import math
from operator import itemgetter
import pytest
import networkx as nx
from networkx.algorithms.tree import branchings, recognition
np = pytest.importorskip("numpy")
#
# Explicitly discussed examples from Edmonds paper.
#
# Used in Figures A-F.
#
# fmt: off
G_array = np.array([
# 0 1 2 3 4 5 6 7 8
[0, 0, 12, 0, 12, 0, 0, 0, 0], # 0
[4, 0, 0, 0, 0, 13, 0, 0, 0], # 1
[0, 17, 0, 21, 0, 12, 0, 0, 0], # 2
[5, 0, 0, 0, 17, 0, 18, 0, 0], # 3
[0, 0, 0, 0, 0, 0, 0, 12, 0], # 4
[0, 0, 0, 0, 0, 0, 14, 0, 12], # 5
[0, 0, 21, 0, 0, 0, 0, 0, 15], # 6
[0, 0, 0, 19, 0, 0, 15, 0, 0], # 7
[0, 0, 0, 0, 0, 0, 0, 18, 0], # 8
], dtype=int)
# Two copies of the graph from the original paper as disconnected components
G_big_array = np.zeros(np.array(G_array.shape) * 2, dtype=int)
G_big_array[:G_array.shape[0], :G_array.shape[1]] = G_array
G_big_array[G_array.shape[0]:, G_array.shape[1]:] = G_array
# fmt: on
def G1():
G = nx.from_numpy_array(G_array, create_using=nx.MultiDiGraph)
return G
def G2():
# Now we shift all the weights by -10.
# Should not affect optimal arborescence, but does affect optimal branching.
Garr = G_array.copy()
Garr[np.nonzero(Garr)] -= 10
G = nx.from_numpy_array(Garr, create_using=nx.MultiDiGraph)
return G
# An optimal branching for G1 that is also a spanning arborescence. So it is
# also an optimal spanning arborescence.
#
optimal_arborescence_1 = [
(0, 2, 12),
(2, 1, 17),
(2, 3, 21),
(1, 5, 13),
(3, 4, 17),
(3, 6, 18),
(6, 8, 15),
(8, 7, 18),
]
# For G2, the optimal branching of G1 (with shifted weights) is no longer
# an optimal branching, but it is still an optimal spanning arborescence
# (just with shifted weights). An optimal branching for G2 is similar to what
# appears in figure G (this is greedy_subopt_branching_1a below), but with the
# edge (3, 0, 5), which is now (3, 0, -5), removed. Thus, the optimal branching
# is not a spanning arborescence. The code finds optimal_branching_2a.
# An alternative and equivalent branching is optimal_branching_2b. We would
# need to modify the code to iterate through all equivalent optimal branchings.
#
# These are maximal branchings or arborescences.
optimal_branching_2a = [
(5, 6, 4),
(6, 2, 11),
(6, 8, 5),
(8, 7, 8),
(2, 1, 7),
(2, 3, 11),
(3, 4, 7),
]
optimal_branching_2b = [
(8, 7, 8),
(7, 3, 9),
(3, 4, 7),
(3, 6, 8),
(6, 2, 11),
(2, 1, 7),
(1, 5, 3),
]
optimal_arborescence_2 = [
(0, 2, 2),
(2, 1, 7),
(2, 3, 11),
(1, 5, 3),
(3, 4, 7),
(3, 6, 8),
(6, 8, 5),
(8, 7, 8),
]
# Two suboptimal maximal branchings on G1 obtained from a greedy algorithm.
# 1a matches what is shown in Figure G in Edmonds's paper.
greedy_subopt_branching_1a = [
(5, 6, 14),
(6, 2, 21),
(6, 8, 15),
(8, 7, 18),
(2, 1, 17),
(2, 3, 21),
(3, 0, 5),
(3, 4, 17),
]
greedy_subopt_branching_1b = [
(8, 7, 18),
(7, 6, 15),
(6, 2, 21),
(2, 1, 17),
(2, 3, 21),
(1, 5, 13),
(3, 0, 5),
(3, 4, 17),
]
def build_branching(edges, double=False):
G = nx.DiGraph()
for u, v, weight in edges:
G.add_edge(u, v, weight=weight)
if double:
G.add_edge(u + 9, v + 9, weight=weight)
return G
def sorted_edges(G, attr="weight", default=1):
edges = [(u, v, data.get(attr, default)) for (u, v, data) in G.edges(data=True)]
edges = sorted(edges, key=lambda x: (x[2], x[1], x[0]))
return edges
def assert_equal_branchings(G1, G2, attr="weight", default=1):
edges1 = list(G1.edges(data=True))
edges2 = list(G2.edges(data=True))
assert len(edges1) == len(edges2)
