# -*- coding: utf-8 -*-
from fractions import gcd
import networkx as nx
from networkx.utils.decorators import *
"""Algorithms for directed acyclic graphs (DAGs)."""
# Copyright (C) 2006-2011 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__author__ = """\n""".join(['Aric Hagberg <aric.hagberg@gmail.com>',
'Dan Schult (dschult@colgate.edu)',
'Ben Edwards (bedwards@cs.unm.edu)'])
__all__ = ['descendants',
'ancestors',
'topological_sort',
'topological_sort_recursive',
'is_directed_acyclic_graph',
'is_aperiodic',
'transitive_closure',
'antichains',
'dag_longest_path',
'dag_longest_path_length']
[docs]def descendants(G, source):
"""Return all nodes reachable from `source` in G.
Parameters
----------
G : NetworkX DiGraph
source : node in G
Returns
-------
des : set()
The descendants of source in G
"""
if not G.has_node(source):
raise nx.NetworkXError("The node %s is not in the graph." % source)
des = set(nx.shortest_path_length(G, source=source).keys()) - set([source])
return des
[docs]def ancestors(G, source):
"""Return all nodes having a path to `source` in G.
Parameters
----------
G : NetworkX DiGraph
source : node in G
Returns
-------
ancestors : set()
The ancestors of source in G
"""
if not G.has_node(source):
raise nx.NetworkXError("The node %s is not in the graph." % source)
anc = set(nx.shortest_path_length(G, target=source).keys()) - set([source])
return anc
[docs]def is_directed_acyclic_graph(G):
"""Return True if the graph G is a directed acyclic graph (DAG) or
False if not.
Parameters
----------
G : NetworkX graph
A graph
Returns
-------
is_dag : bool
True if G is a DAG, false otherwise
"""
if not G.is_directed():
return False
try:
topological_sort(G, reverse=True)
return True
except nx.NetworkXUnfeasible:
return False
[docs]def topological_sort(G, nbunch=None, reverse=False):
"""Return a list of nodes in topological sort order.
A topological sort is a nonunique permutation of the nodes
such that an edge from u to v implies that u appears before v in the
topological sort order.
Parameters
----------
G : NetworkX digraph
A directed graph
nbunch : container of nodes (optional)
Explore graph in specified order given in nbunch
reverse : bool, optional
Return postorder instead of preorder if True.
Reverse mode is a bit more efficient.
Raises
------
NetworkXError
Topological sort is defined for directed graphs only. If the
graph G is undirected, a NetworkXError is raised.
NetworkXUnfeasible
If G is not a directed acyclic graph (DAG) no topological sort
exists and a NetworkXUnfeasible exception is raised.
Notes
-----
This algorithm is based on a description and proof in
The Algorithm Design Manual [1]_ .
See also
--------
is_directed_acyclic_graph
References
----------
.. [1] Skiena, S. S. The Algorithm Design Manual (Springer-Verlag, 1998).
http://www.amazon.com/exec/obidos/ASIN/0387948600/ref=ase_thealgorithmrepo/
"""
if not G.is_directed():
raise nx.NetworkXError(
"Topological sort not defined on undirected graphs.")
# nonrecursive version
seen = set()
order = []
explored = set()
if nbunch is None:
nbunch = G.nodes_iter()
for v in nbunch: # process all vertices in G
if v in explored:
continue
fringe = [v] # nodes yet to look at
while fringe:
w = fringe[-1] # depth first search
if w in explored: # already looked down this branch
fringe.pop()
continue
seen.add(w) # mark as seen
# Check successors for cycles and for new nodes
new_nodes = []
for n in G[w]:
if n not in explored:
if n in seen: # CYCLE !!
raise nx.NetworkXUnfeasible("Graph contains a cycle.")
new_nodes.append(n)
if new_nodes: # Add new_nodes to fringe
fringe.extend(new_nodes)
else: # No new nodes so w is fully explored
explored.add(w)
order.append(w)
fringe.pop() # done considering this node
if reverse:
return order
else:
return list(reversed(order))
[docs]def topological_sort_recursive(G, nbunch=None, reverse=False):
"""Return a list of nodes in topological sort order.
