Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/approximation/dominating_set.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 for finding node and edge dominating sets. |
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2 |
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3 A `dominating set`_ for an undirected graph *G* with vertex set *V* |
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4 and edge set *E* is a subset *D* of *V* such that every vertex not in |
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5 *D* is adjacent to at least one member of *D*. An `edge dominating set`_ |
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6 is a subset *F* of *E* such that every edge not in *F* is |
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7 incident to an endpoint of at least one edge in *F*. |
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8 |
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9 .. _dominating set: https://en.wikipedia.org/wiki/Dominating_set |
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10 .. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set |
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11 |
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12 """ |
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13 |
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14 from ..matching import maximal_matching |
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15 from ...utils import not_implemented_for |
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16 |
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17 __all__ = ["min_weighted_dominating_set", "min_edge_dominating_set"] |
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18 |
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19 |
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20 # TODO Why doesn't this algorithm work for directed graphs? |
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21 @not_implemented_for("directed") |
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22 def min_weighted_dominating_set(G, weight=None): |
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23 r"""Returns a dominating set that approximates the minimum weight node |
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24 dominating set. |
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25 |
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26 Parameters |
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27 ---------- |
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28 G : NetworkX graph |
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29 Undirected graph. |
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30 |
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31 weight : string |
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32 The node attribute storing the weight of an node. If provided, |
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33 the node attribute with this key must be a number for each |
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34 node. If not provided, each node is assumed to have weight one. |
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35 |
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36 Returns |
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37 ------- |
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38 min_weight_dominating_set : set |
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39 A set of nodes, the sum of whose weights is no more than `(\log |
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40 w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of |
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41 each node in the graph and `w(V^*)` denotes the sum of the |
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42 weights of each node in the minimum weight dominating set. |
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43 |
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44 Notes |
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45 ----- |
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46 This algorithm computes an approximate minimum weighted dominating |
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47 set for the graph `G`. The returned solution has weight `(\log |
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48 w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each |
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49 node in the graph and `w(V^*)` denotes the sum of the weights of |
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50 each node in the minimum weight dominating set for the graph. |
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51 |
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52 This implementation of the algorithm runs in $O(m)$ time, where $m$ |
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53 is the number of edges in the graph. |
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54 |
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55 References |
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56 ---------- |
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57 .. [1] Vazirani, Vijay V. |
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58 *Approximation Algorithms*. |
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59 Springer Science & Business Media, 2001. |
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60 |
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61 """ |
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62 # The unique dominating set for the null graph is the empty set. |
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63 if len(G) == 0: |
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64 return set() |
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65 |
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66 # This is the dominating set that will eventually be returned. |
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67 dom_set = set() |
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68 |
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69 def _cost(node_and_neighborhood): |
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70 """Returns the cost-effectiveness of greedily choosing the given |
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71 node. |
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72 |
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73 `node_and_neighborhood` is a two-tuple comprising a node and its |
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74 closed neighborhood. |
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75 |
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76 """ |
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77 v, neighborhood = node_and_neighborhood |
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78 return G.nodes[v].get(weight, 1) / len(neighborhood - dom_set) |
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79 |
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80 # This is a set of all vertices not already covered by the |
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81 # dominating set. |
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82 vertices = set(G) |
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83 # This is a dictionary mapping each node to the closed neighborhood |
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84 # of that node. |
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85 neighborhoods = {v: {v} | set(G[v]) for v in G} |
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86 |
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87 # Continue until all vertices are adjacent to some node in the |
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88 # dominating set. |
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89 while vertices: |
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90 # Find the most cost-effective node to add, along with its |
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91 # closed neighborhood. |
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92 dom_node, min_set = min(neighborhoods.items(), key=_cost) |
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93 # Add the node to the dominating set and reduce the remaining |
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94 # set of nodes to cover. |
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95 dom_set.add(dom_node) |
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96 del neighborhoods[dom_node] |
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97 vertices -= min_set |
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98 |
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99 return dom_set |
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100 |
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101 |
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102 def min_edge_dominating_set(G): |
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103 r"""Returns minimum cardinality edge dominating set. |
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104 |
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105 Parameters |
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106 ---------- |
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107 G : NetworkX graph |
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108 Undirected graph |
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109 |
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110 Returns |
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111 ------- |
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112 min_edge_dominating_set : set |
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113 Returns a set of dominating edges whose size is no more than 2 * OPT. |
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114 |
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115 Notes |
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116 ----- |
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117 The algorithm computes an approximate solution to the edge dominating set |
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118 problem. The result is no more than 2 * OPT in terms of size of the set. |
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119 Runtime of the algorithm is $O(|E|)$. |
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120 """ |
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121 if not G: |
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122 raise ValueError("Expected non-empty NetworkX graph!") |
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123 return maximal_matching(G) |