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Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

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Source code for networkx.utils.rcm

"""
Cuthill-McKee ordering of graph nodes to produce sparse matrices
"""
from collections import deque
from operator import itemgetter

import networkx as nx
from ..utils import arbitrary_element

__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) [1]_. Parameters ---------- G : graph A NetworkX graph heuristic : function, optional Function to choose starting node for RCM algorithm. If None a node from a pseudo-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) Smallest degree node as heuristic function: >>> def smallest_degree(G): ... return min(G, key=G.degree) >>> 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 [2]_. References ---------- .. [1] 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 .. [2] Steven S. Skiena. 1997. The Algorithm Design Manual. Springer-Verlag New York, Inc., New York, NY, USA. """ for c in nx.connected_components(G): yield from connected_cuthill_mckee_ordering(G.subgraph(c), heuristic)
[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) [1]_. Parameters ---------- G : graph A NetworkX graph heuristic : function, optional Function to choose starting node for RCM algorithm. If None a node from a pseudo-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) Smallest degree node as heuristic function: >>> def smallest_degree(G): ... return min(G, key=G.degree) >>> 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 [2]_. References ---------- .. [1] 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 .. [2] 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) visited = {start} queue = deque([start]) while queue: parent = queue.popleft() yield parent nd = sorted(list(G.degree(set(G[parent]) - visited)), key=itemgetter(1)) children = [n for n, d in nd] visited.update(children) queue.extend(children) 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 = arbitrary_element(G) lp = 0 v = u while True: spl = dict(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), key=itemgetter(1)) return v