spanner of the given graph with the given stretch.
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43
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44 Raises
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45 ------
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46 ValueError
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47 If a stretch less than 1 is given.
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48
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49 Notes
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50 -----
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51 This function implements the spanner algorithm by Baswana and Sen,
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52 see [1].
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53
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54 This algorithm is a randomized las vegas algorithm: The expected
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55 running time is O(km) where k = (stretch + 1) // 2 and m is the
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56 number of edges in G. The returned graph is always a spanner of the
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57 given graph with the specified stretch. For weighted graphs the
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58 number of edges in the spanner is O(k * n^(1 + 1 / k)) where k is
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59 defined as above and n is the number of nodes in G. For unweighted
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60 graphs the number of edges is O(n^(1 + 1 / k) + kn).
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61
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62 References
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63 ----------
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64 [1] S. Baswana, S. Sen. A Simple and Linear Time Randomized
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65 Algorithm for Computing Sparse Spanners in Weighted Graphs.
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66 Random Struct. Algorithms 30(4): 532-563 (2007).
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67 """
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68 if stretch < 1:
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69 raise ValueError('stretch must be at least 1')
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70
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71 k = (stretch + 1) // 2
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72
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73 # initialize spanner H with empty edge set
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74 H = nx.empty_graph()
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75 H.add_nodes_from(G.nodes)
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76
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77 # phase 1: forming the clusters
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78 # the residual graph has V' from the paper as its node set
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79 # and E' from the paper as its edge set
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80 residual_graph = _setup_residual_graph(G, weight)
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81 # clustering is a dictionary that maps nodes in a cluster to the
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82 # cluster center
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83 clustering = {v: v for v in G.nodes}
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84 sample_prob = math.pow(G.number_of_nodes(), - 1 / k)
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85 size_limit = 2 * math.pow(G.number_of_nodes(), 1 + 1 / k)
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86
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87 i = 0
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88 while i < k - 1:
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89 # step 1: sample centers
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90 sampled_centers = set()
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91 for center in set(clustering.values()):
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92 if seed.random() < sample_prob:
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93 sampled_centers.add(center)
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94
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95 # combined loop for steps 2 and 3
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96 edges_to_add = set()
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97 edges_to_remove = set()
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98 new_clustering = {}
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99 for v in residual_graph.nodes:
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100 if clustering[v] in sampled_centers:
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101 continue
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102
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103 # step 2: find neighboring (sampled) clusters and
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104 # lightest edges to them
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105 lightest_edge_neighbor, lightest_edge_weight =\
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106 _lightest_edge_dicts(residual_graph, clustering, v)
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107 neighboring_sampled_centers =\
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108 set(lightest_edge_weight.keys()) & sampled_centers
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109
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110 # step 3: add edges to spanner
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111 if not neighboring_sampled_centers:
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112 # connect to each neighboring center via lightest edge
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113 for neighbor in lightest_edge_neighbor.values():
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114 edges_to_add.add((v, neighbor))
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115 # remove all incident edges
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116 for neighbor in residual_graph.adj[v]:
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117 edges_to_remove.add((v, neighbor))
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118
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119 else: # there is a neighboring sampled center
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120 closest_center = min(neighboring_sampled_centers,
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121 key=lightest_edge_weight.get)
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122 closest_center_weight = lightest_edge_weight[closest_center]
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123 closest_center_neighbor =\
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124 lightest_edge_neighbor[closest_center]
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125
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126 edges_to_add.add((v, closest_center_neighbor))
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127 new_clustering[v] = closest_center
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128
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129 # connect to centers with edge weight less than
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130 # closest_center_weight
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131 for center, edge_weight in lightest_edge_weight.items():
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132 if edge_weight < closest_center_weight:
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133 neighbor = lightest_edge_neighbor[center]
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134 edges_to_add.add((v, neighbor))
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135
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136 # remove edges to centers with edge weight less than
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137 # closest_center_weight
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138 for neighbor in residual_graph.adj[v]:
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139 neighbor_cluster = clustering[neighbor]
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140 neighbor_weight = lightest_edge_weight[neighbor_cluster]
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141 if neighbor_cluster == closest_center or neighbor_weight < closest_center_weight:
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142 edges_to_remove.add((v, neighbor))
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143
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144 # check whether iteration added too many edges to spanner,
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145 # if so repeat
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146 if len(edges_to_add) > size_limit:
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147 # an iteration is repeated O(1) times on expectation
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148 continue
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149
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150 # iteration succeeded
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151 i = i + 1
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152
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153 # actually add edges to spanner
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154 for u, v in edges_to_add:
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155 _add_edge_to_spanner(H, residual_graph, u, v, weight)
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156
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157 # actually delete edges from residual graph
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158 residual_graph.remove_edges_from(edges_to_remove)
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159
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160 # copy old clustering data to new_clustering
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161 for node, center in clustering.items():
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162 if center in sampled_centers:
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163 new_clustering[node] = center
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164 clustering = new_clustering
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165
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166 # step 4: remove intra-cluster edges
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167 for u in residual_graph.nodes:
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168 for v in list(residual_graph.adj[u]):
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169 if clustering[u] == clustering[v]:
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170 residual_graph.remove_edge(u, v)
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171
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172 # update residual graph node set
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173 for v in list(residual_graph.nodes):
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174 if v not in clustering:
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175 residual_graph.remove_node(v)
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176
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177 # phase 2: vertex-cluster joining
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178 for v in residual_graph.nodes:
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179 lightest_edge_neighbor, _ =\
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180 _lightest_edge_dicts(residual_graph, clustering, v)
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181 for neighbor in lightest_edge_neighbor.values():
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182 _add_edge_to_spanner(H, residual_graph, v, neighbor, weight)
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183
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184 return H
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185
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186
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187 def _setup_residual_graph(G, weight):
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188 """Setup residual graph as a copy of G with unique edges weights.
