906 lines
27 KiB
Python
906 lines
27 KiB
Python
"""Algorithms for directed acyclic graphs (DAGs).
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Note that most of these functions are only guaranteed to work for DAGs.
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In general, these functions do not check for acyclic-ness, so it is up
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to the user to check for that.
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"""
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from collections import deque
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from math import gcd
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from functools import partial
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from itertools import chain
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from itertools import product
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from itertools import starmap
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import heapq
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import networkx as nx
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from networkx.algorithms.traversal.breadth_first_search import descendants_at_distance
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from networkx.generators.trees import NIL
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from networkx.utils import arbitrary_element
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from networkx.utils import consume
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from networkx.utils import pairwise
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from networkx.utils import not_implemented_for
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__all__ = [
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"descendants",
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"ancestors",
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"topological_sort",
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"lexicographical_topological_sort",
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"all_topological_sorts",
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"is_directed_acyclic_graph",
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"is_aperiodic",
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"transitive_closure",
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"transitive_closure_dag",
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"transitive_reduction",
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"antichains",
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"dag_longest_path",
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"dag_longest_path_length",
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"dag_to_branching",
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]
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chaini = chain.from_iterable
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def descendants(G, source):
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"""Returns all nodes reachable from `source` in `G`.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed acyclic graph (DAG)
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source : node in `G`
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Returns
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-------
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set()
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The descendants of `source` in `G`
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"""
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if not G.has_node(source):
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raise nx.NetworkXError(f"The node {source} is not in the graph.")
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des = {n for n, d in nx.shortest_path_length(G, source=source).items()}
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return des - {source}
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def ancestors(G, source):
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"""Returns all nodes having a path to `source` in `G`.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed acyclic graph (DAG)
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source : node in `G`
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Returns
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-------
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set()
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The ancestors of source in G
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"""
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if not G.has_node(source):
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raise nx.NetworkXError(f"The node {source} is not in the graph.")
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anc = {n for n, d in nx.shortest_path_length(G, target=source).items()}
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return anc - {source}
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def has_cycle(G):
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"""Decides whether the directed graph has a cycle."""
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try:
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consume(topological_sort(G))
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except nx.NetworkXUnfeasible:
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return True
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else:
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return False
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def is_directed_acyclic_graph(G):
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"""Returns True if the graph `G` is a directed acyclic graph (DAG) or
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False if not.
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Parameters
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----------
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G : NetworkX graph
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Returns
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-------
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bool
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True if `G` is a DAG, False otherwise
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"""
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return G.is_directed() and not has_cycle(G)
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def topological_sort(G):
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"""Returns a generator of nodes in topologically sorted order.
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A topological sort is a nonunique permutation of the nodes such that an
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edge from u to v implies that u appears before v in the topological sort
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order.
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Parameters
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----------
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G : NetworkX digraph
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A directed acyclic graph (DAG)
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Returns
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-------
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iterable
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An iterable of node names in topological sorted order.
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Raises
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------
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NetworkXError
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Topological sort is defined for directed graphs only. If the graph `G`
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is undirected, a :exc:`NetworkXError` is raised.
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NetworkXUnfeasible
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If `G` is not a directed acyclic graph (DAG) no topological sort exists
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and a :exc:`NetworkXUnfeasible` exception is raised. This can also be
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raised if `G` is changed while the returned iterator is being processed
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RuntimeError
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If `G` is changed while the returned iterator is being processed.
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Examples
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--------
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To get the reverse order of the topological sort:
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>>> DG = nx.DiGraph([(1, 2), (2, 3)])
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>>> list(reversed(list(nx.topological_sort(DG))))
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[3, 2, 1]
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If your DiGraph naturally has the edges representing tasks/inputs
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and nodes representing people/processes that initiate tasks, then
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topological_sort is not quite what you need. You will have to change
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the tasks to nodes with dependence reflected by edges. The result is
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a kind of topological sort of the edges. This can be done
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with :func:`networkx.line_graph` as follows:
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>>> list(nx.topological_sort(nx.line_graph(DG)))
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[(1, 2), (2, 3)]
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Notes
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-----
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This algorithm is based on a description and proof in
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"Introduction to Algorithms: A Creative Approach" [1]_ .
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See also
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--------
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is_directed_acyclic_graph, lexicographical_topological_sort
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References
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----------
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.. [1] Manber, U. (1989).
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*Introduction to Algorithms - A Creative Approach.* Addison-Wesley.