# Grab the weights only.
e1 = sorted_edges(G1, attr, default)
e2 = sorted_edges(G2, attr, default)
for a, b in zip(e1, e2):
assert a[:2] == b[:2]
np.testing.assert_almost_equal(a[2], b[2])
################
def test_optimal_branching1():
G = build_branching(optimal_arborescence_1)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 131
def test_optimal_branching2a():
G = build_branching(optimal_branching_2a)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 53
def test_optimal_branching2b():
G = build_branching(optimal_branching_2b)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 53
def test_optimal_arborescence2():
G = build_branching(optimal_arborescence_2)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 51
def test_greedy_suboptimal_branching1a():
G = build_branching(greedy_subopt_branching_1a)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 128
def test_greedy_suboptimal_branching1b():
G = build_branching(greedy_subopt_branching_1b)
assert recognition.is_arborescence(G), True
assert branchings.branching_weight(G) == 127
def test_greedy_max1():
# Standard test.
#
G = G1()
B = branchings.greedy_branching(G)
# There are only two possible greedy branchings. The sorting is such
# that it should equal the second suboptimal branching: 1b.
B_ = build_branching(greedy_subopt_branching_1b)
assert_equal_branchings(B, B_)
def test_greedy_branching_kwarg_kind():
G = G1()
with pytest.raises(nx.NetworkXException, match="Unknown value for `kind`."):
B = branchings.greedy_branching(G, kind="lol")
def test_greedy_branching_for_unsortable_nodes():
G = nx.DiGraph()
G.add_weighted_edges_from([((2, 3), 5, 1), (3, "a", 1), (2, 4, 5)])
edges = [(u, v, data.get("weight", 1)) for (u, v, data) in G.edges(data=True)]
with pytest.raises(TypeError):
edges.sort(key=itemgetter(2, 0, 1), reverse=True)
B = branchings.greedy_branching(G, kind="max").edges(data=True)
assert list(B) == [
((2, 3), 5, {"weight": 1}),
(3, "a", {"weight": 1}),
(2, 4, {"weight": 5}),
]
def test_greedy_max2():
# Different default weight.
#
G = G1()
del G[1][0][0]["weight"]
B = branchings.greedy_branching(G, default=6)
# Chosen so that edge (3,0,5) is not selected and (1,0,6) is instead.
edges = [
(1, 0, 6),
(1, 5, 13),
(7, 6, 15),
(2, 1, 17),
(3, 4, 17),
(8, 7, 18),
(2, 3, 21),
(6, 2, 21),
]
B_ = build_branching(edges)
assert_equal_branchings(B, B_)
def test_greedy_max3():
# All equal weights.
#
G = G1()
B = branchings.greedy_branching(G, attr=None)
# This is mostly arbitrary...the output was generated by running the algo.
edges = [
(2, 1, 1),
(3, 0, 1),
(3, 4, 1),
(5, 8, 1),
(6, 2, 1),
(7, 3, 1),
(7, 6, 1),
(8, 7, 1),
]
B_ = build_branching(edges)
assert_equal_branchings(B, B_, default=1)
def test_greedy_min():
G = G1()
B = branchings.greedy_branching(G, kind="min")
edges = [
(1, 0, 4),
(0, 2, 12),
(0, 4, 12),
(2, 5, 12),
(4, 7, 12),
(5, 8, 12),
(5, 6, 14),
(7, 3, 19),
]
B_ = build_branching(edges)
assert_equal_branchings(B, B_)
def test_edmonds1_maxbranch():
G = G1()
x = branchings.maximum_branching(G)
x_ = build_branching(optimal_arborescence_1)
assert_equal_branchings(x, x_)
def test_edmonds1_maxarbor():
G = G1()
x = branchings.maximum_spanning_arborescence(G)
x_ = build_branching(optimal_arborescence_1)
assert_equal_branchings(x, x_)
def test_edmonds1_minimal_branching():
# graph will have something like a minimum arborescence but no spanning one
G = nx.from_numpy_array(G_big_array, create_using=nx.DiGraph)
B = branchings.minimal_branching(G)
edges = [
(3, 0, 5),
(0, 2, 12),
(0, 4, 12),
(2, 5, 12),
(4, 7, 12),
(5, 8, 12),
(5, 6, 14),
(2, 1, 17),
]
B_ = build_branching(edges, double=True)
assert_equal_branchings(B, B_)
def test_edmonds2_maxbranch():
G = G2()
x = branchings.maximum_branching(G)
x_ = build_branching(optimal_branching_2a)
assert_equal_branchings(x, x_)
def test_edmonds2_maxarbor():
G = G2()
x = branchings.maximum_spanning_arborescence(G)
x_ = build_branching(optimal_arborescence_2)
assert_equal_branchings(x, x_)
def test_edmonds2_minarbor():
G = G1()
x = branchings.minimum_spanning_arborescence(G)
