Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.utils.rcm

"""
Cuthill-McKee ordering of graph nodes to produce sparse matrices
"""
#    Copyright (C) 2011-2014 by
#    Aric Hagberg <aric.hagberg@gmail.com>
#    All rights reserved.
#    BSD license.
from operator import itemgetter
import networkx as nx
__author__ = """\n""".join(['Aric Hagberg <aric.hagberg@gmail.com>'])
__all__ = ['cuthill_mckee_ordering',
'reverse_cuthill_mckee_ordering']

[docs]def cuthill_mckee_ordering(G, heuristic=None):
"""Generate an ordering (permutation) of the graph nodes to make
a sparse matrix.

Uses the Cuthill-McKee heuristic (based on breadth-first search) _.

Parameters
----------
G : graph
A NetworkX graph

heuristic : function, optional
Function to choose starting node for RCM algorithm.  If None
a node from a psuedo-peripheral pair is used.  A user-defined function
can be supplied that takes a graph object and returns a single node.

Returns
-------
nodes : generator
Generator of nodes in Cuthill-McKee ordering.

Examples
--------
>>> from networkx.utils import cuthill_mckee_ordering
>>> G = nx.path_graph(4)
>>> rcm = list(cuthill_mckee_ordering(G))
>>> A = nx.adjacency_matrix(G, nodelist=rcm) # doctest: +SKIP

Smallest degree node as heuristic function:

>>> def smallest_degree(G):
...     node,deg = sorted(G.degree().items(), key = lambda x:x)
...     return node
>>> rcm = list(cuthill_mckee_ordering(G, heuristic=smallest_degree))

See Also
--------
reverse_cuthill_mckee_ordering

Notes
-----
The optimal solution the the bandwidth reduction is NP-complete _.

References
----------
..  E. Cuthill and J. McKee.
Reducing the bandwidth of sparse symmetric matrices,
In Proc. 24th Nat. Conf. ACM, pages 157-172, 1969.
http://doi.acm.org/10.1145/800195.805928
..   Steven S. Skiena. 1997. The Algorithm Design Manual.
Springer-Verlag New York, Inc., New York, NY, USA.
"""
for c in nx.connected_components(G):
for n in connected_cuthill_mckee_ordering(G.subgraph(c), heuristic):
yield n

[docs]def reverse_cuthill_mckee_ordering(G, heuristic=None):
"""Generate an ordering (permutation) of the graph nodes to make
a sparse matrix.

Uses the reverse Cuthill-McKee heuristic (based on breadth-first search)
_.

Parameters
----------
G : graph
A NetworkX graph

heuristic : function, optional
Function to choose starting node for RCM algorithm.  If None
a node from a psuedo-peripheral pair is used.  A user-defined function
can be supplied that takes a graph object and returns a single node.

Returns
-------
nodes : generator
Generator of nodes in reverse Cuthill-McKee ordering.

Examples
--------
>>> from networkx.utils import reverse_cuthill_mckee_ordering
>>> G = nx.path_graph(4)
>>> rcm = list(reverse_cuthill_mckee_ordering(G))
>>> A = nx.adjacency_matrix(G, nodelist=rcm) # doctest: +SKIP

Smallest degree node as heuristic function:

>>> def smallest_degree(G):
...     node,deg = sorted(G.degree().items(), key = lambda x:x)
...     return node
>>> rcm = list(reverse_cuthill_mckee_ordering(G, heuristic=smallest_degree))

See Also
--------
cuthill_mckee_ordering

Notes
-----
The optimal solution the the bandwidth reduction is NP-complete _.

References
----------
..  E. Cuthill and J. McKee.
Reducing the bandwidth of sparse symmetric matrices,
In Proc. 24th Nat. Conf. ACM, pages 157-72, 1969.
http://doi.acm.org/10.1145/800195.805928
..   Steven S. Skiena. 1997. The Algorithm Design Manual.
Springer-Verlag New York, Inc., New York, NY, USA.
"""
return reversed(list(cuthill_mckee_ordering(G, heuristic=heuristic)))

def connected_cuthill_mckee_ordering(G, heuristic=None):
# the cuthill mckee algorithm for connected graphs
if heuristic is None:
start = pseudo_peripheral_node(G)
else:
start = heuristic(G)
yield start
visited = set([start])
stack = [(start, iter(G[start]))]
while stack:
parent,children = stack
if parent not in visited:
yield parent
try:
child = next(children)
if child not in visited:
yield child
visited.add(child)
# add children to stack, sorted by degree (lowest first)
nd = sorted(G.degree(G[child]).items(), key=itemgetter(1))
children = (n for n,d in nd)
stack.append((child,children))
except StopIteration:
stack.pop(0)

def pseudo_peripheral_node(G):
# helper for cuthill-mckee to find a node in a "pseudo peripheral pair"
# to use as good starting node
u = next(G.nodes_iter())
lp = 0
v = u
while True:
spl = nx.shortest_path_length(G, v)
l = max(spl.values())
if l <= lp:
break
lp = l
farthest = (n for n,dist in spl.items() if dist==l)
v, deg = min(G.degree(farthest).items(), key=itemgetter(1))
return v