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""" Algorithms for finding k-edge-augmentations A k-edge-augmentation is a set of edges, that once added to a graph, ensures that the graph is k-edge-connected; i.e. the graph cannot be disconnected unless k or more edges are removed. Typically, the goal is to find the augmentation with minimum weight. In general, it is not guaranteed that a k-edge-augmentation exists. See Also -------- :mod:`edge_kcomponents` : algorithms for finding k-edge-connected components :mod:`connectivity` : algorithms for determening edge connectivity. """ import math import itertools as it import networkx as nx from networkx.utils import not_implemented_for, py_random_state from collections import defaultdict, namedtuple __all__ = ["k_edge_augmentation", "is_k_edge_connected", "is_locally_k_edge_connected"] @not_implemented_for("directed") @not_implemented_for("multigraph") def is_k_edge_connected(G, k): """Tests to see if a graph is k-edge-connected. Is it impossible to disconnect the graph by removing fewer than k edges? If so, then G is k-edge-connected. Parameters ---------- G : NetworkX graph An undirected graph. k : integer edge connectivity to test for Returns ------- boolean True if G is k-edge-connected. See Also -------- :func:`is_locally_k_edge_connected` Example ------- >>> G = nx.barbell_graph(10, 0) >>> nx.is_k_edge_connected(G, k=1) True >>> nx.is_k_edge_connected(G, k=2) False """ if k < 1: raise ValueError(f"k must be positive, not {k}") # First try to quickly determine if G is not k-edge-connected if G.number_of_nodes() < k + 1: return False elif any(d < k for n, d in G.degree()): return False else: # Otherwise perform the full check if k == 1: return nx.is_connected(G) elif k == 2: return not nx.has_bridges(G) else: return nx.edge_connectivity(G, cutoff=k) >= k @not_implemented_for("directed") @not_implemented_for("multigraph") def is_locally_k_edge_connected(G, s, t, k): """Tests to see if an edge in a graph is locally k-edge-connected. Is it impossible to disconnect s and t by removing fewer than k edges? If so, then s and t are locally k-edge-connected in G. Parameters ---------- G : NetworkX graph An undirected graph. s : node Source node t : node Target node k : integer local edge connectivity for nodes s and t Returns ------- boolean True if s and t are locally k-edge-connected in G. See Also -------- :func:`is_k_edge_connected` Example ------- >>> from networkx.algorithms.connectivity import is_locally_k_edge_connected >>> G = nx.barbell_graph(10, 0) >>> is_locally_k_edge_connected(G, 5, 15, k=1) True >>> is_locally_k_edge_connected(G, 5, 15, k=2) False >>> is_locally_k_edge_connected(G, 1, 5, k=2) True """ if k < 1: raise ValueError(f"k must be positive, not {k}") # First try to quickly determine s, t is not k-locally-edge-connected in G if G.degree(s) < k or G.degree(t) < k: return False else: # Otherwise perform the full check if k == 1: return nx.has_path(G, s, t) else: localk = nx.connectivity.local_edge_connectivity(G, s, t, cutoff=k) return localk >= k @not_implemented_for("directed") @not_implemented_for("multigraph") def k_edge_augmentation(G, k, avail=None, weight=None, partial=False): """Finds set of edges to k-edge-connect G. Adding edges from the augmentation to G make it impossible to disconnect G unless k or more edges are removed. This function uses the most efficient function available (depending on the value of k and if the problem is weighted or unweighted) to search for a minimum weight subset of available edges that k-edge-connects G. In general, finding a k-edge-augmentation is NP-hard, so solutions are not garuenteed to be minimal. Furthermore, a k-edge-augmentation may not exist. Parameters ---------- G : NetworkX graph An undirected graph. k : integer Desired edge connectivity avail : dict or a set of 2 or 3 tuples The available edges that can be used in the augmentation. If unspecified, then all edges in the complement of G are available. Otherwise, each item is an available edge (with an optional weight). In the unweighted case, each item is an edge ``(u, v)``. In the weighted case, each item is a 3-tuple ``(u, v, d)`` or a dict with items ``(u, v): d``. The third item, ``d``, can be a dictionary or a real number. If ``d`` is a dictionary ``d[weight]`` correspondings to the weight. weight : string key to use to find weights if ``avail`` is a set of 3-tuples where the third item in each tuple is a dictionary. partial : boolean If partial is True and no feasible k-edge-augmentation exists, then all a partial k-edge-augmentation is generated. Adding the edges in a partial augmentation to G, minimizes the number of k-edge-connected