"""Functions for computing sparsifiers of graphs."""
import math
import networkx as nx
from networkx.utils import not_implemented_for, py_random_state
__all__ = ["spanner"]
[docs]@py_random_state(3)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def spanner(G, stretch, weight=None, seed=None):
"""Returns a spanner of the given graph with the given stretch.
A spanner of a graph G = (V, E) with stretch t is a subgraph
H = (V, E_S) such that E_S is a subset of E and the distance between
any pair of nodes in H is at most t times the distance between the
nodes in G.
Parameters
----------
G : NetworkX graph
An undirected simple graph.
stretch : float
The stretch of the spanner.
weight : object
The edge attribute to use as distance.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
NetworkX graph
A spanner of the given graph with the given stretch.
Raises
------
ValueError
If a stretch less than 1 is given.
Notes
-----
This function implements the spanner algorithm by Baswana and Sen,
see [1].
This algorithm is a randomized las vegas algorithm: The expected
running time is O(km) where k = (stretch + 1) // 2 and m is the
number of edges in G. The returned graph is always a spanner of the
given graph with the specified stretch. For weighted graphs the
number of edges in the spanner is O(k * n^(1 + 1 / k)) where k is
defined as above and n is the number of nodes in G. For unweighted
graphs the number of edges is O(n^(1 + 1 / k) + kn).
References
----------
[1] S. Baswana, S. Sen. A Simple and Linear Time Randomized
Algorithm for Computing Sparse Spanners in Weighted Graphs.
Random Struct. Algorithms 30(4): 532-563 (2007).
"""
if stretch < 1:
raise ValueError("stretch must be at least 1")
k = (stretch + 1) // 2
# initialize spanner H with empty edge set
H = nx.empty_graph()
H.add_nodes_from(G.nodes)
# phase 1: forming the clusters
# the residual graph has V' from the paper as its node set
# and E' from the paper as its edge set
residual_graph = _setup_residual_graph(G, weight)
# clustering is a dictionary that maps nodes in a cluster to the
# cluster center
clustering = {v: v for v in G.nodes}
sample_prob = math.pow(G.number_of_nodes(), -1 / k)
size_limit = 2 * math.pow(G.number_of_nodes(), 1 + 1 / k)
i = 0
while i < k - 1:
# step 1: sample centers
sampled_centers = set()
for center in set(clustering.values()):
if seed.random() < sample_prob:
sampled_centers.add(center)
# combined loop for steps 2 and 3
edges_to_add = set()
edges_to_remove = set()
new_clustering = {}
for v in residual_graph.nodes:
if clustering[v] in sampled_centers:
continue
# step 2: find neighboring (sampled) clusters and
# lightest edges to them
lightest_edge_neighbor, lightest_edge_weight = _lightest_edge_dicts(
residual_graph, clustering, v
)
neighboring_sampled_centers = (
set(lightest_edge_weight.keys()) & sampled_centers
)
# step 3: add edges to spanner
if not neighboring_sampled_centers:
# connect to each neighboring center via lightest edge
for neighbor in lightest_edge_neighbor.values():
edges_to_add.add((v, neighbor))
# remove all incident edges
for neighbor in residual_graph.adj[v]:
edges_to_remove.add((v, neighbor))
else: # there is a neighboring sampled center
closest_center = min(
neighboring_sampled_centers, key=lightest_edge_weight.get
)
closest_center_weight = lightest_edge_weight[closest_center]
closest_center_neighbor = lightest_edge_neighbor[closest_center]
edges_to_add.add((v, closest_center_neighbor))
new_clustering[v] = closest_center
# connect to centers with edge weight less than
# closest_center_weight
for center, edge_weight in lightest_edge_weight.items():
if edge_weight < closest_center_weight:
neighbor = lightest_edge_neighbor[center]
edges_to_add.add((v, neighbor))
# remove edges to centers with edge weight less than
# closest_center_weight
for neighbor in residual_graph.adj[v]:
neighbor_cluster = clustering[neighbor]
neighbor_weight = lightest_edge_weight[neighbor_cluster]
if (
neighbor_cluster == closest_center
or neighbor_weight < closest_center_weight