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"""
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if not G.is_directed():
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raise nx.NetworkXError("Topological sort not defined on undirected graphs.")
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indegree_map = {v: d for v, d in G.in_degree() if d > 0}
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# These nodes have zero indegree and ready to be returned.
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zero_indegree = [v for v, d in G.in_degree() if d == 0]
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while zero_indegree:
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node = zero_indegree.pop()
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if node not in G:
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raise RuntimeError("Graph changed during iteration")
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for _, child in G.edges(node):
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try:
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indegree_map[child] -= 1
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except KeyError as e:
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raise RuntimeError("Graph changed during iteration") from e
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if indegree_map[child] == 0:
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zero_indegree.append(child)
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del indegree_map[child]
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yield node
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if indegree_map:
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raise nx.NetworkXUnfeasible(
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"Graph contains a cycle or graph changed " "during iteration"
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)
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def lexicographical_topological_sort(G, key=None):
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"""Returns a generator of nodes in lexicographically topologically sorted
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order.
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A topological sort is a nonunique permutation of the nodes such that an
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edge from u to v implies that u appears before v in the topological sort
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order.
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Parameters
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----------
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G : NetworkX digraph
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A directed acyclic graph (DAG)
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key : function, optional
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This function maps nodes to keys with which to resolve ambiguities in
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the sort order. Defaults to the identity function.
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Returns
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-------
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iterable
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An iterable of node names in lexicographical topological sort order.
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Raises
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------
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NetworkXError
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Topological sort is defined for directed graphs only. If the graph `G`
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is undirected, a :exc:`NetworkXError` is raised.
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NetworkXUnfeasible
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If `G` is not a directed acyclic graph (DAG) no topological sort exists
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and a :exc:`NetworkXUnfeasible` exception is raised. This can also be
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raised if `G` is changed while the returned iterator is being processed
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RuntimeError
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If `G` is changed while the returned iterator is being processed.
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Notes
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-----
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This algorithm is based on a description and proof in
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"Introduction to Algorithms: A Creative Approach" [1]_ .
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See also
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--------
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topological_sort
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References
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----------
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.. [1] Manber, U. (1989).
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*Introduction to Algorithms - A Creative Approach.* Addison-Wesley.
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"""
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if not G.is_directed():
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msg = "Topological sort not defined on undirected graphs."
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raise nx.NetworkXError(msg)
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if key is None:
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def key(node):
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return node
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nodeid_map = {n: i for i, n in enumerate(G)}
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def create_tuple(node):
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return key(node), nodeid_map[node], node
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indegree_map = {v: d for v, d in G.in_degree() if d > 0}
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# These nodes have zero indegree and ready to be returned.
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zero_indegree = [create_tuple(v) for v, d in G.in_degree() if d == 0]
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heapq.heapify(zero_indegree)
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while zero_indegree:
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_, _, node = heapq.heappop(zero_indegree)
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if node not in G:
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raise RuntimeError("Graph changed during iteration")
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for _, child in G.edges(node):
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try:
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indegree_map[child] -= 1
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except KeyError as e:
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raise RuntimeError("Graph changed during iteration") from e
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if indegree_map[child] == 0:
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heapq.heappush(zero_indegree, create_tuple(child))
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del indegree_map[child]
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yield node
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if indegree_map:
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msg = "Graph contains a cycle or graph changed during iteration"
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raise nx.NetworkXUnfeasible(msg)
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@not_implemented_for("undirected")
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def all_topological_sorts(G):
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"""Returns a generator of _all_ topological sorts of the directed graph G.
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A topological sort is a nonunique permutation of the nodes such that an
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edge from u to v implies that u appears before v in the topological sort
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order.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed graph
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Returns
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-------
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generator
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All topological sorts of the digraph G
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Raises
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------
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NetworkXNotImplemented
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If `G` is not directed
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NetworkXUnfeasible
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If `G` is not acyclic
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Examples
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--------
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To enumerate all topological sorts of directed graph:
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>>> DG = nx.DiGraph([(1, 2), (2, 3), (2, 4)])
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>>> list(nx.all_topological_sorts(DG))
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[[1, 2, 4, 3], [1, 2, 3, 4]]
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Notes
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-----
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Implements an iterative version of the algorithm given in [1].
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References
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----------
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.. [1] Knuth, Donald E., Szwarcfiter, Jayme L. (1974).
