def find_route(node_1, node_2):
"""
Function to find wether there is a route between given nodes.
Parameters
----------
node_1: ``Node``
One of the two nodes to find route between.
node_2: ``Node``
Second of the two nodes to find route between.
Time Complexity
---------------
O(N), where N is the number of vertices in graph.
Space Complexity
---------------
O(N), where N is the number of vertices in graph.
"""
print("Checking for path from node_1 to node_2")
graphs.bfs(node_1, node_2)
print("Checking for path from node_2 to node_1")
graphs.bfs(node_2, node_1)
def test_bfs_two(two_comp):
comp1 = set([0, 1, 2])
for i in range(3):
assert bfs(two_comp, i) == comp1
comp2 = set([3, 4])
for i in range(3, 5):
assert bfs(two_comp, i) == comp2
def test_bfs_linear5(self):
"""
Tests BFS on a linear graph with 5 vertices.
"""
g, vertices = generate_linear_graph(5, circular=False)
s = vertices[0]
bfs(g, s)
for v in vertices:
self.assertEqual(v.color, "BLACK")
t = extract_bfs_tree(g)
self.assertEqual(t.count_edges(), g.count_edges() / 2)
self.assertEqual(t.count_vertices(), g.count_vertices())
def colors_bfs(bits):
colors = graphs.bfs(*graphs.tree_to_graph(
graphs.random_spanning_tree(
graphs.grid_graph(1 << bits, 1 << bits, 1 << bits))))
for color in colors:
# yield [c << (8 - bits) for c in color]
yield [c << (8 - bits) for c in cube_interpreter(color)]
def test_bfs(self):
graph = {
'ja': ['tomasz', 'ewa'],
'tomasz': ['natalia', 'adrian'],
'ewa': ['madzia', 'karol'],
'madzia': ['karol', 'natalia', 'adrian'],
'adrian': ['mietek'],
'natalia': [],
'mietek': [],
'karol': []
}
self.assertIs(graphs.bfs(graph, 'ja'), 'mietek')
def test_bfs_complete4(self):
"""
Tests BFS on a complete graph with 4 vertices.
"""
g, vertices = generate_complete_graph(4)
s = vertices[0]
bfs(g, s)
for v in vertices:
self.assertEqual(v.color, "BLACK")
self.assertEqual(s.pi, None)
self.assertEqual(s.d, 0)
for v in vertices[1:]:
self.assertEqual(v.pi, s)
self.assertEqual(v.d, 1)
t = extract_bfs_tree(g)
self.assertEqual(t.count_vertices(), 4)
self.assertEqual(t.count_edges(), 3)
self.assertEqual(len(t.get_outgoing_edges(s)), 3)
def pixels_bfs(image_size):
return graphs.bfs(*graphs.tree_to_graph(
graphs.random_spanning_tree(graphs.grid_graph(image_size,
image_size))))
def test_bfs_isolated(isolated):
for i in range(3):
assert bfs(isolated, i) == set([i])
def test_bfs_single(single):
ans = set([0, 1, 2, 3, 4, 5, 6])
for i in range(7):
assert bfs(single, i) == ans
def edmonds(graph: DirectedGraph, r: any = 'auto') -> None:
# Step 0
if r == 'auto' or r is None:
# We're choosing highest out-degree node
# max_out_degree_entry = max(enumerate(graph.out_degrees), key=lambda e: e[1])
# r = max_out_degree_entry[0]
r, msg = pick_root(graph)
if r is None:
print(msg)
sys.exit(1)
print(f'Chosen root {r}')
if r < 0 or r >= graph.num_nodes:
raise ValueError(
f'Root {r} out of bounds for graph with {graph.num_nodes} vertices.'