components and maximizes the edge connectivity between those components. For details, see :func:`partial_k_edge_augmentation`. Yields ------ edge : tuple Edges that, once added to G, would cause G to become k-edge-connected. If partial is False, an error is raised if this is not possible. Otherwise, generated edges form a partial augmentation, which k-edge-connects any part of G where it is possible, and maximally connects the remaining parts. Raises ------ NetworkXUnfeasible If partial is False and no k-edge-augmentation exists. NetworkXNotImplemented If the input graph is directed or a multigraph. ValueError: If k is less than 1 Notes ----- When k=1 this returns an optimal solution. When k=2 and ``avail`` is None, this returns an optimal solution. Otherwise when k=2, this returns a 2-approximation of the optimal solution. For k>3, this problem is NP-hard and this uses a randomized algorithm that produces a feasible solution, but provides no guarantees on the solution weight. Example ------- >>> # Unweighted cases >>> G = nx.path_graph((1, 2, 3, 4)) >>> G.add_node(5) >>> sorted(nx.k_edge_augmentation(G, k=1)) [(1, 5)] >>> sorted(nx.k_edge_augmentation(G, k=2)) [(1, 5), (5, 4)] >>> sorted(nx.k_edge_augmentation(G, k=3)) [(1, 4), (1, 5), (2, 5), (3, 5), (4, 5)] >>> complement = list(nx.k_edge_augmentation(G, k=5, partial=True)) >>> G.add_edges_from(complement) >>> nx.edge_connectivity(G) 4 Example ------- >>> # Weighted cases >>> G = nx.path_graph((1, 2, 3, 4)) >>> G.add_node(5) >>> # avail can be a tuple with a dict >>> avail = [(1, 5, {"weight": 11}), (2, 5, {"weight": 10})] >>> sorted(nx.k_edge_augmentation(G, k=1, avail=avail, weight="weight")) [(2, 5)] >>> # or avail can be a 3-tuple with a real number >>> avail = [(1, 5, 11), (2, 5, 10), (4, 3, 1), (4, 5, 51)] >>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail)) [(1, 5), (2, 5), (4, 5)] >>> # or avail can be a dict >>> avail = {(1, 5): 11, (2, 5): 10, (4, 3): 1, (4, 5): 51} >>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail)) [(1, 5), (2, 5), (4, 5)] >>> # If augmentation is infeasible, then a partial solution can be found >>> avail = {(1, 5): 11} >>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail, partial=True)) [(1, 5)] """ try: if k <= 0: raise ValueError(f"k must be a positive integer, not {k}") elif G.number_of_nodes() < k + 1: msg = f"impossible to {k} connect in graph with less than {k + 1} nodes" raise nx.NetworkXUnfeasible(msg) elif avail is not None and len(avail) == 0: if not nx.is_k_edge_connected(G, k): raise nx.NetworkXUnfeasible("no available edges") aug_edges = [] elif k == 1: aug_edges = one_edge_augmentation( G, avail=avail, weight=weight, partial=partial ) elif k == 2: aug_edges = bridge_augmentation(G, avail=avail, weight=weight) else: # raise NotImplementedError(f'not implemented for k>2. k={k}') aug_edges = greedy_k_edge_augmentation( G, k=k, avail=avail, weight=weight, seed=0 ) # Do eager evaulation so we can catch any exceptions # Before executing partial code. yield from list(aug_edges) except nx.NetworkXUnfeasible: if partial: # Return all available edges if avail is None: aug_edges = complement_edges(G) else: # If we can't k-edge-connect the entire graph, try to # k-edge-connect as much as possible aug_edges = partial_k_edge_augmentation( G, k=k, avail=avail, weight=weight ) yield from aug_edges else: raise def partial_k_edge_augmentation(G, k, avail, weight=None): """Finds augmentation that k-edge-connects as much of the graph as possible. When a k-edge-augmentation is not possible, we can still try to find a small set of edges that partially k-edge-connects as much of the graph as possible. All possible edges are generated between remaining parts. This minimizes the number of k-edge-connected subgraphs in the resulting graph and maxmizes the edge connectivity between those subgraphs. Parameters ---------- G : NetworkX graph An undirected graph. k : integer Desired edge connectivity avail : dict or a set of 2 or 3 tuples For more details, see :func:`k_edge_augmentation`. weight : string key to use to find weights if ``avail`` is a set of 3-tuples. For more details, see :func:`k_edge_augmentation`. Yields ------ edge : tuple Edges in the partial augmentation of G. These edges k-edge-connect any part of G where it is possible, and maximally connects the remaining parts. In other words, all edges from avail are generated except for those within subgraphs that have already become k-edge-connected. Notes ----- Construct H that augments G with all edges in avail. Find the k-edge-subgraphs of H. For each k-edge-subgraph, if the number of nodes is more than k, then find the k-edge-augmentation of that graph and add it