Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/tournament.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Functions concerning tournament graphs. |
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2 |
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3 A `tournament graph`_ is a complete oriented graph. In other words, it |
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4 is a directed graph in which there is exactly one directed edge joining |
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5 each pair of distinct nodes. For each function in this module that |
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6 accepts a graph as input, you must provide a tournament graph. The |
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7 responsibility is on the caller to ensure that the graph is a tournament |
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8 graph. |
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9 |
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10 To access the functions in this module, you must access them through the |
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11 :mod:`networkx.algorithms.tournament` module:: |
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12 |
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13 >>> from networkx.algorithms import tournament |
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14 >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 0)]) |
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15 >>> tournament.is_tournament(G) |
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16 True |
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17 |
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18 .. _tournament graph: https://en.wikipedia.org/wiki/Tournament_%28graph_theory%29 |
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19 |
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20 """ |
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21 from itertools import combinations |
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22 |
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23 import networkx as nx |
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24 from networkx.algorithms.simple_paths import is_simple_path as is_path |
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25 from networkx.utils import arbitrary_element |
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26 from networkx.utils import not_implemented_for |
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27 from networkx.utils import py_random_state |
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28 |
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29 __all__ = [ |
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30 "hamiltonian_path", |
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31 "is_reachable", |
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32 "is_strongly_connected", |
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33 "is_tournament", |
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34 "random_tournament", |
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35 "score_sequence", |
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36 ] |
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37 |
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38 |
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39 def index_satisfying(iterable, condition): |
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40 """Returns the index of the first element in `iterable` that |
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41 satisfies the given condition. |
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42 |
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43 If no such element is found (that is, when the iterable is |
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44 exhausted), this returns the length of the iterable (that is, one |
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45 greater than the last index of the iterable). |
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46 |
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47 `iterable` must not be empty. If `iterable` is empty, this |
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48 function raises :exc:`ValueError`. |
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49 |
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50 """ |
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51 # Pre-condition: iterable must not be empty. |
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52 for i, x in enumerate(iterable): |
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53 if condition(x): |
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54 return i |
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55 # If we reach the end of the iterable without finding an element |
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56 # that satisfies the condition, return the length of the iterable, |
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57 # which is one greater than the index of its last element. If the |
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58 # iterable was empty, `i` will not be defined, so we raise an |
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59 # exception. |
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60 try: |
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61 return i + 1 |
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62 except NameError as e: |
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63 raise ValueError("iterable must be non-empty") from e |
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64 |
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65 |
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66 @not_implemented_for("undirected") |
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67 @not_implemented_for("multigraph") |
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68 def is_tournament(G): |
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69 """Returns True if and only if `G` is a tournament. |
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70 |
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71 A tournament is a directed graph, with neither self-loops nor |
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72 multi-edges, in which there is exactly one directed edge joining |
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73 each pair of distinct nodes. |
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74 |
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75 Parameters |
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76 ---------- |
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77 G : NetworkX graph |
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78 A directed graph representing a tournament. |
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79 |
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80 Returns |
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81 ------- |
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82 bool |
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83 Whether the given graph is a tournament graph. |
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84 |
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85 Notes |
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86 ----- |
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87 Some definitions require a self-loop on each node, but that is not |
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88 the convention used here. |
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89 |
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90 """ |
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91 # In a tournament, there is exactly one directed edge joining each pair. |
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92 return ( |
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93 all((v in G[u]) ^ (u in G[v]) for u, v in combinations(G, 2)) |
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94 and nx.number_of_selfloops(G) == 0 |
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95 ) |
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96 |
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97 |
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98 @not_implemented_for("undirected") |
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99 @not_implemented_for("multigraph") |
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100 def hamiltonian_path(G): |
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101 """Returns a Hamiltonian path in the given tournament graph. |
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102 |
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103 Each tournament has a Hamiltonian path. If furthermore, the |
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104 tournament is strongly connected, then the returned Hamiltonian path |
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105 is a Hamiltonian cycle (by joining the endpoints of the path). |
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106 |
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107 Parameters |
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108 ---------- |
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109 G : NetworkX graph |
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110 A directed graph representing a tournament. |
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111 |
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112 Returns |
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113 ------- |
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114 bool |
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115 Whether the given graph is a tournament graph. |
