Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/connectivity/stoerwagner.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 """ |
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2 Stoer-Wagner minimum cut algorithm. |
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3 """ |
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4 from itertools import islice |
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5 |
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6 import networkx as nx |
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7 from ...utils import BinaryHeap |
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8 from ...utils import not_implemented_for |
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9 from ...utils import arbitrary_element |
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10 |
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11 __all__ = ["stoer_wagner"] |
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12 |
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13 |
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14 @not_implemented_for("directed") |
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15 @not_implemented_for("multigraph") |
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16 def stoer_wagner(G, weight="weight", heap=BinaryHeap): |
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17 r"""Returns the weighted minimum edge cut using the Stoer-Wagner algorithm. |
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18 |
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19 Determine the minimum edge cut of a connected graph using the |
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20 Stoer-Wagner algorithm. In weighted cases, all weights must be |
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21 nonnegative. |
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22 |
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23 The running time of the algorithm depends on the type of heaps used: |
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24 |
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25 ============== ============================================= |
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26 Type of heap Running time |
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27 ============== ============================================= |
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28 Binary heap $O(n (m + n) \log n)$ |
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29 Fibonacci heap $O(nm + n^2 \log n)$ |
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30 Pairing heap $O(2^{2 \sqrt{\log \log n}} nm + n^2 \log n)$ |
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31 ============== ============================================= |
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32 |
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33 Parameters |
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34 ---------- |
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35 G : NetworkX graph |
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36 Edges of the graph are expected to have an attribute named by the |
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37 weight parameter below. If this attribute is not present, the edge is |
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38 considered to have unit weight. |
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39 |
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40 weight : string |
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41 Name of the weight attribute of the edges. If the attribute is not |
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42 present, unit weight is assumed. Default value: 'weight'. |
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43 |
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44 heap : class |
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45 Type of heap to be used in the algorithm. It should be a subclass of |
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46 :class:`MinHeap` or implement a compatible interface. |
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47 |
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48 If a stock heap implementation is to be used, :class:`BinaryHeap` is |
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49 recommended over :class:`PairingHeap` for Python implementations without |
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50 optimized attribute accesses (e.g., CPython) despite a slower |
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51 asymptotic running time. For Python implementations with optimized |
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52 attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better |
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53 performance. Default value: :class:`BinaryHeap`. |
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54 |
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55 Returns |
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56 ------- |
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57 cut_value : integer or float |
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58 The sum of weights of edges in a minimum cut. |
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59 |
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60 partition : pair of node lists |
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61 A partitioning of the nodes that defines a minimum cut. |
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62 |
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63 Raises |
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64 ------ |
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65 NetworkXNotImplemented |
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66 If the graph is directed or a multigraph. |
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67 |
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68 NetworkXError |
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69 If the graph has less than two nodes, is not connected or has a |
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70 negative-weighted edge. |
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71 |
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72 Examples |
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73 -------- |
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74 >>> G = nx.Graph() |
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75 >>> G.add_edge("x", "a", weight=3) |
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76 >>> G.add_edge("x", "b", weight=1) |
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77 >>> G.add_edge("a", "c", weight=3) |
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78 >>> G.add_edge("b", "c", weight=5) |
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79 >>> G.add_edge("b", "d", weight=4) |
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80 >>> G.add_edge("d", "e", weight=2) |
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81 >>> G.add_edge("c", "y", weight=2) |
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82 >>> G.add_edge("e", "y", weight=3) |
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83 >>> cut_value, partition = nx.stoer_wagner(G) |
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84 >>> cut_value |
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85 4 |
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86 """ |
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87 n = len(G) |
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88 if n < 2: |
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89 raise nx.NetworkXError("graph has less than two nodes.") |
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90 if not nx.is_connected(G): |
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91 raise nx.NetworkXError("graph is not connected.") |
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92 |
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93 # Make a copy of the graph for internal use. |
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94 G = nx.Graph( |
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95 (u, v, {"weight": e.get(weight, 1)}) for u, v, e in G.edges(data=True) if u != v |
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96 ) |
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97 |
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98 for u, v, e in G.edges(data=True): |
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99 if e["weight"] < 0: |
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100 raise nx.NetworkXError("graph has a negative-weighted edge.") |
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101 |
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102 cut_value = float("inf") |
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103 nodes = set(G) |
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104 contractions = [] # contracted node pairs |
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105 |
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106 # Repeatedly pick a pair of nodes to contract until only one node is left. |
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107 for i in range(n - 1): |
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108 # Pick an arbitrary node u and create a set A = {u}. |
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109 u = arbitrary_element(G) |
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110 A = {u} |
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111 # Repeatedly pick the node "most tightly connected" to A and add it to |
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112 # A. The tightness of connectivity of a node not in A is defined by the |
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113 # of edges connecting it to nodes in A. |
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114 h = heap() # min-heap emulating a max-heap |
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115 for v, e in G[u].items(): |
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116 h.insert(v, -e["weight"]) |
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117 # Repeat until all but one node has been added to A. |
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118 for j in range(n - i - 2): |
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119 u = h.pop()[0] |
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120 A.add(u) |
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121 for v, e in G[u].items(): |
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122 if v not in A: |
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123 h.insert(v, h.get(v, 0) - e["weight"]) |
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124 # A and the remaining node v define a "cut of the phase". There is a |
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125 # minimum cut of the original graph that is also a cut of the phase. |
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126 # Due to contractions in earlier phases, v may in fact represent |
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127 # multiple nodes in the original graph. |
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128 v, w = h.min() |
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129 w = -w |
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130 if w < cut_value: |
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131 cut_value = w |
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132 best_phase = i |
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133 # Contract v and the last node added to A. |
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134 contractions.append((u, v)) |
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135 for w, e in G[v].items(): |
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136 if w != u: |
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137 if w not in G[u]: |
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138 G.add_edge(u, w, weight=e["weight"]) |
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139 else: |
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140 G[u][w]["weight"] += e["weight"] |
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141 G.remove_node(v) |
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142 |
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143 # Recover the optimal partitioning from the contractions. |
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144 G = nx.Graph(islice(contractions, best_phase)) |
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145 v = contractions[best_phase][1] |
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146 G.add_node(v) |
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147 reachable = set(nx.single_source_shortest_path_length(G, v)) |
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148 partition = (list(reachable), list(nodes - reachable)) |
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149 |
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150 return cut_value, partition |