1260 lines
38 KiB
Python
1260 lines
38 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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import heapq
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from collections import deque
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from functools import partial
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from itertools import chain, combinations, product, starmap
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from math import gcd
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import networkx as nx
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from networkx.utils import arbitrary_element, not_implemented_for, pairwise
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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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"topological_generations",
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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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"compute_v_structures",
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]
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chaini = chain.from_iterable
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@nx._dispatchable
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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 Graph
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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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Raises
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------
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NetworkXError
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If node `source` is not in `G`.
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Examples
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--------
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>>> DG = nx.path_graph(5, create_using=nx.DiGraph)
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>>> sorted(nx.descendants(DG, 2))
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[3, 4]
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The `source` node is not a descendant of itself, but can be included manually:
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>>> sorted(nx.descendants(DG, 2) | {2})
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[2, 3, 4]
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See also
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--------
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ancestors
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"""
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return {child for parent, child in nx.bfs_edges(G, source)}
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@nx._dispatchable
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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 Graph
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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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Raises
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------
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NetworkXError
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If node `source` is not in `G`.
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Examples
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--------
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>>> DG = nx.path_graph(5, create_using=nx.DiGraph)
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>>> sorted(nx.ancestors(DG, 2))
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[0, 1]
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The `source` node is not an ancestor of itself, but can be included manually:
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>>> sorted(nx.ancestors(DG, 2) | {2})
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[0, 1, 2]
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See also
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--------
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descendants
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"""
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return {child for parent, child in nx.bfs_edges(G, source, reverse=True)}
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@nx._dispatchable
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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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# Feed the entire iterator into a zero-length deque.
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deque(topological_sort(G), maxlen=0)
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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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@nx._dispatchable
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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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Examples
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--------
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Undirected graph::
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>>> G = nx.Graph([(1, 2), (2, 3)])
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>>> nx.is_directed_acyclic_graph(G)
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False
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Directed graph with cycle::
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>>> G = nx.DiGraph([(1, 2), (2, 3), (3, 1)])
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>>> nx.is_directed_acyclic_graph(G)
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False
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Directed acyclic graph::
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>>> G = nx.DiGraph([(1, 2), (2, 3)])
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>>> nx.is_directed_acyclic_graph(G)
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True
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See also
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--------
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topological_sort
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"""
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return G.is_directed() and not has_cycle(G)
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@nx._dispatchable
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def topological_generations(G):
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"""Stratifies a DAG into generations.
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A topological generation is node collection in which ancestors of a node in each
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generation are guaranteed to be in a previous generation, and any descendants of
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a node are guaranteed to be in a following generation. Nodes are guaranteed to
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be in the earliest possible generation that they can belong to.
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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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Yields
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------
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sets of nodes
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Yields sets of nodes representing each generation.
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Raises
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------
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NetworkXError
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Generations are defined for directed graphs only. If the graph
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`G` 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 generations
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exist and a :exc:`NetworkXUnfeasible` exception is raised. This can also
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be 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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>>> DG = nx.DiGraph([(2, 1), (3, 1)])
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>>> [sorted(generation) for generation in nx.topological_generations(DG)]
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[[2, 3], [1]]
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Notes
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-----
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The generation in which a node resides can also be determined by taking the
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max-path-distance from the node to the farthest leaf node. That value can
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be obtained with this function using `enumerate(topological_generations(G))`.
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See also
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--------
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topological_sort
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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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multigraph = G.is_multigraph()
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indegree_map = {v: d for v, d in G.in_degree() if d > 0}
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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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this_generation = zero_indegree
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zero_indegree = []
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for node in this_generation:
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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.neighbors(node):
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try:
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indegree_map[child] -= len(G[node][child]) if multigraph else 1
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except KeyError as err:
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raise RuntimeError("Graph changed during iteration") from err
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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 this_generation
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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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@nx._dispatchable
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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 of a
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directed graph such that an edge from u to v implies that u
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appears before v in the topological sort order. This ordering is
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valid only if the graph has no directed cycles.
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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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Yields
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------
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nodes
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Yields the nodes 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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for generation in nx.topological_generations(G):
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yield from generation
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@nx._dispatchable
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def lexicographical_topological_sort(G, key=None):
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"""Generate the nodes in the unique lexicographical topological sort order.