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116 |
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117 Notes |
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118 ----- |
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119 This is a recursive implementation with an asymptotic running time |
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120 of $O(n^2)$, ignoring multiplicative polylogarithmic factors, where |
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121 $n$ is the number of nodes in the graph. |
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122 |
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123 """ |
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124 if len(G) == 0: |
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125 return [] |
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126 if len(G) == 1: |
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127 return [arbitrary_element(G)] |
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128 v = arbitrary_element(G) |
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129 hampath = hamiltonian_path(G.subgraph(set(G) - {v})) |
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130 # Get the index of the first node in the path that does *not* have |
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131 # an edge to `v`, then insert `v` before that node. |
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132 index = index_satisfying(hampath, lambda u: v not in G[u]) |
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133 hampath.insert(index, v) |
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134 return hampath |
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135 |
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136 |
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137 @py_random_state(1) |
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138 def random_tournament(n, seed=None): |
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139 r"""Returns a random tournament graph on `n` nodes. |
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140 |
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141 Parameters |
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142 ---------- |
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143 n : int |
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144 The number of nodes in the returned graph. |
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145 seed : integer, random_state, or None (default) |
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146 Indicator of random number generation state. |
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147 See :ref:`Randomness<randomness>`. |
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148 |
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149 Returns |
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150 ------- |
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151 bool |
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152 Whether the given graph is a tournament graph. |
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153 |
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154 Notes |
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155 ----- |
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156 This algorithm adds, for each pair of distinct nodes, an edge with |
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157 uniformly random orientation. In other words, `\binom{n}{2}` flips |
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158 of an unbiased coin decide the orientations of the edges in the |
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159 graph. |
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160 |
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161 """ |
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162 # Flip an unbiased coin for each pair of distinct nodes. |
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163 coins = (seed.random() for i in range((n * (n - 1)) // 2)) |
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164 pairs = combinations(range(n), 2) |
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165 edges = ((u, v) if r < 0.5 else (v, u) for (u, v), r in zip(pairs, coins)) |
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166 return nx.DiGraph(edges) |
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167 |
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168 |
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169 @not_implemented_for("undirected") |
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170 @not_implemented_for("multigraph") |
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171 def score_sequence(G): |
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172 """Returns the score sequence for the given tournament graph. |
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173 |
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174 The score sequence is the sorted list of the out-degrees of the |
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175 nodes of the graph. |
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176 |
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177 Parameters |
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178 ---------- |
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179 G : NetworkX graph |
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180 A directed graph representing a tournament. |
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181 |
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182 Returns |
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183 ------- |
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184 list |
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185 A sorted list of the out-degrees of the nodes of `G`. |
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186 |
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187 """ |
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188 return sorted(d for v, d in G.out_degree()) |
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189 |
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190 |
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191 @not_implemented_for("undirected") |
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192 @not_implemented_for("multigraph") |
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193 def tournament_matrix(G): |
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194 r"""Returns the tournament matrix for the given tournament graph. |
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195 |
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196 This function requires SciPy. |
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197 |
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198 The *tournament matrix* of a tournament graph with edge set *E* is |
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199 the matrix *T* defined by |
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200 |
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201 .. math:: |
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202 |
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203 T_{i j} = |
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204 \begin{cases} |
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205 +1 & \text{if } (i, j) \in E \\ |
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206 -1 & \text{if } (j, i) \in E \\ |
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207 0 & \text{if } i == j. |
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208 \end{cases} |
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209 |
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210 An equivalent definition is `T = A - A^T`, where *A* is the |
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211 adjacency matrix of the graph `G`. |
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212 |
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213 Parameters |
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214 ---------- |
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215 G : NetworkX graph |
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216 A directed graph representing a tournament. |
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217 |
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218 Returns |
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219 ------- |
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220 SciPy sparse matrix |
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221 The tournament matrix of the tournament graph `G`. |
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222 |
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223 Raises |
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224 ------ |
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225 ImportError |
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226 If SciPy is not available. |
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227 |
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228 """ |
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229 A = nx.adjacency_matrix(G) |
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230 return A - A.T |
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231 |
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232 |
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233 @not_implemented_for("undirected") |
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234 @not_implemented_for("multigraph") |
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235 def is_reachable(G, s, t): |
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236 """Decides whether there is a path from `s` to `t` in the |