# This was obtained from algorithm. Need to verify it independently.
# Branch weight is: 96
edges = [
(3, 0, 5),
(0, 2, 12),
(0, 4, 12),
(2, 5, 12),
(4, 7, 12),
(5, 8, 12),
(5, 6, 14),
(2, 1, 17),
]
x_ = build_branching(edges)
assert_equal_branchings(x, x_)
def test_edmonds3_minbranch1():
G = G1()
x = branchings.minimum_branching(G)
edges = []
x_ = build_branching(edges)
assert_equal_branchings(x, x_)
def test_edmonds3_minbranch2():
G = G1()
G.add_edge(8, 9, weight=-10)
x = branchings.minimum_branching(G)
edges = [(8, 9, -10)]
x_ = build_branching(edges)
assert_equal_branchings(x, x_)
# Need more tests
def test_mst():
# Make sure we get the same results for undirected graphs.
# Example from: https://en.wikipedia.org/wiki/Kruskal's_algorithm
G = nx.Graph()
edgelist = [
(0, 3, [("weight", 5)]),
(0, 1, [("weight", 7)]),
(1, 3, [("weight", 9)]),
(1, 2, [("weight", 8)]),
(1, 4, [("weight", 7)]),
(3, 4, [("weight", 15)]),
(3, 5, [("weight", 6)]),
(2, 4, [("weight", 5)]),
(4, 5, [("weight", 8)]),
(4, 6, [("weight", 9)]),
(5, 6, [("weight", 11)]),
]
G.add_edges_from(edgelist)
G = G.to_directed()
x = branchings.minimum_spanning_arborescence(G)
edges = [
({0, 1}, 7),
({0, 3}, 5),
({3, 5}, 6),
({1, 4}, 7),
({4, 2}, 5),
({4, 6}, 9),
]
assert x.number_of_edges() == len(edges)
for u, v, d in x.edges(data=True):
assert ({u, v}, d["weight"]) in edges
def test_mixed_nodetypes():
# Smoke test to make sure no TypeError is raised for mixed node types.
G = nx.Graph()
edgelist = [(0, 3, [("weight", 5)]), (0, "1", [("weight", 5)])]
G.add_edges_from(edgelist)
G = G.to_directed()
x = branchings.minimum_spanning_arborescence(G)
def test_edmonds1_minbranch():
# Using -G_array and min should give the same as optimal_arborescence_1,
# but with all edges negative.
edges = [(u, v, -w) for (u, v, w) in optimal_arborescence_1]
G = nx.from_numpy_array(-G_array, create_using=nx.DiGraph)
# Quickly make sure max branching is empty.
x = branchings.maximum_branching(G)
x_ = build_branching([])
assert_equal_branchings(x, x_)
# Now test the min branching.
x = branchings.minimum_branching(G)
x_ = build_branching(edges)
assert_equal_branchings(x, x_)
def test_edge_attribute_preservation_normal_graph():
# Test that edge attributes are preserved when finding an optimum graph
# using the Edmonds class for normal graphs.
G = nx.Graph()
edgelist = [
(0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
(0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
(1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
]
G.add_edges_from(edgelist)
B = branchings.maximum_branching(G, preserve_attrs=True)
assert B[0][1]["otherattr"] == 1
assert B[0][1]["otherattr2"] == 3
def test_edge_attribute_preservation_multigraph():
# Test that edge attributes are preserved when finding an optimum graph
# using the Edmonds class for multigraphs.
G = nx.MultiGraph()
edgelist = [
(0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
(0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
(1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
]
G.add_edges_from(edgelist * 2) # Make sure we have duplicate edge paths
B = branchings.maximum_branching(G, preserve_attrs=True)
assert B[0][1][0]["otherattr"] == 1
assert B[0][1][0]["otherattr2"] == 3
def test_edge_attribute_discard():
# Test that edge attributes are discarded if we do not specify to keep them
G = nx.Graph()
edgelist = [
(0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
(0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
(1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
]
G.add_edges_from(edgelist)
B = branchings.maximum_branching(G, preserve_attrs=False)
edge_dict = B[0][1]
with pytest.raises(KeyError):
_ = edge_dict["otherattr"]
def test_partition_spanning_arborescence():
"""
Test that we can generate minimum spanning arborescences which respect the
given partition.