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Generates a unique ordering of nodes by first sorting topologically (for which there are often
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multiple valid orderings) and then additionally by sorting lexicographically.
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A topological sort arranges the nodes of a directed graph so that the
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upstream node of each directed edge precedes the downstream node.
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It is always possible to find a solution for directed graphs that have no cycles.
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There may be more than one valid solution.
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Lexicographical sorting is just sorting alphabetically. It is used here to break ties in the
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topological sort and to determine a single, unique ordering. This can be useful in comparing
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sort results.
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The lexicographical order can be customized by providing a function to the `key=` parameter.
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The definition of the key function is the same as used in python's built-in `sort()`.
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The function takes a single argument and returns a key to use for sorting purposes.
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Lexicographical sorting can fail if the node names are un-sortable. See the example below.
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The solution is to provide a function to the `key=` argument that returns sortable keys.
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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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A function of one argument that converts a node name to a comparison key.
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It defines and resolves ambiguities in the sort order. Defaults to the identity function.
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Yields
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------
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nodes
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Yields the nodes of G 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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TypeError
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Results from un-sortable node names.
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Consider using `key=` parameter to resolve ambiguities in the sort order.
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Examples
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--------
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>>> DG = nx.DiGraph([(2, 1), (2, 5), (1, 3), (1, 4), (5, 4)])
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>>> list(nx.lexicographical_topological_sort(DG))
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[2, 1, 3, 5, 4]
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>>> list(nx.lexicographical_topological_sort(DG, key=lambda x: -x))
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[2, 5, 1, 4, 3]
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The sort will fail for any graph with integer and string nodes. Comparison of integer to strings
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is not defined in python. Is 3 greater or less than 'red'?
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>>> DG = nx.DiGraph([(1, "red"), (3, "red"), (1, "green"), (2, "blue")])
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>>> list(nx.lexicographical_topological_sort(DG))
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Traceback (most recent call last):
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...
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TypeError: '<' not supported between instances of 'str' and 'int'
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...
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Incomparable nodes can be resolved using a `key` function. This example function
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allows comparison of integers and strings by returning a tuple where the first
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element is True for `str`, False otherwise. The second element is the node name.
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This groups the strings and integers separately so they can be compared only among themselves.
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>>> key = lambda node: (isinstance(node, str), node)
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>>> list(nx.lexicographical_topological_sort(DG, key=key))
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[1, 2, 3, 'blue', 'green', 'red']
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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 err:
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raise RuntimeError("Graph changed during iteration") from err
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if indegree_map[child] == 0:
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try:
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heapq.heappush(zero_indegree, create_tuple(child))
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except TypeError as err:
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raise TypeError(
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f"{err}\nConsider using `key=` parameter to resolve ambiguities in the sort order."
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)
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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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@nx._dispatchable
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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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Yields
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------
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topological_sort_order : list
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a list of nodes in `G`, representing one of the topological sort orders
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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
|
|
# of successors, so count is updated correctly in multigraphs
|
|
for _, j in G.out_edges(q):
|
|
count[j] -= 1
|
|
assert count[j] >= 0
|
|
if count[j] == 0:
|
|
D.append(j)
|
|
current_sort.append(q)
|
|
|
|
# base for current position might _not_ be fixed yet
|
|
if len(bases) < len(current_sort):
|
|
bases.append(q)
|
|
|
|
if len(bases) == 0:
|
|
break
|
|
|
|
|
|
@nx._dispatchable
|
|
def is_aperiodic(G):
|
|
"""Returns True if `G` is aperiodic.
|
|
|
|
A directed graph is aperiodic if there is no integer k > 1 that
|
|
divides the length of every cycle in the graph.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX DiGraph
|
|
A directed graph
|
|
|
|
Returns
|
|
-------
|
|
bool
|
|
True if the graph is aperiodic False otherwise
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If `G` is not directed
|
|
|
|
Examples
|
|
--------
|
|
A graph consisting of one cycle, the length of which is 2. Therefore ``k = 2``
|
|
divides the length of every cycle in the graph and thus the graph
|
|
is *not aperiodic*::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 1)])
|
|
>>> nx.is_aperiodic(DG)
|
|
False
|
|
|
|
A graph consisting of two cycles: one of length 2 and the other of length 3.
