# -*- coding: utf-8 -*-
#
# kernighan_lin.py - Kernighan–Lin bipartition algorithm
#
# Copyright 2011 Ben Edwards <bedwards@cs.unm.edu>.
# Copyright 2011 Aric Hagberg <hagberg@lanl.gov>.
# Copyright 2015 NetworkX developers.
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Functions for computing the Kernighan–Lin bipartition algorithm."""
from __future__ import division
from collections import defaultdict
from itertools import islice
from operator import itemgetter
import random
import networkx as nx
from networkx.utils import not_implemented_for
from networkx.algorithms.community.community_utils import is_partition
__all__ = ['kernighan_lin_bisection']
def _compute_delta(G, A, B, weight):
# helper to compute initial swap deltas for a pass
delta = defaultdict(float)
for u, v, d in G.edges(data=True):
w = d.get(weight, 1)
if u in A:
if v in A:
delta[u] -= w
delta[v] -= w
elif v in B:
delta[u] += w
delta[v] += w
elif u in B:
if v in A:
delta[u] += w
delta[v] += w
elif v in B:
delta[u] -= w
delta[v] -= w
return delta
def _update_delta(delta, G, A, B, u, v, weight):
# helper to update swap deltas during single pass
for _, nbr, d in G.edges(u, data=True):
w = d.get(weight, 1)
if nbr in A:
delta[nbr] += 2 * w
if nbr in B:
delta[nbr] -= 2 * w
for _, nbr, d in G.edges(v, data=True):
w = d.get(weight, 1)
if nbr in A:
delta[nbr] -= 2 * w
if nbr in B:
delta[nbr] += 2 * w
return delta
def _kernighan_lin_pass(G, A, B, weight):
# do a single iteration of Kernighan–Lin algorithm
# returns list of (g_i,u_i,v_i) for i node pairs u_i,v_i
multigraph = G.is_multigraph()
delta = _compute_delta(G, A, B, weight)
swapped = set()
gains = []
while len(swapped) < len(G):
gain = []
for u in A - swapped:
for v in B - swapped:
try:
if multigraph:
w = sum(d.get(weight, 1) for d in G[u][v].values())
else:
w = G[u][v].get(weight, 1)
except KeyError:
w = 0
gain.append((delta[u] + delta[v] - 2 * w, u, v))
if len(gain) == 0:
break
maxg, u, v = max(gain, key=itemgetter(0))
swapped |= {u, v}
gains.append((maxg, u, v))
delta = _update_delta(delta, G, A - swapped, B - swapped, u, v, weight)
return gains
[docs]@not_implemented_for('directed')
def kernighan_lin_bisection(G, partition=None, max_iter=10, weight='weight'):
"""Partition a graph into two blocks using the Kernighan–Lin
algorithm.
This algorithm paritions a network into two sets by iteratively
swapping pairs of nodes to reduce the edge cut between the two sets.
Parameters
----------
G : graph
partition : tuple
Pair of iterables containing an intial partition. If not
specified, a random balanced partition is used.
max_iter : int
Maximum number of times to attempt swaps to find an
improvemement before giving up.
weight : key
Edge data key to use as weight. If None, the weights are all
set to one.
Returns
-------
partition : tuple
A pair of sets of nodes representing the bipartition.
Raises
-------
NetworkXError
If partition is not a valid partition of the nodes of the graph.
References
----------
.. [1] Kernighan, B. W.; Lin, Shen (1970).
"An efficient heuristic procedure for partitioning graphs."
*Bell Systems Technical Journal* 49: 291--307.
Oxford University Press 2011.
"""
# If no partition is provided, split the nodes randomly into a
# balanced partition.
if partition is None:
nodes = list(G)
random.shuffle(nodes)
h = len(nodes) // 2
partition = (nodes[:h], nodes[h:])
# Make a copy of the partition as a pair of sets.
try:
A, B = set(partition[0]), set(partition[1])
except:
raise ValueError('partition must be two sets')
if not is_partition(G, (A, B)):
raise nx.NetworkXError('partition invalid')
for i in range(max_iter):
# `gains` is a list of triples of the form (g, u, v) for each
# node pair (u, v), where `g` is the gain of that node pair.
gains = _kernighan_lin_pass(G, A, B, weight)
csum = list(nx.utils.accumulate(g for g, u, v in gains))
max_cgain = max(csum)
if max_cgain <= 0:
break
# Get the node pairs up to the index of the maximum cumulative
# gain, and collect each `u` into `anodes` and each `v` into
# `bnodes`, for each pair `(u, v)`.
index = csum.index(max_cgain)
nodesets = islice(zip(*gains[:index + 1]), 1, 3)
anodes, bnodes = (set(s) for s in nodesets)
A |= bnodes
A -= anodes
B |= anodes
B -= bnodes
return A, B