)
# Step 1
V = [i for i in range(graph.num_nodes)]
E = graph.edges()
W = [r]
F = []
super_nodes_ = []
all_covered_nodes_ = set()
bfs_numbers = bfs(graph, start=r)
print(f'Starting Edmonson algorithm.')
outer_iter = 0
while True:
print(f'>> Outer iteration {outer_iter} with the following state: ')
print(f'\t\t graph edges E= {E}')
print(f'\t\t new nodes W= {W}')
print(f'\t\t new edges F= {F}')
print(f'\t\t all_covered_nodes: {all_covered_nodes_}')
if set(W).union(all_covered_nodes_) == set(V):
# Step 8:
break
# Step 3: Find node x not in W with max BFS number
bfs_result = bfs(graph, start=r)
print(f'BFS numbers: {bfs_result["numbers"]}')
max_entry = max(filter(
lambda entry: entry[0] not in W and entry[0] not in
all_covered_nodes_, enumerate(bfs_result["numbers"])),
key=lambda entry: entry[1])
x = max_entry[0]
print(f'Node with max number is node {x}')
print(f'Initializing path to empty')
Vp = []
Ep = []
inner_iter = 0
while True:
print(
f'\t ** Inner iteration {inner_iter} started with the following state:'
)
print(f'\t\t\t graph edges E= {E}')
print(f'\t\t\t new nodes W= {W}')
print(f'\t\t\t new edges F= {F}')
print(f'\t\t\t all_covered_nodes: {all_covered_nodes_}')
print(f'\t\t\t X = {x}')
print(f'\t\t\t current_path_nodes Vp= {Vp}')
print(f'\t\t\t current_path_edges Ep= {Ep}')
# Step 4: Form a path P such that V(P) := V(P) U {x} and E(P) := E(P) U {e} where e is the minimum weight edge that ends in x
if x not in Vp: # Happens with supernodes
Vp.append(x)
x_input_edges = list(
filter(
lambda edge: edge[1] == x and x not in all_covered_nodes_,
E))
if not x_input_edges:
print(f'Minimum edge not found not sure what to do.')
sys.exit(1)
e = min(x_input_edges, key=lambda edge: edge[2])
print(f'\t\tMinimum input edge into node x={x} is edge {e}')
Ep.append(e)
e_start_node = e[0]
# Step 5: If the start of edge e is not in Vp U W set it to x and return to step 4
if e_start_node not in set(Vp).union(set(W)):
print(
f'\t\tThis edge starts in node {e_start_node} which is not in Vp U W, therefore we put it as x and continue finding path.'
)
x = e_start_node
inner_iter += 1
continue
# Step 6: If the start of edge e is in Vp then we have a cycle
if e_start_node in Vp:
print(
f"\t\tThis edge starts in node {e_start_node} which is already in our current path, therefore we've got a cycle!."
)
# We have a cycle C (in subgraph Vp, Ep)
# BUT The cycle may only consist of subset of these nodes
previous_appearence_idx = Vp.index(e_start_node)
nodes_in_cycle = Vp[previous_appearence_idx:]
edges_in_cycle = [(u, v, w) for (u, v, w) in Ep
if u in nodes_in_cycle or v in nodes_in_cycle
]
print(
f"\t\t\t The cycle consist of the nodes {nodes_in_cycle}")
print(
f"\t\t\t The cycle consist of the edges {edges_in_cycle}")
super_nodes_.append({
"node_idx": e_start_node,
"covers_nodes": list(nodes_in_cycle),
"cycle_edges": list(edges_in_cycle),
"non_edge": e
})
removed_nodes = list(n for n in nodes_in_cycle
if n != e_start_node)
print(
f"\t\t\t We introduce {e_start_node} as super node that covers nodes {removed_nodes}"
)
all_covered_nodes_ = all_covered_nodes_.union(
set(removed_nodes))
# Now we must modify Vp and Ep
# We remove all `removed_nodes` from Vp
Vp = list(filter(lambda x: x not in removed_nodes, Vp))
print(
f"\t\t\t We remove all covered nodes from current Vp getting {Vp}"
)