|
|
The cycle lengths are coprime, so there is no single value of k where ``k > 1``
|
|
that divides each cycle length and therefore the graph is *aperiodic*::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1), (1, 4), (4, 1)])
|
|
>>> nx.is_aperiodic(DG)
|
|
True
|
|
|
|
A graph consisting of two cycles: one of length 2 and the other of length 4.
|
|
The lengths of the cycles share a common factor ``k = 2``, and therefore
|
|
the graph is *not aperiodic*::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 1), (3, 4), (4, 5), (5, 6), (6, 3)])
|
|
>>> nx.is_aperiodic(DG)
|
|
False
|
|
|
|
An acyclic graph, therefore the graph is *not aperiodic*::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
|
|
>>> nx.is_aperiodic(DG)
|
|
False
|
|
|
|
Notes
|
|
-----
|
|
This uses the method outlined in [1]_, which runs in $O(m)$ time
|
|
given $m$ edges in `G`. Note that a graph is not aperiodic if it is
|
|
acyclic as every integer trivial divides length 0 cycles.
|
|
|
|
References
|
|
----------
|
|
.. [1] Jarvis, J. P.; Shier, D. R. (1996),
|
|
"Graph-theoretic analysis of finite Markov chains,"
|
|
in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling:
|
|
A Multidisciplinary Approach, CRC Press.
|
|
"""
|
|
if not G.is_directed():
|
|
raise nx.NetworkXError("is_aperiodic not defined for undirected graphs")
|
|
if len(G) == 0:
|
|
raise nx.NetworkXPointlessConcept("Graph has no nodes.")
|
|
s = arbitrary_element(G)
|
|
levels = {s: 0}
|
|
this_level = [s]
|
|
g = 0
|
|
lev = 1
|
|
while this_level:
|
|
next_level = []
|
|
for u in this_level:
|
|
for v in G[u]:
|
|
if v in levels: # Non-Tree Edge
|
|
g = gcd(g, levels[u] - levels[v] + 1)
|
|
else: # Tree Edge
|
|
next_level.append(v)
|
|
levels[v] = lev
|
|
this_level = next_level
|
|
lev += 1
|
|
if len(levels) == len(G): # All nodes in tree
|
|
return g == 1
|
|
else:
|
|
return g == 1 and nx.is_aperiodic(G.subgraph(set(G) - set(levels)))
|
|
|
|
|
|
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
|
|
def transitive_closure(G, reflexive=False):
|
|
"""Returns transitive closure of a graph
|
|
|
|
The transitive closure 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 there
|
|
is a path from v to w in G.
|
|
|
|
Handling of paths from v to v has some flexibility within this definition.
|
|
A reflexive transitive closure creates a self-loop for the path
|
|
from v to v of length 0. The usual transitive closure creates a
|
|
self-loop only if a cycle exists (a path from v to v with length > 0).
|
|
We also allow an option for no self-loops.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX Graph
|
|
A directed/undirected graph/multigraph.
|
|
reflexive : Bool or None, optional (default: False)
|
|
Determines when cycles create self-loops in the Transitive Closure.
|
|
If True, trivial cycles (length 0) create self-loops. The result
|
|
is a reflexive transitive closure of G.
|
|
If False (the default) non-trivial cycles create self-loops.
|
|
If None, self-loops are not created.
|
|
|
|
Returns
|
|
-------
|
|
NetworkX graph
|
|
The transitive closure of `G`
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If `reflexive` not in `{None, True, False}`
|
|
|
|
Examples
|
|
--------
|
|
The treatment of trivial (i.e. length 0) cycles is controlled by the
|
|
`reflexive` parameter.