"""
G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
G[3][0]["partition"] = nx.EdgePartition.EXCLUDED
G[2][3]["partition"] = nx.EdgePartition.INCLUDED
G[7][3]["partition"] = nx.EdgePartition.EXCLUDED
G[0][2]["partition"] = nx.EdgePartition.EXCLUDED
G[6][2]["partition"] = nx.EdgePartition.INCLUDED
actual_edges = [
(0, 4, 12),
(1, 0, 4),
(1, 5, 13),
(2, 3, 21),
(4, 7, 12),
(5, 6, 14),
(5, 8, 12),
(6, 2, 21),
]
B = branchings.minimum_spanning_arborescence(G, partition="partition")
assert_equal_branchings(build_branching(actual_edges), B)
def test_arborescence_iterator_min():
"""
Tests the arborescence iterator.
A brute force method found 680 arborescences in this graph.
This test will not verify all of them individually, but will check two
things
* The iterator returns 680 arborescences
* The weight of the arborescences is non-strictly increasing
for more information please visit
https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
"""
G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
arborescence_count = 0
arborescence_weight = -math.inf
for B in branchings.ArborescenceIterator(G):
arborescence_count += 1
new_arborescence_weight = B.size(weight="weight")
assert new_arborescence_weight >= arborescence_weight
arborescence_weight = new_arborescence_weight
assert arborescence_count == 680
def test_arborescence_iterator_max():
"""
Tests the arborescence iterator.
A brute force method found 680 arborescences in this graph.
This test will not verify all of them individually, but will check two
things
* The iterator returns 680 arborescences
* The weight of the arborescences is non-strictly decreasing
for more information please visit
https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
"""
G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
arborescence_count = 0
arborescence_weight = math.inf
for B in branchings.ArborescenceIterator(G, minimum=False):
arborescence_count += 1
new_arborescence_weight = B.size(weight="weight")
assert new_arborescence_weight <= arborescence_weight
arborescence_weight = new_arborescence_weight
assert arborescence_count == 680
def test_arborescence_iterator_initial_partition():
"""
Tests the arborescence iterator with three included edges and three excluded
in the initial partition.
A brute force method similar to the one used in the above tests found that
there are 16 arborescences which contain the included edges and not the
excluded edges.
"""
G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
included_edges = [(1, 0), (5, 6), (8, 7)]
excluded_edges = [(0, 2), (3, 6), (1, 5)]
arborescence_count = 0
arborescence_weight = -math.inf
for B in branchings.ArborescenceIterator(
G, init_partition=(included_edges, excluded_edges)
):
arborescence_count += 1
new_arborescence_weight = B.size(weight="weight")
assert new_arborescence_weight >= arborescence_weight
arborescence_weight = new_arborescence_weight
for e in included_edges:
assert e in B.edges
for e in excluded_edges:
assert e not in B.edges
assert arborescence_count == 16
def test_branchings_with_default_weights():
"""
Tests that various branching algorithms work on graphs without weights.
For more information, see issue #7279.
"""
graph = nx.erdos_renyi_graph(10, p=0.2, directed=True, seed=123)
assert all("weight" not in d for (u, v, d) in graph.edges(data=True)), (
"test is for graphs without a weight attribute"
)
# Calling these functions will modify graph inplace to add weights
# copy the graph to avoid this.
nx.minimum_spanning_arborescence(graph.copy())
nx.maximum_spanning_arborescence(graph.copy())
nx.minimum_branching(graph.copy())
nx.maximum_branching(graph.copy())
nx.algorithms.tree.minimal_branching(graph.copy())
nx.algorithms.tree.branching_weight(graph.copy())
nx.algorithms.tree.greedy_branching(graph.copy())
assert all("weight" not in d for (u, v, d) in graph.edges(data=True)), (
"The above calls should not modify the initial graph in-place"
)