):
edges_to_remove.add((v, neighbor))
# check whether iteration added too many edges to spanner,
# if so repeat
if len(edges_to_add) > size_limit:
# an iteration is repeated O(1) times on expectation
continue
# iteration succeeded
i = i + 1
# actually add edges to spanner
for u, v in edges_to_add:
_add_edge_to_spanner(H, residual_graph, u, v, weight)
# actually delete edges from residual graph
residual_graph.remove_edges_from(edges_to_remove)
# copy old clustering data to new_clustering
for node, center in clustering.items():
if center in sampled_centers:
new_clustering[node] = center
clustering = new_clustering
# step 4: remove intra-cluster edges
for u in residual_graph.nodes:
for v in list(residual_graph.adj[u]):
if clustering[u] == clustering[v]:
residual_graph.remove_edge(u, v)
# update residual graph node set
for v in list(residual_graph.nodes):
if v not in clustering:
residual_graph.remove_node(v)
# phase 2: vertex-cluster joining
for v in residual_graph.nodes:
lightest_edge_neighbor, _ = _lightest_edge_dicts(residual_graph, clustering, v)
for neighbor in lightest_edge_neighbor.values():
_add_edge_to_spanner(H, residual_graph, v, neighbor, weight)
return H
def _setup_residual_graph(G, weight):
"""Setup residual graph as a copy of G with unique edges weights.
The node set of the residual graph corresponds to the set V' from
the Baswana-Sen paper and the edge set corresponds to the set E'
from the paper.
This function associates distinct weights to the edges of the
residual graph (even for unweighted input graphs), as required by
the algorithm.
Parameters
----------
G : NetworkX graph
An undirected simple graph.
weight : object
The edge attribute to use as distance.
Returns
-------
NetworkX graph
The residual graph used for the Baswana-Sen algorithm.
"""
residual_graph = G.copy()
# establish unique edge weights, even for unweighted graphs
for u, v in G.edges():
if not weight:
residual_graph[u][v]["weight"] = (id(u), id(v))
else:
residual_graph[u][v]["weight"] = (G[u][v][weight], id(u), id(v))
return residual_graph
def _lightest_edge_dicts(residual_graph, clustering, node):
"""Find the lightest edge to each cluster.
Searches for the minimum-weight edge to each cluster adjacent to
the given node.
Parameters
----------
residual_graph : NetworkX graph
The residual graph used by the Baswana-Sen algorithm.
clustering : dictionary
The current clustering of the nodes.
node : node
The node from which the search originates.
Returns
-------
lightest_edge_neighbor, lightest_edge_weight : dictionary, dictionary
lightest_edge_neighbor is a dictionary that maps a center C to
a node v in the corresponding cluster such that the edge from
the given node to v is the lightest edge from the given node to
any node in cluster. lightest_edge_weight maps a center C to the
weight of the aforementioned edge.
Notes
-----
If a cluster has no node that is adjacent to the given node in the
residual graph then the center of the cluster is not a key in the
returned dictionaries.
"""
lightest_edge_neighbor = {}
lightest_edge_weight = {}
for neighbor in residual_graph.adj[node]:
neighbor_center = clustering[neighbor]
weight = residual_graph[node][neighbor]["weight"]
if (
neighbor_center not in lightest_edge_weight
or weight < lightest_edge_weight[neighbor_center]
):
lightest_edge_neighbor[neighbor_center] = neighbor
lightest_edge_weight[neighbor_center] = weight
return lightest_edge_neighbor, lightest_edge_weight
def _add_edge_to_spanner(H, residual_graph, u, v, weight):
"""Add the edge {u, v} to the spanner H and take weight from
the residual graph.
Parameters
----------
H : NetworkX graph
The spanner under construction.
residual_graph : NetworkX graph
The residual graph used by the Baswana-Sen algorithm. The weight
for the edge is taken from this graph.
u : node
One endpoint of the edge.
v : node
The other endpoint of the edge.
weight : object
The edge attribute to use as distance.
"""
H.add_edge(u, v)
if weight:
H[u][v][weight] = residual_graph[u][v]["weight"][0]