A topological sort is a nonunique permutation of the nodes such
that an edge from u to v implies that u appears before v in the
topological sort order.
Parameters
----------
G : NetworkX digraph
nbunch : container of nodes (optional)
Explore graph in specified order given in nbunch
reverse : bool, optional
Return postorder instead of preorder if True.
Reverse mode is a bit more efficient.
Raises
------
NetworkXError
Topological sort is defined for directed graphs only. If the
graph G is undirected, a NetworkXError is raised.
NetworkXUnfeasible
If G is not a directed acyclic graph (DAG) no topological sort
exists and a NetworkXUnfeasible exception is raised.
Notes
-----
This is a recursive version of topological sort.
See also
--------
topological_sort
is_directed_acyclic_graph
"""
if not G.is_directed():
raise nx.NetworkXError(
"Topological sort not defined on undirected graphs.")
def _dfs(v):
ancestors.add(v)
for w in G[v]:
if w in ancestors:
raise nx.NetworkXUnfeasible("Graph contains a cycle.")
if w not in explored:
_dfs(w)
ancestors.remove(v)
explored.add(v)
order.append(v)
ancestors = set()
explored = set()
order = []
if nbunch is None:
nbunch = G.nodes_iter()
for v in nbunch:
if v not in explored:
_dfs(v)
if reverse:
return order
else:
return list(reversed(order))
[docs]def is_aperiodic(G):
"""Return 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
Graph
Returns
-------
aperiodic : boolean
True if the graph is aperiodic False otherwise
Raises
------
NetworkXError
If G is not directed
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")
s = next(G.nodes_iter())
levels = {s: 0}
this_level = [s]
g = 0
l = 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] = l
this_level = next_level
l += 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)))
[docs]@not_implemented_for('undirected')
def transitive_closure(G):
""" Returns transitive closure of a directed 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 non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
Graph
Returns
-------
TC : NetworkX DiGraph
Graph
Raises
------
NetworkXNotImplemented
If G is not directed
References
----------
.. [1] http://www.ics.uci.edu/~eppstein/PADS/PartialOrder.py
"""
TC = nx.DiGraph()
TC.add_nodes_from(G.nodes_iter())
TC.add_edges_from(G.edges_iter())
for v in G:
TC.add_edges_from((v, u) for u in nx.dfs_preorder_nodes(G, source=v)
if v != u)
return TC
[docs]@not_implemented_for('undirected')
def antichains(G):
"""Generates antichains from a 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
Graph
Returns
-------
antichain : 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://sage.informatik.uni-goettingen.de/src/combinat/posets/hasse_diagram.py
References
----------
.. [1] Free Lattices, by R. Freese, J. Jezek and J. B. Nation,
AMS, Vol 42, 1995, p. 226.
"""
TC = nx.transitive_closure(G)
antichains_stacks = [([], nx.topological_sort(G, reverse=True))]
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))
[docs]@not_implemented_for('undirected')
def dag_longest_path(G):
"""Returns the longest path in a DAG
Parameters
----------
G : NetworkX DiGraph
Graph
Returns
-------
path : list
Longest path
Raises
------
NetworkXNotImplemented
If G is not directed
See also
--------
dag_longest_path_length
"""
dist = {} # stores [node, distance] pair
for node in nx.topological_sort(G):
# pairs of dist,node for all incoming edges
pairs = [(dist[v][0] + 1, v) for v in G.pred[node]]
if pairs:
dist[node] = max(pairs)
else:
dist[node] = (0, node)
node, (length, _) = max(dist.items(), key=lambda x: x[1])
path = []
while length > 0:
path.append(node)
length, node = dist[node]
return list(reversed(path))
[docs]@not_implemented_for('undirected')
def dag_longest_path_length(G):
"""Returns the longest path length in a DAG
Parameters
----------
G : NetworkX DiGraph
Graph
Returns
-------
path_length : int
Longest path length
Raises
------
NetworkXNotImplemented
If G is not directed
See also
--------
dag_longest_path
"""
path_length = len(nx.dag_longest_path(G)) - 1
return path_length