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189
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190 The node set of the residual graph corresponds to the set V' from
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191 the Baswana-Sen paper and the edge set corresponds to the set E'
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192 from the paper.
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193
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194 This function associates distinct weights to the edges of the
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195 residual graph (even for unweighted input graphs), as required by
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196 the algorithm.
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197
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198 Parameters
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199 ----------
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200 G : NetworkX graph
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201 An undirected simple graph.
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202
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203 weight : object
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204 The edge attribute to use as distance.
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205
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206 Returns
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207 -------
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208 NetworkX graph
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209 The residual graph used for the Baswana-Sen algorithm.
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210 """
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211 residual_graph = G.copy()
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212
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213 # establish unique edge weights, even for unweighted graphs
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214 for u, v in G.edges():
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215 if not weight:
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216 residual_graph[u][v]['weight'] = (id(u), id(v))
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217 else:
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218 residual_graph[u][v]['weight'] = (G[u][v][weight], id(u), id(v))
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219
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220 return residual_graph
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221
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222
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223 def _lightest_edge_dicts(residual_graph, clustering, node):
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224 """Find the lightest edge to each cluster.
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225
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226 Searches for the minimum-weight edge to each cluster adjacent to
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227 the given node.
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228
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229 Parameters
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230 ----------
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231 residual_graph : NetworkX graph
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232 The residual graph used by the Baswana-Sen algorithm.
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233
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234 clustering : dictionary
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235 The current clustering of the nodes.
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236
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237 node : node
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238 The node from which the search originates.
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239
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240 Returns
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241 -------
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242 lightest_edge_neighbor, lightest_edge_weight : dictionary, dictionary
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243 lightest_edge_neighbor is a dictionary that maps a center C to
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244 a node v in the corresponding cluster such that the edge from
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245 the given node to v is the lightest edge from the given node to
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246 any node in cluster. lightest_edge_weight maps a center C to the
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247 weight of the aforementioned edge.
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248
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249 Notes
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250 -----
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251 If a cluster has no node that is adjacent to the given node in the
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252 residual graph then the center of the cluster is not a key in the
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253 returned dictionaries.
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254 """
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255 lightest_edge_neighbor = {}
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256 lightest_edge_weight = {}
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257 for neighbor in residual_graph.adj[node]:
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258 neighbor_center = clustering[neighbor]
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259 weight = residual_graph[node][neighbor]['weight']
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260 if neighbor_center not in lightest_edge_weight or\
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261 weight < lightest_edge_weight[neighbor_center]:
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262 lightest_edge_neighbor[neighbor_center] = neighbor
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263 lightest_edge_weight[neighbor_center] = weight
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264 return lightest_edge_neighbor, lightest_edge_weight
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265
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266
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267 def _add_edge_to_spanner(H, residual_graph, u, v, weight):
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268 """Add the edge {u, v} to the spanner H and take weight from
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269 the residual graph.
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270
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271 Parameters
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272 ----------
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273 H : NetworkX graph
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274 The spanner under construction.
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275
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276 residual_graph : NetworkX graph
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277 The residual graph used by the Baswana-Sen algorithm. The weight
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278 for the edge is taken from this graph.
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279
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280 u : node
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281 One endpoint of the edge.
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282
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283 v : node
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284 The other endpoint of the edge.
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285
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286 weight : object
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287 The edge attribute to use as distance.
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288 """
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289 H.add_edge(u, v)
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290 if weight:
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291 H[u][v][weight] = residual_graph[u][v]['weight'][0]