to the solution. Then add all edges in avail between k-edge subgraphs to the solution. See Also -------- :func:`k_edge_augmentation` Example ------- >>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7)) >>> G.add_node(8) >>> avail = [(1, 3), (1, 4), (1, 5), (2, 4), (2, 5), (3, 5), (1, 8)] >>> sorted(partial_k_edge_augmentation(G, k=2, avail=avail)) [(1, 5), (1, 8)] """ def _edges_between_disjoint(H, only1, only2): """ finds edges between disjoint nodes """ only1_adj = {u: set(H.adj[u]) for u in only1} for u, neighbs in only1_adj.items(): # Find the neighbors of u in only1 that are also in only2 neighbs12 = neighbs.intersection(only2) for v in neighbs12: yield (u, v) avail_uv, avail_w = _unpack_available_edges(avail, weight=weight, G=G) # Find which parts of the graph can be k-edge-connected H = G.copy() H.add_edges_from( ( (u, v, {"weight": w, "generator": (u, v)}) for (u, v), w in zip(avail, avail_w) ) ) k_edge_subgraphs = list(nx.k_edge_subgraphs(H, k=k)) # Generate edges to k-edge-connect internal subgraphs for nodes in k_edge_subgraphs: if len(nodes) > 1: # Get the k-edge-connected subgraph C = H.subgraph(nodes).copy() # Find the internal edges that were available sub_avail = { d["generator"]: d["weight"] for (u, v, d) in C.edges(data=True) if "generator" in d } # Remove potential augmenting edges C.remove_edges_from(sub_avail.keys()) # Find a subset of these edges that makes the compoment # k-edge-connected and ignore the rest yield from nx.k_edge_augmentation(C, k=k, avail=sub_avail) # Generate all edges between CCs that could not be k-edge-connected for cc1, cc2 in it.combinations(k_edge_subgraphs, 2): for (u, v) in _edges_between_disjoint(H, cc1, cc2): d = H.get_edge_data(u, v) edge = d.get("generator", None) if edge is not None: yield edge @not_implemented_for("multigraph") @not_implemented_for("directed") def one_edge_augmentation(G, avail=None, weight=None, partial=False): """Finds minimum weight set of edges to connect G. Equivalent to :func:`k_edge_augmentation` when k=1. Adding the resulting edges to G will make it 1-edge-connected. The solution is optimal for both weighted and non-weighted variants. Parameters ---------- G : NetworkX graph An undirected graph. avail : dict or a set of 2 or 3 tuples For more details, see :func:`k_edge_augmentation`. weight : string key to use to find weights if ``avail`` is a set of 3-tuples. For more details, see :func:`k_edge_augmentation`. partial : boolean If partial is True and no feasible k-edge-augmentation exists, then the augmenting edges minimize the number of connected components. Yields ------ edge : tuple Edges in the one-augmentation of G Raises ------ NetworkXUnfeasible If partial is False and no one-edge-augmentation exists. Notes ----- Uses either :func:`unconstrained_one_edge_augmentation` or :func:`weighted_one_edge_augmentation` depending on whether ``avail`` is specified. Both algorithms are based on finding a minimum spanning tree. As such both algorithms find optimal solutions and run in linear time. See Also -------- :func:`k_edge_augmentation` """ if avail is None: return unconstrained_one_edge_augmentation(G) else: return weighted_one_edge_augmentation( G, avail=avail, weight=weight, partial=partial ) @not_implemented_for("multigraph") @not_implemented_for("directed") def bridge_augmentation(G, avail=None, weight=None): """Finds the a set of edges that bridge connects G. Equivalent to :func:`k_edge_augmentation` when k=2, and partial=False. Adding the resulting edges to G will make it 2-edge-connected. If no constraints are specified the returned set of edges is minimum an optimal, otherwise the solution is approximated. Parameters ---------- G : NetworkX graph An undirected graph. avail : dict or a set of 2 or 3 tuples For more details, see :func:`k_edge_augmentation`. weight : string key to use to find weights if ``avail`` is a set of 3-tuples. For more details, see :func:`k_edge_augmentation`. Yields ------ edge : tuple Edges in the bridge-augmentation of G Raises ------ NetworkXUnfeasible If no bridge-augmentation exists. Notes ----- If there are no constraints the solution can be computed in linear time using :func:`unconstrained_bridge_augmentation`. Otherwise, the problem becomes NP-hard and is the solution is approximated by :func:`weighted_bridge_augmentation`. See Also -------- :func:`k_edge_augmentation` """ if G.number_of_nodes() < 3: raise nx.NetworkXUnfeasible("impossible to bridge connect less than 3 nodes") if avail is None: return unconstrained_bridge_augmentation(G) else: return weighted_bridge_augmentation(G, avail, weight=weight) # --- Algorithms and Helpers --- def _ordered(u, v): """Returns the nodes in an undirected