|
|
|
|
Trivial (i.e. length 0) cycles do not create self-loops when
|
|
``reflexive=False`` (the default)::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
|
|
>>> TC = nx.transitive_closure(DG, reflexive=False)
|
|
>>> TC.edges()
|
|
OutEdgeView([(1, 2), (1, 3), (2, 3)])
|
|
|
|
However, nontrivial (i.e. length greater than 0) cycles create self-loops
|
|
when ``reflexive=False`` (the default)::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1)])
|
|
>>> TC = nx.transitive_closure(DG, reflexive=False)
|
|
>>> TC.edges()
|
|
OutEdgeView([(1, 2), (1, 3), (1, 1), (2, 3), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3)])
|
|
|
|
Trivial cycles (length 0) create self-loops when ``reflexive=True``::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
|
|
>>> TC = nx.transitive_closure(DG, reflexive=True)
|
|
>>> TC.edges()
|
|
OutEdgeView([(1, 2), (1, 1), (1, 3), (2, 3), (2, 2), (3, 3)])
|
|
|
|
And the third option is not to create self-loops at all when ``reflexive=None``::
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1)])
|
|
>>> TC = nx.transitive_closure(DG, reflexive=None)
|
|
>>> TC.edges()
|
|
OutEdgeView([(1, 2), (1, 3), (2, 3), (2, 1), (3, 1), (3, 2)])
|
|
|
|
References
|
|
----------
|
|
.. [1] https://www.ics.uci.edu/~eppstein/PADS/PartialOrder.py
|
|
"""
|
|
TC = G.copy()
|
|
|
|
if reflexive not in {None, True, False}:
|
|
raise nx.NetworkXError("Incorrect value for the parameter `reflexive`")
|
|
|
|
for v in G:
|
|
if reflexive is None:
|
|
TC.add_edges_from((v, u) for u in nx.descendants(G, v) if u not in TC[v])
|
|
elif reflexive is True:
|
|
TC.add_edges_from(
|
|
(v, u) for u in nx.descendants(G, v) | {v} if u not in TC[v]
|
|
)
|
|
elif reflexive is False:
|
|
TC.add_edges_from((v, e[1]) for e in nx.edge_bfs(G, v) if e[1] not in TC[v])
|
|
|
|
return TC
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
|
|
def transitive_closure_dag(G, topo_order=None):
|
|
"""Returns the transitive closure of a directed acyclic graph.
|
|
|
|
This function is faster than the function `transitive_closure`, but fails
|
|
if the graph has a cycle.
|
|
|
|
The transitive closure 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 there
|
|
is a non-null path from v to w in G.
|
|
|
|
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
|
|
-------
|
|
NetworkX DiGraph
|
|
The transitive closure of `G`
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed
|
|
NetworkXUnfeasible
|
|
If `G` has a cycle
|
|
|
|
Examples
|
|
--------
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
|
|
>>> TC = nx.transitive_closure_dag(DG)
|
|
>>> TC.edges()
|
|
OutEdgeView([(1, 2), (1, 3), (2, 3)])
|
|
|
|
Notes
|
|
-----
|
|
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 nx.descendants_at_distance(TC, v, 2))
|
|
|
|
return TC
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
@nx._dispatchable(returns_graph=True)
|
|
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.
|
|
|
|
Examples
|
|
--------
|
|
To perform transitive reduction on a DiGraph:
|
|
|
|
>>> DG = nx.DiGraph([(1, 2), (2, 3), (1, 3)])
|
|
>>> TR = nx.transitive_reduction(DG)
|
|
>>> list(TR.edges)
|
|
[(1, 2), (2, 3)]
|
|
|
|
To avoid unnecessary data copies, this implementation does not return a
|
|
DiGraph with node/edge data.
|
|
To perform transitive reduction on a DiGraph and transfer node/edge data:
|
|
|
|
>>> DG = nx.DiGraph()
|
|
>>> DG.add_edges_from([(1, 2), (2, 3), (1, 3)], color="red")
|
|
>>> TR = nx.transitive_reduction(DG)
|
|
>>> TR.add_nodes_from(DG.nodes(data=True))
|
|
>>> TR.add_edges_from((u, v, DG.edges[u, v]) for u, v in TR.edges)
|
|
>>> list(TR.edges(data=True))
|
|
[(1, 2, {'color': 'red'}), (2, 3, {'color': 'red'})]
|
|
|
|
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")
|
|
@nx._dispatchable
|
|
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)