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"A Structured Program to Generate All Topological Sorting Arrangements"
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Information Processing Letters, Volume 2, Issue 6, 1974, Pages 153-157,
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ISSN 0020-0190,
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https://doi.org/10.1016/0020-0190(74)90001-5.
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Elsevier (North-Holland), Amsterdam
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"""
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if not G.is_directed():
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raise nx.NetworkXError("Topological sort not defined on undirected graphs.")
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# the names of count and D are chosen to match the global variables in [1]
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# number of edges originating in a vertex v
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count = dict(G.in_degree())
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# vertices with indegree 0
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D = deque([v for v, d in G.in_degree() if d == 0])
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# stack of first value chosen at a position k in the topological sort
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bases = []
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current_sort = []
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# do-while construct
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while True:
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assert all([count[v] == 0 for v in D])
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if len(current_sort) == len(G):
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yield list(current_sort)
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# clean-up stack
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while len(current_sort) > 0:
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assert len(bases) == len(current_sort)
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q = current_sort.pop()
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# "restores" all edges (q, x)
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# NOTE: it is important to iterate over edges instead
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# of successors, so count is updated correctly in multigraphs
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for _, j in G.out_edges(q):
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count[j] += 1
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assert count[j] >= 0
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# remove entries from D
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while len(D) > 0 and count[D[-1]] > 0:
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D.pop()
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# corresponds to a circular shift of the values in D
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# if the first value chosen (the base) is in the first
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# position of D again, we are done and need to consider the
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# previous condition
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D.appendleft(q)
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if D[-1] == bases[-1]:
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# all possible values have been chosen at current position
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# remove corresponding marker
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bases.pop()
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else:
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# there are still elements that have not been fixed
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# at the current position in the topological sort
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# stop removing elements, escape inner loop
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break
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else:
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if len(D) == 0:
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raise nx.NetworkXUnfeasible("Graph contains a cycle.")
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# choose next node
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q = D.pop()
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# "erase" all edges (q, x)
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# NOTE: it is important to iterate over edges instead
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# of successors, so count is updated correctly in multigraphs
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for _, j in G.out_edges(q):
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count[j] -= 1
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assert count[j] >= 0
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if count[j] == 0:
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D.append(j)
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current_sort.append(q)
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# base for current position might _not_ be fixed yet
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if len(bases) < len(current_sort):
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bases.append(q)
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if len(bases) == 0:
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break
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def is_aperiodic(G):
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"""Returns True if `G` is aperiodic.
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A directed graph is aperiodic if there is no integer k > 1 that
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divides the length of every cycle in the graph.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed graph
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Returns
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-------
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bool
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True if the graph is aperiodic False otherwise
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Raises
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------
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NetworkXError
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If `G` is not directed
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Notes
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-----
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This uses the method outlined in [1]_, which runs in $O(m)$ time
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given $m$ edges in `G`. Note that a graph is not aperiodic if it is
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acyclic as every integer trivial divides length 0 cycles.
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References
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----------
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.. [1] Jarvis, J. P.; Shier, D. R. (1996),
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"Graph-theoretic analysis of finite Markov chains,"
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in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling:
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A Multidisciplinary Approach, CRC Press.
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"""
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if not G.is_directed():
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raise nx.NetworkXError("is_aperiodic not defined for undirected graphs")
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s = arbitrary_element(G)
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levels = {s: 0}
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this_level = [s]
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g = 0
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lev = 1
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while this_level:
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next_level = []
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for u in this_level:
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for v in G[u]:
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if v in levels: # Non-Tree Edge
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g = gcd(g, levels[u] - levels[v] + 1)
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else: # Tree Edge
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next_level.append(v)
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levels[v] = lev
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this_level = next_level
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lev += 1
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if len(levels) == len(G): # All nodes in tree
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return g == 1
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else:
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return g == 1 and nx.is_aperiodic(G.subgraph(set(G) - set(levels)))
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@not_implemented_for("undirected")
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def transitive_closure(G, reflexive=False):
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""" Returns transitive closure of a directed graph
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The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
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for all v, w in V there is an edge (v, w) in E+ if and only if there
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is a path from v to w in G.
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Handling of paths from v to v has some flexibility within this definition.
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A reflexive transitive closure creates a self-loop for the path
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from v to v of length 0. The usual transitive closure creates a
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self-loop only if a cycle exists (a path from v to v with length > 0).
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We also allow an option for no self-loops.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed graph
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reflexive : Bool or None, optional (default: False)
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Determines when cycles create self-loops in the Transitive Closure.