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237 tournament. |
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238 |
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239 This function is more theoretically efficient than the reachability |
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240 checks than the shortest path algorithms in |
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241 :mod:`networkx.algorithms.shortest_paths`. |
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242 |
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243 The given graph **must** be a tournament, otherwise this function's |
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244 behavior is undefined. |
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245 |
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246 Parameters |
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247 ---------- |
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248 G : NetworkX graph |
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249 A directed graph representing a tournament. |
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250 |
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251 s : node |
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252 A node in the graph. |
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253 |
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254 t : node |
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255 A node in the graph. |
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256 |
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257 Returns |
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258 ------- |
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259 bool |
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260 Whether there is a path from `s` to `t` in `G`. |
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261 |
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262 Notes |
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263 ----- |
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264 Although this function is more theoretically efficient than the |
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265 generic shortest path functions, a speedup requires the use of |
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266 parallelism. Though it may in the future, the current implementation |
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267 does not use parallelism, thus you may not see much of a speedup. |
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268 |
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269 This algorithm comes from [1]. |
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270 |
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271 References |
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272 ---------- |
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273 .. [1] Tantau, Till. |
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274 "A note on the complexity of the reachability problem for |
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275 tournaments." |
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276 *Electronic Colloquium on Computational Complexity*. 2001. |
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277 <http://eccc.hpi-web.de/report/2001/092/> |
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278 |
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279 """ |
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280 |
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281 def two_neighborhood(G, v): |
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282 """Returns the set of nodes at distance at most two from `v`. |
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283 |
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284 `G` must be a graph and `v` a node in that graph. |
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285 |
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286 The returned set includes the nodes at distance zero (that is, |
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287 the node `v` itself), the nodes at distance one (that is, the |
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288 out-neighbors of `v`), and the nodes at distance two. |
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289 |
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290 """ |
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291 # TODO This is trivially parallelizable. |
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292 return { |
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293 x for x in G if x == v or x in G[v] or any(is_path(G, [v, z, x]) for z in G) |
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294 } |
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295 |
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296 def is_closed(G, nodes): |
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297 """Decides whether the given set of nodes is closed. |
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298 |
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299 A set *S* of nodes is *closed* if for each node *u* in the graph |
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300 not in *S* and for each node *v* in *S*, there is an edge from |
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301 *u* to *v*. |
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302 |
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303 """ |
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304 # TODO This is trivially parallelizable. |
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305 return all(v in G[u] for u in set(G) - nodes for v in nodes) |
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306 |
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307 # TODO This is trivially parallelizable. |
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308 neighborhoods = [two_neighborhood(G, v) for v in G] |
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309 return all(not (is_closed(G, S) and s in S and t not in S) for S in neighborhoods) |
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310 |
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311 |
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312 @not_implemented_for("undirected") |
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313 @not_implemented_for("multigraph") |
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314 def is_strongly_connected(G): |
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315 """Decides whether the given tournament is strongly connected. |
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316 |
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317 This function is more theoretically efficient than the |
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318 :func:`~networkx.algorithms.components.is_strongly_connected` |
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319 function. |
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320 |
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321 The given graph **must** be a tournament, otherwise this function's |
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322 behavior is undefined. |
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323 |
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324 Parameters |
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325 ---------- |
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326 G : NetworkX graph |
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327 A directed graph representing a tournament. |
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328 |
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329 Returns |
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330 ------- |
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331 bool |
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332 Whether the tournament is strongly connected. |
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333 |
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334 Notes |
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335 ----- |
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336 Although this function is more theoretically efficient than the |
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337 generic strong connectivity function, a speedup requires the use of |
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338 parallelism. Though it may in the future, the current implementation |
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339 does not use parallelism, thus you may not see much of a speedup. |
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340 |
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341 This algorithm comes from [1]. |
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342 |
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343 References |
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344 ---------- |
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345 .. [1] Tantau, Till. |
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346 "A note on the complexity of the reachability problem for |
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347 tournaments." |
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348 *Electronic Colloquium on Computational Complexity*. 2001. |
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349 <http://eccc.hpi-web.de/report/2001/092/> |
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350 |
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351 """ |
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352 # TODO This is trivially parallelizable. |
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353 return all(is_reachable(G, u, v) for u in G for v in G) |