edge in lower-triangular order""" return (u, v) if u < v else (v, u) def _unpack_available_edges(avail, weight=None, G=None): """Helper to separate avail into edges and corresponding weights""" if weight is None: weight = "weight" if isinstance(avail, dict): avail_uv = list(avail.keys()) avail_w = list(avail.values()) else: def _try_getitem(d): try: return d[weight] except TypeError: return d avail_uv = [tup[0:2] for tup in avail] avail_w = [1 if len(tup) == 2 else _try_getitem(tup[-1]) for tup in avail] if G is not None: # Edges already in the graph are filtered flags = [not G.has_edge(u, v) for u, v in avail_uv] avail_uv = list(it.compress(avail_uv, flags)) avail_w = list(it.compress(avail_w, flags)) return avail_uv, avail_w MetaEdge = namedtuple("MetaEdge", ("meta_uv", "uv", "w")) def _lightest_meta_edges(mapping, avail_uv, avail_w): """Maps available edges in the original graph to edges in the metagraph. Parameters ---------- mapping : dict mapping produced by :func:`collapse`, that maps each node in the original graph to a node in the meta graph avail_uv : list list of edges avail_w : list list of edge weights Notes ----- Each node in the metagraph is a k-edge-connected component in the original graph. We don't care about any edge within the same k-edge-connected component, so we ignore self edges. We also are only intereseted in the minimum weight edge bridging each k-edge-connected component so, we group the edges by meta-edge and take the lightest in each group. Example ------- >>> # Each group represents a meta-node >>> groups = ([1, 2, 3], [4, 5], [6]) >>> mapping = {n: meta_n for meta_n, ns in enumerate(groups) for n in ns} >>> avail_uv = [(1, 2), (3, 6), (1, 4), (5, 2), (6, 1), (2, 6), (3, 1)] >>> avail_w = [20, 99, 20, 15, 50, 99, 20] >>> sorted(_lightest_meta_edges(mapping, avail_uv, avail_w)) [MetaEdge(meta_uv=(0, 1), uv=(5, 2), w=15), MetaEdge(meta_uv=(0, 2), uv=(6, 1), w=50)] """ grouped_wuv = defaultdict(list) for w, (u, v) in zip(avail_w, avail_uv): # Order the meta-edge so it can be used as a dict key meta_uv = _ordered(mapping[u], mapping[v]) # Group each available edge using the meta-edge as a key grouped_wuv[meta_uv].append((w, u, v)) # Now that all available edges are grouped, choose one per group for (mu, mv), choices_wuv in grouped_wuv.items(): # Ignore available edges within the same meta-node if mu != mv: # Choose the lightest available edge belonging to each meta-edge w, u, v = min(choices_wuv) yield MetaEdge((mu, mv), (u, v), w) def unconstrained_one_edge_augmentation(G): """Finds the smallest set of edges to connect G. This is a variant of the unweighted MST problem. If G is not empty, a feasible solution always exists. Parameters ---------- G : NetworkX graph An undirected graph. Yields ------ edge : tuple Edges in the one-edge-augmentation of G See Also -------- :func:`one_edge_augmentation` :func:`k_edge_augmentation` Example ------- >>> G = nx.Graph([(1, 2), (2, 3), (4, 5)]) >>> G.add_nodes_from([6, 7, 8]) >>> sorted(unconstrained_one_edge_augmentation(G)) [(1, 4), (4, 6), (6, 7), (7, 8)] """ ccs1 = list(nx.connected_components(G)) C = collapse(G, ccs1) # When we are not constrained, we can just make a meta graph tree. meta_nodes = list(C.nodes()) # build a path in the metagraph meta_aug = list(zip(meta_nodes, meta_nodes[1:])) # map that path to the original graph inverse = defaultdict(list) for k, v in C.graph["mapping"].items(): inverse[v].append(k) for mu, mv in meta_aug: yield (inverse[mu][0], inverse[mv][0]) def weighted_one_edge_augmentation(G, avail, weight=None, partial=False): """Finds the minimum weight set of edges to connect G if one exists. This is a variant of the weighted MST problem. Parameters ---------- G : NetworkX graph An undirected graph. avail : dict or a set of 2 or 3 tuples For more details, see :func:`k_edge_augmentation`. weight : string key to use to find weights if ``avail`` is a set of 3-tuples. For more details, see :func:`k_edge_augmentation`. partial : boolean If partial is True and no feasible k-edge-augmentation exists, then the augmenting edges minimize the number of connected components. Yields ------ edge : tuple Edges in the subset of avail chosen to connect G. See Also -------- :func:`one_edge_augmentation` :func:`k_edge_augmentation` Example ------- >>> G = nx.Graph([(1, 2), (2, 3), (4, 5)]) >>> G.add_nodes_from([6, 7, 8]) >>> # any edge not in avail has an implicit weight of infinity >>> avail = [(1, 3), (1, 5), (4, 7), (4, 8), (6, 1), (8, 1), (8, 2)] >>> sorted(weighted_one_edge_augmentation(G, avail)) [(1, 5), (4, 7), (6, 1), (8, 1)] >>> # find another solution by giving large weights to edges in the >>> # previous solution (note