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If True, trivial cycles (length 0) create self-loops. The result
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is a reflexive tranistive closure of G.
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If False (the default) non-trivial cycles create self-loops.
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If None, self-loops are not created.
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Returns
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-------
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NetworkX DiGraph
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The transitive closure of `G`
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Raises
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------
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NetworkXNotImplemented
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If `G` is not directed
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References
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----------
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.. [1] http://www.ics.uci.edu/~eppstein/PADS/PartialOrder.py
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TODO this function applies to all directed graphs and is probably misplaced
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here in dag.py
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"""
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if reflexive is None:
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TC = G.copy()
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for v in G:
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edges = ((v, u) for u in nx.dfs_preorder_nodes(G, v) if v != u)
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TC.add_edges_from(edges)
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return TC
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if reflexive is True:
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TC = G.copy()
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for v in G:
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edges = ((v, u) for u in nx.dfs_preorder_nodes(G, v))
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TC.add_edges_from(edges)
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return TC
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# reflexive is False
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TC = G.copy()
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for v in G:
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edges = ((v, w) for u, w in nx.edge_dfs(G, v))
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TC.add_edges_from(edges)
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return TC
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@not_implemented_for("undirected")
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def transitive_closure_dag(G, topo_order=None):
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""" Returns the transitive closure of a directed acyclic graph.
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This function is faster than the function `transitive_closure`, but fails
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if the graph has a cycle.
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The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
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for all v, w in V there is an edge (v, w) in E+ if and only if there
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is a non-null path from v to w in G.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed acyclic graph (DAG)
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topo_order: list or tuple, optional
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A topological order for G (if None, the function will compute one)
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Returns
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-------
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NetworkX DiGraph
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The transitive closure of `G`
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Raises
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------
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NetworkXNotImplemented
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If `G` is not directed
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NetworkXUnfeasible
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If `G` has a cycle
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Notes
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|
-----
|
|
This algorithm is probably simple enough to be well-known but I didn't find
|
|
a mention in the literature.
|
|
"""
|
|
if topo_order is None:
|
|
topo_order = list(topological_sort(G))
|
|
|
|
TC = G.copy()
|
|
|
|
# idea: traverse vertices following a reverse topological order, connecting
|
|
# each vertex to its descendants at distance 2 as we go
|
|
for v in reversed(topo_order):
|
|
TC.add_edges_from((v, u) for u in descendants_at_distance(TC, v, 2))
|
|
|
|
return TC
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
def transitive_reduction(G):
|
|
""" Returns transitive reduction of a directed graph
|
|
|
|
The transitive reduction of G = (V,E) is a graph G- = (V,E-) such that
|
|
for all v,w in V there is an edge (v,w) in E- if and only if (v,w) is
|
|
in E and there is no path from v to w in G with length greater than 1.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX DiGraph
|
|
A directed acyclic graph (DAG)
|
|
|
|
Returns
|
|
-------
|
|
NetworkX DiGraph
|
|
The transitive reduction of `G`
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If `G` is not a directed acyclic graph (DAG) transitive reduction is
|
|
not uniquely defined and a :exc:`NetworkXError` exception is raised.
|
|
|
|
References
|
|
----------
|
|
https://en.wikipedia.org/wiki/Transitive_reduction
|
|
|
|
"""
|
|
if not is_directed_acyclic_graph(G):
|
|
msg = "Directed Acyclic Graph required for transitive_reduction"
|
|
raise nx.NetworkXError(msg)
|
|
TR = nx.DiGraph()
|
|
TR.add_nodes_from(G.nodes())
|
|
descendants = {}
|
|
# count before removing set stored in descendants
|
|
check_count = dict(G.in_degree)
|
|
for u in G:
|
|
u_nbrs = set(G[u])
|
|
for v in G[u]:
|
|
if v in u_nbrs:
|
|
if v not in descendants:
|
|
descendants[v] = {y for x, y in nx.dfs_edges(G, v)}
|
|
u_nbrs -= descendants[v]
|
|
check_count[v] -= 1
|
|
if check_count[v] == 0:
|
|
del descendants[v]
|
|
TR.add_edges_from((u, v) for v in u_nbrs)
|
|
return TR
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
def antichains(G, topo_order=None):
|
|
"""Generates antichains from a directed acyclic graph (DAG).
