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