some of the old edges must be used) >>> avail = [(1, 3), (1, 5, 99), (4, 7, 9), (6, 1, 99), (8, 1, 99), (8, 2)] >>> sorted(weighted_one_edge_augmentation(G, avail)) [(1, 5), (4, 7), (6, 1), (8, 2)] """ avail_uv, avail_w = _unpack_available_edges(avail, weight=weight, G=G) # Collapse CCs in the original graph into nodes in a metagraph # Then find an MST of the metagraph instead of the original graph C = collapse(G, nx.connected_components(G)) mapping = C.graph["mapping"] # Assign each available edge to an edge in the metagraph candidate_mapping = _lightest_meta_edges(mapping, avail_uv, avail_w) # nx.set_edge_attributes(C, name='weight', values=0) C.add_edges_from( (mu, mv, {"weight": w, "generator": uv}) for (mu, mv), uv, w in candidate_mapping ) # Find MST of the meta graph meta_mst = nx.minimum_spanning_tree(C) if not partial and not nx.is_connected(meta_mst): raise nx.NetworkXUnfeasible("Not possible to connect G with available edges") # Yield the edge that generated the meta-edge for mu, mv, d in meta_mst.edges(data=True): if "generator" in d: edge = d["generator"] yield edge def unconstrained_bridge_augmentation(G): """Finds an optimal 2-edge-augmentation of G using the fewest edges. This is an implementation of the algorithm detailed in [1]_. The basic idea is to construct a meta-graph of bridge-ccs, connect leaf nodes of the trees to connect the entire graph, and finally connect the leafs of the tree in dfs-preorder to bridge connect the entire graph. Parameters ---------- G : NetworkX graph An undirected graph. Yields ------ edge : tuple Edges in the bridge augmentation of G Notes ----- Input: a graph G. First find the bridge components of G and collapse each bridge-cc into a node of a metagraph graph C, which is guaranteed to be a forest of trees. C contains p "leafs" --- nodes with exactly one incident edge. C contains q "isolated nodes" --- nodes with no incident edges. Theorem: If p + q > 1, then at least :math:`ceil(p / 2) + q` edges are needed to bridge connect C. This algorithm achieves this min number. The method first adds enough edges to make G into a tree and then pairs leafs in a simple fashion. Let n be the number of trees in C. Let v(i) be an isolated vertex in the i-th tree if one exists, otherwise it is a pair of distinct leafs nodes in the i-th tree. Alternating edges from these sets (i.e. adding edges A1 = [(v(i)[0], v(i + 1)[1]), v(i + 1)[0], v(i + 2)[1])...]) connects C into a tree T. This tree has p' = p + 2q - 2(n -1) leafs and no isolated vertices. A1 has n - 1 edges. The next step finds ceil(p' / 2) edges to biconnect any tree with p' leafs. Convert T into an arborescence T' by picking an arbitrary root node with degree >= 2 and directing all edges away from the root. Note the implementation implicitly constructs T'. The leafs of T are the nodes with no existing edges in T'. Order the leafs of T' by DFS prorder. Then break this list in half and add the zipped pairs to A2. The set A = A1 + A2 is the minimum augmentation in the metagraph. To convert this to edges in the original graph References ---------- .. [1] Eswaran, Kapali P., and R. Endre Tarjan. (1975) Augmentation problems. http://epubs.siam.org/doi/abs/10.1137/0205044 See Also -------- :func:`bridge_augmentation` :func:`k_edge_augmentation` Example ------- >>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7)) >>> sorted(unconstrained_bridge_augmentation(G)) [(1, 7)] >>> G = nx.path_graph((1, 2, 3, 2, 4, 5, 6, 7)) >>> sorted(unconstrained_bridge_augmentation(G)) [(1, 3), (3, 7)] >>> G = nx.Graph([(0, 1), (0, 2), (1, 2)]) >>> G.add_node(4) >>> sorted(unconstrained_bridge_augmentation(G)) [(1, 4), (4, 0)] """ # ----- # Mapping of terms from (Eswaran and Tarjan): # G = G_0 - the input graph # C = G_0' - the bridge condensation of G. (This is a forest of trees) # A1 = A_1 - the edges to connect the forest into a tree # leaf = pendant - a node with degree of 1 # alpha(v) = maps the node v in G to its meta-node in C # beta(x) = maps the meta-node x in C to any node in the bridge # component of G corresponding to x. # find the 2-edge-connected components of G bridge_ccs = list(nx.connectivity.bridge_components(G)) # condense G into an forest C C = collapse(G, bridge_ccs) # Choose pairs of distinct leaf nodes in each tree. If this is not # possible then make a pair using the single isolated node in the tree. vset1 = [ tuple(cc) * 2 # case1: an isolated node if len(cc) == 1 else sorted(cc, key=C.degree)[0:2] # case2: pair of leaf nodes for cc in nx.connected_components(C) ] if len(vset1) > 1: # Use this set to construct edges that connect C into a tree. nodes1 = [vs[0] for vs in vset1] nodes2 = [vs[1] for vs in vset1] A1 = list(zip(nodes1[1:], nodes2)) else: A1 = [] # Connect each tree in the forest to construct an arborescence T = C.copy() T.add_edges_from(A1) # If there are only two leaf nodes, we simply connect them. leafs = [n for n, d in T.degree() if d == 1] if len(leafs) == 1: A2 = [] if len(leafs) == 2: A2 = [tuple(leafs)] else: # Choose an arbitrary non-leaf root try: root = next(n for n, d in T.degree() if d > 1) except StopIteration: # no nodes found with degree > 1 return # order the leaves of C by (induced directed) preorder v2 = [n for n in nx.dfs_preorder_nodes(T, root) if T.degree(n) == 1] # connecting first half of the leafs in pre-order to the second # half will bridge connect the tree with the fewest edges. half = int(math.ceil(len(v2) / 2.0)) A2 = list(zip(v2[:half], v2[-half:])) # collect the edges used to augment the original forest aug_tree_edges = A1 + A2 # Construct the mapping (beta) from meta-nodes to regular nodes inverse = defaultdict(list) for k, v in C.graph["mapping"].items(): inverse[v].append(k) # sort so we choose minimum degree nodes first inverse = { mu: sorted(mapped, key=lambda u: (G.degree(u), u)) for mu, mapped in inverse.items() } # For each meta-edge, map back to an arbitrary pair in the original graph G2 = G.copy() for mu, mv in aug_tree_edges: # Find the first available edge that doesn't exist and return it for u, v in it.product(inverse[mu], inverse[mv]): if not G2.has_edge(u, v): G2.add_edge(u, v) yield u, v break def weighted_bridge_augmentation(G, avail, weight=None): """Finds an approximate min-weight 2-edge-augmentation of G. This is an implementation of the approximation algorithm detailed in [1]_. It chooses a set of edges from avail to add to G that renders it 2-edge-connected if such a subset exists. This is done by finding a minimum spanning arborescence of a specially constructed metagraph. Parameters ---------- G : NetworkX graph An undirected graph. avail : set of 2 or 3 tuples. candidate edges (with optional weights) to choose from weight : string key to use to find weights if avail is a set of 3-tuples where the third item in each tuple is a dictionary. Yields ------ edge : tuple Edges in the subset of avail chosen to bridge augment G. Notes ----- Finding a weighted 2-edge-augmentation is NP-hard. Any edge not in ``avail`` is considered to have a weight of infinity. The approximation factor is 2 if ``G`` is connected and 3 if it is not. Runs in :math:`O(m + n log(n))` time References ---------- .. [1] Khuller, Samir, and Ramakrishna Thurimella. (1993) Approximation algorithms for graph augmentation. http://www.sciencedirect.com/science/article/pii/S0196677483710102 See Also -------- :func:`bridge_augmentation` :func:`k_edge_augmentation` Example ------- >>> G = nx.path_graph((1, 2, 3, 4)) >>> # When the weights are equal, (1, 4) is the best >>> avail = [(1, 4, 1), (1, 3, 1), (2, 4, 1)] >>> sorted(weighted_bridge_augmentation(G, avail)) [(1, 4)] >>> # Giving (1, 4) a high weight makes the two edge solution the best. >>> avail = [(1, 4, 1000), (1, 3, 1), (2, 4, 1)] >>> sorted(weighted_bridge_augmentation(G, avail)) [(1, 3), (2, 4)] >>> # ------ >>> G = nx.path_graph((1, 2, 3, 4)) >>> G.add_node(5) >>> avail = [(1, 5, 11), (2, 5, 10), (4, 3, 1), (4, 5, 1)] >>> sorted(weighted_bridge_augmentation(G, avail=avail)) [(1, 5), (4, 5)] >>> avail = [(1, 5, 11), (2, 5, 10), (4, 3, 1), (4, 5, 51)] >>> sorted(weighted_bridge_augmentation(G, avail=avail)) [(1, 5), (2, 5), (4, 5)] """ if weight is None: weight = "weight" # If input G is not connected the approximation factor increases to 3 if not nx.is_connected(G): H = G.copy() connectors = list(one_edge_augmentation(H, avail=avail, weight=weight)) H.add_edges_from(connectors) yield from connectors else: connectors = [] H = G if len(avail) == 0: if nx.has_bridges(H): raise nx.NetworkXUnfeasible("no augmentation possible") avail_uv, avail_w = _unpack_available_edges(avail, weight=weight, G=H) # Collapse input into a metagraph. Meta nodes are bridge-ccs bridge_ccs = nx.connectivity.bridge_components(H) C = collapse(H, bridge_ccs) # Use the meta graph to shrink avail to a small feasible subset mapping = C.graph["mapping"] # Choose the minimum weight feasible edge in each group meta_to_wuv = { (mu, mv): (w, uv) for (mu, mv), uv, w in _lightest_meta_edges(mapping, avail_uv, avail_w) } # Mapping of terms from (Khuller and Thurimella): # C : G_0 = (V, E^0) # This is the metagraph where each node is a 2-edge-cc in G. # The edges in C represent bridges in the original graph. # (mu, mv) : E - E^0 # they group both avail and given edges in E # T : \Gamma # D : G^D = (V, E_D) # The paper uses ancestor because children point to parents, which is # contrary