|
|
|
|
An antichain is a subset of a partially ordered set such that any
|
|
two elements in the subset are incomparable.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX DiGraph
|
|
A directed acyclic graph (DAG)
|
|
|
|
topo_order: list or tuple, optional
|
|
A topological order for G (if None, the function will compute one)
|
|
|
|
Returns
|
|
-------
|
|
generator object
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed
|
|
|
|
NetworkXUnfeasible
|
|
If `G` contains a cycle
|
|
|
|
Notes
|
|
-----
|
|
This function was originally developed by Peter Jipsen and Franco Saliola
|
|
for the SAGE project. It's included in NetworkX with permission from the
|
|
authors. Original SAGE code at:
|
|
|
|
https://github.com/sagemath/sage/blob/master/src/sage/combinat/posets/hasse_diagram.py
|
|
|
|
References
|
|
----------
|
|
.. [1] Free Lattices, by R. Freese, J. Jezek and J. B. Nation,
|
|
AMS, Vol 42, 1995, p. 226.
|
|
"""
|
|
if topo_order is None:
|
|
topo_order = list(nx.topological_sort(G))
|
|
|
|
TC = nx.transitive_closure_dag(G, topo_order)
|
|
antichains_stacks = [([], list(reversed(topo_order)))]
|
|
|
|
while antichains_stacks:
|
|
(antichain, stack) = antichains_stacks.pop()
|
|
# Invariant:
|
|
# - the elements of antichain are independent
|
|
# - the elements of stack are independent from those of antichain
|
|
yield antichain
|
|
while stack:
|
|
x = stack.pop()
|
|
new_antichain = antichain + [x]
|
|
new_stack = [t for t in stack if not ((t in TC[x]) or (x in TC[t]))]
|
|
antichains_stacks.append((new_antichain, new_stack))
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
def dag_longest_path(G, weight="weight", default_weight=1, topo_order=None):
|
|
"""Returns the longest path in a directed acyclic graph (DAG).
|
|
|
|
If `G` has edges with `weight` attribute the edge data are used as
|
|
weight values.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX DiGraph
|
|
A directed acyclic graph (DAG)
|
|
|
|
weight : str, optional
|
|
Edge data key to use for weight
|
|
|
|
default_weight : int, optional
|
|
The weight of edges that do not have a weight attribute
|
|
|
|
topo_order: list or tuple, optional
|
|
A topological order for G (if None, the function will compute one)
|
|
|
|
Returns
|
|
-------
|
|
list
|
|
Longest path
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed
|
|
|
|
See also
|
|
--------
|
|
dag_longest_path_length
|
|
|
|
"""
|
|
if not G:
|
|
return []
|
|
|
|
if topo_order is None:
|
|
topo_order = nx.topological_sort(G)
|
|
|
|
dist = {} # stores {v : (length, u)}
|
|
for v in topo_order:
|
|
us = [
|
|
(dist[u][0] + data.get(weight, default_weight), u)
|
|
for u, data in G.pred[v].items()
|
|
]
|
|
|
|
# Use the best predecessor if there is one and its distance is
|
|
# non-negative, otherwise terminate.
|
|
maxu = max(us, key=lambda x: x[0]) if us else (0, v)
|
|
dist[v] = maxu if maxu[0] >= 0 else (0, v)
|
|
|
|
u = None
|
|
v = max(dist, key=lambda x: dist[x][0])
|
|
path = []
|
|
while u != v:
|
|
path.append(v)
|
|
u = v
|
|
v = dist[v][1]
|
|
|
|
path.reverse()
|
|
return path
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
def dag_longest_path_length(G, weight="weight", default_weight=1):
|
|
"""Returns the longest path length in a DAG
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX DiGraph
|
|
A directed acyclic graph (DAG)
|
|
|
|
weight : string, optional
|
|
Edge data key to use for weight
|
|
|
|
default_weight : int, optional
|
|
The weight of edges that do not have a weight attribute
|
|
|
|
Returns
|
|
-------
|
|
int
|
|
Longest path length
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed
|
|
|
|
See also
|
|
--------
|
|
dag_longest_path
|
|
"""
|
|
path = nx.dag_longest_path(G, weight, default_weight)
|
|
path_length = 0
|
|
for (u, v) in pairwise(path):
|
|
path_length += G[u][v].get(weight, default_weight)
|
|
|
|
return path_length
|
|
|
|
|
|
def root_to_leaf_paths(G):
|
|
"""Yields root-to-leaf paths in a directed acyclic graph.