|
|
|
|
Yields
|
|
------
|
|
antichain : list
|
|
a list of nodes in `G` representing an antichain
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If `G` is not directed
|
|
|
|
NetworkXUnfeasible
|
|
If `G` contains a cycle
|
|
|
|
Examples
|
|
--------
|
|
>>> DG = nx.DiGraph([(1, 2), (1, 3)])
|
|
>>> list(nx.antichains(DG))
|
|
[[], [3], [2], [2, 3], [1]]
|
|
|
|
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")
|
|
@nx._dispatchable(edge_attrs={"weight": "default_weight"})
|
|
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
|
|
|
|
Examples
|
|
--------
|
|
>>> DG = nx.DiGraph([(0, 1, {"cost": 1}), (1, 2, {"cost": 1}), (0, 2, {"cost": 42})])
|
|
>>> list(nx.all_simple_paths(DG, 0, 2))
|
|
[[0, 1, 2], [0, 2]]
|
|
>>> nx.dag_longest_path(DG)
|
|
[0, 1, 2]
|
|
>>> nx.dag_longest_path(DG, weight="cost")
|
|
[0, 2]
|
|
|
|
In the case where multiple valid topological orderings exist, `topo_order`
|
|
can be used to specify a specific ordering:
|
|
|
|
>>> DG = nx.DiGraph([(0, 1), (0, 2)])
|
|
>>> sorted(nx.all_topological_sorts(DG)) # Valid topological orderings
|
|
[[0, 1, 2], [0, 2, 1]]
|
|
>>> nx.dag_longest_path(DG, topo_order=[0, 1, 2])
|
|
[0, 1]
|
|
>>> nx.dag_longest_path(DG, topo_order=[0, 2, 1])
|
|
[0, 2]
|
|
|
|
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]
|
|
+ (
|
|
max(data.values(), key=lambda x: x.get(weight, default_weight))
|
|
if G.is_multigraph()
|
|
else 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")
|
|
@nx._dispatchable(edge_attrs={"weight": "default_weight"})
|
|
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
|
|
|
|
Examples
|
|
--------
|
|
>>> DG = nx.DiGraph([(0, 1, {"cost": 1}), (1, 2, {"cost": 1}), (0, 2, {"cost": 42})])
|
|
>>> list(nx.all_simple_paths(DG, 0, 2))
|
|
[[0, 1, 2], [0, 2]]
|
|
>>> nx.dag_longest_path_length(DG)
|
|
2
|
|
>>> nx.dag_longest_path_length(DG, weight="cost")
|
|
42
|
|
|
|
See also
|
|
--------
|
|
dag_longest_path
|
|
"""
|
|
path = nx.dag_longest_path(G, weight, default_weight)
|
|
path_length = 0
|
|
if G.is_multigraph():
|
|
for u, v in pairwise(path):
|
|
i = max(G[u][v], key=lambda x: G[u][v][x].get(weight, default_weight))
|
|
path_length += G[u][v][i].get(weight, default_weight)
|
|
else:
|
|
for u, v in pairwise(path):
|
|
path_length += G[u][v].get(weight, default_weight)
|
|
|
|
return path_length
|
|
|
|
|
|
@nx._dispatchable
|
|
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")
|
|
@nx._dispatchable(returns_graph=True)
|
|
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 = nx.prefix_tree(paths)
|
|
# Remove the synthetic `root`(0) and `NIL`(-1) nodes from the tree
|
|
B.remove_node(0)
|
|
B.remove_node(-1)
|
|
return B
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
@nx._dispatchable
|
|
def compute_v_structures(G):
|
|
"""Iterate through the graph to compute all v-structures.
|
|
|
|
V-structures are triples in the directed graph where
|
|
two parent nodes point to the same child and the two parent nodes
|
|
are not adjacent.
|
|
|
|
Parameters
|
|
----------
|
|
G : graph
|
|
A networkx DiGraph.
|
|
|
|
Returns
|
|
-------
|
|
vstructs : iterator of tuples
|
|
The v structures within the graph. Each v structure is a 3-tuple with the
|
|
parent, collider, and other parent.
|
|
|
|
Examples
|
|
--------
|
|
>>> G = nx.DiGraph()
|
|
>>> G.add_edges_from([(1, 2), (0, 5), (3, 1), (2, 4), (3, 1), (4, 5), (1, 5)])
|
|
>>> sorted(nx.compute_v_structures(G))
|
|
[(0, 5, 1), (0, 5, 4), (1, 5, 4)]
|
|
|
|
Notes
|
|
-----
|
|
`Wikipedia: Collider in causal graphs <https://en.wikipedia.org/wiki/Collider_(statistics)>`_
|
|
"""
|
|
for collider, preds in G.pred.items():
|
|
for common_parents in combinations(preds, r=2):
|
|
# ensure that the colliders are the same
|
|
common_parents = sorted(common_parents)
|
|
yield (common_parents[0], collider, common_parents[1])
|