162 lines
6.5 KiB
Python
162 lines
6.5 KiB
Python
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"""Functions for finding chains in a graph."""
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = ["chain_decomposition"]
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def chain_decomposition(G, root=None):
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"""Returns the chain decomposition of a graph.
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The *chain decomposition* of a graph with respect a depth-first
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search tree is a set of cycles or paths derived from the set of
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fundamental cycles of the tree in the following manner. Consider
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each fundamental cycle with respect to the given tree, represented
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as a list of edges beginning with the nontree edge oriented away
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from the root of the tree. For each fundamental cycle, if it
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overlaps with any previous fundamental cycle, just take the initial
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non-overlapping segment, which is a path instead of a cycle. Each
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cycle or path is called a *chain*. For more information, see [1]_.
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Parameters
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----------
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G : undirected graph
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root : node (optional)
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A node in the graph `G`. If specified, only the chain
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decomposition for the connected component containing this node
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will be returned. This node indicates the root of the depth-first
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search tree.
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Yields
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------
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chain : list
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A list of edges representing a chain. There is no guarantee on
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the orientation of the edges in each chain (for example, if a
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chain includes the edge joining nodes 1 and 2, the chain may
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include either (1, 2) or (2, 1)).
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Raises
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------
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NodeNotFound
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If `root` is not in the graph `G`.
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Notes
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-----
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The worst-case running time of this implementation is linear in the
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number of nodes and number of edges [1]_.
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References
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----------
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.. [1] Jens M. Schmidt (2013). "A simple test on 2-vertex-
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and 2-edge-connectivity." *Information Processing Letters*,
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113, 241–244. Elsevier. <https://doi.org/10.1016/j.ipl.2013.01.016>
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"""
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def _dfs_cycle_forest(G, root=None):
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"""Builds a directed graph composed of cycles from the given graph.
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`G` is an undirected simple graph. `root` is a node in the graph
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from which the depth-first search is started.
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This function returns both the depth-first search cycle graph
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(as a :class:`~networkx.DiGraph`) and the list of nodes in
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depth-first preorder. The depth-first search cycle graph is a
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directed graph whose edges are the edges of `G` oriented toward
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the root if the edge is a tree edge and away from the root if
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the edge is a non-tree edge. If `root` is not specified, this
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performs a depth-first search on each connected component of `G`
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and returns a directed forest instead.
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If `root` is not in the graph, this raises :exc:`KeyError`.
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"""
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# Create a directed graph from the depth-first search tree with
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# root node `root` in which tree edges are directed toward the
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# root and nontree edges are directed away from the root. For
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# each node with an incident nontree edge, this creates a
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# directed cycle starting with the nontree edge and returning to
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# that node.
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#
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# The `parent` node attribute stores the parent of each node in
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# the DFS tree. The `nontree` edge attribute indicates whether
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# the edge is a tree edge or a nontree edge.
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#
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# We also store the order of the nodes found in the depth-first
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# search in the `nodes` list.
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H = nx.DiGraph()
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nodes = []
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for u, v, d in nx.dfs_labeled_edges(G, source=root):
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if d == "forward":
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# `dfs_labeled_edges()` yields (root, root, 'forward')
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# if it is beginning the search on a new connected
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# component.
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if u == v:
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H.add_node(v, parent=None)
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nodes.append(v)
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else:
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H.add_node(v, parent=u)
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H.add_edge(v, u, nontree=False)
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nodes.append(v)
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# `dfs_labeled_edges` considers nontree edges in both
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# orientations, so we need to not add the edge if it its
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# other orientation has been added.
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elif d == "nontree" and v not in H[u]:
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H.add_edge(v, u, nontree=True)
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else:
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# Do nothing on 'reverse' edges; we only care about
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# forward and nontree edges.
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pass
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return H, nodes
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def _build_chain(G, u, v, visited):
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"""Generate the chain starting from the given nontree edge.
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`G` is a DFS cycle graph as constructed by
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:func:`_dfs_cycle_graph`. The edge (`u`, `v`) is a nontree edge
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that begins a chain. `visited` is a set representing the nodes
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in `G` that have already been visited.
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This function yields the edges in an initial segment of the
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fundamental cycle of `G` starting with the nontree edge (`u`,
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`v`) that includes all the edges up until the first node that
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appears in `visited`. The tree edges are given by the 'parent'
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node attribute. The `visited` set is updated to add each node in
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an edge yielded by this function.
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"""
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while v not in visited:
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yield u, v
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visited.add(v)
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u, v = v, G.nodes[v]["parent"]
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yield u, v
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# Create a directed version of H that has the DFS edges directed
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# toward the root and the nontree edges directed away from the root
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# (in each connected component).
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H, nodes = _dfs_cycle_forest(G, root)
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# Visit the nodes again in DFS order. For each node, and for each
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# nontree edge leaving that node, compute the fundamental cycle for
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# that nontree edge starting with that edge. If the fundamental
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# cycle overlaps with any visited nodes, just take the prefix of the
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# cycle up to the point of visited nodes.
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#
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# We repeat this process for each connected component (implicitly,
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# since `nodes` already has a list of the nodes grouped by connected
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# component).
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visited = set()
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for u in nodes:
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visited.add(u)
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# For each nontree edge going out of node u...
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edges = ((u, v) for u, v, d in H.out_edges(u, data="nontree") if d)
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for u, v in edges:
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# Create the cycle or cycle prefix starting with the
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# nontree edge.
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chain = list(_build_chain(H, u, v, visited))
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yield chain
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