|
|
|
|
`G` must be a directed acyclic graph. If not, the behavior of this
|
|
function is undefined. A "root" in this graph is a node of in-degree
|
|
zero and a "leaf" a node of out-degree zero.
|
|
|
|
When invoked, this function iterates over each path from any root to
|
|
any leaf. A path is a list of nodes.
|
|
|
|
"""
|
|
roots = (v for v, d in G.in_degree() if d == 0)
|
|
leaves = (v for v, d in G.out_degree() if d == 0)
|
|
all_paths = partial(nx.all_simple_paths, G)
|
|
# TODO In Python 3, this would be better as `yield from ...`.
|
|
return chaini(starmap(all_paths, product(roots, leaves)))
|
|
|
|
|
|
@not_implemented_for("multigraph")
|
|
@not_implemented_for("undirected")
|
|
def dag_to_branching(G):
|
|
"""Returns a branching representing all (overlapping) paths from
|
|
root nodes to leaf nodes in the given directed acyclic graph.
|
|
|
|
As described in :mod:`networkx.algorithms.tree.recognition`, a
|
|
*branching* is a directed forest in which each node has at most one
|
|
parent. In other words, a branching is a disjoint union of
|
|
*arborescences*. For this function, each node of in-degree zero in
|
|
`G` becomes a root of one of the arborescences, and there will be
|
|
one leaf node for each distinct path from that root to a leaf node
|
|
in `G`.
|
|
|
|
Each node `v` in `G` with *k* parents becomes *k* distinct nodes in
|
|
the returned branching, one for each parent, and the sub-DAG rooted
|
|
at `v` is duplicated for each copy. The algorithm then recurses on
|
|
the children of each copy of `v`.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
A directed acyclic graph.
|
|
|
|
Returns
|
|
-------
|
|
DiGraph
|
|
The branching in which there is a bijection between root-to-leaf
|
|
paths in `G` (in which multiple paths may share the same leaf)
|
|
and root-to-leaf paths in the branching (in which there is a
|
|
unique path from a root to a leaf).
|
|
|
|
Each node has an attribute 'source' whose value is the original
|
|
node to which this node corresponds. No other graph, node, or
|
|
edge attributes are copied into this new graph.
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed, or if `G` is a multigraph.
|
|
|
|
HasACycle
|
|
If `G` is not acyclic.
|
|
|
|
Examples
|
|
--------
|
|
To examine which nodes in the returned branching were produced by
|
|
which original node in the directed acyclic graph, we can collect
|
|
the mapping from source node to new nodes into a dictionary. For
|
|
example, consider the directed diamond graph::
|
|
|
|
>>> from collections import defaultdict
|
|
>>> from operator import itemgetter
|
|
>>>
|
|
>>> G = nx.DiGraph(nx.utils.pairwise("abd"))
|
|
>>> G.add_edges_from(nx.utils.pairwise("acd"))
|
|
>>> B = nx.dag_to_branching(G)
|
|
>>>
|
|
>>> sources = defaultdict(set)
|
|
>>> for v, source in B.nodes(data="source"):
|
|
... sources[source].add(v)
|
|
>>> len(sources["a"])
|
|
1
|
|
>>> len(sources["d"])
|
|
2
|
|
|
|
To copy node attributes from the original graph to the new graph,
|
|
you can use a dictionary like the one constructed in the above
|
|
example::
|
|
|
|
>>> for source, nodes in sources.items():
|
|
... for v in nodes:
|
|
... B.nodes[v].update(G.nodes[source])
|
|
|
|
Notes
|
|
-----
|
|
This function is not idempotent in the sense that the node labels in
|
|
the returned branching may be uniquely generated each time the
|
|
function is invoked. In fact, the node labels may not be integers;
|
|
in order to relabel the nodes to be more readable, you can use the
|
|
:func:`networkx.convert_node_labels_to_integers` function.
|
|
|
|
The current implementation of this function uses
|
|
:func:`networkx.prefix_tree`, so it is subject to the limitations of
|
|
that function.
|
|
|
|
"""
|
|
if has_cycle(G):
|
|
msg = "dag_to_branching is only defined for acyclic graphs"
|
|
raise nx.HasACycle(msg)
|
|
paths = root_to_leaf_paths(G)
|
|
B, root = nx.prefix_tree(paths)
|
|
# Remove the synthetic `root` and `NIL` nodes in the prefix tree.
|
|
B.remove_node(root)
|
|
B.remove_node(NIL)
|
|
return B
|