Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/chains.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 chains in a graph.""" |
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2 |
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3 import networkx as nx |
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4 from networkx.utils import not_implemented_for |
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5 |
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6 |
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7 @not_implemented_for("directed") |
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8 @not_implemented_for("multigraph") |
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9 def chain_decomposition(G, root=None): |
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10 """Returns the chain decomposition of a graph. |
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11 |
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12 The *chain decomposition* of a graph with respect a depth-first |
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13 search tree is a set of cycles or paths derived from the set of |
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14 fundamental cycles of the tree in the following manner. Consider |
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15 each fundamental cycle with respect to the given tree, represented |
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16 as a list of edges beginning with the nontree edge oriented away |
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17 from the root of the tree. For each fundamental cycle, if it |
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18 overlaps with any previous fundamental cycle, just take the initial |
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19 non-overlapping segment, which is a path instead of a cycle. Each |
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20 cycle or path is called a *chain*. For more information, see [1]_. |
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21 |
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22 Parameters |
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23 ---------- |
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24 G : undirected graph |
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25 |
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26 root : node (optional) |
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27 A node in the graph `G`. If specified, only the chain |
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28 decomposition for the connected component containing this node |
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29 will be returned. This node indicates the root of the depth-first |
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30 search tree. |
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31 |
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32 Yields |
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33 ------ |
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34 chain : list |
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35 A list of edges representing a chain. There is no guarantee on |
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36 the orientation of the edges in each chain (for example, if a |
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37 chain includes the edge joining nodes 1 and 2, the chain may |
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38 include either (1, 2) or (2, 1)). |
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39 |
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40 Raises |
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41 ------ |
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42 NodeNotFound |
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43 If `root` is not in the graph `G`. |
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44 |
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45 Notes |
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46 ----- |
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47 The worst-case running time of this implementation is linear in the |
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48 number of nodes and number of edges [1]_. |
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49 |
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50 References |
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51 ---------- |
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52 .. [1] Jens M. Schmidt (2013). "A simple test on 2-vertex- |
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53 and 2-edge-connectivity." *Information Processing Letters*, |
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54 113, 241–244. Elsevier. <https://doi.org/10.1016/j.ipl.2013.01.016> |
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55 |
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56 """ |
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57 |
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58 def _dfs_cycle_forest(G, root=None): |
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59 """Builds a directed graph composed of cycles from the given graph. |
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60 |
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61 `G` is an undirected simple graph. `root` is a node in the graph |
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62 from which the depth-first search is started. |
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63 |
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64 This function returns both the depth-first search cycle graph |
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65 (as a :class:`~networkx.DiGraph`) and the list of nodes in |
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66 depth-first preorder. The depth-first search cycle graph is a |
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67 directed graph whose edges are the edges of `G` oriented toward |
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68 the root if the edge is a tree edge and away from the root if |
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69 the edge is a non-tree edge. If `root` is not specified, this |
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70 performs a depth-first search on each connected component of `G` |
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71 and returns a directed forest instead. |
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72 |
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73 If `root` is not in the graph, this raises :exc:`KeyError`. |
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74 |
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75 """ |
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76 # Create a directed graph from the depth-first search tree with |
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77 # root node `root` in which tree edges are directed toward the |
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78 # root and nontree edges are directed away from the root. For |
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79 # each node with an incident nontree edge, this creates a |
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80 # directed cycle starting with the nontree edge and returning to |
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81 # that node. |
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82 # |
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83 # The `parent` node attribute stores the parent of each node in |
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84 # the DFS tree. The `nontree` edge attribute indicates whether |
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85 # the edge is a tree edge or a nontree edge. |
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86 # |
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87 # We also store the order of the nodes found in the depth-first |
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88 # search in the `nodes` list. |
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89 H = nx.DiGraph() |
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90 nodes = [] |
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91 for u, v, d in nx.dfs_labeled_edges(G, source=root): |
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92 if d == "forward": |
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93 # `dfs_labeled_edges()` yields (root, root, 'forward') |
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94 # if it is beginning the search on a new connected |
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95 # component. |
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96 if u == v: |
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97 H.add_node(v, parent=None) |
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98 nodes.append(v) |
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99 else: |
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100 H.add_node(v, parent=u) |
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101 H.add_edge(v, u, nontree=False) |
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102 nodes.append(v) |
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103 # `dfs_labeled_edges` considers nontree edges in both |
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104 # orientations, so we need to not add the edge if it its |
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105 # other orientation has been added. |
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106 elif d == "nontree" and v not in H[u]: |
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107 H.add_edge(v, u, nontree=True) |
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108 else: |
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109 # Do nothing on 'reverse' edges; we only care about |
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110 # forward and nontree edges. |
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111 pass |
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112 return H, nodes |
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113 |
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114 def _build_chain(G, u, v, visited): |
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115 """Generate the chain starting from the given nontree edge. |
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116 |
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117 `G` is a DFS cycle graph as constructed by |
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118 :func:`_dfs_cycle_graph`. The edge (`u`, `v`) is a nontree edge |
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119 that begins a chain. `visited` is a set representing the nodes |
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120 in `G` that have already been visited. |
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121 |
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122 This function yields the edges in an initial segment of the |
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123 fundamental cycle of `G` starting with the nontree edge (`u`, |
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124 `v`) that includes all the edges up until the first node that |
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125 appears in `visited`. The tree edges are given by the 'parent' |
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126 node attribute. The `visited` set is updated to add each node in |
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127 an edge yielded by this function. |
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128 |
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129 """ |
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130 while v not in visited: |
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131 yield u, v |
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132 visited.add(v) |
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133 u, v = v, G.nodes[v]["parent"] |
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134 yield u, v |
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135 |
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136 # Create a directed version of H that has the DFS edges directed |
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137 # toward the root and the nontree edges directed away from the root |
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138 # (in each connected component). |
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139 H, nodes = _dfs_cycle_forest(G, root) |
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140 |
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141 # Visit the nodes again in DFS order. For each node, and for each |
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142 # nontree edge leaving that node, compute the fundamental cycle for |
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143 # that nontree edge starting with that edge. If the fundamental |
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144 # cycle overlaps with any visited nodes, just take the prefix of the |
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145 # cycle up to the point of visited nodes. |
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146 # |
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147 # We repeat this process for each connected component (implicitly, |
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148 # since `nodes` already has a list of the nodes grouped by connected |
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149 # component). |
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150 visited = set() |
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151 for u in nodes: |
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152 visited.add(u) |
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153 # For each nontree edge going out of node u... |
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154 edges = ((u, v) for u, v, d in H.out_edges(u, data="nontree") if d) |
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155 for u, v in edges: |
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156 # Create the cycle or cycle prefix starting with the |
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157 # nontree edge. |
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158 chain = list(_build_chain(H, u, v, visited)) |
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159 yield chain |