# Now, this also effects (I guess) both the Ep edges and the E edges, as other edges are used to construct path
# Therefore we must apply the same edge_replace procedure to them
# It states as follows:
# For all edges f that start outside of C and end in C (edges in edges_in_cycle definitely start within cycle so we dont care about them), replace their destinations with supernode c
# Also modify their weights to we w(f) := w(f) - w(fC) where fC is the input edge to the cycle C
# that has the same destination as edge f
# These affected edges are either edges in Ep that did not participate in the cycle or edges in total graph (E) that are to be added to graph later
print(
f"\t\t\t We now need to update path edges that are currently: {Ep}"
)
Ep, lost_inc1, lost_out1 = update_edge_list(
Ep, e_start_node, nodes_in_cycle, edges_in_cycle)
print(f"\t\t\t After update we have Ep={Ep}")
print(f"\t\t\t We now need to update global edges: {E}")
E, lost_inc2, lost_out2 = update_edge_list(
E, e_start_node, nodes_in_cycle, edges_in_cycle)
print(f"\t\t\t After update we have E={E}")
lost_inc = lost_inc1.union(lost_inc2)
lost_out = lost_out1.union(lost_out2)
super_nodes_[-1]["lost_inc"] = lost_inc
super_nodes_[-1]["lost_out"] = lost_out
# And finally put x = c
x = e_start_node
inner_iter += 1
continue
# Step 7: If the start of edge e is in W then W := W U Vp, F := F U Ep U {e}, and go to step 2
if e_start_node in W:
print(
f"\t\tThis edge starts in node {e_start_node} which is in W (already added to tree). In this case we're done with inner iterations"
)
print(
f"\t\tWe flush the path nodes Vp into W and path edges Ep into F and also add edge {e}"
)
print(f"\t\tAnd we go all the way back to outer iterations.")
W = list(set(W).union(set(Vp)))
F = list(set(F).union(set(Ep)).union(set([e])))
break
outer_iter += 1
print(
'====================================================================================================='
)
print(f'Main algorithm finished.')
print(f'W = {W}')
print(f'F = {F}')
print(f'all_covered_nodes =')
print(all_covered_nodes_)
active_edges = set(F)
# Let's remove all cycle edges that are kept as lost
all_cycles_edges = set()
for super_node in super_nodes_:
tups = map(lambda x: (x[0], x[1]), super_node['cycle_edges'])
all_cycles_edges = all_cycles_edges.union(
set(super_node['cycle_edges']))
# print(f'Cycle edges among all supernodes: ')
# print(all_cycles_edges)
# for super_node in super_nodes_:
# super_node['lost_inc'] = list(filter(lambda edge: edge not in all_cycles_edges ,super_node['lost_inc']))
# super_node['lost_out'] = list(filter(lambda edge: edge not in all_cycles_edges ,super_node['lost_out']))
non_edges = set()
# Step 8: Expand all supernodes starting with the last one that's been added
# Do so by adding cycle_edges that made the node, except for the last one (marked as non-edge in the dict)
print(f'Starting supernode expansion')
for super_node in reversed(super_nodes_):
node_idx = super_node["node_idx"]
print(f'Active edges before expansion: {active_edges}')
print(f'\tExpanding supernode {node_idx}')
print(super_node)
lost_inc = super_node["lost_inc"]
lost_out = super_node["lost_out"]
cycle_edges = super_node["cycle_edges"]
non_edge = super_node["non_edge"]
active_edges = set(
filter(lambda edge: edge[0] != node_idx and edge[1] != node_idx,
active_edges))
for lost_in in lost_inc:
print(f'\t\tAdding inc {lost_in}')
active_edges.add(lost_in)
for lost_out in lost_out:
print(f'\t\tAdding outg {lost_out}')
active_edges.add(lost_out)
for cycle_edge in cycle_edges:
active_edges.add(cycle_edge)
non_edges.add((non_edge[0], non_edge[1]))
active_edges = set(
filter(lambda edge: (edge[0], edge[1]) not in non_edges, active_edges))
result = {
"active_edges": active_edges,
"edge_sum": sum(map(lambda x: x[2], active_edges)),
"root": r
}
print(f'Finally, all active edges are: {sorted(list(active_edges))}')
return result