86 lines
3.1 KiB
Python
86 lines
3.1 KiB
Python
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"""
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Utilities for connectivity package
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"""
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import networkx as nx
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__all__ = ["build_auxiliary_node_connectivity", "build_auxiliary_edge_connectivity"]
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def build_auxiliary_node_connectivity(G):
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r"""Creates a directed graph D from an undirected graph G to compute flow
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based node connectivity.
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For an undirected graph G having `n` nodes and `m` edges we derive a
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directed graph D with `2n` nodes and `2m+n` arcs by replacing each
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original node `v` with two nodes `vA`, `vB` linked by an (internal)
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arc in D. Then for each edge (`u`, `v`) in G we add two arcs (`uB`, `vA`)
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and (`vB`, `uA`) in D. Finally we set the attribute capacity = 1 for each
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arc in D [1]_.
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For a directed graph having `n` nodes and `m` arcs we derive a
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directed graph D with `2n` nodes and `m+n` arcs by replacing each
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original node `v` with two nodes `vA`, `vB` linked by an (internal)
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arc (`vA`, `vB`) in D. Then for each arc (`u`, `v`) in G we add one
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arc (`uB`, `vA`) in D. Finally we set the attribute capacity = 1 for
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each arc in D.
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A dictionary with a mapping between nodes in the original graph and the
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auxiliary digraph is stored as a graph attribute: H.graph['mapping'].
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References
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----------
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.. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and
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Erlebach, 'Network Analysis: Methodological Foundations', Lecture
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Notes in Computer Science, Volume 3418, Springer-Verlag, 2005.
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http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf
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"""
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directed = G.is_directed()
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mapping = {}
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H = nx.DiGraph()
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for i, node in enumerate(G):
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mapping[node] = i
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H.add_node(f"{i}A", id=node)
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H.add_node(f"{i}B", id=node)
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H.add_edge(f"{i}A", f"{i}B", capacity=1)
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edges = []
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for (source, target) in G.edges():
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edges.append((f"{mapping[source]}B", f"{mapping[target]}A"))
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if not directed:
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edges.append((f"{mapping[target]}B", f"{mapping[source]}A"))
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H.add_edges_from(edges, capacity=1)
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# Store mapping as graph attribute
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H.graph["mapping"] = mapping
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return H
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def build_auxiliary_edge_connectivity(G):
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"""Auxiliary digraph for computing flow based edge connectivity
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If the input graph is undirected, we replace each edge (`u`,`v`) with
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two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute
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'capacity' for each arc to 1. If the input graph is directed we simply
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add the 'capacity' attribute. Part of algorithm 1 in [1]_ .
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References
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----------
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.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. (this is a
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chapter, look for the reference of the book).
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http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
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"""
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if G.is_directed():
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H = nx.DiGraph()
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H.add_nodes_from(G.nodes())
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H.add_edges_from(G.edges(), capacity=1)
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return H
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else:
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H = nx.DiGraph()
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H.add_nodes_from(G.nodes())
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for (source, target) in G.edges():
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H.add_edges_from([(source, target), (target, source)], capacity=1)
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return H
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