to networkx standards. So, we actually need to run # nx.least_common_ancestor on the reversed Tree. # Pick an arbitrary leaf from C as the root try: root = next(n for n, d in C.degree() if d == 1) except StopIteration: # no nodes found with degree == 1 return # Root C into a tree TR by directing all edges away from the root # Note in their paper T directs edges towards the root TR = nx.dfs_tree(C, root) # Add to D the directed edges of T and set their weight to zero # This indicates that it costs nothing to use edges that were given. D = nx.reverse(TR).copy() nx.set_edge_attributes(D, name="weight", values=0) # The LCA of mu and mv in T is the shared ancestor of mu and mv that is # located farthest from the root. lca_gen = nx.tree_all_pairs_lowest_common_ancestor( TR, root=root, pairs=meta_to_wuv.keys() ) for (mu, mv), lca in lca_gen: w, uv = meta_to_wuv[(mu, mv)] if lca == mu: # If u is an ancestor of v in TR, then add edge u->v to D D.add_edge(lca, mv, weight=w, generator=uv) elif lca == mv: # If v is an ancestor of u in TR, then add edge v->u to D D.add_edge(lca, mu, weight=w, generator=uv) else: # If neither u nor v is a ancestor of the other in TR # let t = lca(TR, u, v) and add edges t->u and t->v # Track the original edge that GENERATED these edges. D.add_edge(lca, mu, weight=w, generator=uv) D.add_edge(lca, mv, weight=w, generator=uv) # Then compute a minimum rooted branching try: # Note the original edges must be directed towards to root for the # branching to give us a bridge-augmentation. A = _minimum_rooted_branching(D, root) except nx.NetworkXException as e: # If there is no branching then augmentation is not possible raise nx.NetworkXUnfeasible("no 2-edge-augmentation possible") from e # For each edge e, in the branching that did not belong to the directed # tree T, add the corresponding edge that **GENERATED** it (this is not # necesarilly e itself!) # ensure the third case does not generate edges twice bridge_connectors = set() for mu, mv in A.edges(): data = D.get_edge_data(mu, mv) if "generator" in data: # Add the avail edge that generated the branching edge. edge = data["generator"] bridge_connectors.add(edge) yield from bridge_connectors def _minimum_rooted_branching(D, root): """Helper function to compute a minimum rooted branching (aka rooted arborescence) Before the branching can be computed, the directed graph must be rooted by removing the predecessors of root. A branching / arborescence of rooted graph G is a subgraph that contains a directed path from the root to every other vertex. It is the directed analog of the minimum spanning tree problem. References ---------- [1] Khuller, Samir (2002) Advanced Algorithms Lecture 24 Notes. https://www.cs.umd.edu/class/spring2011/cmsc651/lec07.pdf """ rooted = D.copy() # root the graph by removing all predecessors to `root`. rooted.remove_edges_from([(u, root) for u in D.predecessors(root)]) # Then compute the branching / arborescence. A = nx.minimum_spanning_arborescence(rooted) return A def collapse(G, grouped_nodes): """Collapses each group of nodes into a single node. This is similar to condensation, but works on undirected graphs. Parameters ---------- G : NetworkX Graph grouped_nodes: list or generator Grouping of nodes to collapse. The grouping must be disjoint. If grouped_nodes are strongly_connected_components then this is equivalent to :func:`condensation`. Returns ------- C : NetworkX Graph The collapsed graph C of G with respect to the node grouping. The node labels are integers corresponding to the index of the component in the list of grouped_nodes. C has a graph attribute named 'mapping' with a dictionary mapping the original nodes to the nodes in C to which they belong. Each node in C also has a node attribute 'members' with the set of original nodes in G that form the group that the node in C represents. Examples -------- >>> # Collapses a graph using disjoint groups, but not necesarilly connected >>> G = nx.Graph([(1, 0), (2, 3), (3, 1), (3, 4), (4, 5), (5, 6), (5, 7)]) >>> G.add_node("A") >>> grouped_nodes = [{0, 1, 2, 3}, {5, 6, 7}] >>> C = collapse(G, grouped_nodes) >>> members = nx.get_node_attributes(C, "members") >>> sorted(members.keys()) [0, 1, 2, 3] >>> member_values = set(map(frozenset, members.values())) >>> assert {0, 1, 2, 3} in member_values >>> assert {4} in member_values >>> assert {5, 6, 7} in member_values >>> assert {"A"} in member_values """ mapping = {} members = {} C = G.