def test_tree_all_pairs_lowest_common_ancestor8(self):
"""Raises right errors if not a tree."""
# Cycle
G = nx.DiGraph()
G.add_cycle((1, 2))
assert_raises(nx.NetworkXError, list,
nx.tree_all_pairs_lowest_common_ancestor(G))
# DAG
G = nx.DiGraph([(0, 2), (1, 2)])
assert_raises(nx.NetworkXError, list,
nx.tree_all_pairs_lowest_common_ancestor(G))
def test_tree_all_pairs_lowest_common_ancestor7(self):
"""Works on disconnected nodes."""
G = nx.DiGraph()
G.add_node(1)
assert_equal({(1, 1): 1}, dict(nx.tree_all_pairs_lowest_common_ancestor(G)))
G.add_node(0)
assert_equal({(1, 1): 1}, dict(nx.tree_all_pairs_lowest_common_ancestor(G, 1)))
assert_equal({(0, 0): 0}, dict(nx.tree_all_pairs_lowest_common_ancestor(G, 0)))
assert_raises(nx.NetworkXError, list,
nx.tree_all_pairs_lowest_common_ancestor(G))
def test_tree_all_pairs_lowest_common_ancestor5(self):
"""Handles invalid input correctly."""
empty_digraph = nx.tree_all_pairs_lowest_common_ancestor(nx.DiGraph())
assert_raises(nx.NetworkXPointlessConcept, list, empty_digraph)
empty_graph = nx.tree_all_pairs_lowest_common_ancestor(nx.Graph())
assert_raises(nx.NetworkXNotImplemented, list, empty_graph)
nonempty_graph = nx.tree_all_pairs_lowest_common_ancestor(self.DG.to_undirected())
assert_raises(nx.NetworkXNotImplemented, list, nonempty_graph)
bad_pairs_digraph = nx.tree_all_pairs_lowest_common_ancestor(self.DG,
pairs=[(-1, -2)])
assert_raises(nx.NetworkXError, list, bad_pairs_digraph)
def test_tree_all_pairs_lowest_common_ancestor3(self):
"""Specifying no pairs same as specifying all."""
all_pairs = itertools.chain(itertools.combinations(self.DG.nodes(), 2),
((node, node) for node in self.DG.nodes_iter()))
ans = dict(nx.tree_all_pairs_lowest_common_ancestor(self.DG, 0, all_pairs))
self.assert_has_same_pairs(ans, self.ans)
def setUp(self):
self.DG = nx.DiGraph()
self.DG.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
self.ans = dict(nx.tree_all_pairs_lowest_common_ancestor(self.DG, 0))
gold = dict([((n, n), n) for n in self.DG])
gold.update(dict(((0, i), 0) for i in range(1, 7)))
gold.update({(1, 2): 0,
(1, 3): 1,
(1, 4): 1,
(1, 5): 0,
(1, 6): 0,
(2, 3): 0,
(2, 4): 0,
(2, 5): 2,
(2, 6): 2,
(3, 4): 1,
(3, 5): 0,
(3, 6): 0,
(4, 5): 0,
(4, 6): 0,
(5, 6): 2})
self.gold = gold
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)
for edge in connectors:
yield edge
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
root = next(n for n in C.nodes() if C.degree(n) == 1)
# 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:
# If there is no branching then augmentation is not possible
raise nx.NetworkXUnfeasible('no 2-edge-augmentation possible')
# For each edge e, in the branching that did not belong to the directed
# tree T, add the correponding 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)
for edge in bridge_connectors:
yield edge
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)
for edge in connectors:
yield edge
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
root = next(n for n in C.nodes() if C.degree(n) == 1)
# 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:
# If there is no branching then augmentation is not possible
raise nx.NetworkXUnfeasible('no 2-edge-augmentation possible')
# For each edge e, in the branching that did not belong to the directed
# tree T, add the correponding 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)
for edge in bridge_connectors:
yield edge
def test_tree_all_pairs_lowest_common_ancestor6(self):
"""Works on subtrees."""
ans = dict(nx.tree_all_pairs_lowest_common_ancestor(self.DG, 1))
gold = dict(((pair, lca) for (pair, lca) in self.gold.iteritems()
if all((n in (1, 3, 4) for n in pair))))
self.assert_has_same_pairs(gold, ans)
def test_tree_all_pairs_lowest_common_ancestor2(self): """Specifying only some pairs gives only those pairs.""" some_pairs = dict(nx.tree_all_pairs_lowest_common_ancestor(self.DG, 0, [(0, 1), (0, 1), (1, 0)])) assert_true((0, 1) in some_pairs or (1, 0) in some_pairs) assert_equal(len(some_pairs), 1)
def test_tree_all_pairs_lowest_common_ancestor1(self): """Specifying the root is optional.""" assert_equal(self.ans, dict(nx.tree_all_pairs_lowest_common_ancestor(self.DG)))
def __init__(self, T, T_label_to_node, T_node_to_labels):
self.LCA_dict = dict()
self.LCA_label_dict = T_label_to_node
self.LCA_node2lbl = T_node_to_labels
for lca in nx.tree_all_pairs_lowest_common_ancestor(T):
self.LCA_dict[(lca[0][0], lca[0][1])] = lca[1]
