Source code for networkx.algorithms.sparsifiers

"""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] @not_implemented_for("directed") @not_implemented_for("multigraph") @py_random_state(3) @nx._dispatchable(edge_attrs="weight", returns_graph=True) 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]: nbr_cluster = clustering[neighbor] nbr_weight = lightest_edge_weight[nbr_cluster] if ( nbr_cluster == closest_center or nbr_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]: nbr_center = clustering[neighbor] weight = residual_graph[node][neighbor]["weight"] if ( nbr_center not in lightest_edge_weight or weight < lightest_edge_weight[nbr_center] ): lightest_edge_neighbor[nbr_center] = neighbor lightest_edge_weight[nbr_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]