__class__() i = 0 # required if G is empty remaining = set(G.nodes()) for i, group in enumerate(grouped_nodes): group = set(group) assert remaining.issuperset( group ), "grouped nodes must exist in G and be disjoint" remaining.difference_update(group) members[i] = group mapping.update((n, i) for n in group) # remaining nodes are in their own group for i, node in enumerate(remaining, start=i + 1): group = {node} members[i] = group mapping.update((n, i) for n in group) number_of_groups = i + 1 C.add_nodes_from(range(number_of_groups)) C.add_edges_from( (mapping[u], mapping[v]) for u, v in G.edges() if mapping[u] != mapping[v] ) # Add a list of members (ie original nodes) to each node (ie scc) in C. nx.set_node_attributes(C, name="members", values=members) # Add mapping dict as graph attribute C.graph["mapping"] = mapping return C def complement_edges(G): """Returns only the edges in the complement of G Parameters ---------- G : NetworkX Graph Yields ------ edge : tuple Edges in the complement of G Example ------- >>> G = nx.path_graph((1, 2, 3, 4)) >>> sorted(complement_edges(G)) [(1, 3), (1, 4), (2, 4)] >>> G = nx.path_graph((1, 2, 3, 4), nx.DiGraph()) >>> sorted(complement_edges(G)) [(1, 3), (1, 4), (2, 1), (2, 4), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)] >>> G = nx.complete_graph(1000) >>> sorted(complement_edges(G)) [] """ if G.is_directed(): for u, v in it.combinations(G.nodes(), 2): if v not in G.adj[u]: yield (u, v) if u not in G.adj[v]: yield (v, u) else: for u, v in it.combinations(G.nodes(), 2): if v not in G.adj[u]: yield (u, v) def _compat_shuffle(rng, input): """wrapper around rng.shuffle for python 2 compatibility reasons""" rng.shuffle(input) @py_random_state(4) @not_implemented_for("multigraph") @not_implemented_for("directed") def greedy_k_edge_augmentation(G, k, avail=None, weight=None, seed=None): """Greedy algorithm for finding a k-edge-augmentation Parameters ---------- G : NetworkX graph An undirected graph. k : integer Desired edge connectivity avail : dict or a set of 2 or 3 tuples For more details, see :func:`k_edge_augmentation`. weight : string key to use to find weights if ``avail`` is a set of 3-tuples. For more details, see :func:`k_edge_augmentation`. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Yields ------ edge : tuple Edges in the greedy augmentation of G Notes ----- The algorithm is simple. Edges are incrementally added between parts of the graph that are not yet locally k-edge-connected. Then edges are from the augmenting set are pruned as long as local-edge-connectivity is not broken. This algorithm is greedy and does not provide optimality guarantees. It exists only to provide :func:`k_edge_augmentation` with the ability to generate a feasible solution for arbitrary k. See Also -------- :func:`k_edge_augmentation` Example ------- >>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7)) >>> sorted(greedy_k_edge_augmentation(G, k=2)) [(1, 7)] >>> sorted(greedy_k_edge_augmentation(G, k=1, avail=[])) [] >>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7)) >>> avail = {(u, v): 1 for (u, v) in complement_edges(G)} >>> # randomized pruning process can produce different solutions >>> sorted(greedy_k_edge_augmentation(G, k=4, avail=avail, seed=2)) [(1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 4), (2, 6), (3, 7), (5, 7)] >>> sorted(greedy_k_edge_augmentation(G, k=4, avail=avail, seed=3)) [(1, 3), (1, 5), (1, 6), (2, 4), (2, 6), (3, 7), (4, 7), (5, 7)] """ # Result set aug_edges = [] done = is_k_edge_connected(G, k) if done: return if avail is None: # all edges are available avail_uv = list(complement_edges(G)) avail_w = [1] * len(avail_uv) else: # Get the unique set of unweighted edges avail_uv, avail_w = _unpack_available_edges(avail, weight=weight, G=G) # Greedy: order lightest edges. Use degree sum to tie-break tiebreaker = [sum(map(G.degree, uv)) for uv in avail_uv] avail_wduv = sorted(zip(avail_w, tiebreaker, avail_uv)) avail_uv = [uv for w, d, uv in avail_wduv] # Incrementally add edges in until we are k-connected H = G.copy() for (u, v) in avail_uv: done = False if not is_locally_k_edge_connected(H, u, v, k=k): # Only add edges in parts that are not yet locally k-edge-connected aug_edges.append((u, v)) H.add_edge(u, v) # Did adding this edge help? if H.degree(u) >= k and H.degree(v) >= k: done = is_k_edge_connected(H, k) if done: break # Check for feasibility if not done: raise nx.NetworkXUnfeasible("not able to k-edge-connect with available edges") # Randomized attempt to reduce the size of the solution _compat_shuffle(seed, aug_edges) for (u, v) in list(aug_edges): # Don't remove if we know it would break connectivity if H.degree(u) <= k or H.degree(v) <= k: continue H.remove_edge(u, v) aug_edges.remove((u, v)) if not is_k_edge_connected(H, k=k): # If removing this edge breaks feasibility, undo H.add_edge(u, v) aug_edges.append((u, v)) # Generate results yield from aug_edges