networkx.algorithms.connectivity.edge_kcomponents.EdgeComponentAuxGraph¶
- class EdgeComponentAuxGraph[source]¶
A simple algorithm to find all k-edge-connected components in a graph.
Constructing the AuxillaryGraph (which may take some time) allows for the k-edge-ccs to be found in linear time for arbitrary k.
Notes
This implementation is based on [1]. The idea is to construct an auxiliary graph from which the k-edge-ccs can be extracted in linear time. The auxiliary graph is constructed in \(O(|V|\cdot F)\) operations, where F is the complexity of max flow. Querying the components takes an additional \(O(|V|)\) operations. This algorithm can be slow for large graphs, but it handles an arbitrary k and works for both directed and undirected inputs.
The undirected case for k=1 is exactly connected components. The undirected case for k=2 is exactly bridge connected components. The directed case for k=1 is exactly strongly connected components.
References
- 1
Wang, Tianhao, et al. (2015) A simple algorithm for finding all k-edge-connected components. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264
Examples
>>> import itertools as it >>> from networkx.utils import pairwise >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph >>> # Build an interesting graph with multiple levels of k-edge-ccs >>> paths = [ ... (1, 2, 3, 4, 1, 3, 4, 2), # a 3-edge-cc (a 4 clique) ... (5, 6, 7, 5), # a 2-edge-cc (a 3 clique) ... (1, 5), # combine first two ccs into a 1-edge-cc ... (0,), # add an additional disconnected 1-edge-cc ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> # Constructing the AuxGraph takes about O(n ** 4) >>> aux_graph = EdgeComponentAuxGraph.construct(G) >>> # Once constructed, querying takes O(n) >>> sorted(map(sorted, aux_graph.k_edge_components(k=1))) [[0], [1, 2, 3, 4, 5, 6, 7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=2))) [[0], [1, 2, 3, 4], [5, 6, 7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=3))) [[0], [1, 2, 3, 4], [5], [6], [7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=4))) [[0], [1], [2], [3], [4], [5], [6], [7]]
The auxiliary graph is primarilly used for k-edge-ccs but it can also speed up the queries of k-edge-subgraphs by refining the search space.
>>> import itertools as it >>> from networkx.utils import pairwise >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph >>> paths = [ ... (1, 2, 4, 3, 1, 4), ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> aux_graph = EdgeComponentAuxGraph.construct(G) >>> sorted(map(sorted, aux_graph.k_edge_subgraphs(k=3))) [[1], [2], [3], [4]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=3))) [[1, 4], [2], [3]]
- __init__(*args, **kwargs)¶
Initialize self. See help(type(self)) for accurate signature.
Methods
__init__
(*args, **kwargs)Initialize self.
construct
(G)Builds an auxiliary graph encoding edge-connectivity between nodes.
k_edge_components
(k)Queries the auxiliary graph for k-edge-connected components.
k_edge_subgraphs
(k)Queries the auxiliary graph for k-edge-connected subgraphs.