295 lines
9.8 KiB
Python
295 lines
9.8 KiB
Python
|
"""Functions for computing sparsifiers of graphs."""
|
||
|
|
import math
|
||
|
|
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.utils import not_implemented_for, py_random_state
|
||
|
|
|
||
|
|
__all__ = ["spanner"]
|
||
|
|
|
||
|
|
|
||
|
|
@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]
|